The sedimentation of colloidal nanoparticles in solution and its study using quantitative digital photography


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The sedimentation of colloidal nanoparticles in solution
and its study using quantitative digital photography

Johanna Midelet, Afaf El-Sagheer, Tom Brown, Antonios Kanaras, Martinus
H. V. Werts

To cite this version:
Johanna Midelet, Afaf El-Sagheer, Tom Brown, Antonios Kanaras, Martinus H. V. Werts. The
sedimentation of colloidal nanoparticles in solution and its study using quantitative digital photogra-
phy. Particle and Particle Systems Characterization, Wiley-VCH Verlag, 2017, 34 (10), pp.1700095.
�10.1002/ppsc.201700095�. �hal-01637907�

https://hal.archives-ouvertes.fr/hal-01637907
https://hal.archives-ouvertes.fr


The sedimentation of colloidal nanoparticles in
solution and its study using quantitative

digital photography

Johanna Midelet1, Afaf H. El-Sagheer2, Tom Brown2,
Antonios G. Kanaras1, and Martinus H. V. Werts*3

1University of Southampton, Physics and Astronomy, Faculty
of Physical Sciences and Engineering, Southampton SO171BJ,

U.K.
2University of Oxford, Department of Chemistry, 12 Mansfield

Road, Oxford, OX1 3TA, U.K.
3Ecole normale supérieure de Rennes, CNRS, lab. SATIE,

Campus de Ker Lann, F-35170 Bruz, France

author version, accepted manuscript published as
Part. Part. Syst. Charact. 2017, 1700095

DOI: 10.1002/ppsc201700095

Abstract

Sedimentation and diffusion are important aspects of the behaviour
of colloidal nanoparticles in solution, and merit attention during the
synthesis, characterisation and application of nanoparticles. Here we
study the sedimentation of nanoparticles quantitatively using digital
photography and a simple model based on the Mason-Weaver equa-
tion. Good agreement between experimental time-lapse photography
and numerical solutions of the model was found for a series of gold
nanoparticles. The new method was extended to study for the first
time the gravitational sedimentation of DNA-linked gold nanoparticle
dimers as a model system of a higher complexity structure. Addition-
ally we derive simple formulas for estimating suitable parameters for
the preparative centrifugation of nanoparticle solutions.

*martinus.werts@ens-rennes.fr

1



1 Introduction

Observations on the sedimentation of colloidal solutions were important his-
torically in establishing the physical reality of molecules and providing a
molecular basis for thermodynamics in the form of statistical mechanics.[1–
3] Nowadays, there is an intense interest in the development of colloidal
solutions of engineered nanocrystals for a variety of applications. These so-
lutions should display sedimentation behaviour in line with expectations for
these particles on basis of their shape, composition and suspending medium.

For stable and dilute solutions of sufficiently large particles, the sedimen-
tation behaviour (Figure 1) is a result of the particular interplay of nanopar-
ticle hydrodynamics, and gravitational and Brownian forces. Under these
conditions, the surface chemistry of the nanoparticles, while ensuring col-
loidal stability, does not significantly influence the sedimentation behaviour.
For smaller particles (< 10 nm), and in particular ’soft’ particles such as
polymers and proteins, solvation effects on sedimentation may become sig-
nificant, as the shell of solvent molecules around the object influences its
hydrodynamic behaviour.[4–6] Furthermore, the presence of surfactants in-
teracting with colloidal particles may change sedimentation behaviour[7, 8].
These effects are not considered in the present work.

A general understanding of the sedimentation of nanoparticle solutions
is useful as it may give rapid and visual clues about the size distribution
and colloidal stability of newly synthesised colloids. These clues go beyond
the simple assessment of whether a prepared solution is colloidally stable.
Careful monitoring of sedimentation behaviour may be used to verify the
nature of the suspended object in its native medium, and is complemen-
tary to methods that characterise a limited number of specimens deposited
on a substrate, such as electron microsopy. Furthermore, sedimentation,
decantation and controlled centrifugation are often purification steps in wet-
chemical synthesis of nanoparticles and nanoparticle assemblies, for instance
for the purification of gold nanorods.[9] A further example is the purifica-
tion of nanoparticles by flocculation and subsequent sedimentation. Floc-
culation under the influence of depletion forces due to specific flocculatants
produces large aggregates of these nanoparticles that settle rapidly leaving
non-flocculated impurities in the supernatant.[10–12].

When analysing the interaction of nanoparticles with biological systems
in therapeutic and diagnostic applications, their transport properties such as
diffusion and sedimentation need to be taken into account The importance
of sedimentation to in vitro studies of cellular uptake of nanoparticles has
been demonstrated.[13]

There is a clear connection between the gravitational sedimentation of

2



Figure 1 – Sedimentation versus Brownian diffusion in colloidal solutions. (a)
Gravity tends to direct dense particles to the bottom of the cell, whereas the
random Brownian forces tend to disperse the particles throughout the entire
volume. (b) This leads to the establishment of an equilibrium gradient start-
ing from an initial homogeneous distribution of particles. Here we trace the
theoretical initial, transient and equilibrium concentration profiles as a func-
tion of vertical position. (c) Experimental observation of sedimentation over
time of colloidal gold (20, 40, 60 nm) in water using quantitative digital pho-
tography. The cell at the left of each picture contains water only. Photos were
taken at the start, and after 7, 14 and 35 days, respectively. The rightmost
picture represents the cells at equilibrium: a photograph of the Boltzmann
distribution.

colloidal particles and analytical centrifugation techniques, where sedimen-
tation is sped up by increasing g to multiples of the earth gravitational
pull. Analytical ultracentrifugation (AUC), which is well known for its
applications in biochemistry, has been successfully applied to the study of
nanoparticle solutions.[14–17] Optimisation of the measurement using a spec-
troscopic approach in combination with detailed numerical analysis of the
multi-wavelength sedimentation profiles has made AUC extremely versatile
for the analysis of nanoparticle preparations.[18] It has furthermore been
demonstrated that information on the shapes of DNA-based nanoparticle as-
semblies can also be obtained for AUC sedimentation analysis combined with
hydrodynamic modeling.[19]

The rotational speed and centrifugal acceleration in analytical ultracen-
trifuges are extremely high, and adapted to proteins and small bioparti-
cles. As will be further pointed out later on, many assemblies of inorganic
nanoparticles, which have higher density and size, are better analysed with

3



more modest speeds. A relevant recent development in this respect is ana-
lytical centrifugation (AC) with space- and time-resolved extinction profiles
(STEP),[20, 21] which functions at significantly lower speeds than AUC. Its
application to nanoparticle solution analysis is currently emerging.[22, 23]
Another analytical centrifuge technique is differential centrifugal sedimen-
tation (DCS) which is based on the sedimentation of injected samples in a
spinning disk containing a liquid. DCS successfully detects small changes in
particle sedimentation behaviour as a result of their ligand capping,[24] or
particle size variations due to varying synthesis conditions.[25]

For these reasons, it is of fundamental interest and instructive to study
the sedimentation behaviour of nanoparticle solutions in greater detail. Re-
cently, Alexander et al. reported a study of gravitational sedimentation of
gold nanoparticles [26]. They focused on measuring the optical density at a
pre-defined height in the solution as a function of time using an UV-visible
absorption spectrophotometer. This study was then further extended by
making estimates of the size distribution of colloidal gold samples by includ-
ing a multi-wavelength analysis.[27]. Sedimentation has also been instru-
mental for in situ attenuated total internal reflection infrared spectroscopy
by concentrating the particles or their aggregates onto the substrate.[28].
Magnetic-field enhanced sedimentation of superparamagnetic iron oxide par-
ticles has also been investigated.[29]

Here we take advantage of quantitative digital photography[30–32] for
experimentally analyzing the time-evolution of the concentration gradient of
sedimenting nanoparticle solutions. The technique is simple and easily imple-
mented. When properly set up, time-lapse imaging of the entire nanoparticle
concentration gradient is possible (Figure 1c). In particular with nanoparti-
cles of high overall density (such as those having a gold core) sedimentation
may be readily studied without a centrifuge, using only the gravitational
field.

The aim of the present work was to monitor the nanoparticle sedimenta-
tion process and the establishment of a concentration gradient using quan-
titative digital photography, and model this with a simple mathematical
diffusion-sedimentation equation. We also discuss the practical implications
of the theoretical model for the analysis and purification of nanoparticle
assemblies. Furthermore we obtain a photograph of the Boltzmann distribu-
tion (rightmost picture of Figure 1c). This work utilises well-known concepts
from colloid chemistry with an emphasis on their practical application in
purification and characterization of colloidal nanoparticle assemblies.

4



2 Sedimentation vs diffusion:

the Mason-Weaver equation

A simple model for sedimentation can be formulated under the assumptions
that there are no interactions between nanoparticles and that their motion
is only governed by random Brownian forces and a directional gravitational
force. The relevant parameters of the system depend only on the effective
density and overall shape of the particles, and on the density, viscosity and
temperature of the suspending liquid. The surface chemistry of the particles
has no influence, nor does the exact composition of the suspending medium.
However, these chemical parameters may influence the model indirectly, e.g.
by changing density or viscosity of the medium, or by altering particle shape
and effective density. We work under conditions where liquid density and
viscosity are not significantly altered by the dissolved ingredients, and at low
densities of colloid (mass fraction � 1%).

The establishment of a concentration gradient in such a dilute colloidal
solution of independent, non-interacting particles in a homogeneous gravita-
tional field is described by the Mason-Weaver equation,[33] a one-dimensional
partial differential equation. This equation is the direct predecessor to the
more well-known Lamm equation[34, 35] which applies for a centrifugal field
and is extensively used in the analysis of ultracentrifuge data.[36]

The height position in the cell is given by z. The concentration of de-
canting particles c satisfies the Mason-Weaver equation (1), with boundary
conditions (2).

∂c

∂t
= D

∂2c

∂z2
+ sg

∂c

∂z
(1)

D
∂c

∂z
+ sgc = 0(z = zmax,z = 0) (2)

D is the diffusion coefficient, s sedimentation coefficient, g gravitational
constant, and z = zmax and z = 0, are the top and bottom of the cell; re-
spectively. In this work, equations (1) and (2) are solved numerically using a
Crank-Nicolson finite-difference method which is described in the Supporting
Information, Section S.2. The numerical solver accepts an arbitrary concen-
tration profile as the initial condition.

The initial condition at t = 0 that we are interested in is c = c0 for
0 ≤ z < zmax and c = 0 elsewhere. That is, initially the particles are
homogeneously distributed within the cell.

The term sg is the terminal velocity of a particle accelerated by grav-
itational force in a viscous liquid (vterm = sg), and represents the velocity

5



(directed towards the bottom of the cell) this particle would have in absence
of Brownian motion. It is written in terms of the sedimentation coefficient s.
The sedimentation coefficient s for a specific type of particle in a particular
viscous liquid is

s =
mb
f

(3)

This involves the buoyant mass mb (i.e. the mass of the particle minus
the mass of the liquid displaced by the particle) and the frictional coefficient
f. The frictional coefficient depends on the particle’s size and shape and on
the viscosity of the solvent (more on this below). The frictional coefficient f
also is of importance for the diffusion coefficient D.

D =
kBT

f
(4)

For spherical particles we have the well-known results from Stokes’ law
and the Einstein-Smolukowski-Sutherland theory given by equations (5) and
(6).

fsphere = 6πηa mb,sphere =
4
3
πa3(ρ−ρfl) (5)

where a is the particle radius, η the viscosity of the liquid, and ρ and
ρfl the mass density of the particle and the suspending fluid, respectively.
An underlying assumption of these relations is the non-slip condition at the
sphere-liquid interface.

ssphere =
mb

6πηa
=

2

9

a2(ρ−ρfl)
η

Dsphere =
kBT

6πηa
(6)

Table 1 contains diffusion and sedimentation coefficients for gold nanospheres
calculated using Eqns. (6) for selected diameters in water at 277K, which is
the temperature used in the experiments (see Experimental Details). Data
for these particles at 298K can be found in the Supporting Information, Ta-
bles S2, S3, S4 and S5. The table collects further characteristic values related
to sedimentation, which will be discussed below.

At longer times, the solution to the Mason-Weaver equation converges to
the concentration distribution at equilibrium, eqn. (7).

c(z) = Bc0 exp

(
−z
z0

)
(7)

The characteristic height of the equilibrium gradient, z0, is

6



Table 1 – Calculated sedimentation parameters and characteristic values for
gold nanospheres in water at 277K (4 �, η = 1.56 mPa.s). Diffusion (D)
and sedimentation (s) coefficients, characteristic height of concentration gra-
dient at equilibrium (z0), approximate time to equilibrium from completely
dispersed state (t1cmsed ), concentration factor (B

1cm). The two latter values re-
fer to a liquid column height of 1 cm. Further data at 298K can be found in
the Supporting Information.

diam. D s z0 t
1cm
sed B

1cm

nm µm2s−1 10−9 s mm hours
13 19.9 0.110 18.5 2579 1.29
20 13.0 0.260 5.08 1090 2.29
40 6.48 1.04 0.636 272 15.7
50 5.18 1.62 0.325 174 30.7
60 4.32 2.34 0.188 121 53.1
80 3.24 4.16 0.0794 68.1 126
100 2.59 6.50 0.0407 43.5 246
150 1.73 14.6 0.0121 19.4 830

z0 =
D

sg
(8)

Interestingly, by combining (3), (4) and (8), we find that

z0 =
kBT

mbg
(9)

The same result for the equilibrium gradient can be obtained using a sta-
tistical mechanical approach, i.e. calculating the Boltzmann distribution for
dense suspended particles in a gravitational field (Eparticle = mbgz), illustrat-
ing that the digital image of the gold nanoparticle solution at equilibrium
is a photograph of a Boltzmann distribution. The concentration profile at
equilibrium does not depend on the frictional coefficient nor on the viscosity
of the medium, but exclusively on the buoyant mass of the particles. For a
spherical particle this translates into

z0,sphere =
3kBT

4πa3(ρ−ρfl)g
(10)

Some typical values for the characteristic height z0 for gold nanospheres
in water are included in Table 1. For diameters from 13 to 60 nm, the

7



characteristic height falls in the 20 to 0.2 mm range, where the gradient is
readily observed with the naked eye.

Going from the initial dispersed solution to the equilibrium gradient leads
to an increase in concentration of the particles near the bottom of the cell.
In Eqn. (7) this is expressed in the B constant, which represents the factor
by which the particles are concentrated at the bottom. It is obtained by
applying mass conservation to Eqn. (7). With the initial condition being a
uniform concentration c0 throughout the cell (0 ≤ z < zmax), we obtain eqn.
(11). Typical values for gold nanosphere have been added to Table 1.

B =
zmax

z0[1 − exp(−zmax/z0)]
(11)

We found that the time to reach sedimentation equilibrium from the
initial homegeneously dispersed state can be roughly estimated from the
time tsed necessary for a hypothetical particle to traverse the cell entirely
from top to bottom (distance zmax) at its sedimentation velocity sg.

tsed =
zmax
sg

(12)

Comparison of the numerical solutions for various ratios of zmax to sg to
the corresponding equilibrium distributions given by Eqn. (7) showed that
the concentration gradient at tequil ≈ 1.4tsed is in all cases within less than
one percent of the final equilibrium distribution (see Figure S3). This is
a useful result which defines the necessary time window for sedimentation
experiments and simulations. Weaver[37] found analytically that the upper
limit for tequil is 2tsed,

A more sophisticated analysis of the time to reach equilibrium is known
in the literature.[38] In the Supporting Information we demonstrate that
this analysis validates tequil ≈ 1.4tsed as a rough guess (which is already a
30 percent time reduction with respect to the Weaver limit), but in many
cases the time needed to reach equilibrium is actually shorter still, and for
optimisation of experimental procedures the more elaborate formulation can
be helpful. Moreover, the numerical solver described in the present work
enables detailed simulation of the evolution of the concentration profile, and
may be even more complete for optimising the sedimentation and detection
strategy.

In 1 cm high liquid columns, the time needed to reach equilibrium vary
from hours to weeks for typical gold nanospheres (Table 1). In practice,
with the data analysis demonstrated in this paper, it will not be necessary
to run experiments to complete equilibrium, as good estimates for diffusion

8



and sedimentation coefficients can be extracted using the initial progression
of the concentration gradient.

3 Sedimentation of simple gold nanospheres

Experimental concentration profiles of several settling solutions of gold nanopar-
ticles were obtained from digital photographs taken at different moments. A
typical example, using 40 nm gold nanospheres in water, is shown in Figure
2. The photographs for this series (of which 4 are shown in Figure 1) were
taken over a 39 day period, and quantitative vertical optical density profiles
were obtained using the method detailed in the Experimental section. In this
work, we only use the green colour channel of the images, since this produces
the strongest optical response for the red-coloured gold nanoparticles.

In the same figure we also show the solution of the Mason-Weaver equa-
tion c(z,t) at the same times t. The diffusion and sedimentation coeffi-
cients were adjusted independently to obtain best agreement with the ex-
perimental observations. The values obtained, D = 5.1 × 10−12 m2s−1 and
s = 7.9 × 10−10 s, agree within 20% with those expected from the Einstein-
Smolukowski-Sutherland and the Stokes relations for perfect 40 nm diameter
gold spheres in water at 277 K (Table 1).

Diffusion and sedimentation coefficients were obtained for the entire series
of gold nanosphere diameters by fitting the Mason-Weaver solutions to the
optical density profiles evolving in time. These coefficients agree reasonably
well (within 20%) with the predicted values (Figure 3). In addition to the
predictions based on the gold core only, we also calculated the expected
diffusion coefficients for the gold core plus an extra 1 nm of ligand shell
(ρ = 900 kg m−1, dotted curve in Fig. 3). This second theoretical curve
demonstrates that the present simple method can not distinguish between
small differences in overall hydrodynamic radius and density.

These experiments on simple gold nanosphere solutions demonstrate the
feasibility of quantitative sedimentation measurements using digital photog-
raphy. The solutions behave according to the Mason-Weaver model. It is
also illustrated that equilibrium gradients develop for gold nanospheres in
the selected diameter range upon standing undisturbed for prolonged times.
These are observable to the naked eye.

The simplicity of the method and its implementation obviously leads to
a variety of sources of experimental error. The cumulation of these errors
finally leads to an overall error in the diffusion and sedimentation coefficients
that is estimated above to be around 20%, based on the comparison of the
experimental results with the idealized theoretical curves (Figure 3).

9



Figure 2 – Evolution of the vertical particle density gradient in an aqueous
solution of 40 nm diameter gold nanospheres towards sedimentation equilib-
rium, observed using quantitative digital photography at 277K. Photos were
taken at t = 0, 1 h, 21 h, 27 h, 44 h, 52 h, 7 d, 9 d, 11 d, 14 d, 21 d,
23 d, 28 d, 35 d, and 39 days. Top: line-averaged intensities in the green
colour channel, corrected for differences in illumination intensity between dif-
ferent images. Middle: experimental optical density profiles obtained from
the intensity profiles. The arrow indicates the direction of time. Bottom:
theoretical optical density profiles obtained numerically as the solution to the
Mason-Weaver equation, with D = 5.1 × 10−12 m2s−1 and s = 7.9 × 10−10 s.

10



The back-illumination of the observation cells should be homogeneous,
which is not entirely the case in our set-up. The small illumination inho-
mogeneities are corrected for by the baseline correction we apply. Small
deviations in camera positioning from photograph to photograph lead to
changes in optical path lengths across the spectroscopic cells. There are
also fluctuations in illumination intensity; these are small and averaged out
when quantitatively analyzing the time-lapse series using the Mason-Weaver
model, but lead to additional noise. Stray light is present in the optical sys-
tem, i.e. the dark zones of the image are not entirely devoid of light, which
limits the maximum optical density that can be reliably measured. A specific
optical design with dedicated illumination and detection optics may improve
the analysis of the concentration gradient.

The long duration of the sedimentation in some samples calls for effi-
cient sealing of the containers, since solvent evaporation reduces the liquid
column height, introducing further uncertaintity in the analysis. Also, the
the meniscus at the top of the liquid column and the optical artefacts (dark
zone) related to it limit the precision of the measurement of the gradient.

At present, the numerical analysis assumes an ideally monodisperse dis-
tribution of nanoparticles, all being of the same shape and volume. The
diffusion and sedimentation coefficients obtained are effective, average coef-
ficients. We have not attempted to analyse the data in terms of continuous
size distributions, in which case an additional weighting should be applied for
the strong dependence of the optical extinction coefficient on the diameter
of gold nanoparticles.[39]

There is no fundamental impediment to analyzing the time-lapse optical
extinction profile data sets using models based on continuous polydisperse
size distributions, or even multimodal distributions, similarly to what is cur-
rently already done in analytical ultracentrifugation.[15, 20, 36] The numeri-
cal solver presented here may be used to generate time-resolved concentration
profiles for individual nanoparticle diameters in a trial distribution. After
weighting with the effective extinction coefficients these can then be com-
bined into the expected overall optical extinction profiles, which can then be
optimised to fit the experimental data. In this respect, multimodal size dis-
tributions may be challenging if the different populations display unsufficient
contrast in diffusion and sedimentation coefficients.

11



Figure 3 – Diffusion D and sedimentation s coefficients obtained by analysing
the evolving experimental density gradient of settling gold nanosphere solu-
tions (square markers). The solid curves are the expected values for per-
fect golden spheres from the Stokes-Einstein-Sutherland equation (top), and
Stokes’ law (bottom). The dotted curves are for gold spheres with a hypo-
thetical 1 nm thick organic layer.

12



4 Application to nanoparticle assemblies

After the initial demonstration of the quantitative analysis of time-lapse pho-
tography of settling spherical gold nanoparticles in water, we investigated a
sample of purified DNA-linked dimers of 13 nm diameter gold nanospheres
(Figure 4). This experiment illustrates that studying sedimentation can
aid in the chemistry and characterization of biomolecularly-scaffolded multi-
nanoparticle assemblies.

Figure 4 – Top left: transmission electron micrograph of purified DNA-linked
dimer sample. Top right: composed image of the time-lapse photography of
sedimenting DNA-linked gold nanosphere dimers (at 277 K). The rightmost
cell is a sample of monomeric 13 nm gold nanospheres. Bottom: optical density
traces as a function of vertical position in the cell, taken at various points in
time (red solid lines); the black dotted curves are the solution to the Mason-
Weaver equation with D = 5.8 × 10−12 m2 s−1 and s = 9.0 × 10−11 s

.

By adjusting the curves to the experimental data, we obtain D = 5.8 ×
10−12 m2 s−1 and s = 9.0 × 10−11 s, for the diffusion and sedimentation coef-
ficients, respectively. The sedimentation coefficient of DNA-linked dimers is
slightly smaller than that of a bare 13 nm monomer sphere, i.e. it sediments
even more slowly. The extra mass from the second gold sphere is counter-
balanced by more friction with the solvent due to the larger outer surface
area of the dimer. The larger volume comes to a large extent from the DNA

13



which has a much lower density than gold.
Additionally, the diffusion coefficient is lower than that of a 13 nm monomer

as a result of the larger hydrodynamic volume of the object. For the char-
acteristic height (Eqn. 3) we find z0 = 6.6 mm. This value is close to what
we would expect for an object which simply has the double buoyant mass
of a 13 nm monomer. Since gold has a much higher density than water,
whereas the DNA linker has a density comparatively very close to that of
water. Any extra volume taken up by the DNA linkage does not significantly
contribute to the buoyant mass of the object, since the displaced water vol-
ume is replaced with a substance having a density close to that of water. The
buoyant mass of the dimer is therefore determined by the two gold cores. On
the other hand, the bulky DNA structure should indeed have an effect on
the sedimentation and diffusion constants, i.e. sedimentation will be greatly
slowed down compared to an equivalent sphere of the same buoyant mass.

The lower sedimentation coefficients leads to slower establishment of the
gradient. However, in combination with the lower diffusion coefficient, it
finally leads to a more pronounced, shorter density gradient. A further anal-
ysis of the hydrodynamic behaviour and the resulting combination of D and
s of this type of assemblies is not within the scope of this work, but has
received recent attention in the literature.[19]

It is also interesting to note that the estimated time to obtain the final
equilibrium gradient (at 277K, with 2 cm liquid height) is on the order of 200
days for both the monomers and the DNA-linked dimers. For analysing the
transient concentration profile it is not necessary to wait that long, which
demonstrates the interest of having the numerical solution to the Mason-
Weaver at hand. Nonetheless, 30 days is still a long time, and these samples
are better analysed with centrifuge-based techniques.[14, 16, 19, 24, 25, 40].
In this context, it is interesting to note that — as we will see in the following
— the required centrifugal acceleration for many inorganic-core nanoparticles
is well within range of standard laboratory centrifuges, instead of higher-
speed specialised instruments.

5 Centrifugation of nanoparticle solutions

The results from the Mason-Weaver model may be used to generate intial ap-
proximate estimates for preparative and analytical centrifugation. For many
nano-assemblies using high-density inorganic core materials, the centrifugal
acceleration necessary for rapid sedimentation is well within the capabilities
of standard table-top laboratory centrifuges. A centrifugal acceleration that
is much higher than strictly necessary may have deleterious consequences

14



for colloidal stability as the density of nanoparticles in the pellet may be-
come very high, accelerating aggregation. This is one reason why even a
rough estimate is of interest. Furthermore, we anticipate that preparative
purification protocols may be developed for nanoparticle assemblies by using
approximate numerical simulations, using the computer code supplied with
the present work.

As an example we consider the minimal centrifugal acceleration gcfg needed
to obtain sedimentation equilibrium for gold colloid solutions in a centrifuge
within a given time tcfg, i.e. we wish to establish the centrifugal conditions
for a typical nanoparticle ‘centrifugation/re-dispersal’ washing procedure. In
Section 2 of this paper, we found that the time to reach equilibrium is 1.4tsed,
and using Eqn. (12) we can roughly estimate the centrifugal acceleration gcfg
necessary for the chosen centrifugation time tcfg and a liquid height in the
centrifuge tube ztube.

gcfg =
1.4 ×ztube
s× tcfg

(13)

This can be expressed as ‘relative centrifugal force’ (‘times g’, RCF =
gcfg/g, where g = 9.81 m s

−1).

RCF =
1.4 ×ztube
g ×s× tcfg

(14)

As discussed above and detailed in the Supporting Information, the factor
1.4 represents a conservative rough estimate of the time to reach equilibrium.
This factor may become smaller as a function of the particle’s sedimentation
and diffusion coefficients and the container height, using the formula by Van
Holde and Baldwin,[38] or using results from numerical simulation. However,
for the sake of simplicity, we will use 1.4 for the following calculations, as
this is a robust albeit conservative choice.

Table 2 contains the calculated RCF values needed for complete cen-
trifugation of gold colloids in typical Eppendorf-type vials (liquid height
ztube = 3 cm), within tcfg = 30 min. These values were then used in an
experiment in which a selection of colloidal gold solutions were spun for
30 min. at the calculated RCF. The samples were subjected to a typical
centrifugation/re-dispersal cycle, in which 95 vol% of the supernatant was
removed, followed by redispersal of the (relatively fluid, but highly concen-
trated) pellet in fresh water. In most cases the recommended centrifugal
acceleration gave remarkably good results (Supporting Information, Figure
S7).

Measurement of the UV-visible extinction spectra of the resuspended col-
loids confirms the visual impression that the calculated RCF values are indeed

15



Table 2 – Calculated RCF values for complete centrifugation, within 30 min.,
of spherical gold nanoparticles in water, using a liquid height of 3 cm. Ex-
perimentally determined nanoparticle recovery in a typical centrifugation/re-
dispersal washing cycle (n.t. = not tested).

diam. calcd. exp.
(nm) RCF % recovery
13 11671 n.t.
20 4931 94 (a)
40 1233 n.t.
50 789 98
60 548 n.t.
80 308 97
150 88 97
(a) recovery increases to 97% by spinning at 1.2 times
the calculated RCF

sufficient, and higher speeds for centrifugation are not necessary. In the case
of 20 nm diameter gold spheres, centrifugation at slightly higher acceleration
(∼ 20%) was required to concentrate all particles near the bottom of the
tube (Supporting Information, Figure S7).

Centrifugation at a given RCF corresponds to a sedimentation gradient
length scale z

cfg
0 and corresponding concentration factor Bcfg. The gradient

length should be small enough, such that the nanoparticles are concentrated
well in the pellet. The concentration factor gives an estimate of the concen-
tration increase at the bottom of the tube. For spherical particles, coefficient
s and D for Eqn. (15) are obtained from Eqn. (6).

z
cfg
0 =

D

s×g × RCF
Bcfg =

ztube

z
cfg
0 [1 − exp(−ztube/z

cfg
0 )]

(15)

For the conditions in Table 2 z
cfg
0 decreases from 2.1 µm (for 13 nm

diameter gold spheres) to well below 1 µm (larger particles), which indicates
that the particles are well concentrated near the bottom of the tube.

Prolonged centrifugation may concentrate smaller particles of less dense
materials. If we take for instance particles of 8 nm diameter with average
density 4.5×103 kg m−3 in water (a rough estimate for typical[41] CdSe/ZnS
quantum dot particles with a small-molecule capping), then it may be an-
ticipated that these can be concentrated in a liquid pellet less than 300 µm
high (z

cfg
0 ∼ 31µm) by spinning them for 6 hours at 11000 × g (or 12 hours

at 5500 × g for a zcfg0 ∼ 63µm), starting from a 3 cm liquid height (typical
Eppendorf-type vial). Such centrifugation conditions are well in the range of

16



standard laboratory centrifuges.
We may use similar reasoning to find the conditions for minimal centrifu-

gal stress, which will correspond to finding the minimum RCF necessary for
a certain pellet compactness (small z

cfg
0 ), and applying prolonged centrifuga-

tion, the duration in that case given by tcfg.
Here, we were concerned with centrifugation without density gradient,

starting from an initially homogeneous solution. For separation purposes,
density gradient methods may be more adapted.[42] Moreover, we used the
Mason-Weaver model which assumes a constant gravitational field, instead
of the gradient found in centrifugation. Nevertheless, this simple model does
provide a means of predicting the behaviour of nanoparticles in solution in
a standard laboratory centrifuge, and is therefore relevant for the rational
design of nanoparticle purification protocols.

Another implication of this analysis is that the RCFs needed for the
centrifugal sedimentation of inorganic nanoparticles and their assemblies is
significantly lower than those in analytical ultracentrifuges (∼ 100000 × g).
Analytical centrifugation using lower-speed (and simpler) equipment, such
as recent photocentrifuges with space- and time-resolved extinction profiling
(STEP),[20–23] is therefore highly relevant for the characterization of inor-
ganic nanoparticle solutions. Compared to the low-cost digital photography-
based method presented here, such dedicated and optimised equipment pro-
vide more rapid, precise and detailed analysis of nanoparticle samples.

6 Conclusion

This work rationalised observations on the settling of nanoparticles in liquid
solution in the framework of the Mason-Weaver model. This is illustrated by
the quantitative agreement between this model and experimental digital pho-
tography for the time-evolution of the concentration gradient of suspended
nanoparticles submitted to a gravitational field. A simple experimental pro-
tocol for observing the sedimentation process is established, using time-lapse
digital photography of the samples in an undisturbed laboratory fridge in
order to avoid thermally induced convection.

By fitting a numerical solution of the Mason-Weaver equation to the
experimental concentration gradient, the diffusion and sedimentation coeffi-
cients D and s are obtained, without need to wait for complete equilibrium to
be established. Early studies on sedimentation[3, 43] were mostly concerned
with precise measurement of the equilibrium gradient, which only yields the
buoyant mass mb of the particle as the sole parameter, not the separate
contributions of diffusion and sedimentation coefficients.

17



We demonstrated that measuring the evolving concentration gradient of
settling solutions distinguishes between individual monomeric nanoparticles
and their dimeric assemblies, by giving a distinct combination of diffusion
and sedimentation coefficients for each type of nano-object.

In our experiments on gold nanospheres in dilute solution, no signifi-
cant deviations from the simple Mason-Waver model were observed. Col-
loidal gold solutions are generally known to be well-behaved in this respect,
and we expect many dilute solutions of other non-aggregating (inorganic)
nanoparticles to behave in a similar way. Any significant deviations from the
predictions made by the Mason-Waver model would point to stronger inter-
particle interactions, aggregation of individual objects, specific solvation[4–6]
or surfactant[7, 8] effects, or changes in the properties of the liquid medium.
It is important to be aware of such deviations as they may affect other as-
pects of the behaviour of the nanoparticles in liquid media, such as their
interaction with biological entities.

The simple mathematical model and the experimental protocol used here
have obvious limitations, and do not substitute advanced analytical centrifu-
gation techniques[14, 16, 19–25, 40]. However, they do provide means for ini-
tial and very simple screening of nanoparticle solutions, and rationalise visual
observations made at the bench, or upon prolonged storage in the laboratory
fridge. The use of a standard digital photo-camera limits the applications
to systems that have detectable optical response in the visible. Many parti-
cle systems do indeed have such a response, for instance plasmonic materials
such as gold, silver, titanium nitride,[44] but also other metal particles, semi-
conductor quantum dots,[17] and paramagnetic iron oxide.[29] Attainment of
equilibrium can take quite some time, and an analytical photocentrifuge[20–
23] may be a time-saving investment, which also will yield more precise anal-
ysis, in particular for broad size distributions and multimodal samples.

Based on the Mason-Weaver model, we obtained simple expressions that
give useful recommendations for the centrifugation of nanoparticle solutions,
so that delicate solutions can be processed with care, instead of spinning at
maximum RCF. The practical insights and simple quantitative expressions
provided in the present work are of direct interest for wet-chemical synthesis,
purification and application of functional nanoparticle assemblies, and will
stimulate further interest in sedimentation analysis[4, 19, 23] for functional
nanoparticle solutions.

18



7 Experimental details

Temperature stabilisation. In a non-thermostated environment such as
a lab shelf, even modest temperature changes may create temperature gra-
dients in the sample that lead to convective motion sufficiently strong to
prevent the sedimentation equilibrium from establishing itself. Such convec-
tive motion plagued early sedimentation experiments, which were concerned
with precise measurement of the equilibrium distribution.[3, 43] Thermally
driven convection offers an explanation as to why, on a non-thermostated
shelf, many colloidal gold solutions seem not to evolve to a sedimentation
equilibrium gradient. In the present work, mechanical and thermal per-
turbations were avoided by carrying out the experiments in a undisturbed
laboratory fridge at 277 K.

Quantitative digital photography. In previous work we used quantita-
tive colour imaging for measuring concentration profiles in microfluidic chan-
nels, using an optical microscope and a dedicated CCD camera.[45] Here we
use a ’consumer-grade’ photo camera. Images were taken using a digital
camera (Nikon Coolpix P7800, Nikon Corp., Japan) providing the output
of unprocessed (’RAW’) image data, where the individual pixel values are
proportional to the detected light intensity for the particular colour chan-
nel. This circumvents problems of linearization generally encountered with
most digital cameras.[31] Typically, four samples were photographed in a
single picture. A black area was also included to serve as the source for
background subtraction.

The ’RAW’ image data were read by ImageJ software[46] using the dcraw
plugin.[47] The three colour channels of the images were separated (ImageJ)
and treated individually as monochrome intensity images. Intensity gradient
profiles Iraw(z) for all samples (and all colour channels) in each image of
the time series were extracted by horizontal averaging over a rectangular
area (typically, the visible optical window of the cells). The pixel values
of the black area were averaged for background substraction, Idark. The
z scaling was calibrated by precise measurements of specific cell features.
Each image frame thus obtained its specific calibration of pixel size. The
digital intensity profiles Iraw(z) were then further treated numerically using
the Python programming language with scientific extensions.[48–50]

The individual profile traces with their specific z step sizes were re-
sampled at a standard higher resolution in order to have profiles with identical
step sizes, facilitating their processing and comparison with theory. Subse-
quently, background-corrected image profiles I(z) were obtained by subtrac-
tion of the near-zero dark background.

19



I(z) = Iraw(z) − Idark (16)

The top area of each extracted z profile, which does not contain liquid,
was used for calculating I0 by averaging. This corrects slight frame-to-frame
variations in illumination intensity. A linear baseline correction, ODbase(z) =
k1z +k2, was applied globally to all time-frames for each sample. In all cases,
the baseline correction was only modest and not necessary to obtain useful
results.

The final corrected optical density is obtained using eqn. (17). We note
that we apply this Beer-Lambert-Bouguer formulation despite the condition
of monochromatic light not being rigorously fulfilled: the filters used in colour
cameras define large spectral bands (width ∼ 100 nm). As has been discussed
previously,[45] a linear response of the optical density as a function of concen-
tration is still obtained, provided that the extinction spectra of the samples
are sufficiently large and their optical density is sufficiently low (under OD
1).

OD(z) = log10

(
I0
I(z)

)
− ODbase(z) (17)

Comparison of experiment with theory is achieved by converting the con-
centration profiles c(z) from the Mason-Weaver model into modeled optical
density profiles ODmodel(z) using an ”effective extinction coefficient” which
we refer here to as k3 and can be adapted to rescale the model concentration
profile to fit the experimental values.

ODmodel(z) = k3c(z) (18)

Centrifugation. A thermostated bench-top laboratory centrifuge (Hettich
Mikro 220r with 1195-A rotor) was used. The relation between rotational
speed (rounds per minute, RPM) and relative centrifugal force (RCF) is

RCF = 1.12 × 10−3 ×R× (RPM)2 (19)

The rotor radius R for the 1195-A rotor used is 0.087 m. The maximum
speed for this rotor is 18000 RPM, corresponding to an RCF of 31500 × g.
Samples were generally centrifuged with the centrifuge thermostat at 298K.

Colloidal gold solutions. For the study of the sedimentation of gold
nanospheres (Section 3) we used standard aqueous solutions of colloidal gold
from commercial sources (BBI Solutions, UK, and Sigma-Aldrich, France)
and samples synthesised and characterised according to literature procedures.[51]

20



All colloids are stabilised with negatively charged carboxylate ligands, and
the particles have negative zeta potential. The samples were diluted with
pure water where necessary. Typical optical densities at the extinction max-
imum were in the range 0.3 . . . 1.

The carboxylate ligand layers around the particles are thin compared to
the particle diameter (d > 20 nm), and do not contribute significantly to
the buoyant mass. For smaller particles (< 10 nm) and larger ligands such
effects may become significant, and will then show up as a deviation from
the idealised predicted behaviour, and result in a change in effective buoyant
mass, and effective diffusion and sedimentation coefficents. Furthermore, sol-
vation effects on sedimentation may become significant, as the shell of solvent
molecules around the object influences its hydrodynamic behaviour.[4–6]

DNA-linked dimers. The dimers were constructed from 13 nm gold nano-
Gold nanoparticles of diameter 13 nm were synthesised according to the

published protocol.[52, 53] Briefly, a solution of sodium tetrachloroaurate (50
mL, 1 mM) was heated up to 100 °C under stirring. Once boiling, a solution of
sodium citrate (5 mL, 2 wt.%) was added. After appearance of the typical red
colour, boiling and stirring were maintained for 15 additional minutes before
letting cool down to room temperature. To stabilise the AuNPs, BSPP was
added (15 mg) and the solution was left to stir overnight. TEM images were
obtained with a Hitachi H7000 transmission electron microscope, operating
at 75 kV bias. All samples preparation involved deposition and evaporation
of a specimen droplet on a carbon film-coated 400 Mesh copper grid. TEM
analysis is shown in the Supporting Information, Fig. S2.

Dimers of 13 nm AuNPs were synthesised using DNA hybridisation.[54]
BSPP-coated AuNPs of 13 nm (5 pmol) were flocculated using NaCl and
centrifuged for 5 min at RCF 25 000 ×g (Eppendorf centrifuge 5417R, rotor
FA-45-24-11, 8.8 cm radius). The supernatant was taken off and the particles
were re-dispersed in 100 µL buffer (20 mM phosphate, 5 mM NaCl). DNA
single strands S1 respectively S2 (15 pmol) were added each to a separate
nanoparticle solution to achieve a ratio DNA/particles of 3:1. A solution of
BSSP (10 µL, 1 mg/20 µL) was added and the reaction mixture was shaken
for 1 h, in order to deprotect the thiol group on the DNA and to allow its
attachment to the particles. The AuNPs were then centrifuged for 15 min at
25 000 × g and redispersed in hybridization buffer (50 µL, 6mM phosphate,
80mM NaCl). After mixing the two types of particles (S1 and S2 strands), a
complementary DNA strand S3 (2.5 pmol) was added to create the dimers.
Hybridisation was realized by heating up the solution to 70°C and leaving it
to cool down slowly and gradually to room temperature. The dimers were
purified by agarose gel electrophoresis (1.75% agarose gel in 0.5 x TBE, 1

21



h at 90V). After extraction from the gel the solution was centrifuged for 10
min at 25 000 ×g and the dimers were redisolved in Milli-Q water.

Acknowledgements

This work was supported by Dstl (UK) in the framework of the France-UK
Ph.D. programme. MW acknowledges funding by the Agence Nationale de
la Recherche (France), grant ANR-2010-JCJC-1005-1 (COMONSENS).

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25



Supporting Information

for Part. Part. Syst. Charact. 2017, 1700095
DOI: 10.1002/ppsc201700095

The sedimentation of colloidal nanoparticles in solution and its
study using quantitative digital photography

Johanna Midelet, Afaf H. El-Sagheer, Tom Brown,
Antonios G. Kanaras and Martinus H. V. Werts*

S.1 Synthesis of DNA-linked gold nanosphere dimers

Figure S1 – Schematic of the assembly of DNA-linked dimers and subsequent
’click’ chemistry.

.

S1



Table S1 – Sequences of single DNA strands S1, S2 and S3.

Figure S2 – Citrate-stabilised gold nanoparticles used for the construction of
DNA-linked dimers. a) TEM image of 13±2 nm spherical gold nanoparticles.
Scale bar is 100 nm. b) Corresponding size distribution histogram, N = 500
particles

.

S2



S.2 Finite-difference solver for the Mason-Weaver equa-
tion

S.2.1 Dimensionless Mason-Weaver equation

A numerical solver has been designed for the dimensionless version of the
Mason-Weaver equation. Conversion between the Mason-Weaver equation
and its dimensionless form can be achieved using:

ζ = z/z0 z0 =
D

sg
(S1)

and

τ = t/t0 t0 =
D

s2g2
(S2)

The aim is now to obtain the time-evolving spatial concentration distribu-
tion c(ζ,τ) obeying the dimensionless Mason-Weaver equation (S3), starting
from an arbitrary initial concentration profile c(ζ, 0).

∂c

∂τ
=
∂2c

∂ζ2
+
∂c

∂ζ
(S3)

∂c

∂ζ
+ c = 0 @ ζ = 0,ζ = ζmax (S4)

A finite-difference scheme for numerically solving this equation has been
proposed previously,[26] but unfortunately, in our hands, this did not yield a
working computer code. In particular, the proposed time coordinate trans-
formation leads to unsurmountable numerical problems in our implementa-
tion. Here, we propose a different finite-difference approximation that yields
a stable, robust numerical solver, that moreover does not suffer from mass
conservation problems reported[26] for the previously proposed scheme.

S.2.2 Crank-Nicolson scheme

Time and space are discretised into N + 1 and J + 1 grid points, respectively.
Space is on an evenly spaced grid between 0 and ζmax.

ζj = j∆ζ (j = 0, 1 . . .J) (S5)

∆ζ =
ζmax
J

(S6)

S3



Time may be on an evenly or unevenly spaced grid. We use an exponen-
tially expanding grid, going from 0 to τmax.

τn = exp(kn) − 1 (n = 0, 1 . . .N) (S7)

k =
ln(τmax + 1)

N

We make the following Crank-Nicolson style finite-difference approxima-
tions for the PDE, where cnj is the value of c(ζj,τn).

∂c

∂τ
→ γ(cn+1j − c

n
j ) (S8)

∂2c

∂ζ2
→ α(cnj+1 − 2c

n
j + c

n
j−1 + c

n+1
j+1 − 2c

n+1
j + c

n+1
j−1 ) (S9)

∂c

∂ζ
→ β(cnj+1 − c

n
j−1 + c

n+1
j+1 − c

n+1
j−1 ) (S10)

where (noting the factors 2 and 4 in α and β)

γ =
1

τn+1 − τn
α =

1

2(∆ζ)2
β =

1

4∆ζ
(S11)

With these finite-difference approximations, and after rearrangement of
the terms, the discrete Mason-Weaver equation can be written as follows
(j = 1, 2 . . .J − 1).

(−α + β)cn+1j−1 + (γ + 2α)c
n+1
j + (−α−β)c

n+1
j+1

= (α−β)cnj−1 + (γ − 2α)c
n
j + (α + β)c

n
j+1 (S12)

The boundary conditions are set as follows. At ζ = 0 we choose

c → 1
4
(cn0 + c

n+1
0 + c

n
1 + c

n+1
1 ) (S13)

∂c

∂ζ
→ 2β(cn1 − c

n
0 + c

n+1
1 − c

n+1
0 ) (S14)

Approximation of c using (S13) (instead of simply taking the values at
c0) leads to much better behaviour in terms of mass conservation (the sum
over all cj), in particular at longer times.

After substitution and rearrangement this results in

S4



(−2β + 1
4
)cn+10 + (2β +

1
4
)cn+11 = (2β −

1
4
)cn0 + (−2β −

1
4
)cn1 (S15)

At ζ = ζmax the equivalent choice was made (using backward difference
instead of forward difference, naturally).

The equations (discretised PDE and BCs) are assembled into a matrix
equation involving tridiagonal matrices. Numerically solving this equation
yields the cn+1j from c

n
j , i.e. the concentration profile at time τn+1 using

the concentration profile from time τn. Repeating the process, starting from
the initial condition at τ0, yields the evolution of the concentration profile,
obeying the Mason-Weaver equation.

S.2.3 Implementation

The finite-difference scheme was implemented using Python (with numpy for
array manipulation and sparse matrix routines from scipy). For brevity,
we only give the solver code without any plotting, processing or storage of
results. It consists of a single loop that generates the successive concentration
profiles cnj at each τn, starting from the initial conditions at τ0.

# -*- coding: utf -8 -*-

import numpy as np

from scipy import sparse

from scipy.sparse import linalg

""" Solver for the adimensional Mason -Weaver equation.

The solution is calculated on an exponentially expanding time

grid (N+1 points), and an evenly spaced space grid (J+1 points ).

Input:

zeta_max (float ): adimensional height of the cell

tau_end (float): calculate solution from tau=0 to tau=tau_end

J (int): number of space points minus 1 on (linear) grid

N (int): number of time points minus 1 on (exponential) grid

Output:

The solution is stored in a two -dimensional array ’c’.

The space grid points are given by the vector ’zeta ’.

The time grid points are given by the vector ’tau ’.

code tested with Python 3.5.2, Anaconda 4.1.1 (64-bit)

"""

def _tridiamatrix(J,ldiagelem ,cdiagelem ,rdiagelem ,

S5



cstart ,rstart ,lend ,cend):

""" utility function that generates a sparse tridiagonal

matrix from given elements """

ldiag = np.empty(J+1)

cdiag = np.empty(J+1)

rdiag = np.empty(J+1)

ldiag.fill(ldiagelem)

cdiag.fill(cdiagelem)

rdiag.fill(rdiagelem)

cdiag [0]= cstart

rdiag [1]= rstart

ldiag [-2]= lend

cdiag [-1]= cend

diag=[ldiag ,cdiag ,rdiag]

N = J+1

return sparse.spdiags(diag ,[-1,0,1],N,N,format="csr")

# set calculation parameters

zeta_max = 10. # height of cell

tau_end = 10. # end tau

J = 400 # number of space points (-1)

N = 200 # number of time points (-1)

c_init = np.ones(J+1) # initial condition

# define time and space grids

k_tau = np.log(tau_end +1)/N

tau = -1.0 + np.exp(k_tau*np.arange(0,N+1))

dltzeta = zeta_max/J

zeta=np.linspace(0,zeta_max ,J+1,endpoint=True)

# create c array for storing result

c = np.zeros ((J+1,N+1))

c[:,0] = c_init

c_n = c_init

dltzeta=dltzeta

tau=tau

# loop generating c^{n+1} from c^{n} starting from c^0

for n in range(0,N):

alpha = 1/(2*( dltzeta **2))

beta = 1/(4* dltzeta)

gamma = 1/(tau[n+1]-tau[n])

# finite difference diagonals Left Hand Side

ldiagelem = -alpha + beta

cdiagelem = gamma + 2* alpha

rdiagelem = -alpha - beta

# boundary conditions LHS

cstart = -2*beta + 0.25

S6



rstart = 2*beta + 0.25

lend = -2*beta + 0.25

cend = 2*beta + 0.25

# create LHS tridiagonal matrix

LHSmat = _tridiamatrix(J,ldiagelem ,cdiagelem ,rdiagelem ,

cstart ,rstart ,lend ,cend)

# finite difference diagonals Right Hand Side

ldiagelem = alpha - beta

cdiagelem = gamma - 2*alpha

rdiagelem = alpha + beta

# boundary conditions RHS

cstart = 2*beta - 0.25

rstart = -2*beta - 0.25

lend = 2*beta - 0.25

cend = -2*beta - 0.25

# create RHS tridiagonal matrix

RHSmat = _tridiamatrix(J,ldiagelem ,cdiagelem ,rdiagelem ,

cstart ,rstart ,lend ,cend)

# construct RHS vector

RHSvec = RHSmat * c_n # c contains concentration profile

# SOLVE the matrix equation , giving c^{n+1} (c_next)

c_next = linalg.spsolve(LHSmat ,RHSvec)

# store c_{n+1} into solution matrix

c[:,n+1] = c_next

# we swap the vectors containing c_n and c_next via cswap

# we cannot simply re-assign c_n = c_next

cswap = c_n

c_n = c_next

c_next = cswap

# plot example

import matplotlib.pyplot as plt

plt.figure (1)

plt.clf()

for i in [0 ,50 ,100 ,150 ,200]:

plt.plot(zeta , c[:,i],label=’tau=’+str(tau[i]))

plt.ylim (0,3)

plt.xlabel(’$\\ zeta$ ’)

plt.ylabel(’$c$’)

plt.legend ()

plt.show()

S7



S.2.4 Validation of the numerical solution

The numerical solver was tested extensively for a wide range of ζmax and
τ. Typical values for N and J were on the order of several hundred grid
points. Numerical mass conservation and convergence of the gradient to the
analytical Boltzmann equilibrium solution were verified.

An example of a numerical solution is given in Figure S3, which also
illustrates that the numerical solution faithfully converges to the analytical
equilibrium distribution, for τ = 1.4τsed.

Figure S3 – Left: Numerical solution (N = 200, J = 400) to the dimension-
less Mason-Weaver equation with ζmax = 10 and τ going from 0 to 30. Right:
Comparison of the analytical equilibrium profile and the numerical solution
at longer times, demonstrating (1) that the numerical solution converges to
the analytical equilibrium (Boltzmann) distribution and (2) that at 1.4τsed the
system is virtually at equilibrium.

Mass conservation is observed in the numerical scheme, as illustrated in
Figure S4. A slight increase of the total numerical mass is observed, but the
numerical solution does not diverge at longer simulation times, as a result of
our choice for the boundary conditions, cf. Equation (S13).

S8



Figure S4 – Evolution of the total mass in the numerically simulated system
(N = 200, J = 400, ζmax = 10, τ = 0 . . . 30), obtained by summing all space
grid points. A slight increase (< 2%) of the total numerical mass is observed,
but the total mass does not diverge for long simulation times.

S9



S.3 Sedimentation parameters for gold nanospheres in
water

Table S2 – Calculated diffusion D and sedimentation s coefficients for gold
nanoparticles in water at two different temperatures.

298 K 277 K
0.89 mPa.s 1.56 mPa.s

diam. D s D s
(nm) 10−12 m2s−1 10−9 s 10−12 m2s−1 10−9 s
13 37.8 0.193 19.9 0.110
20 24.5 0.457 13.0 0.260
40 12.3 1.83 6.48 1.04
50 9.82 2.86 5.18 1.62
60 8.18 4.12 4.32 2.34
80 6.14 7.32 3.24 4.16
100 4.91 11.4 2.59 6.50
150 3.27 25.7 1.73 14.6

Table S3 – Values of equilibrium gradient length scale z0 (in mm) for spherical
gold nanoparticles in water, at two different temperatures (room temperature
and fridge)

diam. z0 (mm)
(nm) 298 K 277 K
13 19.9 18.5
20 5.147 5.08
40 0.684 0.636
50 0.350 0.325
60 0.203 0.188
80 0.0855 0.0794
100 0.0438 0.0407
150 0.0130 0.0121

S10



Table S4 – Calculated concentration factor B for spherical gold nanoparticles
in water at 277K (left columns) and 298K (right columns), for cell heights
(zmax) of 1 mm and 1 cm

277K 298K
diam. B
(nm) 1 mm 1 cm 1 mm 1 cm
13 1.03 1.29 1.03 1.27
20 1.10 2.29 1.09 2.18
40 1.99 15.7 1.90 14.6
50 3.22 30.7 3.03 28.6
60 5.34 53.1 4.97 49.4
80 12.6 126 11.7 117
100 24.6 246 22.9 229
150 83.0 830 77.1 771

Table S5 – Calculated sedimentation times (in hours) for gold nanoparticles
in water for a cell with zmax = 1 cm. A good rough estimate for the time to
reach equilibrium is tequil ≈ 1.4tsed. These times are directly proportional to
cell height.

diam. tsed(h)
(nm) 298 K 277 K
13 1465 2580
20 619 1090
40 155 272
50 99.0 174
60 68.8 121
80 38.7 68.1
100 24.8 43.6
150 11.0 19.4

S11



S.4 The time needed to reach sedimentation equilib-
rium

In the main text we state that a simple but robust approximation for the
time to reach sedimentation equilibrium is 1.4 times the time tsed needed by
a particle to traverse the liquid column from top to bottom at its terminal
sedimentation velocity. This approximation enables a simple estimate of
the time needed for centrifugation of nanoparticles at a certain centrifugal
acceleration.

A more sophisticated approach to this question is known in the ultra-
centrifugation literature.[38] This approach by Van Holde and Baldwin[38]
yields the time to arrive near equilibrium within a ’distance’ determined by
parameter �. The choice for this parameter made in [38] is based on the
difference in particle concentration between the top and the bottom of the
cell, ∆c = c(0) − c(zmax).

� = 1 −
∆c(t)

∆ceq
(S16)

At the beginning, when the concentration is homogenous, � = 1. As
sedimentation equilibrium is reached � will tend to zero. A close approach to
equilibrium is typically obtained for � < 0.02. The time to come within this
’distance’ � from equilibrium can then be approximated with the following
expression.

tequil ≈
z2max
D

F(α) (S17)

F(α) = −
1

π2U(α)
ln

(
π2U2(α)�

4[1 + cosh(1/2α)]

)
(S18)

U(α) = 1 +
1

4π2α2
(S19)

The time to reach equilibrium is expressed as a function of α, which, in
the context of the present work, is given by

α =
kBT

mbgzmax
(S20)

An interesting point is that the parameter α can also be expressed as the
ratio from the length scale of the equilibrium gradient z0 to the height of the
liquid column zmax.

S12



α =
kBT

mbgzmax
=

z0
zmax

(S21)

It is thus a measure of how the equilibrium gradient ’fits’ in the height
of the liquid column. For ’oversized’ containers (where all particles will fi-
nally be concentrated near the bottom of the container), the attainment of
equilibrium for initial homogeneous distribution will be governed by the time
needed for sedimentation, whereas containers much smaller than the equilib-
rium gradient will not show much of the gradient, i.e. when α becomes very
large, the gradient will be undetectable.

Equation (S17) may be expressed as the ratio of tequil and tsed. It can be
shown that this ratio is a pure function of α. It thus depends only on the
ratio z0/zmax, not on D and s individually.

tequil
tsed

=
F(α)

α
(S22)

It is important to note that Equation (S17) is an approximation for
α > 0.1,[38] and that it is not valid at smaller values. For small α our
approximation tequil ≈ 1.4tsed is valid and safe. For α > 0.1, we find that
the time predicted by Eqn (S17) is always shorter than the rough 1.4tsed
estimation (see Figure S5)

Figure S5 – Left: The ratio of the ‘time-to-equilibrium’ tequil and the ‘sedi-
mentation time’ tsed as a function of parameter α. Curves (a) are calculated
using Eqn. (S17), valid for α > 0.1. We chose � = 0.018, such that the
maximum of the curve is at 1.4. Curves (b) use the constant 1.4 ratio from
this work. Right: Absolute sedimentation times in 2 cm high water column,
calculated for spherical gold particles. T = 277.15 K was used. Small values
of α (left) correspond to large particle diameters (right).

We note that our numerical solver enables full simulation of the sedimen-
tation process. This can be used to study in more detail the evolution of the
concentration gradient and the attainment of equilibrium. A different metric

S13



for the distance to equilibrium may be used instead of the difference between
the two outermost points, such as the root-mean-square deviation over the
entire curve, or a metric that is representative to the particular measurement
used.

We performed numerical simulations of the attainment of sedimentation
equilibrium for different values of α. The numerical solver allows us to study
also the case of α < 0.1 not covered by Equation (S17). Figure S6, shows
how the ‘distance to equilibrium’ � (Eqn. S16) evolves over time (relative
to the sedimentation time tsed). For all values of α, equilibration is near
completion at t ≈ 1.4tsed, even for the ‘worst case’ α = 0.1. As predicted by
Eqn. (S17), equilibrium is attained relatively faster (t < 1.4tsed) for larger α
(smaller particles).

Figure S6 – Approach to sedimentation equilibrium, obtained using the nu-
merical Mason-Weaver solver, for different values of α. Equilibration is always
virtually complete at t = 1.4tsed. For large α, time to equilibrium becomes
shorter.

This brief analysis shows that tequil ≈ 1.4tsed is useful as a rough approx-
imation. It never underestimates the time to equilibrium. However, it does
overestimate the time to equilibrium for the smaller particles, in which case
Equation (S22) becomes relevant.

S14



S.5 Experimental test of the centrifugation parame-
ters recommmended by the Mason-Weaver model

Figure S7 – Centrifugation of gold nanospheres in water, for 30 min. at the
centrifugal acceleration recommended by the Mason-Weaver model. Diame-
ters: (a) 20 nm, (b) 50 nm, (c) 80 nm, (d) 150 nm. Photographs on the left:
centrifuge vials before and after centrifugation. Relative centrifugal force: (a)
4931 × g, (b) 789 × g, (c) 308 × g, (d) 88 × g. UV-visible extinction spectra
on the right: sample before centrifugation (black) and after (blue) re-dispersal
of the liquid pellet (5% of the initial volume); red: supernatant (95% of the
initial volume).

S15



Figure S8 – Centrifugation of 20 nm diameter gold nanospheres in water, for
30 min., at 6000×g, i.e. 20% higher than the Mason-Weaver recommendation.

S16