Intensity – Volume – Number
Which Size is Correct?
Introduction
A key parameter of colloidal systems
is their particle size. Yet there are
many different methods by which a
particle size can be obtained. So
which one should be chosen?
Comparing stated dimensions from
one method with those from another
can be confusing and should be
approached with caution: distributions
in terms of volume, number or
scattering intensity often produce
vastly differing results – despite
expressing exactly the same physical
characteristics of a sample.
Differences between the three
distributions are highlighted using a
titanium dioxide sample as an
example.
What is a Distribution?
Many things in nature have
characteristics that show variations. A
distribution is a description of the
probability of encountering a certain
value of a variable. For example, an
age distribution of a group of people
might express the number of
members of that group in each age
category. This type of distribution
would correctly be denoted as an ‘age
distribution by number’. When particle
size distributions are of interest, three
types of distributions are commonly
encountered.
The number distribution shows the
number of particles in the different
size bins. The volume distribution
shows the total volume of particles in
the different size bins. The intensity
distribution describes how much light
is scattered by the particles in the
different size bins. Are these three
distributions different? Yes they are!
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As a simple example, consider a
distribution consisting of just two
particles, particle species one with
size a and particle species two of size
b. If there are Na particles with the
size a and Nb particles with size b
then in terms of number, their relative
contributions compared to the other
particle are
%Na =
100 ⋅ N a
N a + Nb
(1)
where %Na is the relative number of
the particle with size a. Assuming that
the particles are spherical, the particle
volume is proportional to the size to
the third power. Thus, in terms of
volume, the relative contribution from
a is
%Va =
100 ⋅ N a a 3
N a a 3 + N bb 3
(2)
where %Va is the relative volume of
the particle with size a. Finally, for
small, isotropic particles the scattering
intensity from a spherical particle is
proportional to the size to the sixth
power. Thus, in terms of intensity, the
relative contribution from a is
%Ia =
100 ⋅ N a a 6
N a a 6 + N bb 6
For particles that are much smaller
than the wavelength of the
illuminating light (typically 632.8nm in
the Zetasizer Nano), the above
isotropic assumption holds true. A
complete description of the more
complicated scattering behavior
involves the application of Mie
scattering. This theory describes how
much light is scattered by a particle of
any size when the complete optical
properties (refractive index and
absorption) are known. The intensity
scattered according to Mie theory is
not simply proportional to the size to
the sixth power, but it is modified by a
size dependent Mie function M(a) .
The generalized version of equation 3
then turns into
%Ia =
100 ⋅ N a M (a )
N a M (a ) + N b M (b)
(4)
The direct data obtained from a light
scattering experiment correspond to
these %intensity values for the
intensity distribution.
(3)
where %Ia is the relative amount of
intensity of the particle with size a.
For real-life distributions, the situation
can be modeled in a similar way to
the simplified two-particle model
employed above as an example.
In effect, this typically means that
intensity distributions emphasize the
larger particles in the distribution,
whereas the number distributions
Zetasizer Nano technical note
emphasize the smaller particles in the
distribution. However, it is important to
note that both are just different
representations of the same physical
reality of a distribution of different
sizes.
Size Distribution by Intensity
The intensity size distribution from
TiO2 particles in water is shown in
figure 1. In order to point out the
discrete nature of the regularization
algorithm, the histogram (steps) rather
than the frequency curve (smooth)
representation is displayed. The mean
size is 87nm with a standard deviation
of 32nm – by intensity. The key size
parameters obtained are listed in
Table 1.
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Size Distribution by Volume
The volume size distribution from the
sample TiO2 particles in water is
shown in figure 2. Here the
mechanism as described in equation
4 is used to back-calculate the
distribution that explains the
measured intensity as seen in figure
1. The shape of the distribution is
somewhat different, and appears to
be biased towards smaller sizes. The
mean size is now 65nm with a
standard deviation of 25nm – these
values are by volume.
Figure 1: Size distribution by intensity, %intensity versus size in nm
Size Distribution by Number
The intensity size distribution from the
sample TiO2 may also be transposed
into a number distribution, as shown
in figure 3. The shape of the
distribution is markedly different, and
definitely enhances the magnitude of
the smaller size bands. The mean
size obtained from this representation
is 50nm with a standard deviation of
14nm – these values are by number.
Table 1: Mean size and standard
deviation obtained from a sample of
TiO2 in water. The values depend on
the distribution chosen.
Mean [nm]
Width [nm]
Intensity
87
32
Volume
65
25
Number
50
14
Which is the Correct Size?
The easiest answer to this question is
that all three are correct. As long as it
is clearly stated how the size was
obtained and what distribution
representation was considered, any of
the size distribution provides useful
information about the sample.
For practical applications it is
worthwhile noting that comparison of
different samples should always be
done with the same distribution
analysis. In other words, two different
samples, measured by light
2
Figure 2: Size distribution by volume, %volume versus size in nm
Figure 3: Size distribution by number, %number versus size in nm
scattering, should only be compared
by analyzing the same distribution –
preferably the intensity as those data
are closest to what is really
measured, and do not involve any
assumptions about the Mie scattering
function. For small (<300nm)
Zetasizer Nano technical note
particles, and those where the
parameters for the Mie function are
well-known, reliable volume and
number distributions can be
compared and overlaid, provided the
‘raw’ data quality is good enough, so
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that the distribution conversion
intensity Æ volume Æ number is valid.
Comparison with Other Techniques
When comparing light scattering data
with other sizing techniques, the
detection mechanism of those
technologies must be taken into
account. For example, a UV or RI
detector coupled to chromatography
detects the concentration of material
in the eluted phase. Thus, the
distribution obtained from such a
measurement is effectively a
concentration, or in other words a
volume or – for the case of a
homogeneous population of material
of common density – a mass
distribution. Consequently, the volume
distribution from light scattering would
compare best with UV or RI
chromatography data.
In (electron) microscopy particles are
displayed as numbers in each size
bin, i.e. the obtained average size is
one which is closest related to the
number distribution from light
scattering.
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Zetasizer Nano technical note
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