3.1. Characterization of Non-stabilized AZO Nanocrystal Dispersions
The aim in this section is to investigate the aggregation behavior, the size range of the stabilizable primary particles, and its mass-fractal structures of non-stabilized
nanocrystals being able to make important predictions about the stabilization behavior of
. In this context, the particle structure and the fractal properties of
nanocrystals in an ethanol-based dispersion after synthesis, particle preparation (see
Section 2.1), and before stabilization are examined in more detail using
,
, and
.
Figure 1 shows the
curve (black dots) of a non-stabilized initial
dispersion with a gravimetrically determined initial particle mass concentration
, wherein
is plotted twice logarithmically over
.
In
Figure 1, the qualitative course of
shows the presence of three structural levels
within the particle system examined depending on the
-range:
(dotted) in
,
(dashed) in
and
(dotted/dashed) in
. To each level, one local Guinier area (curved line) can generally be assigned from Equation (5) including the associated local Power-Law (straight line) from Equation (7). The Guinier area of
, indicated by the Power-Law of
and expected in the q-range
, was not detectable due to the lower resolution limit
of the
camera used. Therefore, the Unified-Fit model according to Beaucage from Equation (4) was approximated with a high degree of certainty (
) to the fully resolvable levels
and
of the
curve in
Figure 1, resulting in important structure parameters of the non-stabilized
system such as the two local radii of gyration
,
as well as the local powers
,
. By assuming spherical structures within all levels
, the respective volume-weighted geometric mean diameters
and
were calculated with
. The
image in
Figure 2a is used to assign the structural parameters of the various levels from the Unified-Fit model obtained with
to the actually existing particle structures of the examined non-stabilized
nanocrystals.
Figure 2a shows hexagonally shaped mesocrystalline particle structures of
(solid line) with remaining internal grain boundaries (dashed line) arising due to non-classical crystal growth with oriented agglomeration processes during the synthesis process, as we reported previously [
16,
17]. Here, the mesocrystals are arranged in superordinate aggregate structures with clearly visible phase boundaries, resulting in an average mesocrystal size of
from a statistical imaging evaluation of approximately
mesocrystals using
under the assumption of spherical particles. The order of the size of the present aggregate structures of
mesocrystals could be estimated on the basis of a log-normal particle size distribution fit (
using
data in
Figure 2b to a volume-weighted mean sphere-equivalent diameter
, which is more than
times larger than the
-based mesocrystal size. Using the
and
results in
Figure 2, it is possible to clearly assign the structural levels, obtained from
in
Figure 1 to the actual particle levels of the investigated
nanoparticle system. By comparing the respective orders of magnitude, the local Guinier fit of
with
can be assigned to the mesocrystal level with
), while
with
corresponds to the visible substructures of the incompletely fused mesocrystals on primary particle level in
Figure 2a. The non-resolvable Guinier range of
in
Figure 1 can be solely associated with the aggregation level of the
mesocrystals having the size
. Based on the currently known particle hierarchy of the non-stabilized
dispersion, fractal properties can be derived for the structure levels
and
using the power laws obtained from the Unified-Fit model in
Figure 1. While the mass fractals in
with
(see Equation (8)) describe a high relative aggregation density in the range
[
31],
contains surface fractals with
(see Equation (9)), indicating smooth crystal surfaces on primary particle level in accordance to our prior work [
16]. Since
could not be fully analyzed due to the limited resolution of our
-camera, a precise determination of the fractal properties using the Unified-Fit model is not possible. However, based on the
and
data, mass fractals in the range
could be generally predicted from the aggregation behavior of the
mesocrystals in
Figure 2.
Hence, the existence of three structural levels within the non-stabilized particle system was verified using predominantly measurement technology. In addition to the precise assignment of the individual levels to the present system, their magnitude and fractal properties could also be estimated and additionally validated by and . The knowledge gained here results in comprehensive objectives for the subsequent stabilization process: the aggregate structure in of approximately is to be broken up and is to be stabilized on the mesocrystal level to a mean size of approximately maximum , while no post-synthetic stabilization is expected at the irreversibly intergrown primary crystal level of approximately .
3.2. Qualitative Dispersion Stability Criteria of Stabilized AZO Nanocrystals
The stability in the gravitational field is an important qualitative stability criterion of dispersions utilized for the production of particulate thin films, since sedimentation processes can cause inhomogeneities in the layer formation during coating and drying processes. In this context, sterically stabilized
dispersions with
(see
Section 2.2) were examined by time-resolved sedimentation analysis in the Earth’s gravity field using our
laboratory camera. Assuming that all quantities except
are constant in Equation (3) during a sedimentation process, the linear relationship
allows conclusions to be drawn about the relative temporal change in
by normalizing the scatter intensity to the maximum value at constant
. For this purpose, the relative
of stabilized
dispersions using
in the initial concentration range of
were recorded and finally plotted in
Figure 3a over a sedimentation time period of
.
Irrespective of
, the curves show a decrease in
with progressive sedimentation time up to a maximum loss of approximately
within
at
. Thus, an ongoing sedimentation process in the Earth’s gravity field could be observed using
despite previous stabilization. To verify single grain sedimentation according to Stokes for the investigated particle concentration range of
, the solid volume concentration
can be calculated from the solid mass concentration
using Equation (11) by assuming ideal spheres having the particle density of pure zinc oxide
.
The volume concentrations calculated from Equation (11) for the samples used in
Figure 3 are compared in
Table 1.
In the literature, the Stokes approach applies i.a. to
, wherein the sedimentation behavior is considered independent of the solid volume concentration [
32,
33]. This fact could explain the similar sedimentation behavior observed for the initial concentration range of
in
Figure 3a, whereas the sedimentation rate of
with
is significantly increased. A slight increase in concentration relative to
leads to the formation of particle clusters, which result in a lower sedimentation resistance leading to the previously observed faster sedimentation compared to the Stokes approach [
34]. However, a further increase in concentration would in turn cause a slowdown of the sedimentation due to growing interactions of counter flowing fluid particles, which is considered in the approach of Richardson and Zaki [
35]. Consequently, the sedimentation process observed here within a time period of
can be attributed to an unstable phase fraction within the stabilized
dispersion consisting of aggregate structures that clearly exceed the order of magnitude of the primary crystal and the mesocrystal levels with
, as previously obtained in
Section 3.1. After separation of the unstable phase by centrifugation at the defined separation grain limit of
at the previously observed mesocrystal level using Equation (2), no further sedimentation process could be detected in
Figure 3b for all stable
phases, consisting solely of
and
. The fluctuations of the values in
Figure 3b within the range of approximately
are due to the increasing influence of the Brownian molecular motion on the particle movement in the nanoscale range below
. Finally, the long-term stability of the stable phase over a period of
was successfully demonstrated by
analysis after separating the unstable proportion in the centrifugal field.
As a further qualitative stability criterion regarding optimal application properties of particulate thin films, the highest possible particle occupancy density within the thin film must be ensured by stabilization at the lowest possible structural level of the
system. After the previously shown successful separation at the aggregate level, a structural investigation of the stable
phase is carried out using
in analogy to
Section 3.1. In this context,
Figure 4 shows the
curve of the stable
phase from the initial
dispersion in
Figure 1, including the approximated Unified-Fit model (
) according to Beaucage from Equation (4) with its respective local Guinier- and Power-Laws of two structural levels
(dotted) and
(dashed).
Here, the Unified-Fit results in
with
for
and
with
for
. By comparing these parameters with the non-stabilized
system in
Figure 1 from
Section 3.1, for
and
they are all in the same order of magnitude, whereby they can be clearly assigned to the already known primary crystal (
) and mesocrystal (
) levels having similar fractal properties. This fact shows that the essential structural properties of the
system, such as size and fractal properties, were not significantly influenced by steric stabilization using
. The presence of the third structural level, which was originally assigned to the aggregate structure in
Section 3.1, is no longer detectable for the stable phase in
Figure 4, indicating a successful steric stabilization of the
system at the mesocrystal level
. Obviously, the stabilization at the primary crystal level
could not be achieved due to irreversible adhesions and intergrowth during synthesis, as expected and reported in our previous works [
16,
17]. Possibly, the intergrowth of the primary crystals into highly-ordered, regularly shaped hexagonal
mesocrystals, having a high aggregation density with
, could have a positive effect on the particle occupancy density and thus on the application properties of the final thin films (e.g., lower interfacial resistances), since a comparable density could probably not be generated by self-arrangement of stable particles during the coating and drying processes by stabilization at the primary crystal level alone. To validate the
result in
Figure 4 with respect to the successful stabilization at the mesocrystal level, the volume-weighted size density distribution of the stable
phase from
data in
Figure 5 is used, which is additionally compared to the non-stabilized
dispersion in
Figure 2b.
Due to the stabilization, on the one hand, the log-normal particle size distribution of the stable phase shows a clear shift towards a smaller mean sphere equivalent diameter from originally
to
), being in very good agreement with the mesocrystal level
from
data (
) in
Figure 4 and
data (
) in
Figure 2a. On the other hand, the significant reduction of the density distribution width is a further qualitative indication for the removal of aggregated structures. To summarize this section, the qualitative stability criteria of the stable
dispersion using
offer very good prerequisites for producing homogeneous, particulate
thin films, having the highest possible particle occupancy density, both because of the long-term stability towards sedimentation and due to the stability at the minimum possible mesocrystal level
of approximately
.
3.3. Quantitative Dispersion Stability Criteria of Stabilized AZO Nanocrystals
In addition to the previously investigated qualitative stability criteria, a quantitative evaluation of the stability success by non-invasive determination of the particle concentration of
, both in the non-stabilized system and in the stable phase using
, is reported in this section. Such a quantification is necessary since the particle concentration in the stable phase also significantly influence the layer thickness depending on the selected coating process, which in turn affects their application properties. The stability success
is introduced in Equation (12), which relates the particle concentration in the stable phase
to the initial concentration
, enabling a quantitative evaluation of the particle fraction successfully transferred from the non-stabilized initial
system into the stable phase using
.
For the non-invasive quantitative determination of particle concentrations by means of
, the local Guinier-law in Equation (5) from the Unified-Fit model is used, resulting in Equation (6) and the linear relationship
. Since most of the parameters in Equation (6) are empirically difficult to determine, a theoretical prediction of concentrations based on measured
from scattering curves is limited. Consequently, a calibration of our
laboratory camera was necessary for the
system, describing the linear relationship between
and
from Equation (6) by leaving all particle properties, obtained in
Section 3.1 and
Section 3.2, constant. For this purpose, stable
phases in the range of
were gravimetrically evaluated and their corresponding scattering curves at the mesocrystal level
(
) including corresponding Unified-Fit approximations (
) are shown in
Figure 6a.
Here, the scattering intensity in the Guinier area of
increases with rising particle concentration for constant particle properties (arrow). Based on the Unified-Fit models in
Figure 6a, the exponential pre-factors
of
were determined for each concentration from the respective local Guinier-Law in Equation (5) and show a linear correlation concerning the
concentration with a very high coefficient of determination (
) in
Figure 6b. This linear relationship, confirmed by Equation (6), shows the validity of our
laboratory camera used. Using linear regression, a calibration function could be obtained for the non-invasive determination of the
concentration
by means of
at
from
data in Equation (13).
This calibration function can be generally used for the structural level
in the stable as well as in the non-stabilized
phases, since the
technique is capable of providing this information from Guinier areas independent of the aggregation state. To verify the validity of Equation (13) also for the non-stabilized
phase, after determining the exponential pre-factor of
with
from the Unified-Fit model of the non-stabilized initial
dispersion in
Figure 1 using
, the initial concentration
could be calculated, which corresponds very well to the gravimetric result (
) in
Section 3.1. Similarly, extracting
of the stable
phase from
Figure 4, the corresponding concentration of the stable phase
could also be calculated using Equation (13). Finally, with both concentration values obtained here from
data, a stabilization efficiency of
was derived from Equation (12) for the
system investigated in this work.
To summarize this section, a new method for quantitative determination of the concentration, both for the non-stabilized phase and for the stabilized phase using is successfully demonstrated, leading to an evaluation of the quantitative stability criterion . On the one hand, the non-invasive X-ray scattering offers an advantage to the invasive, non-reversible gravimetric method concerning the further processing of the stable phases into functional thin films. On the other hand, covers a significantly higher concentration range, being particularly relevant for coating applications, for determining the concentration of particulate systems in contrast to, e.g., UV-Vis methods, in which even concentrations of could no longer be described using the Lambert–Beer law.