Having studied the initial state of the precursor solution at 25 °C, we then focused on elucidating the formation of ZnO particles induced by heating. Typically, the reaction is carried out at 60 °C, as ZnAc₂ hydrolyzation and mass transport during the ZnO formation are accelerated at this temperature. Herein we chose to study the ZnO kinetics at T = 58 °C and 63 °C. The reason for that is the significantly slower deposition below 50 °C, as confirmed in preliminary studies, while the upper limitation for the reaction temperature is given by the boiling point of methanol at 65 °C. Figure 2 shows the time-dependent slit-smeared USAXS curves starting from 25 °C (t = 0 min) followed by rapid heating to T = 58 °C (a) or 63 °C (b).

During heating and temperature equilibration the precursor solution to T = 58 °C (Figure 2a), which took about 10 min, the scattering curves are dominated by the scattering of the polymer network with the above-mentioned features of a Guinier region at low q, followed by a power law decay. The slight decrease in the intensity at low q might relate to the heating with an inhomogeneous temperature distribution inside the solution. Potential over subtraction of solvent scattering, which was measured at T = 58 °C, might be another explanation for this decrease.

Subsequently, for reaction times t > 13 min, the scattering intensities at low q increase continuously, which can be attributed to the form factor contribution of newly formed ZnO species, while above q ≥ 0.1 Å⁻¹, the scattering is dominated by the PVP scattering. As the reaction progresses, the form factor minimum becomes increasingly apparent and shifts to lower q values, suggesting the growth of the ZnO species. Finally, the increase in the low q scattering intensity levels off, indicating that the precursor ZnAc₂ has been consumed; therefore, the data acquisition was stopped after 87 min. Contrary to what Lipowsky et al. [33] have seen when ZnO films are formed in the presence of PVP via CBD, no additional contribution of needle-like species with a length scale between 20 and 50 nm was observed in the scattering curves.

A similar behavior of the scattering curves can be seen during the formation of ZnO particles at T = 63 °C in Figure 2b. Compared to the formation at 58 °C, the decrease in the forward scattering observed in the curves recorded at the early stages of the process is even more pronounced, most likely due to over subtraction of the background, measured at the higher final temperature. The earlier appearance of the form factor signature and the earlier leveling off of the increase in forward scattering after 75 min clearly indicate that the reaction is faster at higher temperatures. This can be explained by the enhanced mass transport with increasing temperature, which accelerates the formation of ZnO.

A comparison of the first (t = 0 min, T = 25 °C) and the last (t = 87 min, T = 58 °C) time-resolved USAXS/SAXS scattering curves are shown by means of the ZnO particles’ formation at T = 58 °C (Figure 3, left). This comparison highlights the presence of the additional form factor contribution (q ≤ 0.08 Å⁻¹) in the scattering curve recorded at the process end (t = 87 min). A closer look also reveals a weak shoulder around q~0.15 Å⁻¹. One exemplary image of complementary TEM studies taken from a separately prepared solution after an analogous heating procedure starting from 25 °C to 58 °C is shown in the center of Figure 3. Therein, two species are visible: smaller ones with a radius of R_ZnO,TEM = 1.3 ± 0.7 nm as well as agglomerates of those with an average radius of R_{ZnO-cluster,TEM} = 8.5 ± 1.4 nm. We will refer to the smaller particles as individual ZnO particles and to the larger ones as clusters of ZnO particles. Strikingly, the diameter of the individual ZnO particles is only slightly smaller than the mesh size within the PVP network (ξ_PVP ≈ 2.9 nm), indicating initial growth of the ZnO particles in the network cavities before cluster formation occurs. Furthermore, the average cluster radius is only slightly larger than the Guinier radius of PVP in the precursor solution with R_g(PVP) = 6.3 ± 0.3 nm.

Comparing the final cluster radius obtained from the USAXS/SAXS scattering curve analysis (R_{ZnO-cluster,f} = 10.5 ± 1.9 nm) to TEM, one may assume that the clustered particles are surrounded by a PVP layer that is not visible in the TEM, similar to what has been demonstrated for silica-polymethyl methacrylate (PMMA) nanocomposites by SANS [81]. Since we were not able to study such a layer independently with SANS, we used a simple approach for the quantitative analysis of the time-resolved USAXS/SAXS data. In this approach, the scattering contribution of the PVP network described by the flexible cylinder model [70,71] I(q)_flexC was additively combined with the scattering contributions of the individual ZnO particles I(q)_ZnO and their clusters I(q)_ZnO-cluster, each modeled by the form factor of polydisperse spheres [80], as well as a Gaussian I(q)_G for the intermolecular C-C chain interaction peak [66] at high q. Thus, the total intensity is given by where with ϕ_ZnO,i and V_ZnO,i being the volume fractions and average volumes of particles and particle clusters, respectively. $∆{\rho }^2$ is the scattering length density difference between ZnO (ρ_ZnO = 44.8 × 10⁻⁶ Å⁻²) and methanol (ρ_MeOH = 7.5 × 10⁻⁶ Å⁻²) and P(q,R_ZnO,i) = ${\left(\frac{3J_1\left(q{R}_{\mathrm{Z}\mathrm{n}\mathrm{O},\mathrm{i}}\right)}{q{R}_{\mathrm{Z}\mathrm{n}\mathrm{O},\mathrm{i}}}\right)}^2$ the form factor of a sphere with the radius R_ZnO,i. To account for size distribution effects, the form factor was convoluted with a Gaussian size distribution functions f(R_ZnO,i, R_ZnO,i,f, σ_i) which yields the mean radius R_ZnO,i,f and the width σ_i.

Using this approach, we were able to describe the recorded time-resolved scattering curves almost quantitatively, as shown in the left panel of Figure 3. In this regard, the solid line for t = 0 min corresponds to the final fit following Equation (5), with a PVP chain contour length of L = 80 ± 1 nm, a Kuhn length of b = 3.3 ± 0.3 nm, and a cross-section radius of R_C = 0.27 ± 0.01 nm. For the solid line at t = 87 min (T = 58 °C), Equation (6) was used. Here, the previously determined characteristics of the PVP chains with a slightly adjusted Kuhn length of b = 4.0 ± 0.3 nm were applied, while the polydispersity p_i = σ_i/R _ZnO,i0 = 0.18 of both the ZnO particles and clusters was kept constant. Hence, the analysis of the USAXS/SAXS curve measured after the termination of the reaction yields average radii of R_ZnO,f = 1.5 ± 0.3 nm and R_{ZnO-cluster,f} = 10.5 ± 1.9 nm for the individual ZnO particles and their clusters, respectively. While the size of the individual ZnO particles matches that from the TEM images within the measurement error, the size of the clusters is slightly larger, most probably due to a surrounding PVP layer, as discussed before. The respective contributions of Equation (6) to the final fit are shown in Figure S3 in the Supplementary Materials.

Also, for the ZnO formation performed at T = 63 °C, starting from T = 25 °C, the final ZnO particle and cluster size obtained from TEM images (R_ZnO,TEM = 1.7 ± 0.5 nm, R_{ZnO-cluster,TEM} = 9.7 ± 1.9 nm), shown exemplarily in Figure S4 in the Supplementary Materials, are in good accordance with that from USAXS/SAXS (R_ZnO,f = 1.5 ± 0.3 nm, R_{ZnO-cluster,f} = 11.8 ± 2.1 nm). Comparing the size of the clusters obtained from CBD at the two different temperatures, the temperature increase leads to slightly larger ZnO clusters, which is related to accelerated kinetics at higher temperatures. Hereby, a higher number of individual ZnO particles might be formed, which are subsequently consumed faster by the formation of the clusters.

Figure 4 shows selected time-resolved USAXS scattering curves recorded at T = 58 °C (Figure 4a) and T = 63 °C (Figure 4b), which are almost quantitatively described by the combination of the flexible cylinder model [70,71] and the two spherical form factor contributions [80]. It should be noted that for t < 13 min, we could not obtain any information about the ZnO formation due to the dominance of the PVP scattering contribution.

The results of the quantitative analysis of the scattering data according to Equation (6), with the free fitting parameters being the mean radii from the ZnO particles, R_ZnO,f, and that from the clusters R_{ZnO,cluster,f} as well as their volume fractions ϕ_ZnO and ϕ_ZnO-cluster, are shown in Figure 5 and Figure 6. While Figure 5 shows the radius (R) and R³(t) of the ZnO nanoparticles and ZnO clusters as a function of time for T = 58 °C and T = 63 °C, the time evolution of the respective volume fraction for T = 58 °C is shown in Figure 6.

As can be seen in Figure 5a, the size of the formed ZnO species for T = 58 °C remains almost constant during a short period (~between t = 13 and 18 min). Thereafter, cluster growth occurs, where the average size of these clusters initially increases rapidly before slowly approaching the final size due to the consumption of zinc precursor. Performing the reaction at T = 63 °C, a similar but steeper growth profile was obtained, during which the clusters are consistently larger. For comparison, the ZnO particle size appeared to remain constant and was kept fixed at R_ZnO,f = 1.5 ± 0.3 nm (dashed line in Figure 5a).

Interestingly, when plotting R³_{ZnO-cluster,f} as a function of time, as shown in Figure 5b, a linear increase starting from t~20 min is observed, which levels off to a plateau. The former region can be associated with the growth phase of the ZnO clusters by agglomeration of individual ZnO particles. Thus, a reasonable description of the growth phase is given by with k being the temperature-dependent growth rate constant, and t_cluster being the time after which clusters start to grow t_cluster = t − t_cluster,0 and R_ZnO,f the individual ZnO nanoparticle radius. The analysis of R³_{ZnO-cluster,f}(t) with Equation (8) (dotted lines in Figure 5b) yielded k(T = 58 °C) = 0.44 ± 0.01 nm³/s with t_cluster,0 (T = 58 °C) = 23 min and k(T = 63 °C) = 1.05 ± 0.01 nm³/s with t_cluster,0 (T = 63 °C) = 18 min. For comparison, much smaller rate constants in the order of ~10⁻³ nm³/s were found for the formation of ZnO particles in propanol, without the addition of a polymer such as PVP [82,83].

The large growth rates obtained here can be explained by the occurrence of depletion forces [84,85], as observed during ZnS particle formation in the presence of PVP, where cluster growth via depletion forces appeared as soon as the particle surface was too crowded by PVP [86].

While large parts of the ZnO cluster growth phase can reasonably be described by this power law approach, neither the nucleation phase nor the growth deceleration can be described. Furthermore, the occurrence of depletion forces is not considered in this model. Thus, to describe the sigmoidal time evolution of R³(t), the two-step Finke–Watzky (FW) model was used. In this model slow, continuous nucleation $\left(A\stackrel{{ k}_{1 } }{\to }B\right)$ is followed by autocatalytic growth $\left(A+B\stackrel{{ k}_{2 }}{\to }2B\right)$ with the nucleation (k₁) and growth (k₂) constants [62,63,64]. Here, the cube of the radius as a function of time is given by [65] where R_f corresponds to the final cluster size, and [A]₀ to the starting concentration of Zn²⁺. The induction time, here related to the time after which agglomeration of the ZnO particles into clusters is accelerated, is defined by the intersection of the initial and maximal slope and can be determined according to [65]:

Furthermore, the ratio of the growth to nucleation rate constant $S=\frac{k_2}{k_1 }$ (M⁻¹) is a measure of kinetic control, indicating whether nucleation of new particles or growth of the existing clusters is preferred. Thus, larger S values yield narrow size distributions. In addition, larger ratios were shown to be related to larger clusters [87]. Note that the two-step FW model can be extended to include bimolecular aggregation (B + B→C) [88] and autocatalytic agglomeration (B + C→1.5 C) [89,90]. However, this was not performed here to keep the number of free-fitting parameters at a minimum.

As can be seen from the solid lines in Figure 5, the FW model provides a quantitative description of the time evolution of the ZnO cluster size for t ≥ t_ind. The systematic mismatch of the model at t < t_ind could be due, on one hand, to a nucleation rate that is not constant during the heating process but increases strongly. On the other hand, the spherical form factor used in the analysis of the scattering data represents only an approximation, especially for clusters consisting of a few ZnO particles. The obtained parameters, summarized in Table 2, reflect what was already evident from the USAXS data: the induction time shortens from t_ind = 27 min at T = 58 °C to t_ind = 20 min for T = 63 °C, while the growth rate constant k₂ increases considerably by a factor of 50% at higher temperatures to k₂[A]₀ = 0.17 s⁻¹. The nucleation rate, however, does not change. Consequently, the ratio of the growth-to-nucleation rate constant S is higher for T = 63 °C, which also confirms the previously mentioned assumption of a faster growth rate for higher temperatures.

The FW model assumes continuous nucleation of ZnO particles. This can be indeed observed in the volume fraction profile of the ZnO particles () for T = 58 °C in Figure 6, where an initial increase is followed by reaching an almost constant value, while the volume fraction of the cluster () follows a sigmoidal profile. These results suggest that new ZnO particles are formed continuously throughout the reaction, feeding the clusters, which in turn grow simultaneously. Hence, at intermediate times, an equilibrium is established between ZnO particles’ formation and consumption.

Since R³ is proportional to the volume fraction, Equation (9) can also be used to model ϕ_ZnO-cluster(t)_I(q) () (see solid line in Figure 6), yielding nearly identical values for the FW parameters as compared to the R³(t)-data with k₁(T = 58 °C) = 8.5 × 10⁻⁴ 1/s, k₂[A]₀ (T = 58 °C) = 0.11 1/s with t_ind(T = 58 °C) = 29 min.

In a complementary approach, we determined the volume fraction of ZnO clusters ϕ_ZnO-cluster from the shape- and size-independent scattering invariant Q of the time-resolved desmeared USAXS scattering curves for q ≤ 0.04 Å⁻¹. It should be noted that the invariant was only determined in the low q-part, since on the one hand, the scattering in the large q-part is dominated by the scattering of PVP, and on the other hand, the USAXS data are noisy in this range. To account only for the scattering of the ZnO clusters, the scattering invariant determined for t = 1 min was subtracted from all other data, resulting in a reduced invariant Q*, according to where $∆{\rho }^2$ = |ρ_ZnO − ρ_MeOH |². In Figure 6, the obtained cluster volume fraction ϕ_{ZnO-cluster,inv}(t) () for T = 58 °C is plotted as a function of time. Comparing the cluster volume fractions obtained from the model analysis of the USAXS data and the invariant, i.e., ϕ_{ZnO-cluster,I(q)} and ϕ_{ZnO-cluster,inv}, slightly lower values are found for the latter, which might be a consequence of the finite q-range used for the invariant determination as well as the approximation of spherical clusters with compact packing of ZnO particles and a homogeneous scattering length density distribution.

The profiles of the cluster volume fraction ϕ_{ZnO-cluster,inv}(t) were also described using the FW model (in Figure S5 of the Supplementary Materials, the profiles obtained for T = 58 °C and T = 63 °C are shown for comparison). A comparison of the parameters (Table 2) obtained from the analysis of the R³(t)-data and ϕ_{ZnO-cluster,inv}(t)-data, respectively, reveal the same growth rate constants k₂ and induction times t_ind, which are similar within the measurement uncertainty. The smaller value of the nucleation rate constant for the ϕ_{ZnO-cluster,inv}(t)-data might also be a consequence of the finite q-range used for the invariant determination. Consequently, larger S-values in the analysis of the ϕ_{ZnO-cluster,inv}(t)-data are obtained.

To conclude the in situ study of ZnO particle formation and growth, we found that shortly after heating the precursor solution, ZnO nanoparticles of the order of R_ZnO ≈ 1.5 nm are initially formed, followed by the formation of clusters. An increase in the reaction temperature from 58 °C to 63 °C resulted in the acceleration of the reaction and the formation of larger clusters. From the time evolution of the cube of radius, a linear relationship is apparent at intermediate times. Depletion forces caused by the high volume fraction of PVP might explain the relatively high growth rate. A quantitative description of the time evolution of the ZnO nanoparticle cluster radius, the corresponding cube of the radius, as well as the volume fraction, could be achieved by the Finke–Watzky model [62,63,64,65] with slow, continuous nucleation of individual ZnO particles followed by autocatalytic growth of the clusters, as illustrated in Figure 7. From the analysis of the kinetic data, the higher ratio of growth to nucleation rate explains that slightly larger clusters are formed at the higher temperature of 63 °C.