a Schematic illustrate the in situ crystallization of perovskite nanocrystals in polymeric matrix and time-lapse fluorescence imaging of nanocrystals with SIM. b Fluorescence image of perovskite nanocrystals acquired using SIM. c, d Temporal evolution of fluorescence intensity (normalized) of a bright perovskite nanocrystal and ensembles, respectively.

Figure 1b presents the high spatial resolution SIM image of perovskite nanocrystals. Single nanocrystal can be resolved from the acquired SIM image by tracking the fluorescence intensity of individual particles. We also observed fluorescent blinking of these nanocrystals with time-dependent fluorescence intensity (Supplementary Figs. 4, 5 and Movie 3). The fluorescence blinking hardly changes the overall growth time of an individual nanocrystal. The individual existence of perovskite nanocrystals in polymeric matrix was also confirmed by high-resolution TEM observation (Supplementary Fig. 6). Figure 1c shows a typical time-lapse trace of fluorescence intensity (normalized) of an individual perovskite nanocrystal. The fluorescence intensity curves of individual nanocrystals exhibit “T”-shaped curves. Figure 1d shows the time-lapse traces of normalized fluorescence intensities that summarized over a number of perovskite nanocrystals on an area of 307 pixels × 307 pixels (10?m×10?m). The collective fluorescence intensity curve of about 700 nanocrystals shows a feature of “S” profile. The observed “T” and “S” shaped fluorescence intensity curves are very similar with the nucleation-growth kinetic curves of Avrami model^29,30,31, suggesting the potential of fluorescence intensity in determine the crystallization of perovskite nanocrystals.

According to previous works, the concentration of precursors is one of the key factors affecting nucleation and growth^4,28. In this work, we prepared three precursor solutions with different concentrations of 0.018 mol L⁻¹ (C₀), 0.009 mol L⁻¹ (0.5C₀) and 0.007 mol L⁻¹ (0.4C₀), and we obtained three samples with different sizes. As shown in the TEM images (see below and Supplementary Fig. 6), the average diameters of the resultant perovskite nanocrystals are 3.0 ± 0.4, 4.4 ± 0.6, and 5.1 ± 0.6 nm with standard deviation (SD) of 13%, 14% and 12%, respectively. In order to determine the relationship between fluorescence intensity and nanocrystal size, the steady-state and time-resolved fluorescence spectra of perovskite nanocrystals with different sizes were measured. A mathematical expression of fluorescence intensity related with nanocrystal radius was derived (Supplementary Note 2). The relationship between fluorescence intensity and particle size was derived based on Brus equation. By using this expression, the plot of fluorescence intensity versus time can be transformed into nanocrystal radius versus time, enabling us to analyze the nanocrystal growth via temporal fluorescence intensity.

Crystallization, typical of first-order phase transition from solution to crystalline solid, is a complicated process with nucleation and growth stages^32,33. The theory of crystallization can be divided into kinetics and thermodynamics. The former concerns kinetics rate of nucleation and growth, while the latter focuses on thermodynamic characteristics such as free energy and critical size. According to crystallization theory, these two stages are likely to time overlapping (coupling) with each other during formation of ensemble nanocrystals^34,35. Based on the mathematical relationship between fluorescence intensity and nanocrystal radius, the recorded “S” shaped curve of ensemble nanocrystals shows the collective feature of perovskite formation, which illustrates the coupling between nucleation and growth. As shown in Fig. 1d, it appears as an initial slow kinetics rate then followed by a rapid crystallization that was induced by continuous nucleation and rapid growth of nanocrystals, and next the kinetics rate slows down and becomes constant with the consumption of perovskite precursors. In Fig. 1c, the “T” shaped curve of an individual nanocrystal shows the growth kinetics of a fixed nucleus beginning with a fast growth rate. We first simulate the growth kinetics of perovskite nanocrystals using traditional Avrami equation and Fick’s first law-based growth model. Subsequently, a modified thermodynamics model was developed to simulate the coupling between nucleation and growth.

According to Avrami equation, nucleation and growth can be described by correlating the transformation fraction and time using Eq. (1)

where ? represents volume fraction of formed crystal, k is overall crystallization rate constant and ?0 is time of initial crystallization. n represents the crystallization exponent with a wide range of values depending on nucleation rate, growth dimensionality and growth mechanisms (Supplementary Note 3). Because of the size dependent fluorescence intensity of perovskite nanocrystals, the volume fraction can be estimated by the fluorescence intensity³⁶. Supplementary Fig. 10 show the fitted “S”-shaped crystallization kinetic curves of perovskite nanocrystals at different precursor concentration. The extract crystallization exponent n varied from 1.8 to 4.0, which shows the coupling between nucleation and growth of perovskite nanocrystals at different precursor concentration. n = 4 can be explained to the finite size effects in non-uniform nucleation, and/or three-dimensional anisotropic growth and/or size-dependent growth effect in transient nucleation^37,38. The analysis at single-particle level can isolate growth of individual nanocrystals to illustrate the coupling between nucleation and growth. By fitting the growth kinetic curve using Avrami equation, the growth characteristics of individual nanocrystals can be obtained (Supplementary Figs. 11, 12). The extracted Avrami exponent n of 1.5 is characteristics of diffusion-controlled three-dimensional (sphere) growth.

Growth kinetics of individual nanocrystals

In situ fluorescence spectra of perovskite nanocrystals was measured by spin-coating the precursor solution on glass substrate. The SIM system can detect the signal of perovskite nanocrystals with wavelength over 500 nm. As shown in Fig. 2a, the fluorescence emission peak of perovskite nanocrystals shifts from ~500 nm to 520 nm corresponding to size increasing. Figure 2b shows the plot of the derived function relating fluorescence intensity and particle radius in a range of 0.2–3.0 nm. The temporal evolution of nanocrystal radius is in good agreement with LaMer diffusion-controlled growth theory which accounts for growth of an individual nanocrystal (Supplementary Note 4)^7,39. Figure 2c shows the experimental temporal evolution of an individual nanocrystal radius (purple dots) and theory fitting using LaMer model (orange line). The fitting parameter of diffusion coefficient is on the order of magnitude of ~10⁻¹³ ?2/? (Supplementary Table 3), which is close to that of water in PVDF polymer (~10⁻¹² ?2/?)⁴⁰. Due to the existence of polymer in precursor solution, the mass diffusion rate is much slower than reported diffusion coefficient of perovskite precursors in organic solvent (~10⁻¹⁰ ?2/?)⁴¹. The growth of individual nanocrystals shows a fast growth at the beginning followed by a slow growth with an obvious transition of growth rate. According to Sugimoto’s size-focusing theory, the narrow size distribution is significantly affected by growth kinetics. Figure 2d shows the histogram of growth time distribution of individual perovskite nanocrystals, and as observed that the fast growth process takes only a few seconds (4–8 s). After the fast growth stage, a slow growth stage was followed with time duration of tens of seconds. A few represented growth kinetics curves of individual perovskite nanocrystals with different size are provided in Supplementary Fig. 13. Based on the above results, a fast growth at the beginning followed by a slow growth is critical for the formation of monodisperse perovskite nanocrystals, even the growth is coupled with continuous nucleation. Figure 2e shows the TEM images of the perovskite nanocrystals at precursor concentration of C₀. The perovskite nanocrystals have particle size of 3.0 ± 0.4 nm with SD of 13%. The high-resolution TEM image of an individual perovskite nanocrystal presents obvious crystal lattice fringes with a spacing of 0.20 nm corresponding to (200) plane. The Fast Fourier Transform (FFT) image with obvious crystal diffraction pattern also indicates a good crystallinity of the perovskite nanocrystals.

Fig. 2: Monitoring the growth of individual perovskite nanocrystals and growth kinetics simulation.

a In situ fluorescence spectra of perovskite nanocrystals in polymeric matrix. Insert is plot of Brus equation. The red line with arrow was plotted to indicate the size increasing. b Plot of fluorescence intensity (log) vs. particle radius of perovskite nanocrystal using the derived equation. Insert is flowsheet of the relationship (fluorescence intensity and particle radius) derivation. c Temporal evolution of an individual perovskite nanocrystal radius at precursor concentration of C₀. The blue line represents the calculated curves from the diffusion-controlled growth model. The model is schematically shown in insert. d Statistical histogram of the fast- and slow-growth time duration of the individual perovskite nanocrystals. 119 nanocrystals were counted. 80% of the final size was defined as the segmentation point for the fast and slow growth. e TEM image of the perovskite nanocrystals at precursor concentration of C₀ and histogram of the particle size distribution (insert), as well as the high-resolution TEM image of an individual perovskite nanocrystal and the corresponding FFT image.

Coupling between nucleation and growth

Figure 3a schematically illustrates the comparison between continuous nucleation and burst nucleation. Generally, nuclei start to form when the precursor solution reaches a critical concentration (minimum supersaturated concentration ?min∗). For burst nucleation, a large number of nuclei form at once and all the nucleus grow simultaneously to produce monodisperse particles with narrow size distribution. For continuous nucleation the nuclei number continuously increases until the solution concentration drops below the critical concentration. Once a nucleus forms it grows rapidly and then the growth rate gradually slow down. The continuous nucleation can be analyzed by counting the evolution of particle number on a specific area with time prolonging. Figure 3b shows the temporal evolution of particle numbers recorded on a fixed imaging area (10?m×10?m) for nanocrystals at different precursor concentration of C₀ (red dots), 0.5C₀ (blue dots) and 0.4C₀ (purple dots). The nucleation kinetics can be varied by the precursor concentration. With the precursor concentration decreasing, the nucleation rate (??/??, ? is particle number) decreases with the nucleation time increasing and the final particle number decreasing (Supplementary Note 5). In continuous nucleation of perovskite nanocrystals, the formation time of nuclei lasts for tens of seconds. The continuous nucleation of perovskite nanocrystals was also observed in a different polymeric matrix (Supplementary Note 6). Due to diffusion-controlled growth, perovskite nanocrystals can achieve nearly monodisperse size distribution. This is different from the burst nucleation of LaMer model that a large number of nuclei are initially formed, followed by a slow diffusion-controlled growth in a reaction system. Figure 3c shows that the perovskite nanocrystals have particle size distribution with SD of ~14–17% at nucleation time prolonging of 19^st, 21^th, 24^th, 28^th, and 47^th second (as noted by the orange dot in Fig. 3b).

Fig. 3: Observation of continuous nucleation and comparison with burst nucleation.

a Schematic illustration of monodisperse nanocrystals with continuous nucleation and burst nucleation. b Temporal evolution of particle number at precursor concentration of C₀ (red line), 0.5C₀ (blue line) and 0.4C₀ (orange line) on an area of 307 pixels × 307 pixels (10?m×10?m) in fluorescence images. c Histograms of particle radius distribution and their Gauss fitting at different nucleation time prolonging at precursor concentration of C₀.

The coupling between nucleation and growth was further analyzed by simulating the temporal evolution of free energy during crystallization. According to the well-known Gibbs free energy theory, total free energy of an individual particle can be described by Eq. (2)^32,42

where ? is the particle radius, σ is surface free energy per unit area and Δ?? is bulk free energy per unit volume. In the crystallization process of a reaction system, the free energy of all the nanocrystals can be divided into nucleation free energy and growth free energy. The calculation of the nucleation free energy of the nanocrystals only needs to consider a fixed radius (critical nucleation radius) for all the particles even at different nucleation time prolonging. The calculation of growth free energy is more complicated because it must take into account a variety of particle sizes at each moment considering different particle size at different growth time. A mathematical model of coupled nucleation-and-growth was built and the program was accomplished on Matlab software (Supplementary Note 7). Figure 4a shows the temporal evolution of nucleation (purple curve) and growth free energy (blue curve) of the perovskite nanocrystals at precursor concentration of C₀. The nucleation free energy, as a resisting force to crystallization, is always positive and gradually rises in a “S” profile. The growth free energy is initially positive and then negative. The negative free energy is a driving force to promote nanocrystal grow into large particle. The total free energy of the nanocrystals was also plotted and as expected it shows a critical transition time/size where the total free energy value transforms from positive to negative (red curve). After the critical transition time, the nuclei number increase rapidly and the negative growth free energy dominates the total free energy. The coupling between nucleation and growth shows a time overlap of about 60% for nanocrystals at precursor concentration of C₀ as shown in Fig. 4b. With the decrease of precursor concentration, the nucleation time increases, resulting in longer time overlap between nucleation and growth (Fig. 4b and Supplementary Fig. 16). Figure 4c shows the enlarged view of the free energy curves in the dashed red frame in Fig. 4a. The total free energy has a positive-to-negative transition time at 3.8 s corresponding to a transition radius (??) of 1.2 nm for nanocrystal ensembles in one reaction system. After this transition point, the size distribution of the perovskite nanocrystals becomes narrower. In comparison with the free energy of continuous nucleation, the free energy of growth is dominated in the crystallization of perovskite nanocrystals.

Fig. 4: Simulation of the coupled continuous nucleation-and-growth of perovskite nanocrystals.

a Temporal evolution of perovskite nanocrystal free energy at precursor concentration of C₀. Nucleation free energy (purple curve), growth free energy (blue curve) and total (mathematical sum) free energy (red curve). b Time overlap shows the coupling between nucleation and grow process of perovskite nanocrystals. c The enlarged view of the free energy curve in the red dashed frame in a. The transition size and time refer to the nanocrystal ensembles in a reaction system. d Diffusion-controlled growth rate (??/??) as a function of ? with continuous nucleation of perovskite nanocrystals.

The perovskite nanocrystals have a particle size distribution with SD < 20% (Supplementary Note 8). The narrow particle size distribution of perovskite nanocrystals in polymeric matrix was attributed to the size-dependent growth rate in diffusion-controlled growth mechanism. According to Sugimoto diffusion-controlled growth model¹², the particle size distribution becomes narrow due to the size-dependent growth rate (Supplementary Note 9). Here we simulated the growth rate with size increasing with Eq. (3)

where ?? is a constant for a certain concentration of precursor solution. By using the calculated ??, the growth rate curves were calculated. Figure 4d shows the growth rate curve of perovskite nanocrystals at ?0. The growth rate decreases continuously when particle size is larger than transition radius of 1.2 nm. The smaller nanocrystals grow faster than the larger ones resulting size focusing during nanocrystal crystallization, which is in line with the Sugimoto size-focusing model. The growth dominated crystallization in coupled nucleation-and-growth process leads to particle size narrowing.

We show that super-resolution fluorescence imaging can facilitate high-spatiotemporal-resolution observation of the nucleation and growth of perovskite nanocrystal at single-particle level. The visualization of perovskite crystallization at single-particle level provides important information to isolate the growth kinetics from the coupled nucleation-and-growth. Our finding confirms that the crystallization of perovskite nanocrystals in polymeric matrix shows a typical diffusion-controlled growth with a fast growth at the beginning followed by a slow growth. For the in situ formation of perovskite nanocrystals in polymeric matrix, the nucleation time lasts for tens of seconds. With the precursor concentration decreasing, the nucleation rate decreases and nucleation time increases. By combining the experimental observation and theory simulation, we demonstrate the evolution of free energy for nucleation and growth. In the continuous nucleation of perovskite nanocrystals, free energy of growth is dominated during the crystallization process. The work not only provides a technique to illustrate the crystallization process of nanocrystals, but also describes a coupled nucleation-and-growth model to guide the optimization of nanocrystal synthesis.