Theoretical framework and experimental methodology to elucidate the supersaturation dynamics of nanocrystal growth

Abstract

Supersaturation is the fundamental parameter driving crystal formation, yet its dynamics in the growth of colloidal nanocrystals (NCs) remain poorly understood. Here, we demonstrate an approach to characterize supersaturation during classical NC growth. We develop a framework that relates noninvasive measurements of the temporal, size-dependent optical properties of growing NCs to the supersaturation dynamics underlying their growth. Using this approach, we investigate the seed-mediated growth of colloidal Au nanocubes, identifying a triphasic sequence of supersaturation dynamics: rapid monomer consumption, sustained supersaturation, and then gradual monomer depletion. These NCs undergo different shape evolutions in different phases of the supersaturation dynamics. As shown with the Au nanocubes, elucidated supersaturation profiles enable the prediction of growth profiles of NCs. We then apply these insights to rationally modulate NC shape evolutions, decreasing the yield of impurity products. Our findings reveal that the supersaturation dynamics of NC growth can be more complex than previously understood. As our approach is applicable to many types of NCs undergoing classical growth, this work presents an initial step towards more deeply interpreting the phenomena governing nanoscale crystal growth and provides insight for the rational design of NCs.

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New concepts

Despite significant interest, supersaturation remains difficult to measure directly in colloidal nanocrystal (NC) syntheses. As such, the field of nanoscience's understanding of how supersaturation progresses during NC growth is based on models, such as the LaMer model and Finke–Watzky model. By developing and then using a methodology to noninvasively characterize supersaturation throughout NC growth, we find that even well-studied reactions, such as the seed-mediated growth of colloidal Au nanocubes, can involve multi-phase supersaturation dynamics that are more complex than previous models indicate. Moreover, we show that elucidated supersaturation profiles can be used to model, and predict, the temporal growth profiles of NCs as well as to identify supersaturation-associated shape evolutions during synthesis. Based on these insights, supersaturation can be altered in these reactions to rationally modulate NC shape evolutions. While this study focuses experimentally on Au NCs, our methodological approach is facile and can be extended to many types of NCs. Thus, this work presents conceptual and methodological advances in understanding and engineering the complex interplay of supersaturation, NC growth, and NC shape evolutions.

Introduction

Nanocrystals (NCs) are ordered materials with at least one dimension in the range of 1–100 nm and present physicochemical, electrical, magnetic, and optical properties that can be superior or unavailable to larger crystals.^1–3 For colloidal NCs, these properties depend on their size, shape, crystal habit, surface ligands, and elemental composition.^3–8 Using the mechanistic insight gained by elucidating reaction pathways, synthesis methods can grow colloidal NCs with precise control over these structural parameters,^9–11 potentiating the development of next-generation technologies in heterogeneous catalysis, chemical processing, photonics, sensing, medicine, and environmental remediation.^12–19 Hence, a key focus in the fields of chemistry, materials science, and nanoscience has been to understand how colloidal NCs grow.

Crystal formation can occur classically or nonclassically, and involves nucleation and growth processes. Primary nucleation refers to the initial formation of condensed phases from monomeric units, whereas growth comprises the integration of monomers into already nucleated crystals.²⁰ Nucleation and growth occur sequentially in crystallization reactions.²¹ On the other hand, seed-mediated syntheses spatiotemporally separate growth from nucleation.²² In classical nucleation, monomeric units aggregate stochastically in the bulk (vapor, melt, or solution) phase of the reaction and encounter a thermodynamic balance between surface energy and cohesive energy.²³ Because the surface-to-volume ratio decreases as size increases, cohesive energy overcomes surface energy above a critical radius, leading to thermodynamically stable nuclei.²³ In classical growth, monomers progressively integrate into crystal facets described by the terrace-step-kink (TSK) model.^20,23–27 In contrast, crystals nucleate or grow nonclassically when monomers first form intermediate clusters which then coalesce into nuclei or existing crystals, respectively.^28,29

While the distinction between classical and nonclassical growth is clear (progressive integration of monomers vs. discrete coalescence of intermediate clusters) and presents a general scheme to classify crystal growth, the growth mechanisms of many colloidal NCs remain unclear. That is, how monomers integrate, or how intermediates attach and coalesce, into NCs is poorly understood, based in part on limited insight into the phenomena underlying NC growth.^30,31

Supersaturation is the fundamental, non-equilibrium parameter that drives crystal formation and influences the growth mechanism, shape, exposed facets, crystal habit, size, and uniformity of NCs.^8,32–35 It is defined as

where C is the monomer concentration, C₀ is the saturation concentration of the monomer, Δμ is the chemical-potential difference between monomeric units and integrated units in the crystal, and k_BT is the thermodynamic temperature. To form NCs, synthesis formulations modulate C above the saturation (enabling growth if nuclei are present) or nucleation limits. For substrate-bound NCs, experiments can use open reactions and introduce vapor or liquid sources with fixed supersaturations to uncover mechanistic insight into growth.^35–38 Conversely, colloidal syntheses typically use closed conditions, and precursor is converted to monomer throughout the reaction (Fig. 1a). These processes are dynamic, and previous approaches have required analytical techniques, such as synchrotron-based time-resolved X-ray absorption near edge spectroscopy (XANES), and insight into the molecular structure of the monomer to analyze supersaturation during growth.³⁹

Fig. 1 Precursor-limited homogeneous crystallization of colloidal NCs. (a) Schemes of the general reaction pathways to synthesize NCs composed of noble metals, metal chalcogenides, or metal halide perovskites. M = metal, E, X = counterions, R = reactant, y = oxidation state, k_conv is the rate constant for the conversion of precursor to monomer, and k_g is the rate constant for NC formation. The pathway for perovskites was represented by the oleate injection method for ternary NCs. (b) Qualitative LaMer model of the monomer concentration [M]_i during NC crystallization, in which the monomer concentration reaches saturation and increases in supersaturation (I) before the two-step process for homogeneous nucleation (II) and growth (III).

Thus, supersaturation dynamics have yet to be quantitatively characterized during the formation of many types of colloidal NCs, hindering mechanistic understanding. The LaMer model has, since 1950, been one classical interpretation that has guided a qualitative, but incomplete, understanding of how monomer tends to progress during the crystallization of colloidal NCs (Fig. 1b).⁴⁰ The Finke–Watzky model describes another crystallization process, in which slow, continuous nucleation occurs below the LaMer nucleation limit for monomer concentration (C_nu in Fig. 1b) before autocatalytic surface growth.⁴¹ While Finke–Watzky nucleation deviates mechanistically from classical nucleation, these three-dimensional nuclei still form above a critical size described by classical theory.^20,41 Moreover, both models describe growth classically; even when growth is kinetically controlled, the NCs grow via the progressive integration of monomeric units into facets which have TSK structures.^{20,23–25,40,41} Unlike during nucleation, these models generally assume that the bulk, or local, supersaturation dynamics during growth follow a simple, steadily declining profile as the NCs grow (see stage III in Fig. 1b). However, as revealed in this study, the supersaturation dynamics of colloidal NC growth can be more complex than described by these models.

Here, we develop a framework to characterize supersaturation throughout the growth of colloidal NCs. To avoid experimental measurements that disturb the reaction, this approach noninvasively analyzes the temporal, size-dependent optical properties of growing NCs to determine quantitative profiles of the underlying supersaturation dynamics. These profiles facilitate theoretical modeling of, and mechanistic insight into, NC growth. We use our approach to study the growth of Au nanocubes, finding supersaturation dynamics that are more complex than previously understood via the LaMer or Finke–Watzky models. We then input the supersaturation profiles into theoretical equations to predict the temporal growth profile of the nanocubes and identify supersaturation-associated NC shape evolutions. Based on these insights, we rationally modulate the shape evolutions of the nanocubes during growth.

Results and discussion

Framework to characterize supersaturation during NC growth

To develop the framework, we first consider the monomer dynamics of colloidal NC growth, as summarized in Fig. 2. Colloidal NCs undergoing classical growth typically involve precursor-limited reactions,^21,42,43 in which monomer is converted from precursor before incorporating into a NC (Fig. 1a). This predominantly occurs through one of two pathways: conversion occurs in the bulk phase and monomers diffuse through the stagnant layer of liquid surrounding the NC, or precursors diffuse through the stagnant layer so that the NC surface can catalyze the conversion to monomer.^20,40,41,44

Fig. 2 Schematic representations of monomer dynamics during colloidal NC growth. (a) The kinetic processes governing classical growth for a colloidal NC of radius r. Monomer or precursor diffuses from the bulk phase through the stagnant layer before reaching the NC, where monomer can incorporate into the crystal. [M]_p represents the precursor, [M]_i represents the monomer, k_conv is the rate constant for conversion of precursor to monomer, D is the bulk diffusion coefficient of the monomer, k_g is the rate constant for growth, k_dis is the rate constant for dissolution, and δ is the thickness of the stagnant layer surrounding the NC. For autocatalytic surface growth, conversion of [M]_p to [M]_i occurs at the NC surface. (b) Monomer concentration relative to the NC surface (striped surface). The rate-controlling step for growth can be the integration of monomer into the crystal (kinetic regime) or the diffusion of monomer to the NC (diffusion regime). C is the bulk concentration of precursor (for autocatalytic surface growth) or monomer, C_i is the monomer concentration at the NC interface, C_∞ is the bulk saturation concentration, and C_∞(r) is the local saturation concentration at NC interface, as influenced by the Gibbs–Thomson effect.

At the interface of the bulk phase with a NC, the local saturation concentration of monomer increases relative to that of a macroscopic crystal, which is described by the Gibbs–Thomson equation:²³

where r is the radius of a spherical NC, ν_c is the molar volume of the monomer, γ is the surface energy, and R is the gas constant (all parameters used to develop the theoretical framework are summarized in Table S1, ESI†). The monomer concentration at the NC interface C_i is then⁴⁵where D is the bulk diffusion coefficient of the monomer or precursor that traverses the stagnant layer, k^g₀ is the rate constant for growth of a flat (r → ∞) interface, and δ is the thickness of the stagnant layer. For NCs, δ tends to be much larger than r, and C_i can be considered in two regimes. In the diffusion regime (D ≪ rk^g₀), the rate-controlling step for growth is the diffusion of monomer or precursor through the stagnant layer, and C_i ≈ C₀(r). Conversely, in the kinetic regime (D ≫ rk_g), the rate-controlling step is the incorporation of monomer into the crystal, and C_i ≈ C (Fig. 2b).

Since directly measuring the concentration of monomer in colloidal syntheses is experimentally difficult and can disturb the reaction, we sought to find another approach to characterize supersaturation. In classical growth, the rate at which monomers incorporate into NCs, R_NC(t), is a function of the supersaturation,²³ which has been demonstrated previously in colloidal NC syntheses.^21,39,42 Moreover, colloidal NCs present a broad range of size-dependent optical properties, such as light scattering, absorbance, and fluorescence, which can be measured in real time by facile, noninvasive techniques.^42,46–49 Our framework will interpret the optical progression of growing NCs to determine R_NC(t) and then σ(t).

To obtain an expression for R_NC(t), we start with an expression based on eqn (2) and (3). As derived by Sugimoto⁴⁵ and expanded by Talapin et al.,⁵⁰ the radial rate of classical growth for a colloidal NC is described by

where α is the transfer coefficient of growth in the reaction. Fick's first law relates growth rate to the total flux of monomer onto the surface of a single NC:⁴⁵which accounts for the surface-area dimensionality of monomer flux during NC growth. Thus, the intensive form of the molar rate of monomer consumption can be formulated aswhere n_NC(t) is the number of NCs undergoing growth and V_s is the volume of the synthesis formulation. During nucleation reactions, n_NC(t) changes as nuclei emerge or disappear. On the other hand, seed-mediated syntheses separate growth from nucleation, and these growth reactions may take n_NC as the number of seeds added, a constant parameter for a given reaction.

In the diffusion regime, combining eqn (4)–(6) leads to an expression for supersaturation:

The term 2ν_cγ/RT tends to be on the order of 1 nm, meaning r > 10 nm leads to a small contribution from the Gibbs–Thomson effect. In this case, NCs grow at rates comparable to macroscopic crystals andIn the kinetic regime, supersaturation can be derived similarly asEqn (7)–(9) express σ(t) as a function of R_NC(t). As mentioned previously, we can determine a second expression for R_NC(t) from the size-dependent optical properties of growing NCs. For instance, the optical attenuation through a suspension of metal NCs can be measured during a reaction and is described by the Beer–Lambert Law:

A = clε (10)

where c represents either the concentration of particles (n_NC/V_s) or the concentration of monomer in the NCs (C_NC) depending on the reaction, l is the sample path length, and ε is the extinction coefficient, which is composed of the absorption (ε_abs ∝ r³) and scattering (ε_sc ∝ r⁶) coefficients.⁵¹

Synthesis can affect both c and ε. When changes in ε are small, such as during some crystallization reactions with minor changes in NC size,⁵²ε may be approximated as a constant value,^21,42 meaning that c = C_NC is the sole variable during the reaction, so that A corresponds directly with C_NC. On another hand, seed-mediated syntheses involve a constant c = n_NC/V_s but often grow NCs to larger sizes, meaning that ε changes over the reaction. When absorption predominates the overall extinction, ε is proportional to the volume of the NCs,⁵¹ which depends on the amount of consumed monomer C_NC. Thus, A corresponds with C_NC in both cases.

When a synthesis reaction is run to completion, the final monomer concentration approaches C₀. If the total concentration of usable monomer added to the closed reaction is C_T, then the final concentration of consumed monomer is approximately C_T − C₀. When colloidal NC syntheses directly introduce monomer or quantitatively convert precursor to monomer before seed addition, some reactions can follow pseudo-first-order kinetics, especially when n_NC is large.^43,53–55 In both cases discussed above, A corresponds with C_NC, and the pseudo-first-order kinetics can be described empirically as

C_NC(t) = (C_T − C₀) − (C_T − C₀) exp(−k_gt). (11)

where k_g is the rate constant of NC growth, which is fitted on the temporal profile of absorbance measurements taken throughout synthesis, andIn this case, monomer is found either in the bulk phase, stagnant layer, or NC. Hence, C(t) = C_T − C_NC(t) in these pseudo-first-order reactions, and the remaining monomer concentration can be described by

C(t) = (C_T − C₀) exp(−k_gt) + C₀ (13)

When seed-mediated reactions have an induction period, during which precursor is converted into monomer such as in autocatalytic reactions, they often show sigmoidal kinetics.^56–58 These kinetics can be represented empirically by the Boltzmann sigmoid function,⁵⁹

where t₀ denotes the sigmoidal inflection point, andThus, supersaturation can be characterized via one of eqn (7)–(9), in which R_NC(t) is specified by eqn (12) for pseudo-first-order reactions, eqn (15) for low-order reactions with an induction period, or higher-order equations for reactions with greater kinetic complexity.^41,43

We can further formulate an expression to predict the final size of NCs in the reaction. The concentration of integrated monomer in a population of spherical NCs can be approximated as 4πr³n_NC/(3ν_cV_s), where r is the volume–average radius. Since C_NC(t → completion) = C_T − C₀, the average NC size after seed-mediated growth can be estimated with

where r₂ and r₁ are final and seed radii, respectively.

In summary, our framework analyzes the progression of a size-dependent optical property to quantify the underlying supersaturation dynamics in the classical growth of NCs, particularly those with simple geometries in its current form. To implement our approach, the optical measurements should be conducted throughout the entire reaction and are coupled with knowledge of the experimental parameters; these required parameters are listed in Table S2 (ESI†). In its implementation in this study, our framework measures absorbance as this optical property and considers reactions in which absorption predominates the overall extinction. The latter condition is typically valid when colloidal NCs remain smaller than 50–60 nm.^46,60 In-line spectrophotometers can rapidly and noninvasively measure absorbance through a transparent synthesis vial, such that high-frequency absorbance measurements yield high-resolution supersaturation dynamics. Moreover, since we characterize σ(t) based on R_NC(t), our approach requires neither insight into the source of the monomer nor its molecular structure, limiting the requirements of a priori knowledge and expanding the utility of the methodology. As such, it should be applicable to reactions whether the conversion of precursor to monomer occurs in the bulk phase or is catalyzed by the NC surface.

Our approach can also be extended to additional types of experimental apparatuses and types of NCs. The other size-dependent optical properties of NCs, including light scattering and fluorescence, can also be related with R_NC in the framework.^46–48 Additionally, the framework can be built upon to interpret anisotropic NC growth more accurately by modifying the dimensionality of eqn (4) and (5) or to interpret crystallization reactions by consider temporal changes in n_NC/V_s during nucleation. Lastly, the noninvasive optical methods can be used alongside complementary analytical techniques for additional insight into NC growth, such as in situ X-ray diffraction for the evolution of crystal phases and multimetallic compositions.⁶¹

Elucidating supersaturation dynamics throughout the growth of colloidal Au nanocubes

We used our framework to investigate the growth of colloidal Au nanocubes. To assess how supersaturation dynamics influence the shape evolutions of NCs, we chose a seed-mediated synthesis formulation with cetyltrimethylammonium chloride (CTAC) that predominantly grows nanocubes but also grows a small proportion of impurity shapes. To experimentally analyze NCs throughout the aqueous reaction, we arrested growth at various times using ligand exchange with thiol-terminated polystyrene and solvent transfer via tetrahydrofuran into toluene.⁶² We imaged these samples using high-angle annular darkfield scanning transmission electron microscopy (HAADF-STEM) (Fig. 3a), which showed that the cuboctahedral seeds primarily evolved to truncated cubes and then to cubes along growth (Fig. 3b).

Fig. 3 Characterization of the supersaturation-dependent growth of colloidal Au nanocubes. (a) HAADF-STEM images of the NCs found throughout colloidal nanocube growth. Times denote when growth was arrested relative to the initiation of growth. (b) HAADF-STEM images of single NCs from the above samples depicting the transition of cuboctahedra to cubes. The scale bar applies to each micrograph. (c) Time-resolved extinction spectra taken throughout colloidal NC growth. (d) Kinetics of NC growth analyzed via the peak nanoplasmonic extinction. The data were fitted to eqn (14) (r² > 0.99). (e) Temporal profile of supersaturation dynamics throughout colloidal NC growth, as estimated by eqn (8) and (15). The dashed lines delineate the triphasic sequence of rapid monomer consumption (i), sustained supersaturation (ii), and then gradual depletion (iii). (f) Experimental and theoretically predicted growth of colloidal nanocubes. Theoretical growth was modeled using the supersaturation profile from (e) and eqn (4). Dots and bars represent the mean and SD, respectively.

Towards a high-resolution assessment of supersaturation, we took high-frequency absorbance measurements throughout the reaction (Fig. 3c). The absorbance kinetics had an induction period, and we fitted them to eqn (14) (the equation constants are summarized in Table S2 (ESI†); see Section S1 in the ESI† for additional details). As shown in Fig. 3d, this model showed a strong agreement with the experimental data (coefficient of determination, r² > 0.99). Since the absorbance of the NCs leads to visually discernible colors, we also took images of the synthesis formulation throughout growth and analyzed the colorimetric development (Fig. S1a and b, ESI†). Imaging and spectrophotometry yielded similar results in their analyses of the absorbance kinetics (Fig. S1c, ESI†), indicating that both approaches can be used to determine the variable parameters of eqn (14).

We applied these fitted parameters to characterize the supersaturation dynamics in the reaction. When D ≪ 1, colloidal NC growth tends to occur in the diffusion regime.²⁰ For our synthesis formulation, D is on the order of 10⁻¹⁰ m² s⁻¹ (ref. 63), so we used eqn (8) and (15) to determine the supersaturation dynamics (Fig. 3e, the equation constants are summarized in Table S2, ESI†). This revealed a triphasic sequence: an initial phase of rapid monomer consumption (0.0–1.5 min) followed by phases of sustained supersaturation (1.5–5.0 min) and then gradual monomer depletion (>5.0 min). This profile indicates that monomer consumption outpaced the conversion of precursor to monomer in the initial 1.5 min of NC growth. Sustained supersaturation in the second phase suggests that the NCs grew between 1.5–5.0 min while the conversion of precursor to monomer occurred at a rate similar to the rate of monomer consumption. Lastly, the supply of supersaturated monomer was gradually exhausted between 5–20 min as this reaction completed.

Although seed-mediated syntheses of colloidal NCs have been studied heavily, the supersaturation dynamics of these reactions have been assumed to follow the descriptions of the growth stages of the LaMer (see Section III in Fig. 1b) or Finke–Watsky models, in which supersaturation follows a simple, steadily declining profile as the NCs grow.^40,41 Notably, the elucidated supersaturation profile for our colloidal nanocubes showed a triphasic sequence with distinct phases of monomer consumption (Fig. 3e and Fig. S2, ESI†), revealing that the supersaturation dynamics of colloidal NC growth can be more complex than previously understood.

We next used the supersaturation profile (Fig. 3e) to model the growth of the colloidal nanocubes using eqn (4) (see Section S2 in the ESI† for additional details). This model estimated that the isometric spherical growth in eqn (4) approximated the isometric cubic growth in our NCs. We also used the HAADF-STEM images of the NCs throughout the reaction to analyze the experimental sizes of the cuboctahedra and cubes (Fig. S3 (ESI†) summarizes the sizing measurements, and the raw sizing data are included in Fig. S4 (ESI†)). As shown in Fig. 3f, the theoretical model demonstrated a strong agreement with the experimental results, predicting the temporal growth profile of the NCs. According to this profile, the NCs grew from 10.0 nm (the seed size) to 29.0 nm in the initial phase of the supersaturation dynamics (0.0–1.5 min), then to 46.5 nm in the intermediate phase (1.5–5.0 min), and finally to approximately 61.2 nm (5.0 min to completion).

To further investigate the influence of the supersaturation dynamics on growth, we artificially simulated a supersaturation profile (Fig. S5a, ESI†) which lacks the initial phase of rapid monomer consumption and more resembles that expected when considering the LaMer model (see Section S3 in the ESI† for additional details). We inputted the simulated supersaturation into eqn (4) for another theoretical growth model, which predicted the final size of the NCs but did not conform to the growth profile during the earlier stages of the reaction (Fig. S5b, ESI†). This distinction reinforced that this reaction involves the complex supersaturation dynamics found in Fig. 3e and further supported the utility of our framework.

Supersaturation-associated shape evolutions of nanocubes during growth

With the supersaturation profile in hand, we sought additional insight into growth by investigating the relationship between the supersaturation dynamics and shape evolutions of the NCs. The HAADF-STEM images and additional scanning electron (SE)-STEM images identified six NC shapes present in this reaction (Fig. 4a). Three of these shapes (cuboctahedra, truncated cubes, and cubes) were involved in the primary growth pathway (Fig. 3b), but the other three shapes (overgrown cubes, tetrahedra, and a small proportion of other, undefined shapes) were undesired. While cuboctahedral seeds grow into tetrahedra and the observed undefined shapes as kinetic impurities of synthesis,^64–66 it was unclear how cube overgrowth occurred.

Fig. 4 Supersaturation-associated shape evolutions of nanocubes. (a) SE-STEM images showing the NC shapes found throughout colloidal nanocube growth. Cuboctahedra (green), truncated cubes (blue), cubes (red), overgrown cubes (yellow), tetrahedra (grey) and other shapes (grey). The micrographs are not to scale. (b) Temporal profile of supersaturation dynamics taken from Fig. 3e (black curve) overlaid on the distribution of shapes (bars) at 0.5, 2, 4, 7, and 15 min during colloidal growth. These NCs were grown using 0.1 equiv. of NaOH. Shapes are denoted by the outline colors in (a). The triphasic sequence of supersaturation dynamics from Fig. 3e were denoted with i, ii, and iii. (c) Summary of the associations between the supersaturation phases and shape evolutions.

To study the relationship between the supersaturation dynamics and overgrowth, we used the SE-STEM images to analyze the distribution of NC shapes throughout growth (Table S3, ESI†) and overlaid them on the supersaturation profile (Fig. 4b). By doing so, we found associations, which were not necessarily causative, between shape evolutions and supersaturation dynamics. This analysis showed that the population of cuboctahedra predominantly evolved to cubic shapes (truncated cubes, cubes, or overgrown cubes) during the initial phase of rapid monomer consumption. Truncated cubes were the dominant shape shortly after this transition but decreased in proportion as the reaction progressed. The overall yield of NCs with cubic shapes remained consistent throughout growth (92.9, 92.3, 92.1, and 93.0% at 2.0, 4.0, 7.0, and 15 min, respectively) (Table S3, ESI†), suggesting that cubes and overgrown cubes emerged from the truncated cubes rather than directly from cuboctahedra.

Fig. 4b also showed that cubes and overgrown cubes emerged in different phases of monomer consumption, as summarized in Fig. 4c. Cubes primarily emerged in the third phase, gradual monomer depletion. Conversely, overgrowth primarily occurred in the first and second phases. By 2 min, 10.2% of the NCs were overgrown cubes. By 4 and 7 min, this proportion increased to 15.0 and 21.9%, respectively. Between 7–15 min, however, it only increased to 24.4%. That is, cube overgrowth was largely associated with the high supersaturation of the earlier phases of growth (Fig. 4b and c). As described by Fick's first law,^45,50

J = 4πDr(C − C_i) (17)

higher supersaturation leads to higher rates of monomer flux to the NCs and, as our reaction occurred in the diffusion regime, higher rates of monomer incorporation. Excessive monomer incorporation into the NC surface is a plausible kinetic mechanism for overgrowth. As such, we rationalized that decreasing supersaturation in the earlier stages of the reaction would diminish the yield of overgrown cubes.

Modulating supersaturation to influence nanocube shape evolutions

In our synthesis, the reduction of Au precursor to monomer produces H⁺ and Cl⁻ (Fig. S6, ESI†). The addition of HCl to this reaction should shift the equilibrium and reduce the monomer concentration before the addition of seed. Moreover, H⁺ and Cl⁻ are labile, weakly binding ions on Au surfaces, and do not promote anisotropic growth.^30,67 We hypothesized that adding HCl before the addition of seed would diminish monomer conversion in the early stages of growth and thus limit cube overgrowth. The synthesis formulation for the NCs shown in Fig. 3 and 4 used 0.1 molar equivalents (equiv., relative to Au added) of NaOH. We compared these NCs to the NCs from two syntheses that replaced NaOH with 1.0 or 2.0 equiv. of HCl, small amounts of HCl intended to slightly diminish monomer conversion in the early stages of growth.

Fig. 5a–c shows the NCs produced by these syntheses, with larger-area SE-STEM images shown in Fig. S7 (ESI†). As done previously, we took high-frequency absorbance measurements throughout these reactions and analyzed the absorbance kinetics (Fig. S8a–c, ESI†). Fitting them to eqn (14) characterized the monomer conversion kinetics, which showed that the addition of HCl indeed diminished the monomer conversion and supersaturation in the early stages of growth (Fig. 5d and Fig. S8d, ESI†) while yielding nanocubes of similar size (Table S4, ESI†).

Fig. 5 Supersaturation dynamics mediate shape evolutions during NC growth. (a–c) SE-STEM image of the NCs (left) and SE-STEM (top right) and HAADF-STEM (bottom right) image of an individual nanocube when grown with (a) 0.1 equivalents (equiv.) of NaOH, (b) 1 equiv. of HCl or (c) 2 equiv. of HCl. The arrows point to nanocube corners, which increasingly truncate with greater amounts of HCl during growth. (d) Conversion kinetics for colloidal nanocube growth. Inset, early kinetics. The arrow denotes decreasing monomer conversion in early growth. (e) Shape yields for the synthesis formulations. (f) Comparison of the corner sharpness of nanocubes from the various formulations. Inset, schematic representation of the measurements and sharpness index (S). S is depicted in units of nm and {111} interplanar distances. Lines and whiskers represent means and 95% CIs, respectively.

To investigate the influence of these different supersaturation dynamics, we analyzed the shape distribution for each sample. The addition of HCl indeed coincided with lower yields of overgrown cubes (Fig. 5e). With 0.1 equiv. of NaOH added, 17.1% of the NCs were overgrown nanocubes. This proportion diminished to 10.2% for 1 equiv. of HCl and then to 4.4% for 2 equiv. of HCl. Moreover, the addition of HCl also coincided with increasingly truncated cubes (Fig. 5a–c and e). Previous research has indicated that decreasing supersaturation during synthesis is expected to dull NC corners.⁶⁸ To further quantify this in our samples, we measured a sharpness index (S), as defined in the inset of Fig. 5f, for individual nanocubes. With 0.1 equiv. of NaOH added, the mean estimate of S was 16.3 (95% confidence interval [CI]: 15.9–16.6) nm, whereas it was 15.3 (15.0–15.6) nm for 1 equiv. of HCl and 13.3 (12.9-13.7) nm for 2 equiv. of HCl. Thus, diminishing early monomer conversion, especially in the first stage of growth (Fig. 5d), was associated with both lower yields of overgrown cubes (Fig. 5e) and decreased nanocube sharpness (Fig. 5f). Taken together, these results show that altering the supersaturation dynamics during growth can modulate the features as well as shape evolutions of colloidal NCs.

Conclusions

This work developed a theoretical framework and the accompanying experimental methodology to characterize the supersaturation dynamics underlying colloidal NC growth. Our framework relates the temporal, size-dependent optical properties of growing NCs, such as absorbance, light scattering, or fluorescence, to supersaturation throughout the reaction. As such, this approach uses real-time optical measurements, which can be conducted noninvasively, without requiring a priori knowledge into the source of the monomer or its molecular structure. Elucidated supersaturation dynamics can be used to model temporal profiles of, and identify supersaturation-associated shape evolutions during, NC growth. These profiles show the relationships between growth rates and shape changes with the time course of reaction, supersaturation levels, and supersaturation dynamics, revealing specific mechanistic parameters to tune for synthesis development. This also finds concurrent outcomes—aspects of growth or shape evolutions that coincide with the same phase of the reaction—that may not be resolved by tuning supersaturation alone. Reagents and other reaction parameters can then be adjusted rationally based on these insights, including to increase product yields or modulate NC features.

By using this approach, our study found that the growth of CTAC-capped Au nanocubes involves a triphasic sequence of supersaturation dynamics. This result demonstrates that the supersaturation dynamics of colloidal NC growth can be more complex than previously understood, including through the LaMer or Finke-Watzky models.^40,41 Our analyses also showed that the emergence of cubes and overgrown cubes from truncated cubes were primarily associated with different phases of the supersaturation dynamics: overgrown cubes emerged in the early phases when monomer consumption occurred more rapidly, whereas cubes emerged in the later phase as monomer was gradually depleted. These insights led us to decrease early monomer conversion to modulate the sharpness of nanocube corners and reduce the yield of overgrown cubes. In conclusion, this work presents an initial step towards uncovering the complex thermodynamic and kinetic phenomena underlying NC growth and may guide rational design at the nanoscale.

Experimental

All experimental details for this study are available in the ESI.†

Author contributions

P. Z. C., conceptualization, formal analysis, investigation, software, visualization, methodology, writing – original draft, writing – review and editing; A. J. C., formal analysis, investigation, writing – review and editing; F. X. G., funding acquisition, supervision, writing – review and editing.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank J. Watchorn (UToronto) for helpful comments regarding this manuscript. STEM images were taken at the Ontario Center for the Characterization of Advanced Materials (OCCAM) at the University of Toronto. This work was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant (06441). P. Z. C. was supported by the NSERC Vanier Canada Graduate Scholarship. A. J. C. was supported by the Ontario Graduate Scholarship. F. X. G. was supported by the NSERC Senior Industrial Research Chair program.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1nh00572c

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