Ligand-Mediated Nanocrystal Growth - ACS Publications - American

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Ligand-mediated nanocrystal growth Stefano Lazzari, Pius M. Theiler, Yi Shen, Connor W. Coley, Andreas Stemmer, and Klavs F. Jensen Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00076 • Publication Date (Web): 11 Feb 2018 Downloaded from http://pubs.acs.org on February 16, 2018

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Ligand-mediated nanocrystal growth Stefano Lazzari,†,¶ Pius M. Theiler,‡,†,¶ Yi Shen,† Connor W. Coley,† Andreas Stemmer,‡ and Klavs F. Jensen∗,† †MIT, Department of Chemical Engineering, 77, Massachusetts Avenue, 02139 Cambridge, MA, USA ‡ETH Zurich, Nanotechnology Group, S¨aumerstrasse 4, 8803 R¨ uschlikon, Switzerland ¶These authors contributed equally E-mail: [email protected]

A microfluidic platform combined with a deterministic model accounting for surface ligands reveals precious insights into the nanocrystal formation process. The comparison of on-line kinetic information with model predictions enables the derivation of temperaturedependent kinetic parameters for the CdSe model system. This fully generalizable approach represents a step forward toward a quantitative prediction of the nanocrystal size distribution, enabling the control and optimization of process performance and material properties.

Introduction The tunable optoelectronic properties that materials exhibit at the nanoscale have enabled applications in the fields of energy, medicine and electronics. 1–9 Advances in the synthesis of a great variety of nanocrystals (metal, semiconductor, oxide) have allowed a precise control over their composition, average size and distribution width. 4,5,7,10–12 Precursors give rise to monomers that in turn nucleate and grow, either by simple monomer addition or by ripening

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and coalescence phenomena. In all these processes, the surface ligands play a crucial role in stabilizing the various structures. 4,7 Despite the significant knowledge gathered, nanocrystal production processes still rely on expensive time-consuming trial and error approaches. 13,14 No theoretical framework allows for the description -nor the control- of the nanocrystal size distribution as a function of synthesis parameters. Therefore, the final optoelectronic properties cannot be optimized. The lack of a systematic comparison between large experimental data sets and deterministic models has prevented the development of this very desirable framework. 13–17 The aim of this work is to develop a microfluidic platform that is able to automatically provide large kinetic data sets and to couple it with a deterministic model accounting for the relevant role of surface active ligands. As one of the most well-studied nanomaterials, CdSe quantum dots (QDs) are employed as our model system, but our approach is fully generalizable to other nanocrystal syntheses. Notably, many experimental platforms allow the automatic gathering of large experimental data sets, 9,18–20 but only few systematic comparisons between model predictions and experimental data have been performed. 7,16,17,21,22 This is related to the limited availability of reliable kinetic models. Kinetic models describe either average properties of the QDs size distribution, or their full size distribution through population balance equations (PBE). Average properties-based models disregard how monomers and ligands are distributed among the QDs population, and assume that the differently sized QDs possess the same reactivity, surface accessibility, stability and coverage. 23 PBE models naturally accommodate size- and surface dependent reaction parameters, and thus represent the ideal framework to describe QDs size distributions and identify reliable temperature-dependent kinetic parameters. Notably, PBE-based models proved that nanocrystal growth is reaction-limited. 15,21 Moreover, they were able to describe the formation kinetics and size distribution evolution of the QDs. 16,17,21 All these models 15–17,21 give valuable insights into the kinetics of QD formation. However, they can

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not provide robust, temperature-dependent kinetic parameters describing the evolution of the QD size distribution, in part because surface-stabilizing ligands are not considered explicitly in the models. 15–17,21 Ligands are known to strongly affect the QDs size distribution, their shape, and their stability towards ripening and aggregation. 4,24,25 A model accounting for ligands would enable the precise tuning of particle size distributions (and hence material properties) through manipulation of reaction conditions. The scope of our paper is threefold: i) to collect on-line spectroscopic data of QDs formation kinetics under different synthetic conditions, ii) to establish a link between the online, high-temperature, spectroscopic information and properties of the QDs size distribution, and iii) to develop and validate a kinetic model, accounting for the QDs size and surface ligand distribution.

Experimental details The employed experimental set up is an evolution of a previously described platform. 26 The full details are reported in the Electronic Supplementary Information (ESI) section S2, while the essence of the approach is summarized in what follows.

Materials Cadmium oxide (CdO, 99.99%), Oleic acid (OA, 90%), Oleylamine (OAm, 70%), selenium powder (Se, >99.5%), 1-Octadecene (ODE, 70%) were purchased from Sigma Aldrich. Trioctylphosphine (TOP, 97%) was purchased from Strem Chemicals. All chemicals were used without additional purification.

Cadmium precursor solution 256.9 mg CdO and 1.6 ml OA were mixed in 48 ml ODE. The mixture was degassed at 100◦ C for 1 hour and then heated up at 230◦ C under Argon, until a colorless and clear solution was 3


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obtained. The mixture was then cooled down and 0.5 ml oleylamine were injected. The 40 mM Cd-precursor solution was degassed again by vacuum at 110◦ C and transferred into the glovebox. The stock solution was 40 mM Cd : 100 mM OA : 30 mM OAm and was diluted to the desired concentrations with ODE prior to use.

Selenium precursor solution 1.74 g Se powder were mixed with 10 ml TOP and stirred in the glovebox until a clear solution was obtained. The TOPSe stock solution (2.2 M) was diluted to the desired concentrations with ODE prior to use.

Synthesis procedure The entire experimental procedure (Figure 1a) was initiated and controlled through an inhouse LabVIEW code. The precursor solutions employed were always in a 1:1 Cd:Se molar ratio, and two initial total precursor concentrations were explored: 20 mM and 40 mM. A 10 µl droplet of each precursor solution was mixed in a cross junction with two syringe pumps (PHD 2200, Harvard Apparatus) in order to obtain a slug of 20 µl. This well mixed slug was brought into the reactor by means of a gas slug added by a third pump (400 µl/min). The 12 cm quartz reactor (I.D. 1 mm, O.D. 3 mm) was surrounded by an aluminum chuck, that ensured a rapid heat transfer. Once the droplet was in the detecting zone, the gas pump was stopped so that the slug remained positioned in the light path between a fiber-coupled light source and fiber-coupled UV-Vis spectrometer (HR2000, Ocean Optics). Growth kinetics were not affected by keeping the droplet stationary, as it was uniform in temperature and composition. An average of 25 spectra with an integration time of 40 ms were recorded every 5 s till the reaction was completed. In contrast to the previously described oscillatory reactor, 26,27 this experimental platform enabled us to achieve better time resolution during the QD growth process, as more spectra can be obtained within a given time. A typical experimental result (Figure 1b) shows the evolution of QD absorbance spectra 4


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with time. As the QDs grow, their spectra broaden and move towards lower energies (i.e. the QDs red-shift).

Linking experiments to model Relating the absorbance curves of CdSe to average properties of the QDs size distribution is essential to compare model predictions with experimental data. Information on the QDs concentration, average size, and polydispersity can be extracted using the first absorption peak height, position, and width, respectively. 11 However, these correlations are available only at room temperature 11 and need to be adjusted in order to account for the temperature effect on absorbance measurements (Figure 1c). As expected, 28 a red-shift (i.e. a shift towards lower energies) is observed for the band edge peak position, and the peak height is also significantly affected by temperature. This shifting behavior is reversible, suggesting that the change is not due to aggregation of the particles at elevated temperature (cf. ESI Figure S4 ). To quantify the impact of temperature on the first absorption peak height, position, and width, the absorption of three QDs samples (of sizes 2.4, 2.7 and 3.1 nm) was measured at different temperatures. This allowed the extension of literature correlations 11 to the synthesis temperatures employed, enabling a more thorough on-line study of the QDs formation kinetics. Detailed explanation of the automated absorbance curves processing, and on the temperature-dependent correlations, are available in the ESI section S3 and S4, respectively. Notably, the peak width turns out to be temperature independent, as opposed to the first absorption maximum A1,T and its position En,T (T in Kelvin): A1,T A1,T =293 K

= 1.7733 − 0.0027 × T

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(1)

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a) Step 1: Quartz Reactor Inline UV-Vis Sample collection and BPR

Argon

Sample collection and BPR

Argon

Step 2:

Cd precursor

Se precursor

precursor mixture / QDs

Argon

Figure 1: a) Sketch of the experimental set-up. The precursor streams are mixed and the resulting droplet is “held” in front of the inline UV-Vis. BPR is the short form for backpressure regulator. b) Typical experimental output: absorbance growing towards larger wavelengths, reflecting the QDs growth process. The dashed line tracks the time-evolution of the first absorption maximum. The continuous lines correspond to spectra measured at t = 500, 750 and 1000 s. c) Absorbance of the same QDs population as a function of temperature. The temperature increases from 20◦ C in even increments to 220◦ C along the direction of the arrows, from light yellow to dark red.

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En,T0 =293K = En,T + αT (T − T0 )

(2)

αT = −0.51 ± 0.12 meV/K A full list of symbols is provided in ESI section S1.

The model Kinetic scheme All kinetic models describing QD formation by means of population balance equations (PBE) have so far neglected the fundamental role of surface active ligands and focused on the QDs size distribution. 15–17,21 The present framework aims at explicitly accounting for the QDs size- and surface ligand distributions. To describe the ligand-mediated QDs formation, a 2-D population balance equation (PBE) is introduced, where Qi,j (t) represents the concentration of QDs consisting of i monomers, stabilized by j ligands, at time t. Note that both i and j belong to the set of natural numbers. For the sake of brevity, the time-dependence in the notation is dropped in the following equations for all species. For the sake of simplicity only one type of ligand L is considered here. As a result, the Cd and Se precursor species considered are CdL2 and SeL, accounting for the fact that Cd typically offers two binding sites (e.g. Cd(OA)2 ), whereas Se only one (e.g. T OP Se). The kinetic scheme considered involves precursor conversion, reversible nucleation, reversible growth, and reversible ligand binding events: (cf. scheme 3-6).

k

CdL2 + SeL −−P→ M L2 + L

k

N −−− −− −− * nM L2 ) − Qn,2n

kD,n,2n

7


(3)

(4)

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kG,i,j

−− −− −− * Qi,j + M L2 − ) − Qi+1,j+2 kD,j+2

kA,i,j

−−− −− −− * Qi,j + L ) − Qi,j+1 kE,j+1

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(5)

(6)

It should be stressed that the monomeric species formed through precursor conversion (kinetic rate kP ) is M L2 , hence a monomeric CdSe unit (M ) is assumed to be always bound to two ligands L. Therefore, whenever a monomer adsorbs on or desorbs from a QD, the number of ligands on the QD change. Nucleation has been assumed to be a second order process, following a “molecular approach” discussed in the literature. 29 The kinetic rate constants for growth (kG,i,j ) and ligand adsorption (kA,i,j ) are assumed to be mass- and ligand-dependent. The dissociation (kD,j ) and ligand desorption (kE,j ) rates are considered a function of solely the number of ligands on the QDs surface:

kG,i,j = kG (Ns,i − j)

(7)

kD,j = kD j

(8)

kA,i,j = kA (Ns,i − j)

(9)

kE,i,j = kE j

(10)

Notably, dissociation and ligand desorption are indirectly a function of the QDs size i too, as the j available ligands on an i-sized QD are a function of the total available surface sites Ns,i : Ns,i = b3.86i2/3 c

(11)

The derivation of the prefactor in equation 11 is related to the CdSe crystal structure and is described in the ESI section S5. Note that when the number of surface ligands j on an i-sized QD approaches the total number of available surface sites Ns,i , both the growth and 8


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ligand adsorption kinetic constants (and therefore their rates) tend to zero (cf. equation 7 and 9).

The equation set For the sake of brevity, only the balance on the generic QDs of size i and bearing j ligands has been reported for a well-mixed, isothermal batch reactor. All other balances on precursors (CdL2 , SeL), monomers (M L2 ), ligands (L), and nuclei (Qn,2n ) are reported in ESI section S6.

d [Qi,j ] =kG [M L2 ] [Qi−1,j−2 ](Ns,i−1 − j + 2) dt − kG [M L2 ] [Qi,j ](Ns,i − j) + kD [Qi+1,j+2 ](j + 2) − kD [Qi,j ]j + kA [L][Qi,j−1 ](Ns,i − j + 1)

(12)

− kA [L][Qi,j ](Ns,i − j) + kE [Qi,j+1 ](j + 1) − kE [Qi,j ]j ∀ i, j ∈ N ∀i>n To avoid the solution of the full 2-D PBE (equation 12), the method of moments is applied on the second internal coordinate of the balance, as discussed in ESI section S7. Instead of solving one 2-D PBE, two 1-D PBE are solved. This simplification is possible under the assumption that the j ligands distribute on an i-sized QD according to a binomial distribution. A similar assumption was already employed in the literature 30 and reflects the probabilistic nature of the growth and adsorption processes. The likelihood for a monomer or a ligand to dock the surface of a QD is related to the probability to encounter a free or an occupied site, suggesting that these events can be described by a binomial distribution. The 9


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simplified equation set (cf. equations S23-S33) is solved using the Matlab function “ode113”, enabling computation of i) the concentration of QDs of size i and ii) the total concentration of ligands on an i−sized QD. There are eight necessary model parameters (Table 1), comparable to previously developed frameworks. 15–17 The present framework has been derived for isotropic growth of Table 1: Overview of model parameters Reaction Precursor conversion Reversible nucleation Reversible growth Reversible ligand association

Parameter kP n, kN , kDn,2n kG , kD kA , kE

spherical nanocrystals, but it is extendable to non-isotropic growth (e.g. of nanowires) by a few simple geometric considerations (e.g. a re-definition of the available surface sites Ns,i ). Successful examples of PBE describing anisotropic assembly are already present in the literature. 31,32

Extraction of properties from the model A link between model results and experimental information needs to be established for model validation. Following a number of transformations 17,33 reported in ESI section S8, it is possible to extract Q(D, t)dD and L(D, t)dD. Q(D, t)dD represents the concentration of QDs having a diameter between D and D + dD at time t. L(D, t)dD is the concentration of ligands on QDs belonging to the aforementioned size interval at time t. It is now possible to define the generic moments of order k: ∞

Z µk (t) =

Dk Q(D, t)dD

(13)

Dk L(D, t)dD

(14)

0

Z λk (t) =

∞

0

10


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From the moments equations 13 and 14, the average properties of the size distributions are defined: ¯ = µ0 (t) C(t)

(15)

¯ D(t) = µ1 (t)/µ0 (t) 2 µ1 (t) µ2 (t) 2 − σ ¯ (t) = µ0 (t) µ0 (t)

(16)

LB (t) = λ0 (t)

(18)

ρ¯(t) =

0.452/3 λ0 (t) 2 DM µ2/3 (t)

(17)

(19)

Here DM = 0.471 nm is the monomer diameter assumed, while the origin of the prefactor ¯ 0.45 in equation 19 is detailed in ESI section S8. 17 Notably, C(t) is the total concentration ¯ of QDs at time t, D(t) is their average diameter and σ ¯ 2 (t) is the average variance of the QDs sample. ρ¯(t) is the average number of ligands per total surface area and LB is the total concentration of ligands bound on the QDs surface.

Results Parametric studies Before discussing model fit and comparison with experimental results, the model features are explored through parametric studies. The focus is set on ligand-related variables, as extensive parametric studies have been conducted on nucleation-, growth- and dissociationrelated variables. 15–17,21 Overall, three model parameters are related to ligands: i) the ligand association rate constant kA , ii) the ligand elimination rate constant kE , and iii) the initial free ligand concentration [L0 ]. For the sake of brevity, the impact of kA and kE is reported in ESI section S10, while the effect of [L0 ] is detailed in Figure 2. The parameters used for these simulations -reported in ESI section S9- have been selected in the range of the fitted parameters obtained by comparing the model to the experimental data, as discussed 11


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in a subsequent section. As only the ligand concentration L0 has been changed, the QDs properties reported in this section reflect realistic synthetic scenarios.

Figure 2: Impact of initial ligand concentration [L0 ], increasing along arrow direction. a) Q(D, t = 1200 s) vs D, b) ρ¯ vs time, c) Monomer concentration [M L2 ] (left axis) and bound ligand concentration [LB ] (right axis) vs time, d) total QDs concentration, C¯ vs time. Figure 2a) shows how increasing the initial ligand concentration [L0 ] shifts the distribution Q(D, t = 1200s) towards the left. A larger ligand availability implies that more ligands adsorb on the QDs surface. As a result, more of the available surface sites are occupied by ligands, reducing the likelihood of monomer addition, thus reducing QDs growth. In a limiting case, if all QDs surfaces were fully covered with ligands, growth would stop, and the QDs size distribution would reach an equilibrium. Figure 2b) shows how the average ligand density ρ¯ on the QDs increases for larger values of the initial ligand concentration, [L0 ]. It is the first time that this intuitive result has been captured by a deterministic model. Moreover, the values of the ligand density fall within the reported experimental values of 1-5 per nm2 . 24,34,35 The trends of ρ¯ in time suggest that at the very beginning, the QDs have a relatively large surface coverage. This is related to the initial large number of nuclei, described as consisting of n monomer units and 2n ligands in our kinetic scheme (cf. equation 4), resulting in a high ligand density. Successively, as growth proceeds and the QDs surfaces become larger, the ligand density 12


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drops until it reaches an equilibrium towards the end of the reaction when ligand association and dissociation offset one another. Such equilibrium is reached faster the larger [L0 ] is, as growth becomes increasingly hindered. Larger values of [L0 ] promote therefore a larger value of the total concentration of bound ligands LB (cf. right y-axis of Figure 2c), that slow down the growth process (cf. Figure 2a). As the growth process is slowed down, larger monomer concentrations are observed in solution (cf. left y-axis in Figure 2c). These larger monomer concentrations increase the nucleation rates, and therefore the total amount of QDs present, as can be seen in Figure 2d). This brief analysis shows the tremendous importance of ligands in the QDs growth formation, and how the present model reproduces experimentally observed phenomena. Similar considerations hold when increasing the ligand association rate constant kA and when decreasing the elimination rate constant kE , as discussed in ESI section S10. Notably, for larger values of [L0 ], a “roughness” in the distribution is observed (cf. Figure 2a). This reflects a further feature of the present model: it is able to describe the accumulation of QDs of specific sizes, the so-called “magic-sized” QDs. 36 This effect is related to the maximum available surface sites Ns,i on an i-sized QDs. Ns,i is proportional to bi2/3 c, i.e. it can take only integer values (Figure 3e). As a result, differently sized QDs may possess the same maximum number of surface sites. In such an ensemble (different sizes, same maximum surface sites), smaller QDs are more likely to have fewer surface ligands (i.e. more “docking” space) than larger ones, as each growth event adds two ligands on the QDs surface (cf. equation 5). Therefore, in such an ensemble, smaller QDs are more likely to grow than larger ones, promoting the accumulation of the larger QDs of the ensemble. Clearly, several such ensembles exist, therefore justifying the observed “roughness” in 2a). This behavior is more pronounced at larger ligand coverages, as growth becomes particularly hindered. The sizes of the accumulating QDs were related to the function bi2/3 c, and in particular to its discontinuities. To further prove that the observed roughness is indeed related to differ-

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ently sized QDs possessing the same number of ligands, a simulation has been performed where the number of surface sites Ns,i was proportional to i2/3 , allowing, non-physically, for non-integer numbers of surface sites (Figure 3f). In this scenario, larger QDs always exhibited more available surface sites than smaller ones. The resulting distributions were always smooth, see Figure 3 b), d) and f) vs Figure 3a), c) and e).

Figure 3: Effect of a discontinuous/continuous Ns,i . Comparison of Ns,i functions proportional to discontinuous bi2/3 c (a,c,e) and continuous i2/3 (b,d,f). White regions in the contour plots (a,d) represent high, dark ones low concentrations. If Ns,i is an integer number, the concentration profiles becomes serrated (a,c) whereas the peak positions correspond to positions of the rounding plateaus (e). These features disappear (b,d) if Ns,i is smooth (f).

Model fit In order to estimate whether the proposed model quantitatively captures the QDs growth process, the eight model parameters (cf. Table 1) are fitted against the 40 mM experimen¯ the average diameter D ¯ and the tal data set. In particular, the average concentration C, average polydispersity σ ¯ predicted by the model (cf. equation 15-17) were compared with their corresponding quantities extracted from the absorbance curves, as detailed in the experimental section and in ESI section S12. The fitting results are reported in Figure 4, and 14


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the optimized parameters are reported in ESI section S13.

¯ vs time, c) Monomer conFigure 4: Fitting results on the 40 mM data set. a) C¯ vs time, b) D ¯ centration σ ¯ /D vs time, d) Q(D, t = 320 s). The temperature increases along the direction of the arrows, from light yellow to dark red. The model describes well all average properties of interest (cf. Figure 4a) - 4c) for all temperatures explored (170◦ C−220◦ C). The distributions shown at t = 320 s (cf. Figure 4d) exhibit the accumulation of specific QDs sizes. As previously discussed, this is related to a high degree of ligand coverage that can be appreciated by inspecting the ligand density in time ρ¯ (cf. ESI section S11). Notably, only the concentration profile at the highest considered temperature (Figure 4a) (dark red curve) is not well-captured by the model. This issue is discussed in a subsequent section. The global error between model and experiment is about 2% for the average diameter, around 7% for the distribution width and around 15% for the concentration. These uncertainties are well within the experimental error intrinsically present when converting absorbance signals to size distribution information. 11,17 A large error is observed when obtaining the experimental distribution width, due to the underlying uncertainties in extracting such property from the high-temperature absorbance signal (ESI sections S3 and S4 ). However, this experimental information can hardly be underestimated, as it is crucial in defining the shape of the distribution. 17 The retrieved parameters scale according to Arrhenius law (cf. ESI section S14) with activation energies and prefactors

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comparable to literature ones, 37 opening up the possibility to predict QDs growth at different temperatures than the investigated ones. Notably, the nuclei size n, is found to increase at larger temperatures (cf. ESI Figure S9a), in agreement with literature observations on nuclei dissociation and nanocrystals melting temperatures. 38,39

Model validation The fitted parameters from the 40 mM data set have been used to verify the model predictive abilities against the 20 mM data set, unused in the fitting procedure. The results (Figure 5),

¯ vs time, c) Figure 5: Model prediction results on the 20 mM data set. a) C¯ vs time, b) D ¯ Monomer concentration σ ¯ /D vs time, d) Q(D, t = 320 s). The temperature increases along the direction of the arrows, from light yellow to dark red. show very good agreement between model predictions and experimental data. This suggests that the model is indeed able to describe the ligand-mediated QDs growth. Once more, the only property that exhibits deviation from the model prediction is the concentration of the QDs synthesized at higher temperatures (cf Figure 5a). Discrepancies between model predictions and experimental values can be related to the simplifying model assumptions, and suggest at least three possible improvement directions: i) accounting for different ligand species, ii) accounting for the non-stoichiometric QDs composition, and iii) improving the link between size distribution and absorbance. Notably, all 16


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of the aforementioned physical phenomena (ligand binding, QDs stoichiometry and the link between size distribution and absorbance) strongly depend on the temperature, and could significantly impact the way QD concentrations are computed from both an experimental and a modeling perspective. Notably, three different ligands (oleic acid, oleylamine and tri-octyl-phosphine) were employed in our synthesis, but only one generic ligand species L was considered in the model. Knowing the competitive ligand adsorption kinetics could guide model improvement, especially if experimental insights into how ligand densities change with QDs size were available. Distinguishing among different ligands would require additional fitting parameters, but these could be estimated separately from ligand binding experiments using non-growing QDs. 40 More information on the ligand binding kinetics could also help unveil the role of ligands in reversible nucleation and reversible growth. In the present model, ligands are hypothesized to mediate both the addition and the dissociation of monomers, only accounting for the number of available sites. A more thorough description of ligand-mediated reversible growth should eventually include the role of QDs size, ligand polarity, medium polarity, and ligand chain length. The same considerations hold for reversible nucleation, which has been assumed to be ligand-independent as no explicit information was available in the literature. 17 Distinguishing ligands is closely related to the specific binding of ligands with surface atoms, their exchange reactions, and the fact that non-stoichiometric metal to non-metal compositions are observed in QDs. 24 A further improvement on the modeling side, could be distinguishing between the different types of atoms in the QDs, that control the ligand population on the surface, that in turn regulates the entire QDs formation process. The link between absorbance curves and size-distributions is a critical aspect of model validation. Several approaches exist to convert spectroscopic information into QDs size distributions, 11,22,41–43 but improvement is needed for: i) having reliable on-line tools that allow relating size distributions to high-temperature absorbance spectra, ii) improving the quality of the current relations and extending these relations to different types of QDs.

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This complex picture and the many unanswered questions place the aforementioned discrepancy and the overall good performance of the model in perspective. For the first time, a theoretical framework based on PBE was used to quantitatively predict QDs formation using independently fitted parameters. Compared to existing literature approaches 15–17,21 this represents a clear improvement, owing to the inclusion of the ligand binding kinetics in mediating the QDs formation.

Conclusion In the present work, ligand-mediated QDs growth process has been described with a theoretical framework, and validated against experimental data. In particular: • new data sets on CdSe QDs formation have been obtained, varying synthesis temperatures and initial precursor concentrations. QDs growth kinetics was monitored on-line with an automated microfluidic platform through absorbance measurements; • literature correlations linking absorbance spectra to average size distribution properties have been refined in order to enable their utilization on-line, at synthesis-relevant temperature; • a new theoretical framework based on population balances has been developed, accounting for the size distribution of the QDs and for the ligand distribution on the QDs surface; • parametric studies, and a comparison of the model with experimental data have demonstrated its descriptive and predictive abilities, and enabled the identification of temperaturedependent kinetic parameters. Overall, this theoretical framework represents a clear step forward compared to previous literature models. For the first time, QDs formation has been modeled accounting for the full distributions of QDs sizes and surface ligands. To further improve the model predictive 18


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ability, a distinction between different ligand types and non-stoichiometric QDs composition would represent desirable evolutions.

Acknowledgments The authors acknowledge Hendrik Utzat for the useful discussions on QDs spectroscopy and Prof. Michael Strano for sharing the stationary UV-Vis spectrometer. The experimental part of the work was supported by the National Science Foundation under grant number ECCS1449291. S.L. gratefully acknowledges the Swiss National Science Foundation (SNSF) for financial support (grant number P300P2 167683). P.T. gratefully acknowledges the Swiss Study Foundation for financial support.

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