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Growth Mechanisms in Nanocrystalline Lead Sulfide by Stopped-Flow Kinetic Analysis Allison L. Brazeau and Nathan D. Jones* Department of Chemistry, The UniVersity of Western Ontario, London, Ontario, Canada, N6A 5B7 ReceiVed: July 29, 2009; ReVised Manuscript ReceiVed: October 8, 2009
The growth mechanisms of thiolate-capped lead sulfide nanocrystals have been elucidated using stoppedflow UV-visible absorption spectroscopy. Spectra were recorded over the range of 0-45 °C at 48 ms intervals during the first 144 s of reaction between Pb(NO3)2 and Na2S in an aqueous solution containing thioglycerol and dithioglycerol capping ligands at pH 11. The spectra were deconvoluted by fitting to three Gaussian curves; the lowest-energy of these corresponded to the first excitonic transition and was used to approximate the average particle radius using a linear relationship between the band gap and the particle diameter for spherical PbS clusters. Kinetic models for oriented attachment (OA) and Ostwald ripening (OR) growth mechanisms were used to interpret the evolution of particle radius with time. Least-squares fitting analysis revealed that OR was the single dominant mechanism of growth in the low temperature range (0-25 °C), while at higher temperatures (30-45 °C) the mechanism was time dependent: within the first 20 s, growth was predominantly by OR; OA was favored thereafter. Parametric optimization showed that OR was controlled by the dissolution kinetics at the particle-matrix interface. The enthalpy (∆H‡) and entropy (∆S‡) of activation for the OA process were determined by Eyring treatment to be 12 ( 3 and -160 ( 10 J K-1 mol-1, respectively; the resulting Gibbs free energy of activation (∆G‡) was 62 ( 6 kJ mol-1. The very early stages of nanocrystal growth (90%, Acros Organics) were used as received. The general procedure for the synthesis of thiolate-capped PbS in water was based on a procedure by Kumacheva and co-workers.14 Stopped-flow data were collected using a BioLogic SFM-300 with a TIDAS diode array. Data acquisition was triggered by the hard stop signal of the SFM-300 with an approximate dead time of 2.1 ms. Three thousand spectra were recorded over 144 s with an integration time of 48 ms and spanning a wavelength range of 302-1148 nm. The spectrum of the lamp was recorded before and after the experiment to ensure that the intensity was not decreased due to deposition of solid materials on the cell windows. Preparation of the Lead-Containing Reagent. Lead nitrate (83 mg, 0.25 mmol) was dissolved in distilled H2O (15 mL). TG (130 µL, 1.5 mmol) and DTG (50 µL, 0.5 mmol) were added with rapid stirring, causing a bright yellow precipitate to deposit from the colorless solution. The pH of the mixture was adjusted to approximately 11 by addition of Et3N (ca. 0.5 mL), whereupon the yellow precipitate redissolved to regenerate a colorless solution, which was thoroughly deoxygenated by vigorous N2 purge for 30 min.
The procedure used in this study for the synthesis of watersoluble PbS NC was essentially the same as that reported by Kumacheva and co-workers14 but employed Pb(NO3)2 instead of Pb(OAc)2 as the source of Pb2+. The capping ligands were TG and DTG, which were administered in the previously optimized 6:2 molar ratio; higher concentrations of DTG inhibited NC growth, while lower concentrations gave uncontrolled growth and precipitation of bulk material.14,19 The basic reaction is shown in Scheme 1. In situ stopped-flow UV-visible absorption spectroscopy was used to measure the kinetics of growth of the PbS NC produced by the reaction in Scheme 1. Spectra were acquired every 48 ms using a diode-array detector in the wavelength range of 302-1148 nm. Selected results for the temperature range of 0-45 °C are shown in Figure 1. In general, the spectra were broad and lacked clearly discernible features; after 144 s, reaction solutions were brown-black and opaque to the naked eye. These findings were in line with previous studies of PbS colloids by UV-visible absorption spectroscopy reported both by us20 and others14,19 and have been attributed to broad particle size distributions and surface defects, which tend to smear distinct electronic transitions. For any given temperature, the absorption maximum shifted to longer wavelength and there was a broadening of the peak over the time interval of 144 s.
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Figure 1. UV-visible absorption spectra for the growth of PbS NC over 144 s at (a) 0, (b) 25, and (c) 45 °C. A peak at 362 nm (associated with free thiolate) has been subtracted from all spectra, which gave rise to the artifact of negative absorption at short times in some cases.
Figure 2. Sample of Gaussian multiple peak fitting for absorption spectra of the reaction between Pb(NO3)2 and Na2S at 25 °C for t ) (a) 0.5, (b) 60, and (c) 144 s. A peak centered at 362 nm due to thiolate capping ligands has been subtracted from all spectra (see Supporting Information for all temperatures).
As the reaction temperature was decreased, the spectra became narrower, which suggested a more focused size distribution, as expected. The spectra were deconvoluted by fitting the broad absorption to three overlapped Gaussian curves the maxima of which were optimized to fall at ∼600-700, 530, and 400 nm (Figure 2). These curves are attributed to the 1Se-1Sh, the 1Se-1Ph, and the 1Pe-1Ph transitions of the quantum states, respectively.34-36 Similar peaks were observed by Wise and co-workers (at 600, 400, and 300 nm) for poly(vinylalcohol)-coated PbS particles of 4.3 nm diameter that had been deposited as a thin film.34 The primary peak of interest was the lowest energy absorption (manifested in a shoulder at ∼600-700 nm), which corresponded to the first excitonic transition and was used as an approximation of the band gap (BG). The energy (in eV) of this shoulder was estimated by Gaussian curve fitting to approximate the average size of the PbS nanoparticles using
BG ) 0.435 + 24.89/D
(1)
where D is the diameter of the particle (in nm). This empirical linear relationship, developed by Silbey and co-workers,37 correlated much better with experimental data than the effective mass approximation inherent in the Brus equation which significantly overestimated the diameters of small PbS particles.37,38 As presented in this work, eq 1 has been corrected for temperature because our experiments were conducted at near ambient temperatures and not at 4.2 K; this is valid on the assumption that the band gap of bulk PbS does not vary significantly over the range of temperatures studied herein, i.e., within 25 degrees of 300 K.
Variation in the calculated approximate average particle diameter with time for a variety of temperatures is shown in Figure 3. The average radius of the PbS nanocrystals was weakly correlated with the temperature of their synthesis: the difference in diameter after 144 s between particles grown at 0 °C and those grown at 45 °C was limited to ∼0.4 nm. The calculated diameter for particles grown over the higher temperature range (30-45 °C) were almost identical, and all of these curves had the same general shape of a quick rise in the first 20 s and a slower steady growth from 20 s onward. On the other hand, the radii of particles grown at lower temperatures (0-25 °C) varied more substantially, and increased quickly in the first 5 s, then steadily over the remaining of the 144 s time frame. Kinetic models for OR and OA growth mechanisms were applied to the time-evolved particle diameters over all temperatures. OR processes for particle growth are typically described by the theory developed by Lifshitz, Slyozov29 and Wagner,28 also known as the LSW model:
D - D0 ) k(t - t0)1/n
(2)
where D0 is the radius of the particle at t0, D is the radius at time t, k is a temperature-dependent material constant and n is a parameter between 2 and 4 that depends on how crystal growth is controlled: diffusion of ions at the solution-particle boundary is indicated by n ) 2, volume diffusion of ions in the solution by n ) 3, and dissolution kinetics of the smaller particle at the particle-solution interface by n ) 4.28,30,32 In our case, values of t0, D0 and k were optimized freely, while n was iterated over integer values in the range 2 e n e 4; in every case, n ) 4 gave the fit with the lowest value for the sum of the squared residuals. The results of iterative fitting are listed in Table 1.
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Figure 3. Variation in approximate average particle radius (calculated from absorption data using eq 1) with time for thiolate-capped PbS nanocrystals grown by bottom-up reactions of Pb2+ and S2- in water over the temperature range (a) 0-25 and (b) 30-45 °C. Symbols represent experimental measurements, while lines refer to calculated values for OR (s, eq 2) and OA ( · · · , eq 3) mechanisms.
TABLE 1: Least-Squares Residuals (R2) for OA and OR Models for Growth of PbS NCs at Temperatures between 0 and 45 °C over 0.5 e t e 144 sa OR T (°C) 45 40 35 30 25 15 5 0
-1/n
k (nm s
)
0.20 0.21 0.22 0.24 0.15 0.13 0.12 0.11
OA
D0 (nm)
t0 (s)
b
n
R
1.32 1.27 1.21 1.13 1.34 1.26 1.22 1.14
0.528 0.528 0.528 0.486 0.035 0.52 0.52 0.49
4 4 4 4 4 4 4 4
0.0477 0.0648 0.0635 0.0430 0.0004 0.0036 0.0009 0.0008
a The listed parameters are those given by eqs 2 and 3. optimized with the constraint 0 et0 e 0.528 s.
b
-1
k (s )
D0 (nm)
R2
0.111 0.114 0.103 0.081 0.038 0.077 0.063 0.037
1.52 1.50 1.47 1.45 1.49 1.35 1.30 1.22
0.0999 0.1138 0.1956 0.1905 0.0036 0.0130 0.0060 0.0052
Only experimentally determined data for t g 0.5 s were included, but t0 was
We chose the kinetic model reported by Huang et al.10 to describe the OA mechanism for particle growth. This kinetic model describes the growth of nanoparticles by two primary nanoparticles combining to form a larger secondary nanoparticle without dissolution of either particle. This is the simplest model to describe the OA mechanism. It uses the assumption that dispersed nanoparticles can be treated as molecules or molecular clusters in solution and is given by 3
D0(√2kt + 1) D) (kt + 1)
2
(3)
where k is the rate constant and D0 is the initial particle diameter. Both of these parameters were optimized by least-squares fitting of the function to the experimental data (Table 1). Equations 2 and 3 have often been applied to the growth of nanoparticles by hydrothermal coarsening.10,12,39 However, it has also been demonstrated recently that they provide satisfactory explanations of the early stages of growth of, for example, ZnS nanoparticles made using bottom-up protocols at ambient temperatures.9 The present study clearly demonstrates that it is also legitimate to use them to model kinetic data for early growth of thiolate-capped PbS nanoparticles in aqueous solution. Figure 3 shows superimposed calculated and experimentally determined data. In the lower temperature range (0-25 °C), the growth data were best fit over the entire 144 s by the OR model (0.0004 e R2 e 0.0036 vs 0.0036 e R2 e 0.0130 for OA; see Table 1).
For all temperatures in this range, n assumed an optimized value of 4, which indicated that OR was controlled by dissolution kinetics at the interface between the small particles and the solution.10,30,32 This treatment is shown graphically in Figure 3a. Conversely, a single mechanism could not be used to fit the experimental growth data between 30 and 45 °C, as neither the OR or OA model alone gave a good fit (Table 1, and see Figures S6 and S7 in the Supporting Information). However, if the data were broken down into two intervals, “early” (0.5 e t e 20 s) and “late” (5 e t e 144 s) with the breakpoint between the intervals depending inversely on temperature (Table 2), the OR model could be used successfully to fit the early data (0.0008 e R2 e 0.0039), while the late interval was better fitted by the OA model (0.0001 e R2 e 0.0017). This treatment is shown graphically in Figure 3b. Time-dependent growth mechanisms have also been proposed in the syntheses of PbS and ZnS nanocrystals by hydrothermal coarsening, but in these cases OA dominated at early times while the growth process was predominantly by OR at later times.10,12,39,40 Our system differed from these in that the previously reported studies of hydrothermal coarsening were conducted on isolated, preformed nanocrystals; ours involved instead the bottom-up growth of PbS nanocrystals. A dominant OR growth mechanism for PbS at lower temperatures was consistent with the mechanisms previously proposed for the production of unsupported ZnS nanoparticles, which favored OR at low temperatures and OA at high, but in that case the particle growth was extremely rapid (127 ms),
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TABLE 2: Least-Squares Residuals (R2) for OA and OR Models for Growth of PbS NCs at Temperatures between 30 and 45 °C over “Early” (0.5 e t e 20 s) and “Late” (5 e t e 144 s) Intervalsb OR
OA
T (°C)
interval (s) “early”
k (nm s-1/n)
D0 (nm)
t0 (s)a
n
R2
interval (s) “late”
k (s-1)
D0 (nm)
R2
45 40 35 30
0.5-5 0.5-5 0.5-10 0.5-20
0.51 0.62 0.52 0.39
0.882 0.707 0.767 0.878
0 0 0 0
4 4 4 4
0.0015 0.0008 0.0011 0.0039
5-144 5-144 10-144 20-144
0.072 0.069 0.061 0.032
1.56 1.56 1.54 1.55
0.0010 0.0017 0.0013 0.0001
a The listed parameters are those given by eqs 2 and 3. b Only experimentally determined data for t g 0.5 s were included, but t0 was optimized with the constraint 0 e t0 e 0.528 s. In every case, the best fits were achieved for t0 ) 0 s.
owing to the absence of capping ligand, and the transition between mechanisms occurred between 10 and 20 °C.9 The fact that growth by OA became important at higher temperatures in both PbS and ZnS systems could be rationalized by considering that for it to occur, two crystals must collide with appropriate crystallographic orientation. This would be favored at high temperature where the particles have enhanced Brownian motion and therefore a higher statistical and energetic probability of productive collision (it should be noted that there was no mechanical stirring during our stopped-flow experiments). Here, the difference with our ligand stabilized nanocrystals was that they are not only temperature dependent but also time dependent above 30 °C. The values of k determined by least-squares fitting of eq 3 to stage 2 of the high-temperature experimental data were used in an Arrhenius plot (Figure S8) to obtain the activation energy for the aqueous synthesis of thiolate-capped PbS nanoparticles by OA. A value of 14 ( 3 kJ mol-1 was determined. The enthalpy (∆H‡) and entropy (∆S‡) of activation for the OA reaction were determined by Eyring treatment of the same rate constants (Figure S9) to be 12 ( 3 kJ mol-1 and -160 ( 10 J K-1 mol-1, respectively; the resulting Gibbs free energy of activation (∆G‡) was 62 ( 6 kJ mol-1. The small magnitude of ∆H‡ may have derived from the enthalpic gain resulting from formation of lead-sulfide interactions being offset by the cost of dissociation of bound ligand by lead-thiolate bond cleavage and surface reorganization prior to coalescence. The negative value of ∆S‡ presumably reflected a decrease in the degrees of freedom associated with the coalescence of two distinct particles into one, with the entropy increase associated with free thiolate being offset by solvent organization around the charged species. To the best of our knowledge, these values represent the first reported thermodynamic parameters for activation in OA processes, although straightforward Eyring analysis of available literature data permits comparison to related systems. For example, using the rate constants reported by Tiemann,9 we could determine ∆H‡ and ∆S‡ values of 23 kJ mol-1 and -117 J K-1 mol-1, respectively, for the bottom-up synthesis of unsupported ZnS nanoparticles. We now turn our discussion to the subject of the very early stages of nanocrystalline growth. By applying OA and OR models, the growth of thiolate-capped PbS nanocrystals could be described accurately only for t > 0.5 s; at earlier times, the data could not be fit using either eq 2 or 3 and the mechanisms remain difficult to discern. This may be expected because OR, which necessitates crystal dissolution, would be disfavored until the nutrient ions have been substantially depleted, especially for highly insoluble salts like PbS. Furthermore, OA may not occur before there is a sufficiently high population of nanocrystals to participate in the process. See the Supporting Information for additional data and discussion covering this period.
Conclusions The rate of water self-exchange for Pb2+(aq) is on the order of 108 s-1,41 which approaches the diffusion limit. The lead dication is thus one of the most labile metallic ions in aqueous solution, and it is expected that growth of PbS nanocrystals should be very rapid. This is all the more true when one considers that the likely molecular species from which the nanocrystals are presumably assembled in this system are lead-thiolate clusters that already contain multiple Pb2+ ions in close proximity.42 Therefore, it remains a considerable experimental challenge to determine with accuracy the pathway of bottom-up nanocrystalline growth. As this study suggests, the overall mechanism is complicated. It probably involves temporally overlapped nucleation, OA and OR phases. However, we were successful in determining the predominant mechanism in specific time and temperature regimes. At times greater than 0.5 s, growth was mainly by OR in the low temperature regime (0 e T e 25 °C). At higher temperature (30 e T e 45 °C), growth was predominantly by OR at early times (0.5 < t < 20) and mainly by OA thereafter. At the very early stages of crystal formation (t < 0.5 s), nucleation processes were undoubtedly also at play and the growth curves could not be fit to either OA or OR mechanisms. Acknowledgment. We gratefully acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Foundation for Innovation (CFI) for financial support of this work through Discovery and New Opportunities grants, respectively, to N.D.J. A.L.B. thanks the Province of Ontario for a Graduate Scholarship (OGS) and NSERC for a Canada Graduate Scholarship, Doctoral (CGS-D). Supporting Information Available: Raw spectra prior to subtraction of peak corresponding to thiolate ligands, timeevolved UV-visible absorption spectra for all temperatures, Gaussian peak fittings for all temperatures, shift in second and third Gaussian fitted peaks over time, Arrhenius and Eyring plots, and additional data and discussion pertaining to growth at very early times. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Semiconductor and metal nanocrystals: synthesis and electronic and optical properties; Klimov, V. I., Ed.; Marcel Dekker, Inc.: New York, 2004. (2) Wise, F. W. Acc. Chem. Res. 2000, 33, 773. (3) Schaller, R. D.; Agranovich, V. M.; Klimov, V. I. Nat. Phys. 2005, 1, 189. (4) Günes, S.; Fritz, K. P.; Neugebauer, H.; Sariciftci, N. S.; Kumar, S.; Scholes, G. D. Sol. Energy Mater. Sol. Cells 2007, 91, 420. (5) Klimov, V. I.; Mikhailovsky, A. A.; McBranch, D. W.; Leatherdale, C. A.; Bawendi, M. G. Science 2000, 287, 1011. (6) Colvin, V. L.; Schlamp, M. C.; Allvisatos, A. P. Nature 1994, 370, 354.
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