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Crystallization kinetics of CdSe nanocrystals synthesized via the TOP–TOPO–HDA route
ARTICLE IN PRESS Journal of Crystal Growth 310 (2008) 3504– 3507
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Crystallization kinetics of CdSe nanocrystals synthesized via the TOP– TOPO– HDA route Xie Chuang a,b, Hao Hongxun a, Chen Wei a, Wang Jingkang a, a b
School of Chemical Engineering and Technology, Tianjin University, No. 92 Weijin Road, Tianjin 300072, PR China Post-Doctoral Station of Tianjin Economic and Technological Development Area, Tianjin 300457, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 29 March 2008 Accepted 21 April 2008 Communicated by K. Nakajima Available online 3 May 2008
An effective trioctylphosphine–trioctylphosphine oxide–hexadecylamine (TOP–TOPO–HDA) route was used to investigate the crystallization kinetics of CdSe nanocrystals. A new diffusion-controlled growth model was founded to simulate the CdSe nanocrystals (NCs) growth process. It was found that the population density of the CdSe NCs in solution remained constant during the early stage of the growth before ripening. About 28% of the Cd was consumed during nucleation. The concentration of Cd decreased after nucleation and the Cd monomer was almost exhausted after 300 s growth. The diffusion coefficient was found to be 150 nm2/s. The apparent critical nucleus was estimated to be 1.64 nm by extrapolation of the modeling curve. & 2008 Elsevier B.V. All rights reserved.
PACS: 81.07.Bc 81.10.Aj 81.10.Dn Keywords: A1. Growth models A2. Growth from solutions B1. Nanomaterials B2. Semiconducting cadmium compounds
1. Introduction High-quality CdSe nanocrystals (NCs) have been extensively investigated [1–7] in recent years due to their great prospects in various technical applications, such as light-emitting diodes (LEDs) [8], biomedical tags [9], lasers [10], solar cells [11], etc. Since the TOP–TOPO route was developed by Bawendi’s group [12], this kind of organometallic route has led to the synthesis of high-quality CdSe NCs. The substitution of CdO [13] for the conventional precursor Cd(CH3)2, and the introduction of HAD [4] have provided greener, reproducible and more economical routes to obtain CdSe NCs with higher luminescence and better monodispersity in size and shape. Although some other nonaqueous routes including new materials [14–18] were provided, the trioctylphosphine–trioctylphosphine oxide–hexadecylamine (TOP–TOPO–HDA) route performed the highest photoluminescence quantum yield up to now and was used frequently. By this time, the crystallization kinetics of CdSe NCs synthesized via TOP–TOPO–HDA route was determined qualitatively without modeling [19]. The systematic investigation [20] on CdSe sizing curve and extinction coefficient provided us an available method to determine the kinetics directly. Recently, the nucleation and growth kinetics of CdSe NCs in octadecene [21] and in Corresponding author. Tel.: +86 22 27405754; fax: +86 22 27374971.
E-mail address: [email protected] (W. Jingkang). 0022-0248/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2008.04.029
oleic acid/dodecylamine [18] have been determined by using the method mentioned above. These encourage us to determine the crystallization kinetics of this route qualitatively. In this work, the TOP–TOPO–HDA route developed by Qu et al. [19] was used to investigate the crystallization kinetics of CdSe NCs. A new diffusion-controlled growth model was introduced and the crystallization kinetics of this system was quantitively determined.
2. Experiments 2.1. Materials TOPO (99%), HDA (90%) and Se powder (99.5+%) were purchased from ACROS. Stearic acid (SA) (95%) was purchased from TCI, and tech-grade TOP was purchased from Aldrich. CdO (GR) and chloroform (HPLC) were purchased in China. All chemicals were directly used in experimental preparations without further purification. 2.2. Synthesis and characterization The CdSe NCs were synthesized by the procedure developed by Quand and Peng [5] with minor modifications. 0.1 mmol CdO and 0.4 mmol of SA were heated to 150 1C under Ar flow until CdO was
ARTICLE IN PRESS X. Chuang et al. / Journal of Crystal Growth 310 (2008) 3504–3507
completely dissolved. The mixture was dissolved in 1 mL TOP and the solution was transferred into a syringe in the glovebox. TOPO and HDA, 1.94 g of each, and the Se solution containing 1 mmol of Se dissolved in 2.92 g TOP were added to the flask at room temperature. The mixture was heated to 300 1C under Ar flow to form an optically clear solution. At this temperature, the Cd stock solution was swiftly injected under vigorous stirring. After the injection, the temperature was set at 250 1C for the growth of the NCs. Aliquots with a needle-tip amount of the reaction mixture were removed at various time intervals and diluted by 4.0 mL chloroform. Such samples were used directly in further optical measurements. The masses of the aliquots were precisely weighed in order to calculate the concentration of the NCs in the reaction solution. The reaction mixture was mixed with about 15 mL of chloroform at about 30–50 1C before it was completely solidified. UV–vis absorption spectra of the samples were collected at room temperature on a Varian Cary 100 spectrometer. Photoluminescence (PL) experiments were carried out on a Cary Eclipse Fluorescence Spectrophotometer. Fluorescence spectra were collected at room temperature with 377 nm excitation. A TECHNAI G2 F20 electron microscope operating at 200 kV was used for transmission electron microscopy (TEM). The element analysis of the NCs was performed on the EDX detector accessorial to the TEM. The concentration of CdSe NCs (and thus, the population density of NCs) was evaluated according to the method developed by Yu et al. [20].
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of the CdSe NCs. The molar ratio of Cd to Se, 1.2, was obtained by EDX analysis, which is very consistent with the result determined by Taylor et al. [22]. All of the characterization results indicate that our experiment is valid to synthesize high-quality CdSe NCs. Fig. 3 gives the diameter–time curve during the synthesis process described in the experiment section. The diameters were calculated from the first UV absorption peak positions using the sizing curve [20]. The NCs grew up gradually after a burst of nucleation and reached a ‘‘maximum’’ diameter after about 300 s. This could be regarded as the termination of the experiment. Further holding at high temperature could result in Ostwald ripening, as a result of which the size distribution of the NCs would broaden. This ripening kinetics has been studied by Peng et al. [1] and Talapin et al. [23] both experimentally and theoretically. The data on the NCs concentration versus time are also presented in Fig. 3. The concentrations of the CdSe NCs were
3. Results and discussion A rapid color change of the solution took place as soon as Cd stock solution was injected. The solution developed from orange into red color in the first 10 s. The phenomena are quite consistent with the description of Ref. [5]. Fig. 1 presents the temporal evolution of the first absorption peak position of CdSe NCs after the burst of nucleation. The PL spectra (Fig. 1 inset) indicate very high luminescence CdSe NCs. The relative standard deviation of the NCs obtained in this work without size sorting is about 17%, which could be regarded as approximately monodisperse, as shown in Fig. 2. The lattice fringes on the particles (Fig. 2(a) inset) indicate high crystallinity
Fig. 1. Temporal evolution of UV–vis absorption spectra of CdSe NCs during the synthesis; inset: absorption and photoluminescence spectrum of 4.39 nm CdSe NCs.
Fig. 2. (a) The TEM and (b) the size distribution of the CdSe NCs. The sample was prepared from the first aliquot.
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Fig. 3. Diameter and concentration of the CdSe NCs vs. time curves for the preparation as in Fig. 1; the diameters and the concentrations of NCs were calculated by the method mentioned in the experiment section.
Fig. 4. Free Cd concentration vs. time curve and the percent of deposited Cd as a function of time for the preparation as in Fig. 1.
calculated also by the method developed by Yu et al. [20]. The NCs concentration remained of the same magnitude, and could be regarded as a constant, 0.008 mmol/g (mmol NCs per gram of reaction solution), within the experiment error just like in some former works [18,21]. It indicates a clean separation between nucleation and growth. This verified the assumption of the kinetics described below. The population density N0, 4.8 1015 No./cm3, could be obtained by multiplying the concentration by Avogadro constant considering the solution density (0.9 g/cm3). In Fig. 4, the Cd concentrations in bulk solution at time t, [Cd]bulk, are plotted, which are equal to the initial concentration, [Cd]0, less the amount that was already deposited, as shown in
where x is the distance from the center of the particle, D is the diffusion coefficient and [Cd] is the concentration of free Cd. The integration of [Cd] from r+d to r gives [25]
½Cdbulk ¼ ½Cd0
4pN0 r 3V m
3
(1)
where r is the mean radius of the NCs at time t, and Vm is the molar volume of the bulk CdSe (32.99 cm3/mol). The calculation above is based on the hypotheses of monodispersity, spherical symmetry and separation between nucleation and growth. The percent of Cd that was already deposited, [Cd]con%, could be calculated from Eq. (2) and is also plotted in Fig. 4: ½Cd0 ½Cdbulk 100% ½Cdcon % ¼ ½Cd0
(2)
It was found that about 28% of the Cd was consumed during nucleation. [Cd]bulk decreased from the initial injection and the Cd monomers were almost exhausted after 300 s growth, which was constant with that described in the Ref. [19]. Qu et al. [19] determined the kinetics of TOP–TOPO–HDA scheme qualitatively without modeling. Some discussions were based on the assumption of a diffusion-controlled growth process, which was verified indirectly by their experimental and theoretical results [1,24]. For the diffusion-controlled mechanism, a theoretical result of the mathematic models has been suggested by Sugimoto [25] and was applied by Peng et al. [1]. Talapin et al. [23] modified the model under the consideration of both Gibbs–Thomson effect and the magnitude of ‘‘capillary length’’. In this paper, another diffusion-controlled growth model was introduced on the basis of the theoretical and experimental results mentioned above. The diffusion of monomers, J, from bulk solution toward the particle surface is written by Fick’s first law as d½Cd J ¼ 4px2 D (3) dx xXr
J¼
4prðr þ dÞD ½Cdbulk ½Cdreaction d
(4)
where d is the thickness of the diffusion layer, and [Cd]reaction is the Cd concentration at the NCs surface. The size of NC could be negligible compared to the thickness of the diffusion layer [3,23]. [Cd]reaction could be replaced with the equilibrium concentration [Cd]eq, since we assume that the growth is a diffusion-controlled process. Thus, Eq. (4) can be simplified: J ¼ 4pDrð½Cdbulk ½Cdeq Þ
(5)
On the other hand, J is related with growth rate dr/dt as [25] J¼
4pr 2 dr V m dt
(6)
Hence dr DV m ¼ ½Cdbulk ½Cdeq dt r With Eqs. (1) and (8) can be obtained as dr DV m 4pN0 3 D 3 ¼ ½Cd0 ½Cdeq ¼ ða3 b r 3 Þ r dt r r 3V m
(7)
(8)
where a3 ¼ Vm([Cd]0[Cd]eq) and b3 ¼ 4/3pN0. Here, we assume [Cd]eq as constant despite the Gibbs–Thomson effect because the magnitude estimation result shows that [Cd]eq contributes much less than 4/3pN0, and such treatment can dramatically simplify the computation. Integration of Eq. (8) yields time as a function of radius: # ( " ) 2 pffiffiffi 1 b r 2 þ abr þ a2 a þ 2br t¼ ln 2 3 arctan pffiffiffi þc 2 2 3a 6Dab ðbr aÞ (9) Suitable values of the parameters including N0 from Fig. 4 and Vm (32.99 cm3/mol) could guide the curve fitting. The curve fitting gives satisfactory precision as shown in Fig. 5, and the parameters obtained are reasonable. The values of a and b could be examined by rmax ¼ (a/b), which can be derived from Eq. (8) when dr/dt ¼ 0. The (a/b) obtained, 2.885, is very close to the experimental result of rmax, 2.89 nm. The diffusion coefficient, D, is 150 nm2/s, which is almost 7 orders of magnitude less than a general one.
ARTICLE IN PRESS X. Chuang et al. / Journal of Crystal Growth 310 (2008) 3504–3507
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the Ostwald ripening. The extrapolation of the fitting curve to t ¼ 0 can result in r0 ¼ 1.64 nm, which represents the apparent ‘‘critical size’’ nucleus formed in the nucleation process as soon as the monomers were injected.
4. Conclusions A TOP–TOPO–HDA route was used to synthesize high-quality CdSe NCs and the crystallization kinetics of this process was determined. A new diffusion-controlled growth model was developed and employed to model the growth process of CdSe NCs. The population density of the NCs remained constant before Ostwald ripening. The temporal evolution of the CdSe NCs concentration was obtained with a small diffusion coefficient. The size of apparent critical nucleus and the consumption of free Cd in nucleation were also found. Fig. 5. Fit to time–radius curve using Eq. (9).
Acknowledgments The authors thank Professor Paul Mulvaney of the University of Melbourne for the helpful assistance on diffusion-controlled crystal growth. The useful discussions according to curve fitting with Dr. Liu Yong are acknowledged. This work was financially supported by the Natural Science Foundation of Tianjin (China) under Grant no. 05YFJZJC02000. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] Fig. 6. The numerical solution of Eq. (10) when D is given.
The results above are based on assuming that the [Cd]eq is as constant. To verify the rationality of this hypothesis, the Gibbs–Thomson effect is taken into account and Eq. (8) is reformed as dr D 2sV m 4pN0 3 ¼ V m ½Cd0 V m ½Cd1 exp r (10) dt r rRT 3 Eq. (10) does not have analytical solutions but could be numerically solved when D is given. The results are plotted in Fig. 6. It could be found that the Gibbs–Thomson effect did bring some changes but the difference is not distinct. It should be noted that all the discussion above about the growth mode is limited in the early period of the growth before
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
X. Peng, J. Wickham, A.P. Alivisatos, J. Am. Chem. Soc. 120 (1998) 5343. L. Manna, E.C. Scher, A.P. Alivisatos, J. Am. Chem. Soc. 122 (2000) 12700. Z.A. Peng, X. Peng, J. Am. Chem. Soc. 123 (2001) 1389. D.V. Talapin, A.L. Rogach, A. Kornowski, M. Haase, H. Weller, Nano Lett. 1 (2001) 207. L. Qu, X. Peng, J. Am. Chem. Soc. 124 (2001) 2049. D.V. Talapin, A.L. Rogach, I. Mekis, S. Haubold, A. Kornowski, M. Haase, H. Weller, Colloids Surf. A Physicochem. Eng. Aspects 202 (2002) 145. X. Peng, Chem. A Eur. J. 8 (2002) 334. S. Coe, W.-K. Woo, M. Bawendi, V. Bulovic, Nature 420 (2002) 800. S.K. Poznyak, D.V. Talapin, E.V. Shevchenko, H. Weller, Nano Lett. 4 (2004) 693. M.V. Artemyev, U. Woggon, R. Wannemacher, H. Jaschinski, W. Langbein, Nano Lett. 1 (2001) 309. D.V. Talapin, S.K. Poznyak, N.P. Gaponik, A.L. Rogach, A. Eychmullera, Physica E 14 (2002) 237. C.B. Murray, D.J. Norris, M.G. Bawendi, J. Am. Chem. Soc. 115 (1993) 8706. Z.A. Peng, X. Peng, J. Am. Chem. Soc. 123 (2001) 183. W.W. Yu, X. Peng, Angew. Chem. Int. Ed. 41 (2002) 2368. Z.T. Deng, L. Cao, F.Q. Tang, B.S. Zou, J. Phys. Chem. B 109 (2005) 16671. S. Asokan, K.M. Krueger, A. Alkhawaldeh, A.R. Carreon, ZuzeMu, V.L. Colvin, N.V. Mantzaris, M.S. Wong, Nanotechnology 16 (2005) 2000. D.G. Wu, M.E. Kordesch, P.G. Van Patten, Chem. Mater. 17 (2005) 6436. B.F. Pan, R. He, F. Gao, D.X. Cui, Y.F. Zhang, J. Crystal Growth 286 (2006) 318. L. Qu, W.W. Yu, X. Peng, Nano Lett. 4 (2004) 465. W.W. Yu, L. Qu, W. Guo, X. Peng, Chem. Mater. 15 (2003) 2854. C.R. Bullen, P. Mulvaney, Nano Lett. 4 (2004) 2303. J. Taylor, T. Kippeny, S.J. Rosenthal, J. Cluster Sci. 12 (2001) 571. D.V. Talapin, A.L. Rogach, M. Haase, H. Weller, J. Phys. Chem. B 105 (2001) 12278. B.D. Dickerson, D.M. Irving, E. Herz, R.O. Claus, J.W.B. Spillman, K.E. Meissner, Appl. Phys. Lett. 86 (2005) 1719151. T. Sugimoto, Adv. Colloid Interface Sci. 28 (1987) 65.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Crystallization kinetics of CdSe nanocrystals synthesized via the TOP– TOPO– HDA route Xie Chuang a,b, Hao Hongxun a, Chen Wei a, Wang Jingkang a, a b
School of Chemical Engineering and Technology, Tianjin University, No. 92 Weijin Road, Tianjin 300072, PR China Post-Doctoral Station of Tianjin Economic and Technological Development Area, Tianjin 300457, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 29 March 2008 Accepted 21 April 2008 Communicated by K. Nakajima Available online 3 May 2008
An effective trioctylphosphine–trioctylphosphine oxide–hexadecylamine (TOP–TOPO–HDA) route was used to investigate the crystallization kinetics of CdSe nanocrystals. A new diffusion-controlled growth model was founded to simulate the CdSe nanocrystals (NCs) growth process. It was found that the population density of the CdSe NCs in solution remained constant during the early stage of the growth before ripening. About 28% of the Cd was consumed during nucleation. The concentration of Cd decreased after nucleation and the Cd monomer was almost exhausted after 300 s growth. The diffusion coefficient was found to be 150 nm2/s. The apparent critical nucleus was estimated to be 1.64 nm by extrapolation of the modeling curve. & 2008 Elsevier B.V. All rights reserved.
PACS: 81.07.Bc 81.10.Aj 81.10.Dn Keywords: A1. Growth models A2. Growth from solutions B1. Nanomaterials B2. Semiconducting cadmium compounds
1. Introduction High-quality CdSe nanocrystals (NCs) have been extensively investigated [1–7] in recent years due to their great prospects in various technical applications, such as light-emitting diodes (LEDs) [8], biomedical tags [9], lasers [10], solar cells [11], etc. Since the TOP–TOPO route was developed by Bawendi’s group [12], this kind of organometallic route has led to the synthesis of high-quality CdSe NCs. The substitution of CdO [13] for the conventional precursor Cd(CH3)2, and the introduction of HAD [4] have provided greener, reproducible and more economical routes to obtain CdSe NCs with higher luminescence and better monodispersity in size and shape. Although some other nonaqueous routes including new materials [14–18] were provided, the trioctylphosphine–trioctylphosphine oxide–hexadecylamine (TOP–TOPO–HDA) route performed the highest photoluminescence quantum yield up to now and was used frequently. By this time, the crystallization kinetics of CdSe NCs synthesized via TOP–TOPO–HDA route was determined qualitatively without modeling [19]. The systematic investigation [20] on CdSe sizing curve and extinction coefficient provided us an available method to determine the kinetics directly. Recently, the nucleation and growth kinetics of CdSe NCs in octadecene [21] and in Corresponding author. Tel.: +86 22 27405754; fax: +86 22 27374971.
E-mail address: [email protected] (W. Jingkang). 0022-0248/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2008.04.029
oleic acid/dodecylamine [18] have been determined by using the method mentioned above. These encourage us to determine the crystallization kinetics of this route qualitatively. In this work, the TOP–TOPO–HDA route developed by Qu et al. [19] was used to investigate the crystallization kinetics of CdSe NCs. A new diffusion-controlled growth model was introduced and the crystallization kinetics of this system was quantitively determined.
2. Experiments 2.1. Materials TOPO (99%), HDA (90%) and Se powder (99.5+%) were purchased from ACROS. Stearic acid (SA) (95%) was purchased from TCI, and tech-grade TOP was purchased from Aldrich. CdO (GR) and chloroform (HPLC) were purchased in China. All chemicals were directly used in experimental preparations without further purification. 2.2. Synthesis and characterization The CdSe NCs were synthesized by the procedure developed by Quand and Peng [5] with minor modifications. 0.1 mmol CdO and 0.4 mmol of SA were heated to 150 1C under Ar flow until CdO was
ARTICLE IN PRESS X. Chuang et al. / Journal of Crystal Growth 310 (2008) 3504–3507
completely dissolved. The mixture was dissolved in 1 mL TOP and the solution was transferred into a syringe in the glovebox. TOPO and HDA, 1.94 g of each, and the Se solution containing 1 mmol of Se dissolved in 2.92 g TOP were added to the flask at room temperature. The mixture was heated to 300 1C under Ar flow to form an optically clear solution. At this temperature, the Cd stock solution was swiftly injected under vigorous stirring. After the injection, the temperature was set at 250 1C for the growth of the NCs. Aliquots with a needle-tip amount of the reaction mixture were removed at various time intervals and diluted by 4.0 mL chloroform. Such samples were used directly in further optical measurements. The masses of the aliquots were precisely weighed in order to calculate the concentration of the NCs in the reaction solution. The reaction mixture was mixed with about 15 mL of chloroform at about 30–50 1C before it was completely solidified. UV–vis absorption spectra of the samples were collected at room temperature on a Varian Cary 100 spectrometer. Photoluminescence (PL) experiments were carried out on a Cary Eclipse Fluorescence Spectrophotometer. Fluorescence spectra were collected at room temperature with 377 nm excitation. A TECHNAI G2 F20 electron microscope operating at 200 kV was used for transmission electron microscopy (TEM). The element analysis of the NCs was performed on the EDX detector accessorial to the TEM. The concentration of CdSe NCs (and thus, the population density of NCs) was evaluated according to the method developed by Yu et al. [20].
3505
of the CdSe NCs. The molar ratio of Cd to Se, 1.2, was obtained by EDX analysis, which is very consistent with the result determined by Taylor et al. [22]. All of the characterization results indicate that our experiment is valid to synthesize high-quality CdSe NCs. Fig. 3 gives the diameter–time curve during the synthesis process described in the experiment section. The diameters were calculated from the first UV absorption peak positions using the sizing curve [20]. The NCs grew up gradually after a burst of nucleation and reached a ‘‘maximum’’ diameter after about 300 s. This could be regarded as the termination of the experiment. Further holding at high temperature could result in Ostwald ripening, as a result of which the size distribution of the NCs would broaden. This ripening kinetics has been studied by Peng et al. [1] and Talapin et al. [23] both experimentally and theoretically. The data on the NCs concentration versus time are also presented in Fig. 3. The concentrations of the CdSe NCs were
3. Results and discussion A rapid color change of the solution took place as soon as Cd stock solution was injected. The solution developed from orange into red color in the first 10 s. The phenomena are quite consistent with the description of Ref. [5]. Fig. 1 presents the temporal evolution of the first absorption peak position of CdSe NCs after the burst of nucleation. The PL spectra (Fig. 1 inset) indicate very high luminescence CdSe NCs. The relative standard deviation of the NCs obtained in this work without size sorting is about 17%, which could be regarded as approximately monodisperse, as shown in Fig. 2. The lattice fringes on the particles (Fig. 2(a) inset) indicate high crystallinity
Fig. 1. Temporal evolution of UV–vis absorption spectra of CdSe NCs during the synthesis; inset: absorption and photoluminescence spectrum of 4.39 nm CdSe NCs.
Fig. 2. (a) The TEM and (b) the size distribution of the CdSe NCs. The sample was prepared from the first aliquot.
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Fig. 3. Diameter and concentration of the CdSe NCs vs. time curves for the preparation as in Fig. 1; the diameters and the concentrations of NCs were calculated by the method mentioned in the experiment section.
Fig. 4. Free Cd concentration vs. time curve and the percent of deposited Cd as a function of time for the preparation as in Fig. 1.
calculated also by the method developed by Yu et al. [20]. The NCs concentration remained of the same magnitude, and could be regarded as a constant, 0.008 mmol/g (mmol NCs per gram of reaction solution), within the experiment error just like in some former works [18,21]. It indicates a clean separation between nucleation and growth. This verified the assumption of the kinetics described below. The population density N0, 4.8 1015 No./cm3, could be obtained by multiplying the concentration by Avogadro constant considering the solution density (0.9 g/cm3). In Fig. 4, the Cd concentrations in bulk solution at time t, [Cd]bulk, are plotted, which are equal to the initial concentration, [Cd]0, less the amount that was already deposited, as shown in
where x is the distance from the center of the particle, D is the diffusion coefficient and [Cd] is the concentration of free Cd. The integration of [Cd] from r+d to r gives [25]
½Cdbulk ¼ ½Cd0
4pN0 r 3V m
3
(1)
where r is the mean radius of the NCs at time t, and Vm is the molar volume of the bulk CdSe (32.99 cm3/mol). The calculation above is based on the hypotheses of monodispersity, spherical symmetry and separation between nucleation and growth. The percent of Cd that was already deposited, [Cd]con%, could be calculated from Eq. (2) and is also plotted in Fig. 4: ½Cd0 ½Cdbulk 100% ½Cdcon % ¼ ½Cd0
(2)
It was found that about 28% of the Cd was consumed during nucleation. [Cd]bulk decreased from the initial injection and the Cd monomers were almost exhausted after 300 s growth, which was constant with that described in the Ref. [19]. Qu et al. [19] determined the kinetics of TOP–TOPO–HDA scheme qualitatively without modeling. Some discussions were based on the assumption of a diffusion-controlled growth process, which was verified indirectly by their experimental and theoretical results [1,24]. For the diffusion-controlled mechanism, a theoretical result of the mathematic models has been suggested by Sugimoto [25] and was applied by Peng et al. [1]. Talapin et al. [23] modified the model under the consideration of both Gibbs–Thomson effect and the magnitude of ‘‘capillary length’’. In this paper, another diffusion-controlled growth model was introduced on the basis of the theoretical and experimental results mentioned above. The diffusion of monomers, J, from bulk solution toward the particle surface is written by Fick’s first law as d½Cd J ¼ 4px2 D (3) dx xXr
J¼
4prðr þ dÞD ½Cdbulk ½Cdreaction d
(4)
where d is the thickness of the diffusion layer, and [Cd]reaction is the Cd concentration at the NCs surface. The size of NC could be negligible compared to the thickness of the diffusion layer [3,23]. [Cd]reaction could be replaced with the equilibrium concentration [Cd]eq, since we assume that the growth is a diffusion-controlled process. Thus, Eq. (4) can be simplified: J ¼ 4pDrð½Cdbulk ½Cdeq Þ
(5)
On the other hand, J is related with growth rate dr/dt as [25] J¼
4pr 2 dr V m dt
(6)
Hence dr DV m ¼ ½Cdbulk ½Cdeq dt r With Eqs. (1) and (8) can be obtained as dr DV m 4pN0 3 D 3 ¼ ½Cd0 ½Cdeq ¼ ða3 b r 3 Þ r dt r r 3V m
(7)
(8)
where a3 ¼ Vm([Cd]0[Cd]eq) and b3 ¼ 4/3pN0. Here, we assume [Cd]eq as constant despite the Gibbs–Thomson effect because the magnitude estimation result shows that [Cd]eq contributes much less than 4/3pN0, and such treatment can dramatically simplify the computation. Integration of Eq. (8) yields time as a function of radius: # ( " ) 2 pffiffiffi 1 b r 2 þ abr þ a2 a þ 2br t¼ ln 2 3 arctan pffiffiffi þc 2 2 3a 6Dab ðbr aÞ (9) Suitable values of the parameters including N0 from Fig. 4 and Vm (32.99 cm3/mol) could guide the curve fitting. The curve fitting gives satisfactory precision as shown in Fig. 5, and the parameters obtained are reasonable. The values of a and b could be examined by rmax ¼ (a/b), which can be derived from Eq. (8) when dr/dt ¼ 0. The (a/b) obtained, 2.885, is very close to the experimental result of rmax, 2.89 nm. The diffusion coefficient, D, is 150 nm2/s, which is almost 7 orders of magnitude less than a general one.
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the Ostwald ripening. The extrapolation of the fitting curve to t ¼ 0 can result in r0 ¼ 1.64 nm, which represents the apparent ‘‘critical size’’ nucleus formed in the nucleation process as soon as the monomers were injected.
4. Conclusions A TOP–TOPO–HDA route was used to synthesize high-quality CdSe NCs and the crystallization kinetics of this process was determined. A new diffusion-controlled growth model was developed and employed to model the growth process of CdSe NCs. The population density of the NCs remained constant before Ostwald ripening. The temporal evolution of the CdSe NCs concentration was obtained with a small diffusion coefficient. The size of apparent critical nucleus and the consumption of free Cd in nucleation were also found. Fig. 5. Fit to time–radius curve using Eq. (9).
Acknowledgments The authors thank Professor Paul Mulvaney of the University of Melbourne for the helpful assistance on diffusion-controlled crystal growth. The useful discussions according to curve fitting with Dr. Liu Yong are acknowledged. This work was financially supported by the Natural Science Foundation of Tianjin (China) under Grant no. 05YFJZJC02000. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] Fig. 6. The numerical solution of Eq. (10) when D is given.
The results above are based on assuming that the [Cd]eq is as constant. To verify the rationality of this hypothesis, the Gibbs–Thomson effect is taken into account and Eq. (8) is reformed as dr D 2sV m 4pN0 3 ¼ V m ½Cd0 V m ½Cd1 exp r (10) dt r rRT 3 Eq. (10) does not have analytical solutions but could be numerically solved when D is given. The results are plotted in Fig. 6. It could be found that the Gibbs–Thomson effect did bring some changes but the difference is not distinct. It should be noted that all the discussion above about the growth mode is limited in the early period of the growth before
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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