# Study on Growth Kinetics of CdSe Nanocrystals with a New Model

- Huaipeng Su
^{1}, - John David Dixon
^{2}, - Andrew Y Wang
^{2}, - Jeremy Low
^{3}, - Jian Xu
^{3}and - Jingkang Wang
^{1}Email author

**Received: **4 November 2009

**Accepted: **24 February 2010

**Published: **25 March 2010

## Abstract

A model which involves both bulk diffusion process and surface reaction process has been developed for describing the growth behaviour of nanoparticles. When the model is employed, hypothesising that either of the processes alone dominates the overall growth process is unnecessary. Conversely, the relative magnitude of contributions from both processes could be obtained from the model. Using this model in our system, the growth process of CdSe QDs demonstrated two different growth stages. During the first stage, the growth of CdSe QDs was dominated by bulk diffusion, whereas, neither the bulk diffusion process nor the surface reaction process could be neglected during the later stage. At last, we successfully modelled the Ostwald ripening of CdSe QDs with LSW theories.

### Keywords

Growth kinetics Semiconductor nanocrystals Model CdSe Ostwald ripening## Introduction

A lot of work has been devoted to the preparation of colloidal semiconductor nanocrystals, also called quantum dots (QDs), and the investigation of their optical properties and applications during the past decade. Highly monodisperse II–VI [1–4] and III–V [5–7] QDs have been prepared and applied in various applications, such as light-emitting diodes [8, 9], biological labels [10, 11], solar cells [12]. There have, however, been relatively few papers on the crystallization kinetics of QDs [13].

With a quantitative understanding of the kinetics during crystallization, crystal growth could be optimized by adjusting some experimental parameters, so that high-quality crystals are reproducibly prepared. Several methods have been developed to prepare high-quality CdSe QDs, which provides an ideal system to verify the kinetic model of particle growth in solution. If the level of solution supersaturation could be measured, experimental growth rates could then be easily correlated with the classic growth model. Unfortunately, in the case of CdSe QDs, the supersaturation is generally unable to be measured accurately. Nevertheless, Yu et al. [14] found that the extinction coefficient per mole of CdSe at the first excitonic absorption peak was depended strongly on the size, and not dependent on synthesis methods, surface ligands, etc. We used this method to assess the size and concentration of CdSe QDs from the peak position and absorbance contained within the absorption spectrum.

The crystallization kinetics of CdSe QDs has been reported in several papers. Peng et al. observed “focusing” and “defocusing” of size distribution during the growth of several II–VI and III–V QDs [15]. They explained the behaviour by a diffusion-controlled process. Xie et al. reported another diffusion-controlled growth model to study the crystallization kinetics of CdSe synthesized via the TOP–TOPO–HDA route [16]. Bullen and Mulvaney, on the other hand, suggested a surface reaction-controlled model to explain the growth behaviour [17].

A colloidal particle grows by a sequence consisting of bulk diffusion of crystallizing materials towards the crystal surface and integration of crystallizing materials into the crystal lattice (also called surface reaction), so the growth of particles should be determined by both diffusion and reaction processes. The published investigations on growth kinetics of CdSe QDs, however, were almost based on the hypothesis that the growth rate was determined by either diffusion process or reaction process.

In this report, a new growth model that involves both crystallizing materials diffusion in bulk solution and surface reaction processes was developed and employed in the growth process of CdSe QDs. Instead of hypothesising, the possible growth determining process could be elucidated with the new model. At last we modelled the Ostwald ripening of CdSe QDs with LSW theories.

## Experimental Section

### Preparation of CdSe QDs

In a typical procedure, 0.0314 g of CdO, 0.1835 g of ODPA and 9.8 g of ODE were loaded into a 100-mL three-neck flask and heated to 310°C. The solution was cooled down to room temperature after becoming colourless. Then, 9.0 g of ODA and 1.5 g of TOPO were added into the solution and heated under nitrogen flow to 280°C, at which point 1.1 mL of Se/TBP (2.0 M) solution was swiftly injected into the reaction flask. The injection of Se stock solution resulted in immediate nucleation of CdSe QDs. The solution was cooled quickly to 230°C for the growth of CdSe QDs. Strong coordinating alkylphosphonic acid was used to prepare cadmium precursor in order to maintain control over the growth.

Samples were taken at various time intervals and diluted immediately with toluene to quench the QDs growth. The masses of the samples and toluene were precisely weighed in order to calculate the QDs concentration in the reaction solution. UV–vis absorption and photoluminescence (PL) spectra were collected on a Perkin Elmer Lambda 45 UV/vis spectrophotometer and a Fluoromax-3 spectrophotometer, respectively.

### Simulation Model

As soon as nuclei are formed in a supersaturated solution, they begin to grow into crystals. Generally, two main processes occur sequentially during the growth of crystals from solution [18]: the mass transport of crystallizing materials from the bulk supersaturated solution to the crystal-solution interface by bulk diffusion and the incorporation of crystallizing materials into the crystal lattice through the surface reaction process.

*δ*, the total flux of solute (crystallizing materials),

*J*, passing through a spherical surface with radius

*x*can be derived by Fick’s law as

*D*is the diffusion coefficient and

*C*is the solute concentration at

*x*(

*r*≤

*x*≤

*δ*). Under steady-state growth conditions, integration of

*C*(

*x*) from

*r*+ δ to

*r*gives

where *C*
_{
b
} is the bulk solute concentration and *C*
_{
i
} is the solute concentration at the interface.

*J*, can be written as

where *k* is the rate constant and *C*
_{
eq
} is the equilibrium concentration of the particle of radius *r*.

where *V*
_{
m
} is the molar volume of the solid.

*r*<< δ), Eq. (5) can be reduced to

We define a dimensionless variable *H* = *D*/(*kr*) as a representative of the relative magnitude of contributions from bulk diffusion and surface reaction. When *H* >> 1, the mass transport to the surface has a much higher rate than that of surface reaction, therefore the rate determining process is the surface reaction process. In this case, particle growth can be regarded as a surface reaction-controlled growth. Conversely, if *H* << 1, bulk diffusion process will be the rate determining process and particle growth can be regarded as a diffusion-controlled growth.

## Results and Discussion

Nucleation took place so rapidly that it is very difficult to study the nucleation kinetics. From the aforementioned discussion, however, we know that the nucleation and the subsequent processes (viz. growth process and Ostwald ripening) were well separated. Thereby, we have simulated only the kinetics of growth and Ostwald ripening.

### Growth Process

*N*

_{0}calculated from Fig. 2 was 3.92 × 10

^{16}no./cm

^{3}. In our reaction, the selenium concentration was excessive so that the concentration of cadmium in solution changed significantly. We can describe the particle growth in terms of cadmium concentration. The bulk solute concentration in terms of cadmium in solution during the growth process is given by

where
is the equilibrium concentration of cadmium. Xie et al. [16] estimated that
contributes much less than 4/3*πN*
_{0}, so we treat
as a constant here.

Eq. (10) has the same form as that deduced by Bullen et al. [17] based on the surface reaction-controlled model.

Eq. (11) has the same form as that developed by Xie et al. [16] based on the diffusion-controlled model.

From Eq. (8), the maximum radius of QDs during the growth process is given as *r*
_{max.} = *a*
_{2}/*b*
_{2} = 1.24 nm, which is very close to the experimental result. Two different fitting stages mean the growth of CdSe QDs follows different mechanisms. Because the radius was kept at the same magnitude, and using the average radius during the two growth stages, we obtained H_{1} = 0.00049 and H_{2} = 0.95. Hence, during the first growth stage, bulk diffusion was estimated to be more important, whereas, neither the bulk diffusion process nor the surface reaction process can be neglected during the later stage. We have discussed previously that there were some narrowing and broadening of the size distribution of CdSe QDs during the growth process. On the premise of the fitting results, the experimental change of size distribution of CdSe QDs matches well with crystallization theories. According to classic theories, at high supersaturation, the Gibbs–Thomson effect is weak, and crystal growth is usually dominated by the bulk diffusion process in which small crystals grow at a higher rate than large ones so that the particle size distribution is focused [13, 21, 22]. Sugimoto showed broadening of distribution of colloidal particles takes place all the time in the surface reaction-controlled growth due to the Gibbs–Thomson effect [23]. The fitting results show the influence of surface reaction in the second stage of growth of CdSe QDs cannot be neglected, so some broadening of distribution occurred.

### Ostwald Ripening

where
is the mean particle radius,
is the initial particle radius and *K* is the ripening parameter.

The curve fitting gives satisfactory precision with a related coefficient of 0.997. The parameter values are *K* = 8.32 × 10^{−5} nm^{3}/s and
which is very close to the radius maximum during the growth process.

## Conclusions

A new model which involves both bulk diffusion process and surface reaction process was developed. When the developed model is employed, there is no need to hypothesise possible determining process. Conversely, the relative magnitude of contributions from the two processes could be indicated from the model. Using this model, the growth process of CdSe QDs was simulated, and two different growth stages were demonstrated. During the first stage, the growth of CdSe was dominated by bulk diffusion process, whereas, both processes markedly influenced the kinetics of crystal growth during the latter growth stage. At last, we successfully modelled the Ostwald ripening of CdSe QDs with LSW theories.

## Declarations

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

## Authors’ Affiliations

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