The grand challenge in synthesizing nanocrystals of high quality (crystallinity, monodispersity etc.) is to understand the interplay of the reaction factors (e.g. precursor reactivity, precursor concentration and reaction temperature) and the growth process in detail. For many years, the formation process of colloidal systems was described by basic theoretical models which are explained in the following paragraphs.
2.1.1 LaMer Theory
The formation process of inorganic nanocrystals according to the model of LaMer and Dinegar[27]
has been known since 1950 and can be divided in three parts which are illustrated in figure 1:
I) The free monomer concentration in solution increases rapidly with proceeding reaction time.
After reaching the supersaturation level CS, homogenous nucleation is possible but effectively infinite.
II) After exceeding a critical nucleation concentration Cmin level the system has to surpass a high energy barrier for the self-nucleation. The burst nucleation reduces the free monomer concentration which drops below the critical level and no additional nucleation occurs.
III) Nanocrystals grow under the control of monomer diffusion towards pre-existing nuclei in solution. The reaction on stable nuclei surfaces results in discrete particles under the consumption of monomers. For the preparation of monodisperse nanoparticles the separation of nucleation and growth processes is essential and a high nucleation rate is followed by a slow growth process.[18]
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Figure 1: Schematic representation of the LaMer model. The qualitative plotted curve describes the monomer concentration as a function of time.
2.1.2 Classical Nucleation
The basic idea of the separation of nucleation and growth[18] can be interpreted as the division into a homogenous and heterogeneous phase. The energy barrier for the generation of nuclei from solution (homogenous nucleation) compared to the growth process at the expense of monomers onto existing stable nucleating surfaces (heterogeneous nucleation) is much higher. The classical nucleation[28] theory is the expression for the thermodynamically driven formation process which is depicted in figure 2.
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Figure 2: Free energy ∆G diagram of the surface ∆GS, bulk ∆GB,and total free energy ∆GT contributions as function of the particle radius r. The critical radius rc describes the energy barrier for the smallest stable particle during the nucleation according the classical nucleation theory.
The homogenous nucleation is described by summing at the total free energy of a system ∆𝐺𝑇 of spherical particles with radius r. The principal aim of the model is to estimate the formation of a stable nucleus in which the total free energy of ∆𝐺𝑇 is in general expressed as the sum of the surface free energy ∆GS with surface energy 𝛾 and bulk free energy ∆GB, which is dependent upon temperature T, Boltzmann´s constant 𝑘𝐵, supersaturation of the solution S and molar volume v of the system.
∆𝐺𝑇 = 4𝜋𝑟2𝛾 + 4
3 𝜋𝑟3∆𝐺𝐵 (2.1)
∆𝐺𝐵 = −𝑘𝐵𝑇 ln (𝑆) 𝑣⁄ (2.2) Changing the experimental parameters like temperature T, supersaturation S and surface free energy ∆GS due to variation of stabilizing ligands has strong effects on the system.[18] The contribution from the bulk free energy term ∆GB is always negative and favours nuclei formation in a supersaturated system. The positive term of the surface free energy ∆GS describes the unfavourable formation of new surfaces. The total free energy ∆GT as a function of radius r represents a maximum free energy which a nucleus needs to become stable, without getting redissolved.
∆𝐺𝐶 = 4 3⁄ 𝜋𝛾𝑟𝐶2 (2.3)
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This energy barrier is called the critical free energy ∆GC which is required to obtain stable particles with the critical radius 𝑟𝐶.
𝑟𝐶 = −2𝛾 ∆𝐺
𝐵 (2.4) The critical particle radius corresponds to the minimum size particles which persist in solution before growing further.
2.1.3 Classical Growth
According to the classical nucleation model[28], nuclei are formed which act as templates for further crystal growth. Each growing particle can be treated as a spherical and independent particle which is surrounded by a concentration gradient with spherical symmetry.[29] The growth process can be described according the classical growth theory[18][30] and consists of monomer diffusion to the surface and then reaction at the surface. Both the number of nuclei and free monomers control the growth process.
The diffusion mechanism can be described according Fick’s first law[29], where the monomer transport J is proportional to the diffusion coefficient D and x is the distance to the centre of the particle. The monomer concentration gradient dC
dx is the driving force and is also proportional to the flux of the monomers to the particles J.
𝐽 = −4𝜋𝑥2𝐷𝑑𝐶
𝑑𝑥 (2.5) For particles in solution, where δ is the distance from the particle surface to the bulk monomer concentration CB and CI the monomer concentration at the solid/liquid interface, Fick´s first law can be rewritten as
𝐽 = −4𝜋𝐷𝑟(𝑟+𝛿)
𝛿 (𝐶𝐵− 𝐶𝐼) (2.6) For nanosized particles, the particle radius r is smaller than the diffusion layer δ (r ≪ δ) and can be neglected. After the diffusion to the particle, the surface reaction can be written as
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𝐽 = −4𝜋𝑟2𝑘(𝐶𝐵− 𝐶𝑅) (2.7) Where the rate of the surface reaction, k, is independent of the particle size, and CR is the concentration at the surface of the particle. For the classical growth model, either the monomer diffusion to the particle surface or the monomer reaction at the surface is the limiting factor. If the monomer-surface reaction is faster than the diffusion step, it can be ascribed as a diffusion controlled (D << kr, kr: rate of surface reaction) process. The opposite case describes a surface reaction rate controlled (D >> kr) process, where the diffusion rate is much higher than the reaction rate.
2.1.4 Growth Mechanism of Nanoparticles
Different existing models describe the growth mechanisms that produce nanocrystals and which dictate the final morphology of nanoparticles. Modern in-situ characterization techniques[31] allow the evaluation of the growth process based on new concepts of non-classical growth of inorganic nanoparticles.[32]
Monomer attachment describes the initial growth process after the nucleation stage.[32] The mechanism includes two similar processes: coalescence[33] and orientated attachment[34][35] which differ in the orientation of the crystal lattice. The coalescence describes the non-preference for grain attachment among neighbouring grains which leads to randomly orientated lattices planes.[31] The orientated attachment, also known as the grain rotation-induced grain coalescence mechanism (GRIGC), is the perfect crystallographic alignment of the lattice planes and coalescence of neighbouring grains, eliminating a common grain boundary. The rotation of grains during the attachment is driven by low energy configurations.[36]
In 1950 the widely used nucleation and growth model of LaMer was described the first time and it is also interpreted for the synthesis of nanocrystals. According to this model, the diffusion and the consumption of monomers result in particle growth. Oriented attachment is a new approach to explain the growth process of nanocrystals and does not match with the classical model by LaMer.
The approach was increasingly described in the last twenty years and explains the self-assembly into single-crystalline nanostructures from individual particles.[34][35] There is no current model for quantifying this growth process and modelling the growth kinetics.
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Ostwald ripening occurs in the later growth stages.[31] The thermodynamic driven mechanism describes the growth process of larger particles at the expense of smaller particles in solution.[37]
The atoms from smaller particles undergo dissolution because of high solubility and surface energy, resulting in larger particles. Digestive ripening[38] can be explained as the inverse Ostwald ripening process. Colloidal particles are transformed in smaller particles at the expense of large ones by the reduction of the interfacial free energy.[39]
The controlled synthesis of high quality semiconductor nanocrystals has been an important material chemistry research topic. The shape control of CdSe nanocrystals was described by Peng and co-workers as interparticle growth process.[40] The interplay of chemical potential of the bulk solution and the surface energy of the particle facets is responsible for the diffusion-controlled growth process of the monomer along the nanocrystal surface.