We seek to understand the stability of a particular nanoparticle structure of a certain size. For a given temperature and pressure, this requires minimization of the Gibbs free energy (G(p, T) for bulk materials). The original calculation of the surface contribution to the free energy of a small particle is due to Gibbs.1415 In a similar treatment we can write an expression for G(p, T, γ), where γ is an interfacial free energy. If the nanoparticles are kinetically stable against coarsening or dissolution, we can write:

G(p, T, γ) = E - TS + (p + p')V + γA + other terms     (1)

where T and p are ambient temperature and pressure, p' is an excess internal pressure associated with strain at the surface;2 A is the surface area. The other terms include additional contributions at the surface, such as edge or defect free energies. An expansion γ = c₁ + c₂/r, where c₁ and c₂ are constants and r is the particle radius, is sometimes adopted to empirically include the additional terms.8 From eqn. (1), it is clear that an evaluation of γ is required for consideration of phase stability at small particle sizes. Furthermore, changes in the nanoparticle environment can be directly included in this description by modification of γ.

Experimental determination of interfacial free energy of solids is difficult,17 and theoretical approaches are an attractive alternative. In MD calculations, the interactions between atoms are usually parametrized as pair potentials.16 Static (low temperature) or dynamic (300 K or higher) energetic relaxation of the common termination surfaces of bulk crystals allows calculation of surface energy. 181920 The errors in equating surface energy with surface free energy are expected to be small.17 Combining these values with known, or assumed crystal habits provides average surface free energy values for different phases that can be used to predict the variation in phase stability with particle size.

Empirical and theoretical studies (below) have shown that the size dependence of the relative phase stability of polymorphs often follows the trend given in Fig. 1. Below a critical size, D_c, one of the bulk metastable phases becomes thermodynamically favored. For ZnS, bulk sphalerite is stable relative to wurtzite by ΔG_bulk ≈ 13 kJ mol^-1, and a stability inversion is predicted to occur for particle sizes below D_c ≈ 7 nm (in vacuum).18 For TiO₂, bulk rutile is stable relative to anatase by ΔG_bulk ≈ 67 kJ mol^-1, and the predicted D_c ≈ 14 nm,22 in good agreement with experimental observation.23 An additional important result of MD simulations is that the free energy curves may be modified in the presence of alternative solvents or vacuum. The simulations show that, in the presence of surface water, the calculated phase stability inversion for ZnS is shifted to smaller particle sizes, or absent all together.

Due to the extremely low solubility of TiO₂ and ZnS, suspensions of capped (organic ligand coated) or uncapped nanoparticles are kinetically stabilized against dissolution-based coarsening at room temperature. For example, the size of aqueous ZnS nanoparticles in pure water at room temperature is unchanged in XRD determination for at least six months. (However, increasing the ionic strength of the solution leads to much more rapid coarsening.) Uncapped particles carrying surface charge (and, generally, capped nanoparticles) are kinetically or thermodynamically stable against aggregation.24 If the ionic strength of the room temperature solution containing ZnS nanoparticles is increased, aggregation occurs and is accompanied by coarsening. At elevated temperatures, or if aggregation can occur, nanoparticle growth may be observed, with coarsening kinetics characteristic of the pathway.

For nanoparticles in solution, the mechanism of crystal growth may be surface precipitation of solvated atoms or few-atom clusters (diffusion-based), or crystallographically-specific aggregation-based growth (oriented aggregation) involving whole nanoparticles.12 For ZnS, hydrated species may be Zn₁S₁,25 or Zn₃S₃ and Zn₄S₆.26 Diffusion-based growth is called Ostwald ripening, and the kinetics are described further below. Aggregation-based growth has been described mathematically by Smoluchowksi.27 It includes consideration of whether successful coalescence accompanies every contact, or whether the probability of coalescence is less than unity per collision, in which case additional constraints (activation barriers; orientation dependence) may apply. This is a current topic for large-scale molecular modeling.2829 However, the crystallographic requirements and interfacial reactions implicit in growth via oriented aggregation have not been fully explored and the kinetic consequences have only been formulated in a few examples.

If structure transformation accompanies growth, the rate of transformation may be limited by the growth rate, or it may be affected by the aggregation state. The universal phenomenological Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation is widely used in the former situation.30 Aggregation-based transformation pathways are introduced with specific examples below.

TiO₂ is prepared by controlled hydrolysis, in which a titanium precursor (titanium alkoxide, titanium chloride or titanium isopropoxide) is hydrolyzed in an aqueous solution. The results are dependent on temperature, solution composition, stirring conditions, etc. Using these methods, we have prepared (1) 5–7 nm nanocrystalline anatase based on a sol–gel method31 (dried as pieces of membranes, rather than as a powder);33 (2) 8–21 nm nanocrystalline anatase samples, tuning the particle size by heat treatment of nanometer-sized amorphous TiO₂;3435 (3) ~3 nm nanocrystalline anatase by dripping a mixture of the titanium isopropoxide and ethanol into a 1.1 pH solution at 5°C.36 The samples have been further processed under hydrothermal conditions (T = 100–250°C, pH = 2–12) to obtain TiO₂ nanoparticle products that differ widely in morphology, phase and size distribution.

ZnS is prepared by controlled precipitation. We have prepared ~3 nm ZnS nanoparticles by reaction of zinc chloride and sodium sulfide in an aqueous solution with and without capping surfactant (e.g. mercaptoethanol),37 or in a non-aqueous solvent such as methanol, without surfactant.49 The ZnS nanoparticles may subsequently be treated in hydrothermal conditions.

The phase composition of the prepared samples is characterized by X-ray diffraction (XRD) and high resolution transmission electron microscopy (TEM). Particle size and size distribution is studied by TEM, UV absorption spectroscopy, small angle X-ray scattering (SAXS) and dynamic light scattering (DLS). Note that particle sizes given in this paper refer to the mean particle diameter. TEM imaging additionally shows particle morphology. Additional structural studies include synchrotron-based wide angle X-ray scattering (WAXS), X-ray adsorption near-edge structure (XANES) and extended X-ray adsorption fine structure (EXAFS). BET (Brunauer–Emmett–Teller) and BJH (Barrett–Johner–Halenda) measurements provide surface area and micro-pore size distribution by nitrogen adsorption.

Ostwald-ripening (O-R) refers to a dissolution–precipitation based mechanism for particle growth (coarsening) in solution. The chemical potential of surface atoms of small solid nanoparticles is elevated with respect to larger particles due to the presence of the excess energy associated with the surface. This increases the relative solubility of small particles, and larger particles grow preferentially. The kinetics of O-R crystal growth can usually be described by the following power law,30

D(t) = D₀ + k·t^1/n

where D₀ is the initial particle size, D(t) is the size at time t, k is a rate constant for the limiting step, and the exponent, n, is determined by the nature of the rate limiting step. The rate of the growth may be controlled by diffusion in solution (n ≈ 1), diffusion at the particle surface (n ≈ 2), or the interface dissolution/precipitation step (n ≈ 3). Diffusion-based growth can occur without a solvent matrix (sintering), given particle contact, by atomic diffusion from smaller particles to bigger ones.

For systems of nanoparticles, in addition to O-R, the nanoparticles can themselves act as the building blocks for crystal growth. An oriented pair of nanoparticles can attach to each other, eliminating the attaching free surfaces and releasing the surface energies associated with the free surfaces. This new mechanism for nanoparticle growth through oriented-attachment (OA) was first discovered in samples of hydrothermally treated nanocrystalline TiO₂. 383940 Natural nanoparticles of iron oxy-/hydroxyl oxides were also found to grow by OA under certain geochemical conditions.41 Imperfect OA can produce defects (e.g. edge dislocations, stacking faults, twins) in the OA-grown crystal, that often mark the original boundaries between component nanocrystals growing via the OA pathway.42

Evidence for OA-based growth comes both from interpretation of microstructure, and the deviation of the growth kinetics from the O-R model. OA growth does not follow Ostwald ripening kinetics. During hydrothermal coarsening of ~3 nm uncapped ZnS nanoparticles,43 TEM and growth kinetics indicate that O-R occurs throughout (Fig. 2a), but that OA dominates in the early stages. If the nanoparticles are capped by mercaptoethanol, a distinct initial growth stage is observed (Fig. 2b), during which O-R is suppressed and pure OA-based growth occurs. The capping layer provides a barrier to dissolution before it is thermally desorbed. In the second growth stage, both OA and O-R contribute to the growth of ZnS nanoparticles. In the analysis of the first growth stage, assuming particle volume doubles after OA events, the evolution of particle size is well described by the following expression:

D(t) = D₀(kt + 1)/(kt + 1)

where D, k and t are as defined above.

The occurrence of OA-based growth in hydrothermally coarsened TiO₂ depends on pH, as expected. Away from the zero point of charge (pH_zpc ≈ 6 for nanocrystalline anatase31), the presence of surface charge prevents aggregation and promotes dissolution/precipitation based growth.32

OA-based growth has been observed during the thermal coarsening of dry samples. For nanoparticles suspended in solution, thermal motion coupled with interparticle electrostatic interactions may promote particle contact, oriented alignment and hence growth. In some cases, if the nanoparticles are dry, or tightly aggregated in solution, thermal energy will not be sufficient to explore many configurations. Nevertheless, a subset of the vast number of contact points may be sufficiently well aligned to allow joining and interface elimination.

During the heating in air of nanoscale amorphous TiO₂, the formation of nanocrystalline anatase was accompanied by OA-based growth. The time-evolution of the phase composition, the average particle size and the particle size distribution (Fig. 3) of anatase nanoparticles were described quantitatively using the Smoluchowski equation.34

During the heating of nanocrystalline anatase in air, the formation of rutile accompanies the coarsening of anatase, but only after the anatase particle size exceeds 10–15 nm. The transformation temperature is 465°C, a few hundred degrees below that for bulk materials.33 Partially transformed anatase nanoparticles are seldom seen in TEM, suggesting rapid nuclei growth rate following nucleation. The kinetics of the transformation was observed by quantitative XRD phase analysis. None of the available kinetic models derived for phase transformation in bulk materials, including the JMAK equation, were suitable for the experimental data.

Thermodynamic analysis (illustrated by the scheme in Fig. 1) showed that at the particle size of D_c ≈ 14 nm, the free energies of anatase and rutile are equal.2 This may explain why nanocrystalline anatase is the majority phase of the nanocrystalline TiO₂ products in most syntheses, and rutile forms only after anatase coarsens above 10–15 nm.22233 The relative phase stability of nanophases is the driving force for phase transformation.

The unusual kinetic behavior of the transformation was explained with a microscopic model. High resolution TEM analysis showed that rutile-like structural elements can be produced in nanocrystalline anatase twin {121} interfaces.45 Such structural elements may serve as nuclei for transformation from nanocrystalline anatase to rutile. Nucleation at preexisting structural sites would require much lower activation energy (transformation temperature) than for nucleation on surface or in the bulk.

Analysis of the kinetic data lead to a rate law for a phase transformation that explictly includes particle size. This novel approach showed that the kinetics of the transformation is second order with respect to the number of anatase nanoparticles (Fig. 4). This result is consistent with a model in which the number of anatase particles transformed to rutile in a unit time is proportional to the chances of contact between anatase nanoparticles. In a dry powder, this model requires the making and breaking of interparticle contacts. Successful (oriented) contacts lead to rutile nucleation and rapid transformation of the joined nanoparticle pair. Thus, structural and kinetic analysis lead to the development of a new kinetic model, termed interface nucleation (IN) model.46 The activation energy for IN in nanocrystalline anatase was derived to be ~166 kJ mol^-1. Interface nucleation is dominant at lower temperatures (< ~600°C) for phase transformation of nanocrystalline anatase, At higher temperatures (> 600°C), both surface nucleation and interface nucleation govern the phase transformation.47

Subsequent experiments are consistent with the IN model. Dilution of anatase nanoparticles by mixing them with inert nanoparticles (e.g. A1₂O₃) reduced the the transformation rate.47 Conversely, in denser nanocrystalline anatase samples, the transformation rate was enhanced.48 Due to the lack of large interfaces in bulk materials, phase transformation from bulk anatase to rutile mainly happens via surface nucleation and/or bulk nucleation at high temperatures,47 both of which have higher activation energies.

Experimental measurements of the transformation temperature from nano-sphalerite to nano- wurtzite agree very well with the phase boundary predicted by thermodynamic analysis.18 In TEM observations of reacted sphalerite samples (larger than 10 nm), wurtzite domains are always observed to be sandwiched between sphalerite structures. Such observations suggest that the wurtzite nucleation occurs at interfaces of two sphalerite particles, i.e. interface nucleation. Stacking faults, twining and polytype structures are readily formed, as illustrated in the TEM image shown in Fig. 5b.

Experimentally we observe that wurtzite is formed during the heating of initially ~3 nm sphalerite at 350°C for 2 h in vacuum. In contrast, no wurtzite formation is observed when the same initial material was heated at 350°C for 2 h in air. This is consistent with the MD results, because nanoparticles exposed to the ambient air readily adsorb water moisture. Water may be able to chemically bind with ZnS nanoparticles even at this temperature.

Transformation from sphalerite nanoparticles to wurtzite was also investigated under hydrothermal conditions (140–225°C in water).50 As shown in Fig. 5a, TEM examinations of the nanoparticles reacted for different lengths of time found wurtzite domains always at the nanoparticle surface, and never sandwiched between sphalerite domains, in striking contrast to the results of vacuum coarsening experiments (Fig. 5b). During hydrothermal treatment, the interfacial area between the outer wurtzite phase and the inner sphalerite phase (≈ 20 nm²) is invariant with time. This distinctive phenomenon suggests that the activation energy of the sphalerite → wurtzite transformation has a dependence on the reaction area (i.e. the wurtzite/ sphalerite interface area).

As experiment and simulation indicate that the presence of water can invert the sphalerite–wurtzite stability relations in nanosize materials, we investigated the possibility that water binding may induce a structural transition. From eqn. (1), at a fixed particle size, modifications to the free energy via the surface term, γ A, may be achieved if γ_vacuum ≠ γ_water.

Experimentally we synthesized ~3 nm nanocrystalline ZnS in anhydrous methanol, and performed in situ structural analyses, such as WAXS, Fig. 6, before and after the addition of water (50 μl H₂O/ml methanol). The ZnS nanoparticles in methanol retain a structure based on ZnS₄ tetrahedra, but diffraction peak broadening renders definite phase analysis (i.e. sphalerite vs. wurtzite) very difficult. Diffraction peak broadening is due to small particle size and the presence of substantial lattice strain. It was not possible to model the data by the inclusion of stacking faults to otherwise undistorted nanoparticles. As shown in Fig. 6, the addition of water had a profound effect on nanoparticle structure. Water addition causes a significant removal of lattice strain, and a sphalerite structure is observed, with no trace of wurtzite. UV absorption and TEM analysis showed no associated growth.

MD simulations yielded nanoparticle structures in good agreement with the WAXS data,49 providing an interpretation of the effect of water on nanoparticle structure. In the absence of any strong surface interactions, nanocrystalline ZnS in methanol is highly distorted. Atomic positions and tetrahedral bond angles are significantly distributed around equilibrium bulk values. The distortion is greatest at the surface, but all interior atoms are affected. Water binding relieves distortion, resulting in a more crystalline sphalerite-like structure. MD simulations indicate a strong enthalpy of adsorption (including structural stabilisation energy), which is consistent with the fact that the water induced transition cannot presently be reversed. Nevertheless, other work on surface interactions, such as methanol desorption,49 and aggregation in methanol,51 observed reversible structural transitions.