Thermodynamics of Nanoparticles: Experimental Protocol Based on

Letter pubs.acs.org/NanoLett

Thermodynamics of Nanoparticles: Experimental Protocol Based on a Comprehensive Ginzburg-Landau Interpretation Denis Machon,*,† Lucas Piot,† Dimitri Hapiuk,† Bruno Masenelli,‡ Frédéric Demoisson,∥ Romain Piolet,∥ Moustapha Ariane,∥ Shashank Mishra,⊥ Stéphane Daniele,⊥ Mongia Hosni,¶ Noureddine Jouini,¶ Samir Farhat,¶ and Patrice Mélinon† †

Institut Lumière Matière, UMR 5306 Université Lyon 1-CNRS, Université de Lyon 69622 Villeurbanne cedex, France Université de Lyon, F-69000 Lyon, France and Institut des Nanotechnologies de Lyon, UMR 5270 CNRS, INSA Lyon, 7 avenue Jean Capelle, 69621 Villeurbanne Cedex, France ∥ Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université de Bourgogne, 9 Av. A. Savary, BP 47 870, F-21078 Dijon Cedex, France ⊥ IRCELYON, UMR 5256 Université Lyon 1-CNRS, 2 Avenue A. Einstein, 69626 Villeurbanne Cedex, France ¶ LSPM−CNRS, Université Paris Nord, 99 avenue J.-B. Clément, 93430 Villetaneuse, France ‡

S Supporting Information *

ABSTRACT: The effects of surface and interface on the thermodynamics of small particles require a deeper understanding. This step is crucial for the development of models that can be used for decision-making support to design nanomaterials with original properties. On the basis of experimental results for phase transitions in compressed ZnO nanoparticles, we show the limitations of classical thermodynamics approaches (Gibbs and Landau). We develop a new model based on the Ginzburg−Landau theory that requires the consideration of several terms, such as the interaction between nanoparticles, pressure gradients, defect density, and so on. This phenomenological approach sheds light on the discrepancies in the literature as it identifies several possible parameters that should be taken into account to properly describe the transformations. For the sake of clarity and standardization, we propose an experimental protocol that must be followed during high-pressure investigations of nanoparticles in order to obtain coherent, reliable data that can be used by the scientific community. KEYWORDS: High pressure, phase transformation, thermodynamics, kinetics, Ginzburg Landau, ZnO

T

instance, ZnO nanoparticles can be quenched in a highpressure rocksalt structure, a situation that cannot happen in bulk ZnO.7 Thus the study of solid−solid structural transition upon pressure is important for developing a deeper understanding of cluster thermodynamics and kinetics.8,9 The seminal work by the Alivisatos group8−11 highlighted two size-dependent effects on pressure-induced phase transitions: (i) The transition pressure tends to increase as the particle size decreases. This is explained by considering the appropriate thermodynamics potential including the surface energy contribution and the tendency of the high-pressure phase surface energy to be higher than the low-pressure phase one. (ii) A kinetics effect broadens the metastability range of the high-pressure phase during the decompression. This may lead to the possibility of recovering the high-pressure phase at ambient conditions.

he synthesis and control of nanostructures allow the elaboration of new materials with original properties. The key point is that for a given physical property with its associated characteristic length, nanostructures of lower sizes classically obey a scaling factor that may possibly have additional quantum effects.1,2 This nanoscience is a new research area to be explored and also a source of economic developments. At the nanoscale, the contribution of the surface energy in the energetic balance plays a decisive role. This additional energy may modify the phase equilibrium and stabilize new structures at ambient conditions with original properties different from the bulk counterparts.2 For instance, bulk TiO2 adopts a rutile structure whereas TiO2 nanoparticles crystallize in anatase structure;3 these nanoparticles are widely used for their photocatalytic properties.4 To better understand the effect of surface energy on phase stability, the combination of pressure and particle size is particularly important as, keeping the particle size constant, pressure allows the energy landscapes of the system to be explored.5 In addition, pressure and size are two parameters that can be used conjointly to stabilize new phases.6 For © XXXX American Chemical Society

Received: October 21, 2013 Revised: December 9, 2013

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dx.doi.org/10.1021/nl4039345 | Nano Lett. XXXX, XXX, XXX−XXX

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thermodynamics approaches and considering the nanoparticle as an isolated nanosingle-crystal. First, we will review the classical treatment of surface energy using the Gibbs description of phase transitions. In a second part, we will develop a new approach based on the Landau theory of phase transitions. Thermodynamics models based on capillarity give a 1/R dependence in the solid−liquid transition and suppose firstorder phase transition with latent heat.19 This is an illustration of the universal relationship

Even though several models have been proposed in the literature, it is difficult to fully validate them because of experimental discrepancies.12−15 In a simple thermodynamic model, the increase or decrease of transition pressure may arise because the nanocrystals change shape upon structural transformation.8 However the thermodynamics approach presents serious failures. Thermodynamics arguments suggest that there is a lower bound on nanocluster size beyond which phase coexistence becomes unstable.16 However, between the onset of coexistence and full transition, several transitions may occur depending on the kinetics (how fast a reaction is) rather than on thermodynamics (is a reaction favored). In small systems, fluctuations are well described in the microcanonical ensemble where the inherent inhomogeneity is suppressed. For the smallest ones (N < 500 atoms, that is, particles of approximately 3 nm of diameter) where fluctuations dominate, static coexistence gives way to dynamic coexistence.16 A common approach to the study of transition is to apply critical droplet theory in the capillarity approximation (canonical ensemble) involving bulk parameters. However, the kinetics of the phase transition is unknown especially at low temperature when the space configuration is not fully probed. In addition, defects are of prime importance as they act as embryos for nucleation centers in the phase transition. Finally, experiments carried out on samples embedded in a medium call into question the validity of the surface concept by introducing interfaces with poorly controlled surface energies. All these parameters have a strong effect on the transition and could explain the contradictory observations reported in the literature.17 Numerical simulations have been developed to better understand the phase transformations and/or the surface effects allowing a better understanding at atomistic scales.18 Our approach is complementary to these works as it corresponds to a thermodynamics and kinetics understanding of phase transitions in nanomaterials. This article presents a phenomenological approach based on the framework of the Ginzburg−Landau theory with a proposed experimental protocol that is needed to clarify the discussion through the illustrative example of ZnO nanoparticles. This material has tremendous potential for applications in optics, solar cells, catalysis, biology, and electronic devices. ZnO is a II−VI compound semiconductor with a particular stability for defects, especially oxygen vacancies. Vacancy density can be monitored by careful control of the sample preparation and defect recovery. ZnO under pressure is also a textbook-type transition between wurtzite and rocksalt. We intend to study the pressure-induced evolution in nanoparticles and to observe their phase transitions. We show that through kinetics the defects govern the phase transition and scramble the thermodynamic transition. These in situ highpressure studies will allow new pressure−particle-size phase diagrams to be established. In addition, this theoretical development with new insights on the kinetics of pressureinduced phase transitions is versatile and presents a large field of applications. First, it applies to any compound and is not restricted to ZnO that is used here as a shining example. Second, the model is also exploitable for bulk materials. This point shows the input obtained by the study of pressureinduced phase transitions in nanoparticles on a more general comprehension of phase transitions and especially on the kinetics aspects. Let us summarize the present state of comprehension of the impact of surface energy on phase transitions using

1/2S ϕ(R ) ⎛ α (R ) ⎞ ⎟ = ⎜1 − ⎝ R ⎠ ϕ∞

where φ is a physical parameter (cohesive energy, melting temperature ...); S is a parameter depending on the statistics, and S = 1/2 for a Fermionic system; the exponent ∞ is related to the bulk phase; and α(R) is a size-dependent parameter related to the material with R being the particle size. However, the associated latent heat is not strictly speaking defined in finite systems with a broadened transition as reported in free clusters.20 Templating liquid−solid phase transition suggests that solid−solid transition obeys the same dependence, the key parameter being the difference in surface energies between the two solid phases assuming a first order transition. The expression of the Gibbs energy including the surface energy term and considering isothermal compression is dG = V dP + γ dA

Expressing and integrating the Maxwell’s relations, we find that the Gibbs energy of a system is

ΔG = VmP +

3Vmγ r

where Vm is the molar volume of the phase and γ is the surface energy. This model is fully consistent with the one developed by Alivisatos’ group to explain the higher transition pressure in nanoparticles.8,9 An illustration of these considerations is given in Figure 1. The Gibbs energy is shown as a function of pressure. We considered a transition between a low-pressure phase (lp) and a high-pressure phase (hp) in the cases of (i) a bulk materials and (ii) nanoparticles of the same composition. Considering γhp− γlp > 0 as proposed by Alivisatos (where γhp and γlp are the surface energies of the high-pressure phase and the low-pressure phase, respectively), we predict an increase of the transition pressure for nanoparticles. Note that in perfect unreconstructed ZnO nanocrystals no clear-cut trend can be drawn because of the spread in the prediction of surface energies, which is related to the set of nonpolar/polar facets in the nanoparticle shape (Table S1 in Supporting Information).21,22 This transition will be strongly dependent on the morphology, as reported by molecular dynamics.23 Note that a transition toward the two expected equilibrium shapes in vacuum given by Wullf’s construction (truncated bypyramid and cube in wurtzite and rocksalt, respectively) always gives γhp − γlp < 0, leading to the prediction of a transition at lower pressure. This is not observed, since many factors are to be considered: • the surface reconstruction;24 • the passivation by surfactant molecules reducing the surface reconstruction; B


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between the primary order parameter and the surface energies. Then solving the equation ∂F/∂η = 0 leads to PC* = PC + 3δ

α0R

(3)

where P*C is the pressure at which the transition is observed. This is the classical dependence on 1/R. Such dependence remains identical in the case of reconstructive first-order transitions such as in the case of the wurtzite-to-rocksalt transitions observed in ZnO under pressure. For γhp − γlp > 0, as reported with ligands, the transition pressure is increased as evidenced by Alivisatos.8,9 Until now, we showed that the classical Gibbs approach allows the size-dependence of pressure-induced transition to be understood. Moreover, we proposed an original approach based on the Landau theory of phase transitions by introducing a coupling between the order parameter and the surface energies. Both approaches agree on a size-dependency. Without strong assumptions, the Landau approach naturally leads to a 1/R dependence of the transition pressure as predicted by other theoretical methods (capillary) and observed experimentally.28,29 To test and validate these results, we carried out a set of high-pressure experiments on ZnO nanoparticles using Raman spectroscopy. First, the phase transition on bulk ZnO is welldocumented and the range of pressure is easily accessible. The wurtzite-to-rocksalt transition occurs at around 8−9 GPa in bulk phase and the transition is sharp in the sense that the coexisting range of pressure is very narrow ( 0, leading to an increase of the transition pressure. Now, it is instructive to define the transition using the Landau approach. The theory of phase transitions developed by Landau was first restricted to transitions for which a group− subgroup relationship is preserved. However, since the seminal work by Tolédano and Dmitriev26 demonstrates the possibility of defining a transcendental order parameter, the Landau approach of phase transitions can be extended to study any kind of phase transition (including reconstructive transitions, that is, without a group-to-subgroup relationship such as in the wurtzite−rocksalt pressure-induced transition, and liquid−solid transitions27). This key point reveals the universality of Landau’s formalism. For the bulk materials, the Landau potential for a second order transition (the formalism can also be extended to a first order transition by using a potential of higher order) is F = F0 + αη2 + βη 4

γhp − γlp

(1)

where η is the order parameter, that is, a physical quantity driving the transition, α = α0(P − PC) changes its sign at the transition pressure PC and β > 0; the equation of state is then given by ∂F/∂η = 0. Now, we consider nanoparticles where the symmetry is broken at the surface. An additional coupling term between the primary order parameter and the surface energies should be considered. Assuming that the volume difference between two phases is neglected, the Landau potential is γhp − γlp F = F0 + αη2 + βη 4 + δη2 S (2) V where S and V are the surface and the volume of the nanoparticle, respectively. Here, δ is the coupling constant C


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Supporting Information). This indicates that the disordering is limited. (ii) The transition pressure is dependent on the synthesis method for a fixed particle size. Thus, the transition to rocksalt starts at higher pressure for Sample 2 than for Sample 1. (iii) Even though no size-dependency seems to apply for Sample 1 when considering the starting transition pressure, the pressure range over which the transition occurs is larger (∼3 GPa) than for the bulk materials (∼0.5 GPa). Below a critical particle size, when a critical concentration of defects is reached, amorphization in nanoparticles can occur even in compounds considered to be poor glass-formers. The source of defects varies; they can be surface defects, radiationinduced defects, mechanically induced defects, and so on.33 In our case, the partial disordering observed in Samples 3 and 4 may result from a higher initial defect concentration (mainly oxygen vacancies) than in Samples 1 and 2 (point (i)). The variation of the transition pressure with the synthesis method (point (ii)) is explained by the use of the interface energy rather than of the surface energy. If the concept of surface energy holds to describe the properties of free nanoparticles, this model can hardly be generalized to describe phase transitions, as the surface state (chemical interaction, defects, etc.) may play a determinant role in the phase equilibria.17 Replacing the surface energy γ used in the previous model by an interface energy parameter Γ including γ and additional terms related to the surface defects and functionalization allows the dependence to be understood with regard to the synthesis process. Point (iii) is more striking. The nanoparticles in Sample 1 have been obtained by low energy cluster beam deposition (LECBD) methods, leading to defect-free nanoparticles.34 The interface energy in this case is nearly equal to the surface energy. As underlined in Supporting Information Table S1, there is no significant jump of the surface energy during the transition. Therefore, the term γhp − γlp in eq 3 is negligible and the transition pressure is predicted to be close to the bulk value. The models developed above need to be reconsidered to understand the magnitude of the range of pressure over which both phases coexist. Usually, such broadening is attributed to kinetics and is not treated by thermodynamics models as thermodynamics deals with relative stability of phases while kinetics refers to activations energies between these different states. Therefore, models purely based on a thermodynamics approach have reached their limit and failed to explain in a unified framework all the features of the transition. In the following, we present an extension of the model to a realistic and practical situation based on the Ginzburg−Landau approach of phase transitions that takes into account kinetics effect. One of our assumptions was to consider a single nanoparticle. However, usually in high-pressure experiments the samples consist of a collection of nanoparticles pressed together. Therefore, a more realistic model consists in considering this collection as a polycrystal. If all the particles are identical (narrow size distribution), we just consider a collection of nanoparticles identical to a polycrystal with a homogeneous distribution of well-oriented grain boundaries. To take into account the domain wall, the square of the strain gradient is included in the free energy to account for three terms: (i) the pressure gradient, which is a common source of errors in all the high-pressure experiments (Kpressure), (ii) the dipolar interaction between two nanoparticles with different structures (wurtzite and rocksalt)

Figure 2. Raman spectra of ZnO nanoparticles (sample 2) with increasing pressure. The initial phase adopts the wurtzite structure and the main Raman peak is centered at 432 cm−1 (label W). Under pressure, a broad peak develops between 500 and 600 cm−1. This band is assigned to the second-order Raman signal of ZnO in the RockSalt phase (label RS). The pressure at which the transition starts (start) corresponds to the observation of a signal associated with the RS phase. The transformation is considered as complete (end) when there is no signal from the wurtzite phase in the Raman spectrum.

Figure 3. Summary of high-pressure experiments on ZnO particles with respect to the synthesis method. Depending on the synthesis, the transition pressure, the coexistence range and the high-pressure structure may be affected.

(i) The high-pressure phase may differ depending on the synthesis method. It can be seen that for Samples 1 and 2, the high-pressure phase is the rocksalt structure, as expected. However, for the other two samples, the Raman signatures do not correspond to the rocksalt phase.31 This can be attributed to a vibrational density of states resulting from a strong disordering described earlier by Shuker and Gammon.32 Complementary X-ray diffraction experiments on Sample 3 at high pressure, however, identified the high-pressure phase as the crystalline rocksalt structure (see Figure S4 in the D


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behaves like the Heaviside function with a clear-cut upstroke at the transition P = Pc (here Pc is defined in eq 3). In the low coupling regime (no gradient of pressure, no interaction between particles), the order parameter is just influenced by the “kinetics part”, which is reduced to Kinterface. This leads to a spreading order parameter depending on the magnitude of Kinterface. This explains why a clear-cut upstroke at the transition is not observed. Increasing the K factor (by taking account of all the Ginzburg terms) introduces a domain where the transition is not clearly defined. This is the origin of the hysteresis, the upstroke− downstroke width being about 4ε. In an ideal case (a single nanoparticle in an ideal medium), the interface between the two phases during the nucleation process is the first origin of phase coexistence. A dipolar effect and pressure gradient participate in phase coexistence in a “true” experiment. The effect of nonhydrostaticity deserves a deeper comment as the effect of the pressure-transmitting medium is central in high-pressure experiments and often the degree of hydrostaticity is only qualitatively determined (hydrostatic, quasihydrostatic, nonhydrostatic). Our theoretical approach includes the effect of nonhydrostaticity. If we push the model to its limit, that is, considering a compression of a bulk sample without interaction, this leads to the applicability of the model to any pressure-induced phase transition whatever the particle size. As the model is analytical, this means that some comparison can be made between the effects of different pressure-transmitting media (PTM) on a phase transitions in a given system. Similarly, using the same PTM, one can compare its effect on different systems. These applications are much wider that the nanoparticles under pressure and all pressure-induced phase transitions can be treated in the framework of this model. Now let us examine the case of nanoparticles with defects. From a thermodynamic point of view, the defects, mainly oxygen vacancies, are randomly distributed in each nanoparticle. One can associate Voronoi cells around defects inside the nanoparticle (these Voronoi cells are analogous to Bloch domains in magnetism). Even though each defect is an embryo for a phase transition, we need a preliminary growth of the Voronoi cells in order to observe a minimum domain for Raman characterization (or diffraction) (lRaman coherence < Voronoi size). In a grain growth process, the grains can grow at the expense of each other and thus the order parameter is not conserved. The system is described by a set of Q nonconserved order parameters within the continuum timedependent Ginzburg−Landau framework where each orderparameter field represents the Voronoi cell. Thus, the Ginzburg−Landau potential is defined as Fnanoparticle = F + Fgrain, where Fgrain is the free energy density due to the orientational degrees of freedom of the Voronoi cells. Fgrain is composed of two terms: the first describes a potential with one degenerate minimum corresponding to nanoparticle orientations; the second one is a gradient energy (N > 0), which is the energy cost of creating an interface (a domain wall) between two Voronoi cells38

(Kdipolar), and (iii) the gradient associated with the disorder induced at the interface between the two phases during nucleation of a phase (rocksalt) inside the other (wurtzite) (and wurtzite inside rocksalt, respectively, during the downstroke) (Kinterface). This last term was already discussed in the literature.8,9 This is an asymptotic case of quasi-martensitic transition where the rocksalt embryo grows by pressureinduced stresses of the w-ZnO lattice showing characteristic diffusionless features. This assumption is valid at room temperature.35 All these terms are referred to as the Ginzburg term and the resultant description is called the Ginzburg− Landau free energy.36 Thus, the Ginzburg−Landau potential is defined as F = F0 + αη2 + βη 4 + δη2 .

γhp − γlp V

S

+ (K interface + K pressure + Kdipolar)(∇η)2

(4)

The Ginzburg−Landau potential is conservative as long as we use a single order parameter that is identical for all the nanoparticles (narrow size distribution). The Ginzburg term corresponds to a kinetic term; the others correspond to the potential term. Using eq 4, one derives the Euler equation that can be solved analytically37 (see Supporting Information). The nontrivial solution in one dimension leading to the expression of the order parameter is37 η(x ) =

2 / ε)x

−1

( 2 / ε)x

+1

e( e

(5)

ε corresponds to the width of the transition region. This function is plotted in Figure 4. In the absence of the Ginzburg term (Kinterface + Kpressure + Kdipolar = 0), ε ∼ 0 and the function

Figure 4. Evolution of the renormalized order parameter as a function of the pressure. The red curve is the upstroke well-defined when there are no coupling and no pressure gradient. The blue window is the domain of coexistence between the two phases. If the pressure is defined at the inflection point, as proposed by Alivisatos et al., ΔP is the apparent pressure for the transition. In mono domains (defect free) the window is directly proportional to the gradient (dipolar anisotropy and pressure). In particles with defects, the window depends to the number of defects.

Q

Fgrain =

Q

Q

Q

∑ (aηi2+bηi4) + c ∑ ∑ (ηi2ηj2) + ∑ N(∇ηi)2 i=1

i=1 j>1

i=1

(6)

The sum is over the Q possible orientations of the Voronoi cells inside the nanoparticle. The equation cannot be solved because the distribution of the defects is unknown. Assuming E


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Figure 5. Different transition scenario depending of the experimental conditions. (A) Collective behavior of a collection of identical nanoparticles well aligned upon time and pressure. (B) Dipolar interaction between two particles in two phases. For the sake of simplicity (100) and (1011̅ ) are facets in rocksalt and wurtzite, respectively. This interaction is zero when particles are in the same phase (no coupling regime). (C) Evolution of the domains in a nanoparticle with defects (from left to right panels): distribution of the defects and the associated Voronoi cells, growth of the domain, and upstroke.

that the Voronoi cells have a narrow size distribution, we can discuss a crude approximation within a conservative order parameter Fgrain = aη2 + (b + c)η 4 + N (∇η)2

F = F0 + αη2 + βη 4 + δη2

γhp − γlp V

Consequently, the transition domain is governed by an experimental error (pressure gradient, which is an offset since this is the same parameter for all the experiments), the coupling between nanoparticles, and the coupling between the defects inside the particle. As reported above, the problem is complex and different parameters may modify the transition pressure. These widely uncontrolled parameters explain the spread in critical pressure reported in the literature. However, being aware of the possible causes leading to the spreading of the transition pressure should encourage the definition of an experimental protocol to minimize the contributions of each factor leading to this inhomogeneity. Moreover, the experimental access to the sum of these factors (through ΔP and consequently ε) allows their determination by dedicated experiments designed to minimize some factors. • Use a well-controlled media and ligand-free nanoparticles, as organic bonding in ligands are poorly isotropic under pressure • Use a single particle or well-separated particles selected in size and composition embedded in an isotropic matrix (LiF for instance). This protocol allows conservative parameters only. • Both intrinsic (Schottky and Frenkel) and extrinsic (stoichiometry) point defects have to be characterized. No defect limits nonconservative parameters. • Size control. • Sample storage environment (avoid physi- or chemisorbed molecules).

(7)

S + aη2 + (b + c)η 4

+ (K interface + K pressure + Kdipolar + N )(∇η)2

(8)

The upstroke pressure is augmented by two factors: (i) the energy required for creating domain walls, and (ii) the energy corresponding to the mutual orientation. The width of the hysteresis curve is also augmented. These two terms are related to the distribution of the defects and widely independent of the interaction among the nanoparticles themselves. This model is fully instructive about the kinetics of the transition. It has been well established that the distribution of the nanoparticles (the domain-growth) in one phase scales with time as t1/3 and t1/2 for conservative and nonconservative order parameters, respectively (Figure 5).39 The strength of the Ginzburg−Landau approach is the class of universality; that is, the phase scales with time independently of the nature of the interactions. This suggests that the nucleation mechanism for the nanoparticles assembly (conservative parameter) is slower than the rearrangement of the multidomains inside the particle (nonconservative parameter). This explains why the kinetics of phase transformation is dependent on the pressure and the temperature.35 F


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• Use a slow pressure slope as kinetics effects and dynamic fluctuations have different time scales. A long time ensures equilibrium. In conclusion, Ginzburg−Landau approach provides a theoretical scheme to explain the coexistence between two phases in a large pressure range. The equation solving takes place in a very simple case when particles are defect-free and interact weakly. For naked and defect-free ZnO nanoparticles, the wurtzite/rocksalt transformation starts at the same pressure as the bulk phase. This phase transition pressure is higher, taking the midpoint of the hysteresis curve. For particles with a low concentration of defects and partially embedded in ligands, an offset is observed leading to a net transition at higher pressure. Particles with a large number of defects lead to Voronoi cells with a size lower than or comparable to the characteristic length (here the Raman coherence length). In this case, a crystal transforms to a disordered phase as observed in ZnO and may ultimately become amorphous.33 More generally, particles with a net dipole interact between them during the transition. This can be ruled out by dispersing the nanoparticles in a matrix (LiF). In addition, we demonstrated that the kinetics of transition shows two time scales depending on the nature of the parameters, and a careful control of the slope of the pressure increase is needed for separating conservative and nonconservative parameters. Our analysis stands for nanoparticles that situate between two well-defined regimes: 1. That of the bulk material for which the phase transitions are mainly governed by the energetic volume contribution. Only the effect of nonhydrostaticity introduced in our model is sufficiently important to be considered. 2. That of very small particles (smaller than typically 2 or 3 nm in diameter) for which the classical thermodynamics do not apply anymore. In this case, phase transitions are governed by the fluctuation effects. In our case, the nanoparticles are sufficiently small to introduce interface effects into the energetic balance and they will be in competition with purely volume contribution. However, their size is large enough to apply a classical thermodynamics analysis. This case covers most of the works on particles showing an original behavior attributed to a size effect. Therefore, our approach in the present work is to determine the effect of parameters other than pure size effect. We demonstrated that the features of the transition (start and end of the transition) are dependent on the synthesis method at a constant particle size. Based on these observations, we developed a new model that incorporates the possible couplings with parameters such as functionalization, defects and dipolar interaction. This model is highly interesting because it allows reproducing all the features of the transition in nanoparticles: size effect when we considered small enough nanoparticles (for instance, phase transitions in CdSe nanoparticles of 4 nm are certainly and prominently affected by size effect contrary to ZnO nanoparticles of 15−20 nm), surface state effect, pressure gradient. This model is universal as it is applicable not only to the specific case of ZnO, treated here as an example, but it can also be used to any phase transitions in nanostructures. Moreover, if we push the model to its limit, that is, considering a compression of a bulk sample without interaction, this leads to the applicability of the model to any pressure-induced phase transitions irrespective of the particle size.

Besides, the broadened pressure range of the transition is always attributed to “kinetics effect” that cannot be expressed analytically in a universal equation. In the Ginzburg−Landau approach, time is a natural parameter and kinetics as well as free energy are introduced in a single relationship. Our model incorporates this effect analytically opening the possibility for numerical estimation of each parameter and comparison between different systems and/or different experimental conditions. This has tremendous application in nanostructure design as it allows determining conditions that may stabilize new polymorphs using the appropriate parameters.

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ASSOCIATED CONTENT

S Supporting Information *

Description of experimental methods, samples preparation, high pressure apparatus, Raman spectra under high pressure of the different samples, and the development of the theoretical model. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the PLYRA for the experimental preparation of the ZnO nanoparticles by LECBD method. We thank Pierre Tolédano and Julius Jellinek for their comments and encouragements.

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REFERENCES

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Nano Letters

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