1. Introduction

Broad applications of nanoparticles (NPs) in many industries call for a comprehensive analysis of their thermal and mechanical stability. One example of such problem is structural stability of core-shell NPs known for their unusual functionalities. However, these particles are not stable against heat treatment: they must undergo alloying and subsequent transformation into bilobed ‘Janus’ structure. Indeed, as one can see from Figure 1, the bilobed structure has significantly smaller amount of the interfacial free energy for the approximately same amount of bulk free energy than the core-shell structure.

Figure 1. Structural transformation in nanoparticles from core-shell to ‘Janus’ structure.

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Another example of NP phase stability is presented by amorphous structures, which appear in different binary systems of sizes below the critical, but are not known in the same materials of sizes above the critical [1,2,3,4,5,6,7,8,9,10,11]. These structures vary in their levels of stability with respect to the thermal/mechanical treatment: after heating and cooling back to room temperature, some structures recrystallize [1,11], others return to the same structures. The latter structures present amorphous (disordered) nanopahses [2,3,4,5,6,7,8,9,10].A natural question arises: What is the origin of this phase and its relation to the bulk phases of the system?

Until recently there was no reasonable answer to this question. Sometimes one can find in the literature [2,3,4,10] or hear at the presentations an argument that the amorphous (disordered) nanophase (ANP) is a metastable phase of the bulk. This argument is false because, if it were the case, then there must be a domain of thermodynamic parameters—temperatures, pressures, and concentrations—where this phase would be the most stable. Indeed, a metastable phase possesses a free energy which can be compared with the free energies of the other stable phases. Such comparison results in the lines of equilibrium, which intersect to form a phase diagram. Topologically, it is clear that such diagram must have a domain of stability where the phase in question is the bulk stable phase, as it is the case in all systems with multiple equilibrium phases, e.g. solid, liquid, and gas or bcc, fcc, and hcp. However, as we pointed out above, this is not the case for the ANPs that is, they have never been observed in bulk samples. Another argument against ANP being a metastable phase of the bulk is that the metastable phase would be allowed to coexist with other bulk phases, which has never been observed experimentally.

In a series of recent publications [12,13,14,15] the author has developed a Landau-type theory of the amorphous nanophases, which claims that the phase that appears in NPs of sizes below certain value is a transition state between the stable bulk phases in the space of the order parameter, which characterizes structures and symmetries of the bulk phases. According to this theory, the transition state gains stability if the control parameters cross the spinodal boundary on the phase diagram of the system or if the experiments are conducted in conditions of conservation—energy, matter, or volume. As known, in large systems with a conservation constraint, in a certain domain of the conserved quantity, the transformation produces a heterostate—two-phase mixture of stable bulk phases. The theory of Refs 12-15 claims that if the size of the system is below critical value, such state is energetically impossible due to high ‘energy cost’ of the phase separating interface, see Figure 2. Then, the two-phase state is replaced by a homogeneous state, which is the transition state between the bulk phases.

Figure 2. Size-composition phase diagram of the binary system described by Eqs.(1-4). Red lines are phase boundaries of large systems; blue lines separate the bulk phases from the domain of stability of the transition state—ANP.

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In Section II of this publication we summarize the main results of the experimental studies of the phase transformations in NPs of Sn-Au-(Sb, Bi) system [2,3,4,5,6,7,8] and review the theory of Ref. 14. In Section III we summarize the experimental results of the studies of the lead telluride subjected to high external pressure [9,10], review the theory of Ref. 13 in more detail, extend it on the case of NPs, and apply it to the lead telluride system. Finally, in Section IV we discuss the theory of ANP formation and identify its successes and failures.

2. Amorphous Phases of Binary Alloys

2.1 Experiment

Yasuda, Mori, et al studied nanoparticles (NP’s) of binary and ternary alloys formed in the Sn-Au-(Sb, Bi, In, Pb, Zn, Cu) system. They found that the NP’s of all binary combinations had very strong size effect of increased solubility of the species compared to the bulk phase diagrams. However, the binary Sn-Au [2,3,4], Au-Sb [5,6], and Sn-Bi [7,8] systems exhibited even more dramatic effect: they produced phases that do not exist in the bulk. Namely, the NP’s of 16-20 nm in diameter and higher had recognizable bulk phases, mainly the solid intermetallics, while the NP’s of less than 10-8 nm were homogeneous amorphous structures of various degree of fluidity. Specifically, in 20nm size particles of Au-38at%Sn at room temperature the intermetallic phases e-AuSn and z’-Au5Sn appeared on HREM images separated by a well recognized interface. However, “with decreasing particle size below 8 nm in diameter, the salt and pepper contrast characteristic of amorphouslike phases appears” [2,3,4]. Moreover, upon annealing at the temperature above the bulk liquidus (~620K) “in a 5-nm-sized Au-36at% Sn alloy particle an amorphouslike phase directly changes into a liquid phase with increasing temperature from 293 to 773K and then the reversible transformation occurs with decreasing temperature from 773 to 293K. It was confirmed that no crystallization took place with increasing or decreasing temperature [2,3,4].” As a result of deposition of antimony on the gold NP’s “an ultra-fine grain phase similar to amorphous-like materials is formed in 4 nm-sized Au-Sb alloy particles which fall in the two-phase region in the bulk alloy phase diagram“ [5,6]. However, deposition of gold on the antimony NP’s produced different result [5,6], which, probably, is due to strong ability of pure antimony to amorphize at the nanoscale. Similar transformation sequences were observed in the Sn-Bi binary system [7,8].

2.2 Theory

In Ref. 14 the author presented a theoretical phase-filed model which explained appearance of an amorphous-like, disordered phase in nanoparticles of binary and multicomponent alloys. The model claims that the phase represents the transition state between the two bulk phases in the space of the order parameter of the alloy system. While this state is completely unstable in the conditions of an open system where exchange of the species between the particle and surrounding is not limited, it is stabilized in the closed system where the species exchange is prohibited. The condition for the stabilization is the material-parameters criterion; it has the following expression for a binary system:

Here V_m is the average molar volume, F_m is the molar Helmholtz free energyof the alloy, [X] is the difference of the equilibrium concentrations of the phases, s is the free energy of the interface, l is the interfacial thickness, which is usually close to 1 nm, and e is the interfacial energy density. The theory of Ref. 9 shows that the disordered phase emerges in the system as the replacement of the two-phase state (heterostate) as the size of the system decreases. The domain of stability of the transition state (see Fig.5 of Ref. 14) is separated from the domain of stability of the heterostate by the line of transition sizes @X = \tilde X\left( V \right) \approx l@, with the transition state being stable in layers of thickness @X < \tilde X@. That result is due to the free energy of the phase separating interface, not the energy of the free surface of the particle, which was excluded from the bifurcational analysis of Ref. 14 by the zero-flux boundary conditions.

Analysis of the 3D model of a particle completely consistent with the phase-field theory developed in Ref. 14, requires us to consider the surface and interface of the particle as smooth distributions of the order parameter, which should be optimized using the free energy functional. This approach will be taken in the next publication. The model presented here assumes that the surface of the particle has tension s and zero volume, the surface tensions of both bulk phases (α and β) are equal to the interfacial free energy of the heterostate, and the surface tension of the disordered phase is zero. The heterostate (h) consists of n_α and n_β moles of the two phases with the molar free energy of F_h (n_α, n_β). Compared to the homogeneous a phase particle it has an additional surface—the phase-separating interface. Then the h→t transition from the heterostate to the disordered phase occurs when the free energy difference between the two states is not enough to accommodate the energy of the particle’s surfaces. For a system where [X] « 1, the condition of the transition h→t is expressed as fo llows:

Then the condition Eq.(2) yield the particle transition size of

Using this results, we can estimate the particle transition size as D_t = 2R_t ≈ 10 nm. Figure 2 expresses the size-composition phase diagram of the binary system. It shows that above the critical size the microstructure of the particle consists of two coexisting domains of two different bulk phases while below the critical size the particles do not ‘phase separate’, forming an amorphous-like “salt and pepper” structure.

2.3 Theoretical explanation of experimental results

Let us analyze specifically the experiments in the binary tin-gold system between the intermetallic phases ζ’-(Au₅Sn; R3) and δ-(AuSn; P6₃/mmc) that lead to the formation of the amorphous-like, disordered phases in nanometer-sized particles [2,3,4]. First, we introduce an order parameter which represents the change of symmetry between the trigonal α=ζ’-(Au₅Sn) and hexagonal β = δ − (AuSn) phases so that η_ζ’ = 0 and η_δ = 1. Such order parameter can be introduced (e.g., see [16,17]) because of the group-subgroup relation between these two crystal classes. Then the theory presented above claims that the amorphous-like phase observed experimentally represents the transition state between the ζ’-(Au₅Sn) and δ-(AuSn) phases. The interfacial thickness can be estimated on the HREM images of Ref. [2,3,4] as few interatomic planes thick that is, l ≈ 1 nm. The interfacial energy can be estimated from other systems and we believe that σ ≈ 0.5 J/m² is a reasonable estimate. The value of the second derivative @{\partial ^2}{\text{F}}_{\text{m}}^{{\text{ζ'}}}/\partial X_2^2@, which Darken called the stability function [18], should have been available from the database modeling (e.g. CalPhaD or ThermoData) or first-principles (ab initio) calculations. Unfortunately, at present, this value is not available in the literature because the intermetallic phases are perceived as stoichiometric (line) compounds with no deviation from the fixed-ratio composition. In this case, formally speaking, the second derivative @{\partial ^2}{\text{F}}_{\text{m}}^{{\text{ζ'}}}/\partial X_2^2@ is infinitely large and the criterion, Eq.(1), is automatically satisfied. However, we realize that the formal argument is not acceptable here. Following the Hillert’s suggestion [19] that a stoichiometric phase may be described by a very steep parabola, we considered a model binary system with equal species self-interaction coefficients, which are smaller than the species cross-interaction ones. According to the presented explanation of the effect, the HREM image in Fig.2b of Ref. 2 is the high-resolution visualization of the transition-state order parameter, which has intermediate character between the order parameters of the bulk phases. Fig.3 of Ref. 2 is analogous to Fig.2 of the present publication. According to the presented theory the order parameter of the amorphous-like structure should depend strongly on the composition of the particle. The theory also predicts that if the disordered particle grows in size while its average composition does not change (this can be realized by depositing controlled amounts of species) then it will start separating into two bulk-like phases as soon as the size of the particle exceeds the critical one. These predictions can be easily verified experimentally by in situ HREM visualization of different particles in the transformation process.

Figure 3. Phase map of lead telluride system in the plane of two order parameters. Circles—stable an unstable equilibrium states; arrows—directions of the transitions in the experiments.

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The explanation of the amorphous phase as a stabilized transition state applies to the simplest possible alloy systems with at least two stable phases. It has an advantage over the explanation presented in Refs. 2-8 because the latter requires an additional bulk metastable amorphous phase existence of which has not been proven experimentally. The transition-state-nanophase theory has another advantage: it can help to explain the fact that some binary systems do produce disordered amorphous phases while others don’t. The latter may be a result of two different circumstances: either the stability moduli of the “unsuccessful” bulk phases are not large enough or the varying symmetry element is not compatible with amorphization.

3. High-Pressure Phase Transformations in Nanoparticles

3.1 Experiment

At ambient pressure, bulk PbTe samples crystallize in a face-centered-cubic structure (fcc, NaCl-type or rock-salt). The rock-salt structure in these samples remains stable up to a pressure of 6 GPa when it transforms to an intermediate, orthorhombic structure (space group: Pnma) and to body-centered-cubic structure (bcc, CsCl-type) at 16 GPa [20]. Although a coherent interface can be constructed between NaCl- and CsCl-type phases, the intermediate, orthorhombic phase makes this construct much easier [16,17]. The 13 nm PbTe NPs remain in the rock-salt structure up to 8 GPa. “Above 8 GPa, the 13 nm PbTe NPs start a phase transformation from rock-salt structure to a mixture of intermediate orthorhombic and CsCl-type structure. Both high pressure phases coexist until ∼14 GPa and then completely transform into a single CsCl-type structure under a maximum pressure of 20 GPa that was applied” [9]. “When the particle size is smaller than 9 nm, … it remains stable in the rock-salt structure to the pressure of 8 GPa and then surprisingly starts a transformation to an amorphous (disordered) phase rather than an orthorhombic or CsCl-type structure. The amorphous phase was preserved upon release of pressure to ambient conditions.” [9] “The 6 nm PbTe NCs underwent a pressure-induced crystalline-to-amorphous phase transformation, rather than the conventional rock salt-to-orthorhombic transformation observed in bulk and in larger PbTe NCs. When the loaded pressure exceeded 15 GPa, the low density (LDA) phase transformed to a high density (HDA) phase.” [10] The X-ray scattering measurements produced the (200) d-spacings, which allow to estimate the variation of molar volume with pressure. Surprisingly, the experimentally measured bulk modulus of the NaCl phase of PbTe particles is significantly greater than that measured for the bulk samples (see Table 2 and [20]), although the theoretical calculations show that a small particle should be softer than its bulk material. “Once the pressure reached above 10 GPa, the d-spacing of (200) peak suddenly shifted to a smaller value in a nonlinear fashion and the pressure-dependent d-spacing curve displayed an apparent discontinuity. This implies the first-order phase transition of PbTe NCs from rock salt to LDA occurs at a pressure over 10 GPa. This LDA phase possesses comparable compressibility with the rock salt phase, as revealed by their similar curve slope from 10 to 15 GPa. The incompressibility became relatively greater as compared with that at pressures below 15 GPa (Figure 4). These typical features suggested that a HAD phase formed above 15 GPa. This amorphous phase is significantly different from the LDA phase that formed at lower pressure.” [10]

Figure 4. Size-pressure phase diagram of lead telluride system. α, β, γ-phases; t—transition state. Overturned triangles—α/LDA transition points; black triangle—LDA/HDA transition point; squares—bulk-phase transition points.

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Table 1. Transition Sizes and Pressures in Lead Telluride.

Initial Structure	Final Structure	Radius (nm)	Pressure (GPa)	Reference
NaCl	Orthorhombic	≥ 20	6	[15]
Orthorhombic	CsCl	≥ 20	16	[15]
NaCl	Orthorhombic/CsCl	≥ 6.5	8	[5]
NaCl	Disordered LDA	5	10	[5]
Disordered LDA	Disordered HDA	3	15.6	[6]
NaCl	Disordered LDA	3	10.4	[6]
NaCl	Disordered LDA	2.5	8	[5]
NaCl	Disordered LDA	> 1.5	0	[5]

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3.2 Theoretical explanation of experimental results

According to the Landau theory of phase transitions [16,17,21,22], the bulk phases are separated from each other by a transition state, which represents a free energy barrier in the space of the order parameter that reflects the symmetries and structures of the phases. As pressure increases, the barrier lowers and the transition state approaches the metastable phase. The merger of the two is called the spinodal point. The order parameter of the transition state is intermediate between those of the bulk phases, hence, may not reflect any particular crystalline symmetry that is, the transition state is disordered. These ideas allow one to treat the NaCl→LDA transition as the spinodal phenomenon and put forth the following scenario of the high-pressure transformations in lead telluride. The NaCl→LDA transition point is a point of the loss of absolute stability (global and local) by the NaCl phase beyond which, that is at the pressures higher than the spinodal pressure P_S, the NaCl/Pnma transition state turns into the disordered (amorphous) phase. Remember that a phase is a homogeneous, stable (including metastable) equilibrium state of a system [22]. At even higher pressures in small particles (<13 nm) this phase is replaced by the Pnma/CsCl transition state, which is the HDA phase. The LDA phase transitions to HDA phase because of the proximity of the order parameters (see Figure 3). At lower pressure but larger particle size (8 GPa, 13 nm) the Pnma/CsCl transition state is replaced by the Pnma/CsCl heterostate because the latter has less total amount of the free energy. Below this scenario is tested quantitatively.

Let us first consider two bulk phases of the PbTe system: high-volume NaCl-type phase α, low-volume orthorhombic phase β, and disregard the lowest-volume CsCl-type phase γ. The total Gibbs free energy of n moles of the α phase shaped in a particle of radius R is

Here g_α is its molar Gibbs free energy, P is the external pressure, and the last term represents the Laplacian pressure effect. The total volume of the particle is

where V_α(P) is the molar volume of the α phase. The total Gibbs free energy of n moles of the particle of the transition state is

where δg_t(P) is the molar Gibbs free energy barrier. Then the original phase α will experience a transition into the t-state if G_t^P < G_α^P and Eqs.(5-7) yield the size-dependent spinodal condition:

The disordered phase (t-state) is stable at R < R^S (P).

In Ref. 13 was also considered a simple theoretical model of a phase transition between two bulk phases, α and β, with equal and pressure-independent coefficient Z=|∂V/∂P|. Using this model the size-dependent spinodal condition, Eq.(8), is expressed as follows:

Here P_E is the equilibrium pressure of the phases α and β, P_S is the spinodal pressure, V_α and V_β are the equilibrium molar volumes of the corresponding phases (at P = P_E), [V] = V_α - V_β, V_αβ is the average molar volume, β_E is the isothermal compressibility of the material, l is the interfacial thickness, and e is the interfacial energy density. Eq.(9), depicted in Figure 4, is the size-pressure phase diagram of the small particles. It allows one to match the theory with the experiments of [9,10] using two adjustable parameters: ΔP_s and l. Using the properties of the system listed above, the matching yielded: ΔP_s ≈ 50 GPa and l ≈ 0.6 nm. For the energy density Eq.(9a) yields: ε ≈ 1.25 GPa. Then the interfacial energy is σ ≈ 0.75 J/m².

Using the model system from Ref. 13 one can obtain the pressure/molar-volume relations of the α phase:

Using the properties obtained above, Eqs.(10, 11) were plotted in Figure 5. The radius of the particle at P₀ = 0 was selected to be R₀=3 nm.The theory is compared with the experimental results of Ref. 10.

Figure 5. Pressure/molar-volume diagram of lead telluride system. Dots—experimental results from Ref. 10; blue line—NaCl structure, Eq.(10); red line—LDA structure, Eq.(11).

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Furthermore, experimentally large particles (R ≥ 6.5 nm) separate into a mixture of bulk phases (half-black square in Fig.4). According to our theory, this is a typical feature of the closed-system (volume-controlled) experiments when the initial phase is pressurized and encapsulated into a set volume where the transformation takes place. Notice that in the conditions of an open system (pressure-controlled) the phase separation is not thermodynamically stable. The closed-system conditions appear in the experiments because the particles form a thin layer squeezed in a small window. According to Eq.(4) and the estimate of l ≈ 0.6 nm, the critical diameter of NPs above which the phase separation takes place is estimated as 6 nm.

4. Conclusion

Above, we described examples of nanophase formation in the systems of very diverse nature, which differ by the production methods, role of the substrates and containers, and, more importantly, by the type of the equilibrium structures that emerge. However, there is a unifying theme in all of these examples: they all entail stable homogeneous nanostructures which are completely unstable in the bulk. These examples inspired us to ask the following questions: What are the origins of the nanophases? What are the connections between the nanophases and bulk phases? In this publication, we attempted to construct a unifying theory that provides a common explanation for these questions.

The model treats the disordered phase as a transition state between the bulk phases in the space of the order parameter, which provides a distinction between the phases. In the case of a transformation in binary nanoparticles, the model claims that the phase represents the transition state between the two bulk phases in the space of the order parameter of the alloy system. While this state is completely unstable in the conditions of an open system where exchange of the species between the particle and surrounding is not limited, it is stabilized in the closed system where the species exchange is prohibited. In the case of high-pressure transition in small particles, the model treats the transition as the spinodal effect that is, an approach of the absolute instability of the initial phase. The initial phase transforms into the disordered phase instead of the other bulk phase because of the proximity of the order parameter. The model explains the molar volume jump at the transition point. The model explains the phase separation of large particles of PbTe (>13nm in diameter) into a mixture of the bulk phases by having a feature of the volume-controlled experimental conditions. The model explains the LDA-HAD transition as a transition between NaCl/orthorhombic and orthorhombic/CsCl barrier states. The model does not explain (so far) the softening of the particles at the transition point.

Acknowledgments

This work was supported by award 70NANB14H012 from NIST Department of Commerce as part of the Center for Hierarchical Materials Design at Northwestern University and NSF HRD 1436120 from Integrated Materials Science Research and Education Laboratory at Fayetteville State University.

Conflict of Interest

There is no conflict of interest related to this document.