Solid metal or ceramic nanoparticles with a diameter in the range of 1–100 nm, when placed in an inert environment, can be considered highly dispersed composite materials (HDCMs). HDCMs exhibit potential for the use under conditions of high temperatures and radiation exposure, making them promising materials for new-generation modular nuclear reactors, advanced charge storage applications, and other emerging nanotechnologies. When HDCMs are exposed to radiation, such as ion bombardment or exposure to high-energy radiation sources, defects (vacancies, interstitials, point defect clusters, voids, and interstitial loops) are created in the crystal lattice because of the displacement of atoms. These defects can significantly alter the structural, mechanical, and electronic properties of materials. This prompts the questions: How do radiation-induced defects influence first-order phase transformations in nanoscale systems? Can radiation-induced defects initiate polymorphic transitions or amorphization in metallic/ceramic nanoparticles, leading to changes in the crystal structure? Is it feasible to establish a fundamental basis to explain the behavior of materials under irradiation?

Most nuclear materials have not been tested beyond an irradiation dose of 200 displacements per atom (dpa) (equivalent to 40 years of service). Under irradiation, the main point defects are vacancies and interstitials. Point defects can develop into clusters of dislocations, stacking faults, or voids. They can also relax onto existing sinks such as dislocation loops, grain boundaries, phase interfaces, and cavities . Experimental studies on Pd have shown that the defect density generally increases with grain size; in grains smaller than 30 nm, no defects were observed , suggesting that large defects (clusters and dislocations) do not exist in small nanoparticles.

One possible explanation is based on the fact that the movement of dislocations is impeded by particle surfaces (grain boundaries) quite rapidly. For example, a transmission electron microscopy study (irradiation with Kr ions at 1 MeV at room temperature and an average defect generation rate of about 2 × 10−3 dpa·s−1) showed that, in nanosilver, a dislocation loop migrates to the free surface of the particle within 0.1 s . This suggests that dislocation loops and interstitials are leveled out fairly quickly in nanoparticles, making vacancies the main defects that affect the material’s properties.

According to experimental data, under operating conditions, HDCMs are susceptible to risks such as radiation-induced vacancy saturation (the accumulation of vacancy-type defects), swelling (an increase in linear dimensions and volume upon irradiation), damage, amorphization (crystal-to-glass transition), and other phase transformations (phase-to-phase transitions) . Despite the interest in these materials, the behavior of nanoparticles under irradiation and their peculiarities are not yet completely understood. One of the important issues is the influence of vacancy saturation on phase changes and phase stability.

A literature review reveals that the stability of materials under irradiation is influenced by numerous factors. Some of these characteristic factors include the elemental composition and chemical structure, the microstructure of the material (including grain boundaries, defects, dislocations, and interfaces), the dose and energy of the radiation source, different types of radiation, environmental conditions, the purity and homogeneity of the material, and the crystal structure and phase stability. Let us briefly consider these publications and highlight characteristic factors to facilitate understanding and subsequent description.

It is important to distinguish between two types of point defects, that is, (i) thermal-equilibrium defects (vacancies and interstitials that exist without irradiation treatment) and (ii) radiation-induced defects. Point defects caused by radiation are formed when a fast moving ion knocks an atom from its initial lattice position. The appearance of point defects increases the energy of the crystal, as energy is required to create each defect. For example, the energy of vacancy formation in the face-centered cubic (fcc) Cu lattice is about 1 eV, and in the body-centered cubic (bcc) Fe lattice it is about 1.5–2.0 eV; the energy of interstitial formation ranges from 2 to 4 eV. It is accepted that interstitials are mobile at room (low) temperature because of significantly less migration energies of 0.01–0.50 eV, whereas vacancies are mobile at very high temperatures. Disorder can arise from the recombination of these defects . In metals, for instance, the equilibrium concentration of thermal vacancies, even at high pre-melting temperatures, reaches values of only about 0.1% . Therefore, in the following, we will focus on radiation-induced vacancies, assuming that the concentration of radiation-induced point defects at characteristic temperatures (far from melting) exceeds the concentration of thermal-equilibrium defects.

The behavior of HDCMs under irradiation highly depends on their size. For example, when TiN nanograins are irradiated with He+ ions, their amorphization leads to a reduction in nanohardness, and this reduction is strongly correlated with the grain size . Phase instability (radiation-induced amorphization) is observed in zirconia nanoparticles (ZrO2) embedded in nanocrystalline composites. ZrO2 nanoparticles can be amorphized at an irradiation dose of 0.9 dpa, whereas bulk zirconia remains stable with respect to amorphization up to 680 dpa . Additionally, experimental data indicate that materials containing coherent-to-the-matrix dispersed particles may experience reduced swelling under irradiation compared to similar materials lacking such precipitates. For example, nanocrystalline solids of ZrO2 or Pd demonstrate high resistance to radiation-induced defect production compared to coarse-grained polycrystals with the same chemical composition . Consequently, a significant increase in radiation tolerance is anticipated in nanocrystalline materials compared to bulk solids with conventional grain sizes. Therefore, it is expected that interfaces such as coherent and incoherent boundaries in HDCMs will act as sinks to promote point defect annihilation .

In many cases, multicomponent alloys and HDCMs demonstrate greater stability compared to nanocrystalline pure materials. For example, nanocrystalline NiFe alloys can withstand a much higher radiation dose, twice that of nanocrystalline Ni, and are more stable under prolonged irradiation compared to nanocrystalline elemental Ni . Additionally, high radiation tolerance was observed in crystalline Fe/amorphous SiOC nanolaminates .

Long-term irradiation treatments have revealed phase transformations in HDCMs, such as the crystallization of an earlier formed amorphous state (re-crystallization) or a change in the basic crystalline state. For example, a two-phase TiCr alloy undergoes phase transformation when irradiated with Kr+ ions with an energy of 1 MeV . In another study, a bcc→fcc phase transition was observed in an α-FeNi alloy due to radiation of self-ions at 673 K and a dose of approximately 1.2 dpa . Furthermore, changes in the structural properties of Ni nanostructures due to ion bombardment have been reported in . Interestingly, certain types of radiation ions have shown a positive effect on the crystal structure, leading to an increase in the degree of crystallinity after the austenitic annealing of defects. However, in cases where the structure is rebuilt, the role of vacancies becomes less obvious. This suggests that other factors must be considered to better understand the transformations in HDCMs under irradiation.

A recent study reported a phase transformation in a 25 nm thick nanocrystalline Au thin film through in situ ion irradiation, observed using atomic-resolution transmission electron microscopy . The gold sample was irradiated with 2.8 MeV Au4+ ions at 200 °C with a fluence of approximately 1014 ions·cm−2 (equivalent to a dose of 10 dpa). A combination of surface- and radiation-induced effects led to a polymorphic phase change, transforming the high-density fcc structure to a low-density hexagonal close-packed crystallographic phase.

The investigation of the radiation stability of nanocrystalline single-phase multicomponent alloys (NiFe, NiCoFe, and NiCoCr) using molecular dynamics simulations reveals that the critical irradiation dose for nanocrystallinity collapse varies among different simulation cells. Not only the size, but also the crystallographic orientation, shape of the grains, and structure of the grain boundaries have a strong impact on the stability of the nanocrystalline phase . In all cells, the grains undergo a phase transition from a pure high-density fcc phase to a mixture of fcc and bcc phases during prolonged irradiation. These simulations confirm that the phase transition occurs because of the ground-state energies of the compositions rather than the irradiation itself. Consequently, it would be intriguing to identify other systems capable of undergoing, for example, polymorphic transformations or amorphization under irradiation, transitioning from a high-density phase to a low-density phase. One such example will be discussed further.

There is a general deficiency in theoretical descriptions, particularly regarding thermodynamic calculations, which could elucidate the phase stability and radiation stability of nanodispersed particles under irradiation. In essence, there is a lack of robust theory to inform studies of HDCMs under irradiation. The authors have identified only a few papers proposing a thermodynamic assessment . Our aim in this work is to fill this gap.

Shen’s proposed qualitative framework suggests that the grain size of a material influences its resistance to amorphization and the removal of radiation defects by altering the Gibbs free energy and kinetic rate theory . Shen delineates five size-dependent regions that govern the material’s response to irradiation. Nevertheless, Shen’s approach remains qualitative, highlighting the need for a more comprehensive thermodynamic assessment to enhance our understanding of HDCMs’ behavior under irradiation.

Recently, we attempted to adapt Shen’s model for polymorphic transformations in nanoscale Fe systems (conference report, providing an initial approximation to the formulation of the problem) . However, certain assumptions raise doubts about the results, namely, (i) the absence of vacancies in the secondary phase, which should be released under irradiation, (ii) the consideration of vacancy diffusion coefficients as constants, rather than exponentially dependent on migration energy and temperature, and (iii) the modeling of thermodynamic parameters, such as the enthalpy and entropy of vacancy formation in Fe.

First-order phase transformations are accompanied by nucleation and the overcoming of energy barriers. To our knowledge, the consideration of nucleation energy barriers and the alteration of surface energies during transformation under irradiation has been largely overlooked. This raises questions about the potential outcomes when the surface energy decreases because of the emergence of a new phase. Other factors contributing to non-uniform behavior and strongly influencing transformation patterns may also warrant investigation. In this regard, solid nanomaterials with phase change and reduction in surface tension serve as suitable systems for elucidation and comparison.

In summary, there is a competition among various energetic factors influencing phase stability and transformations in HDCMs during irradiation. These factors include (i) the bulk thermodynamic stimulus for phase change, (ii) the contribution of surface energy due to a high percentage of surface atoms, (iii) interfaces acting as sinks for radiation-induced point defects, (iv) the accumulation of defects (saturation of vacancies) in the material as a driving force of phase changes, and (v) the nucleation of a new phase. We leverage this competition to develop a fundamental description, employing both a thermodynamic approach based on the calculation of Gibbs free energy and a kinetic approach based on chemical rate theory. Given the complexity arising from multiple factors, it is evident that a simple theoretical description may not suffice. Therefore, our aim is to tackle the problem comprehensively and emphasize the most significant aspects of phase stability under irradiation.

To the best of our knowledge, this is the first work that simultaneously takes into account the above factors in a comprehensive thermodynamic approximation. As our model system, we selected a spherical nanoscale particle in an inert medium, for which we utilized the parameters necessary for calculations. Our aim is to investigate the effects of powder dispersion, surface energies of phases, and vacancy saturation on the radiation stability and first-order phase changes of spherical nanoparticles. Specifically, our objective in this work is to study the phase transitions from the bcc α phase to the fcc β phase under irradiation, as well as the reverse transition from β phase to α phase under irradiation. It is worth noting that the thermodynamic analysis and conclusions drawn from this study are applicable for understanding amorphization and polymorphic phase transitions in both metals and ceramics. In the present study we investigate model systems with structure change, but without composition change, in order to study the effect of radiation on structure change.

The paper is organized as follows: In Section “Theory”, we develop a thermodynamic approach and discuss a kinetic model of steady-state concentrations of radiation-induced defects based on chemical rate theory. Section “Results” focuses on the model of α→β phase transition and the reverse β→α phase transition from a thermodynamics perspective. Finally, section “Discussion” discusses the effects of size and irradiation, justifying their relevance to metals and ceramics, and presents the particular case of Fe nanoparticles.

Figure 1: Model of a HDCM under irradiation providing a schematic representation of the irradiation treatment and first-order phase transformation of a nanopowder in an inert medium. (a) Transition from α phase to β phase and (b) transition from β phase to α phase. The solid nanoparticles are represented as identical spherical balls with a diameter d = 2R.

Beginning with an initially homogeneous nanoscale droplet of the α phase (or the β phase) at a specific temperature, the thermodynamic analysis considers the effects of irradiation, which generate vacancies and interstitial atoms in both the interior and on the surfaces of the nanoparticle. Our approach utilizes thermodynamic calculations to determine the Gibbs free energy of a nanoparticle in various phase states with vacancy-type defects. Additionally, we consider the size-dependence of radiation-induced concentrations of point defects in our analysis. Let us proceed to discuss the thermodynamic calculations.

To start, we calculate the energy change without considering the effects of irradiation. The change in Gibbs free energy of the particle, which is represented by the bulk driving force for the phase transition and the surface energy term, can be expressed as follows:

ΔGbulk represents the bulk Gibbs free energy change, which serves as the bulk thermodynamic stimulus for the phase transition from one phase to another. ΔGsurf denotes the surface energy change during the phase transition. Let N0 be the number of atoms in the nanoparticle with radius R. Then, the Gibbs free energy of an α-phase particle, Gα, is:

The total Gibbs free energy, Gβ, of a β-phase particle is given by:

In these equations, gβ (gα) represent the bulk Gibbs free energy per atom (the bulk energy density) of the β phase (α phase), while σβ (σα) represent the specific surface energy (energy per unit area) of the β phase (α phase). The surface areas of the particles are given by Sα = 4πRα2 and Sβ = 4πRβ2; they depend on the atomic densities of α phase and β phase, respectively.

In the following, the letters “α” and “β” indicate the corresponding phases. For simplicity and convenience, we initially focus on the α→β transition and write all equations accordingly. It is evident that for the β→α phase change, one may replace the subscript “α” with “β” and vice versa.

To describe the energy change under irradiation, we need to incorporate the effects of vacancy saturation caused by radiation-induced point defects. This can be achieved by modifying the expression for the bulk Gibbs free energy change to account for the additional energy associated with vacancy saturation. Let us introduce the change of the Gibbs free energy of a particle under irradiation, ΔG, and the Gibbs free energy for creating point defects in a material, ΔGpd.

The formation of defects alters the initial state and the final stage, resulting in an increase in the energy of the nanoparticle, (Gα + ΔGpd(α)) for the α-phase nanoparticle and (Gβ +ΔGpd(β)) for the β-phase nanoparticle. We assume that vacancies are present in both the initial and secondary phases, which are expected to precipitate under irradiation.

By introducing the notation, ΔGpd = ΔGpd(β) − ΔGpd(α), the last expression can be rewritten as:

Phase transition is thermodynamically possible only when the relationship for the change in Gibbs free energy is fulfilled:

In the following, the condition ΔG < 0 for a nonzero particle diameter d (or N0) indicates the occurrence of the phase transformation and is used as the phase transformation criterion.

Additionally, we investigate the behavior of the bulk under irradiation and the saturation of vacancies, assuming an infinite size where the surface terms are negligible. In this case, as d→∞ (or N0→∞), the surface terms can be neglected (|ΔGbulk| ≫ |ΔGsurf|), and one can find the energy difference, ΔG ≈ ΔG∞, as the combination of the bulk thermodynamic stimulus and the energy for creating point defects ΔGpd:

The phase transition in an irradiated bulk material is thermodynamically favorable if the following condition is met:

The specific effects of irradiation on the energy change depend on the details of the material and the irradiation conditions and may require further analysis or experimental data for accurate characterization. In our case, the energies of point defects, ΔGpd(α) and ΔGpd(β), depend on the vacancy concentrations (denoted as Cvα and Cvβ, respectively) and can be expressed as follows :

where, ΔHf is the enthalpy change for forming of a vacancy, ΔSf is the entropy change for vacancy formation, and ΔHmix is the ideal entropy of vacancy mixing, which may be given as:

Here, T is the absolute temperature, and kB is the Boltzmann constant.

The chemical rate theory approach involves the application of concepts from chemical kinetics to describe the evolution of defects in materials under irradiation. It considers the rates of defect formation, migration, and annihilation processes and aims to predict the steady-state concentrations of these defects under given irradiation conditions. In this approach, the rates of defect formation and annihilation are described by kinetic equations, which may be derived from fundamental principles such as the laws of thermodynamics and statistical mechanics. These equations typically involve parameters such as activation energies and defect concentrations, and they can be solved to obtain the steady-state concentrations of defects.

According to chemical rate theory, which incorporates the effect of particle interface sinks, steady-state concentrations of interstitials and vacancies in a material can be determined by considering two extreme cases, namely, (i) the case of vacancy–interstitial recombination, where defects are annihilated through recombination reactions, and (ii) the case of particle interface sinks, where defects are trapped and annihilated at external boundaries. These two cases represent different mechanisms for defect annihilation and can lead to different steady-state concentrations of defects depending on the material and irradiation conditions .

The time-dependence of the vacancy concentration, Cv, and interstitial concentration Ci can be described by kinetic equations taking into account recombinations :

Here, Kv is the defect generation rate (or atomic displacement rate, displacements per atom per second), Re is the recombination coefficient, Kd is the sink strength at the interface or external boundary (assumed equal for both vacancies and interstitials, Kd = 57.6/d2), and Dv and Di are the diffusion coefficients for vacancies and interstitials, respectively.

In the following, we suggest that defect annihilation in HDCMs is dominated by nanoparticle surface sink effects, where interstitials rapidly migrate to the surface sink and recombine with vacancies located at the particle surface or interphase boundary. (The concentration of interstitials becomes much smaller than the vacancy concentration, while the diffusion coefficient of interstitials is much larger than the diffusion coefficient of vacancies). In this case, the point defects in HDCMs under irradiation are mainly vacancies inside the nanoparticle, and the movement of interstitials from their initial positions to the surfaces is assumed to be rapid . Additionally, the nanoparticles are considered isolated, with no exchange of atoms between them, making the saturation of vacancies inside the nanoparticle the primary factor for irradiation effects in HDCMs. It is also assumed that the concentration of radiation-induced point defects at characteristic temperatures exceeds the concentration of thermal-equilibrium defects and that there are no other reservoirs besides the surface of the particle.

In the steady-state regime for small nanoparticles, recombinations are unimportant, and the time-dependence of the vacancy concentrations Cvα in the α phase and Cvβ in the β phase can be described by kinetic equations :

where, Dvα and Dvβ are the diffusion coefficients for vacancies in the α phase and the β phase, respectively:

Here, D0α and D0β are the kinetic coefficients related to the jump frequencies of vacancies in α phase and β phase, respectively; EAα and EAβ are the vacancy activation energies (usually specified through vacancy migration energies) in α phase and β phase, respectively. The diffusion coefficients for vacancies vary exponentially with the activation energies, EAβ and EAα, divided by temperature. D0α and D0β are determined by the number of neighboring atoms. The diffusion coefficients of vacancies can also be calculated using the self-diffusion coefficients in a monovacancy mechanism mediated by nearest-neighbor vacancy jumps .

It is important to note that Shen’s approach assumes a size-dependence of the vacancy concentration in the steady-state regime (Equations 15 and 16), that is, the concentrations are proportional to the square of the particle size, R2:

Here, C0β = Kv/(57.6Dvβ) and C0α = Kv/(57.6Dvα) are the proportionality factors, and the vacancy diffusion coefficients determine the proportionality factors C0 in Equations 11–14. In the following, we employ the steady-state approach (Equations 17–21) in chemical rate theory.

For relatively small vacancy concentrations (that we usually deal with), the energies of point defects ΔGpd(α) and ΔGpd(β) increase almost linearly with Cvα and Cvβ. Hence, the size-dependent behavior of point defects leads to a size-dependent behavior of ΔGpd(α) and ΔGpd(β). In the steady-state regime, a nanoparticle with larger R should possess greater ΔGpd(α) and ΔGpd(β) because of the increased concentrations of radiation vacancies in the particle interior.

The probability aspect is an important consideration when analyzing phase transitions. The probability of a phase transition to occur can be described by considering the relative stability of different phases and the energy barriers between them. In our case, the transformation of α phase to β phase is indeed a first-order phase transition, and Gibbsian thermodynamics can be used to estimate the probability and energies involved in the transformation. However, this approach does not directly provide information on the kinetics of how the transformation occurs. The probability of the phase transition (referred to as p) can be related to the energy difference between the initial and final states and can be described by an exponential function:

These equations illustrate that the probability of the phase transition decreases exponentially as the energy difference between the initial and final states increases. Therefore, phase transitions with higher energy barriers are less likely to occur at a given temperature. From this, one can consider the different configurations of α→β phase change and calculate the energies and the probabilities. Hence, one can gain insights into the thermodynamic feasibility and likelihood of the phase transition occurring under specific conditions.

For the graphical representation of the results, we introduce energy densities per atom as follows: Δgbulk = gβ − gα = ΔGbulk/N0 represents the bulk Gibbs free energy change, Δgsurf = ΔGsurf/N0 represents the Gibbs free surface energy change, Δgp = ΔGp/N0 represents the total Gibbs free energy change of the particle without irradiation, Δg = ΔG/N0 represents the total Gibbs free energy change of the particle under irradiation, and Δgpd = ΔGpd/N0 represents the energy change of defect formation. Our calculations demonstrate the importance of distinguishing between instability points (where the condition ΔGp = 0 or ΔG = 0 is met) and phase stability zones (or radiation tolerance zones) in the temperature–size phase diagrams discussed here.

Since the presented thermodynamic approach is general, we can consider various possible phase transformations under irradiation. For example, polymorphic transformation involving a change in lattice type and amorphization are two common types of transformations observed in metal nanosystems and ceramic substances, respectively. (As we know, most elements of the periodic table are metals.) Among the metals undergoing polymorphic transformations are Fe, Co, V, W, Ti, Tl, Zr, Sr, Mn, Al, Ga, Sc, Ba, Li, Na, and K. Therefore, it is important to examine a comprehensive range of quantities, including thermodynamic parameters, driving forces, and kinetic characteristics, under irradiation conditions. However, determining surface energies and activation energies of vacancy diffusion poses a particular challenge because of contradictory experimental data from various sources, which may lead to divergent results. Unfortunately, we were unable to find real systems with a complete set of the mentioned parameters. Therefore, we resorted to model approximations to qualitatively demonstrate possible situations.

As an example, in this work we consider a model of an iron-like nanomaterial with a polymorphic phase transition, for which we will subsequently generalize to other cases by varying parameters. Bulk thermodynamic data for pure iron have been sourced from various references to compile the set of parameters . At low and intermediate temperatures, bulk Fe can exist in two crystallographic modifications, that is, the bcc phase (T < 1183 K) and the fcc phase (1183 K < T < 1665K). In this study, the bcc phase represents the model α phase, while the fcc phase represents the model β phase. Therefore, our focus will be on analyzing the transformations from bcc to fcc and from fcc to bcc that occur in an iron-like nanomaterial. We detail the findings for pure iron at the end of the paper.

The enthalpy change for vacancy formation can be estimated from the equilibrium melting temperature, Tm, and is ΔHfα = 3.76·10−19 J for the α phase and ΔHfβ = 3.28·10−19 J for the β phase . The entropy change can be estimated using the Boltzmann constant, with values of ΔSfα = −0.5kB for the α phase and ΔSfβ = 0.2kB for the β phase . Regarding kinetic parameters, the diffusion coefficients are estimated as D0α = 1.03·10−3 m2·s−1 and D0β = 1.07·10−3 m2·s−1, while the vacancy migration energies are taken (in most cases as reference values) as Emα = 4.96·10−19 J (3.1 eV) for the α phase and Emβ = 5.36·10−19 J (3.3 eV) for the β phase (close to ceramics) to demonstrate the vacancy effect clearly . For comparison, in SiC, the energy of silicon vacancy migration is nearly 2.4 eV, and the energy of carbon vacancy migration is about 3.6 eV . We focus on vacancy migration energies at the end of the paper and detail the findings. The surface energies σα and σβ of an iron-like nanoparticle are estimated as 2.21 and 2.17 J·m−2, respectively, while the interphase energy is taken as σαβ = 0.04 J·m−2 according to data . We focus on surface energies at the end of the paper and detail the findings for a pure iron nanoparticle. The volume density of atoms, n, varies within 1–2% (for example, at 1500 K, it is nearly 7.92·1028 m−3 for the β phase and 7.94·1028 m−3 for the α phase). Both the driving force of the transformation, Δgbulk = gβ − gα, and the density, n, are functions of the temperature . The model parameters for irradiation include a defect generation rate Kv set at 10−3 to 10−4 dpa·s−1.

Figure 2: Comprehensive visualization of the energy changes for the model of an iron-like nanomaterial with polymorphic phase transitions. (a) Energy changes Δg for the α→β phase transition, point defects, and surface energy as functions of size (d) and irradiation (Kv). (b) Energy changes Δg (represented by black crosses) and Δgp (by red points), illustrating the α→β phase transition in a spherical α-phase nanoparticle as a function of particle size, demonstrating three distinct zones, namely, zone I (Δg < 0, Δgp < 0), intermediate zone II (Δg > 0, Δgp < 0), and zone III (Δg > 0, Δgp > 0). (c) Effect of low temperature on the shift of the intermediate zone II. Decreasing the temperature leads to the narrowing of zone II. At T = 900 K, zone II disappears.

Zone I – unstable α-phase particle. Transformation can occur for α-phase nanoparticles (up to nearly d1 = 12 nm at T = 1100 K), regardless of whether the material is irradiated or not (indicating instability of the initial bcc phase). In this zone, the dominant mechanism is not radiation but rather surface effects associated with a decrease in surface energy during the phase change. Consequently, both functions Δg and Δgp are negative in zone I.

Zone III – stable α-phase particle. Phase transformation cannot occur, either with or without irradiation, indicating the stability of the initial bcc phase due to the dominant influence of bulk driving force Δgbulk. In this zone, both Δg > 0 and Δgp > 0.

The previous thermodynamic approach may be applied to various cases, such as size-dependent transitions. However, it only considers the initial and final single-phase stages of the transforming system. Another important aspect for nanoscale systems is nucleation, which involves the appearance and growth of a new β-phase nucleus, leading to the formation of a two-phase α+β system.

In the nanoscale case, the nucleation energy criterion is crucial for understanding phase transformations and the formation of new phases in materials. Nucleation represents a first-order phase transition and results in the creation of a new interphase surface, characterized by a corresponding specific interphase energy (σαβ) and area (Sαβ). Due to the interplay between bulk energy stimulus (ΔGbulk) and surface energy terms, the Gibbs free energy required to form a nucleus of a new phase reaches a maximum value, known as ΔG* (referred to as the nucleation barrier or the critical work of nucleus formation). It is important to note that the size of a nucleus is termed the critical size and may not necessarily coincide with the size of the entire particle denoted as d.

The nucleation energy criterion states that for nucleation to occur, the nucleation energy barrier must be surpassed. Referring to classic textbooks , one can formulate the nucleation energy criterion for phase formation as follows:

If the value of ΔG* is very high (greater than approximately 50kBT), then the phase transition is suppressed. Therefore, it is essential to consider nucleation and the nucleation barrier.