Advances in thermodynamic modelling of nanoparticles

ABSTRACT Among all the computational techniques (Density Functional Theory, Molecular Dynamics, Monte-Carlo Simulations, Nanothermodynamics) used to investigate the properties of nanoparticles, nanothermodynamics is the most unusual one. Indeed, most people still thing that thermodynamics does not apply at the nanoscale; nonetheless, thermodynamic concepts can still be applied at the nanoscale to predict various properties of nanoparticles like melting temperature, energy bandgap … In this review, we first introduce the fundamental concepts and methods of nanothermodynamics starting from Hill’s contributions to the most recent developments focusing specifically on the relationship between the material property and the following parameters as quantum statistics (Fermi-Dirac or Bose-Einstein), size and shape of the nanoparticle. Graphical abstract


Introduction
The concept of 'nanotechnology' was first introduced to the world on 29 December 1959 by the theoretical physicist Richard Feynman  during his famous lecture 'There is plenty of room at the bottom' at the annual American Physical Society meeting at Caltech [1]. However, the invention of scanning probe microscope and the discovery of C 60 fullerenes were still two decades away. The plenty of room, Feynman was referring to, came from electron microscopy observations that allowed scientists in 1959 to observe details as small as~1 nm; but that resolution limit was still too large to observe atoms individually [2]. The major problem of electron microscopy at that time was spherical aberration [3], a problem solved in the late 1990s with the presence of quadrupoles and octopoles in aberration-corrected electron microscopes [4].
During the same period, another theoretical physicist, Terrell L. Hill (1917Hill ( -2014, presented a brief introduction of his work on the thermodynamics of small systems at the American Chemical Society meeting at Washington D. C. on 23 March 1962 [5] and he published a first paper on that topic the exact same year in the Journal of Chemical Physics [6]. Then, he refined his ideas and finally published his famous book 'Thermodynamics of Small Systems' in 1964 [7]. Thermodynamics is all about conservation of energy (1 st law of thermodynamics) and as everybody knows it is always true. So, this theory should still be applicable at the nanoscale. His main idea was that the size of a system affects various intensive properties of that system and this modification should be considered into the conservation of energy. His work stayed mostly unknown during 40 years; it is only when Paul Alivisatos, one of the two Editors-in-chief of the freshly launched 'Nano Letters' journal (established in January 2001 by the American Chemical Society), invited him to write a paper in the first volume of 'Nano Letters' that the scientific community became fully aware of his work [8]. Indeed, the progress in the synthesis and processing of nanomaterials has created the demand for greater scientific understanding of the thermodynamics at the nanoscale. This review paper summarizes the advances made by different thermodynamics group around the world to the field of nanotechnology.

Hill's theory
A first approach to discuss the thermodynamics of small systems is by introducing the finite size effects within the macroscopic thermodynamics (Table 1). Hill did that by introducing a new thermodynamic potential called the sub-division potential, U sub , and it is defined as [6][7][8]: where U is the internal energy of the system, S is the entropy of the system, V is the volume of the system, N is the number of particles within the system and N sub is the number of sub-divisions in the system. Therefore, at the nanoscale, the internal energy of the system made of a single component material, U, is given by the Euler's equation: where the temperature, T, and entropy, S, are the thermal conjugate variables; the pressure, p, and volume, V, are the mechanical conjugate variables; the chemical potential, μ, and the number of particles, N, are the chemical conjugate variables. There exist a new pair of conjugate variables at the nanoscale (U sub , N sub ) represented by the sub-division potential and the number of sub-divisions. Generally, the number of sub-divisions is much smaller than the number of particles in the system (N sub <N) [9]. Therefore, the Gibbs free energy of the system at the nanoscale, G, becomes; Equation (3) is often generally simply written as the summation between two contributions, the bulk and the surface ones as G ¼ G 1 þ G AE where G 1 ¼ μN (extensive term) and G AE ¼ U sub N sub (non-extensive term) [10,11]. At the bulk scale, the chemical potential is simply given by μ ¼ @G @N À Á T;p ¼ @U @N À Á S;V ;μ 1 while at the nanoscale the chemical potential becomes μ ¼ μ 1 þ @U sub @N N sub [12,13]. Thus, μ is a function of just two independent variables T and p. However, since the extensive state functions of nanoparticles are not linearly proportional to N, μ is not just a function of T and p, but is also a function of N. Hill's sub-division potential, U sub , can be understood by comparison to Gibbs' chemical potential, μ. μ is the change in energy to take a single particle from a bath of particles into the system, whereas U sub is the change in energy to take a cluster of N interacting particles from a bath of clusters into the system [14,15].

Tsallis' theory
The second approach to discuss the thermodynamics of small systems is to consider their non-extensivity by redefining the entropy (Tsallis' approach) instead of redefining the internal energy (Hill's approach). Indeed, when a physical quantity χ is expressed by χ / N λ , they are classified into two groups: intensive (λ ¼ 0) and extensive ones (λ ¼ 1). A physical quantity is intensive if it is invariant under changes to the system size (λ ¼ 0). A physical quantity is extensive if it scales linearly with the system size (λ ¼ 1). We shall refer to λ as Euler exponent as it is defined through Euler's theorem of homogeneous functions: χ λx; λy; λz ð Þ¼λχ x; y; z ð Þ [16]. By shrinking the size of a system to the nanoscale, its surface-to-volume ratio varies as N À1 = 3 [17][18][19], meaning that λ is neither 0 nor 1, therefore the system behaves non-extensively (Table 1). When the surface to volume ratio increases the non-extensivity becomes more pronounced. And consequently, non-extensity is described by a fractal Euler exponent, λ.
The task of redefining the entropy was undergone in 1988 by Constantino Tsallis (1943-) and this non-additive entropy, S q , is known today as the Tsallis' entropy [20][21][22][23]: where the q-logarithm is defined as ln q ζ ¼ ζ 1Àq À1 1Àq (ζ > 0 and ln 1 ζ ¼ ln ζ); Ω is the total number of microstates; p i is the probability of finding the system in the i th microstate (with the condition that P Ω i¼1 p i ¼ 1); q is the entropic index, a real parameter which characterizes the degree of non-additivity of the entropy (i.e. a measure of how strong the correlations are) and k is the Boltzmann's constant (k B ) when q ¼ 1 otherwise kÞk B when qÞ1.
If a system has weak or no correlations at all, then the number of microstates Ω increases exponentially with the number of elements N i.e. Ω ¼ X N , and the non-additive entropy S q is extensive for q ¼ 1 (S q¼1 ¼ kN ln X) and non-extensive for all qÞ1. Whereas, if the system has strong correlations, the number of microstates Ω increases and can be described by a power law, Ω ¼ N X . The non-additive entropy S q is therefore extensive for q ¼ 1 À 1 X while being non-extensive for all other values of q, including q ¼ 1 (S q¼1 ¼ kX ln N).
By using Penrose's definition of entropic additivity (Equation (5)), we have that the entropy S q is non-additive for all values of q except for q ¼ 1. Indeed, when q ¼ 1, the entropy is additive and the usual Boltzmann-Gibbs entropy is recovered (S q¼1 ¼ k ln Ω) [24].
Where A and B are two independent systems. The case q < 1, q ¼ 1, q > 1 correspond, respectively, to super-extensivity, extensivity and sub-extensivity. The Hill's approach can be deduced from the Tsallis formalism by defining the sub-division potential as [9,13,25,26]: The non-additivity property of the Tsallis entropy forms the basis of the non-extensivity of the sub-division potential.

Thermodynamic limit
The question 'Small systems: when does thermodynamic apply?' was raised 80 years ago in the context of world war II during the development of nuclear weapons [27]; at that time, Weisskopf, Landau and Frenkel were introducing the concept of temperature in nuclear reactions. Nowadays, within the context of nanotechnology, this question has to be asked again [9,28]! The thermodynamic limit is essentially a consequence of the law of large numbers, i.e. when sample size tends to infinity, the sample mean equals to population mean. Consequently, the ratio of the size of the fluctuations to the mean is of order δN = N : 1 ffiffiffiffi N p meaning that the fluctuations vanish when the system becomes macroscopic or the fluctuations become huge at small scales (Table 1).
At the macroscale, the Euler's equation is U À TS þ pV À μN ¼ 0; and consequently, the temperature is defined as the partial derivative of the internal energy with respect to the entropy: However, at the nanoscale the Euler's equation becomes U À TS þ pV À μN ¼ U sub N sub , therefore the temperature is redefined by the following expression: Suppose that we have one system that we separate into two identical smaller systems, the decrease in the temperature of the smaller systems can be understood by the conversion of internal energy into surface energy. The decrease of internal energy decreases the kinetic energy of the particles and consequently the temperature. Therefore, to get an idea of the relative temperature fluctuation δT = T : 1 ffiffiffiffi N p inside a cube made of 10,000 gold atoms (L,3:4 nm), the fluctuation reaches~1% while it reaches~10% for a cube made of 100 gold atoms (L,0:7 nm) [27,29,30]. Consequently, there is no strict thermodynamic validity limit at the nanoscale, the limit you want to set only depends on the magnitude of the fluctuation you are ready to accept on the material property you are studying [30][31][32][33][34]. Thermodynamics can still be applied at the nanoscale because the temperature can be defined for small systems (Equation (8)) by using either a classical or quantum approach. Indeed, classically, the temperature can be defined from the kinetic energy of particles constituting the system; consequently, the temperature can be defined very locally even for one atom. Quantum-mechanically, the temperature is defined from the quanta of vibrations (phonons); therefore, the temperature can only be defined for sizes larger than the phonon mean free path due to Heisenberg's principle [35]. As the phonon mean free path is temperature-dependent, the classical definition will differ only from the quantum one at low temperatures [36,37].

Fermionic or bosonic material property?
All particles in nature are either bosons or fermions. As a general rule, matter is made of fermions, e.g. protons, neutrons and electrons are fermions. Bosons are particles (quanta) associated with interactions, e.g. photons and phonons are bosons. A fermion is a particle that follows Fermi-Dirac statistics while a boson is a particle that follows Bose-Einstein statistics. Therefore, by having only two types of particles in nature, there will be only two types of size-dependent material properties in nature; the ones that follow a Fermi-Dirac statistics and the others that follow a Bose-Einstein statistics. By analyzing the sizedependency of several different material properties like, for example, the melting temperature, Debye temperature, cohesive energy . . . ; it has been found by Guisbiers [38][39][40] that all those size-dependent material properties can be described by using only two scaling laws (one fermionic and one bosonic; Figure 1) which can therefore be unified into one single scaling relationship as: where represents the size/shape-dependent material property and 1 represents the bulk material property, S represents the type of quantum statistics, i.e. the particle obeys to either Fermi-Dirac (S = 1/2) or Bose-Einstein (S = 1).
where A/V is the surface-to-volume ratio, ΔH m,∞ is the bulk melting enthalpy, and γ l and γ s are the surface energies in the liquid and solid states, respectively. Physically, α shape quantifies the magnitude of the size effect; the values indicated in Figure 2 are for spherical nanoparticles. Values of α shape for other polyhedral shapes can be found in Ref [41].
From Equation (9), it is clear that for a given material (i.e. a given α shape parameter) and a given size (D), the size effect on materials properties described by a Fermi-Dirac statistic ('"fermionic properties"') is more pronounced than the size effect on materials properties described by a Bose-Einstein one ('"bosonic properties"'). Moreover, the scaling exponent in Equation (9) exhibits the fractal nature of size effects, S À1 = 2 being either a non-integer equal to ½ or an integer equal to 1. Some phase transitions like order-disorder and superconductivity exhibit a scaling exponent equals to ½ while others exhibit a scaling exponent equal to 1 like melting. Moreover, Xiong et al. also confirmed that for some physico-chemical properties like the Debye temperature (i.e. temperature of a material's highest normal mode of vibration) the sizedependency adopts a ½ scaling exponent following therefore this scaling law [42]. The square root size-dependency of the Debye temperature has also been confirmed by Yang et al. [43]. At frequencies lower than the Debye frequency (kT Debye ¼ " hω Debye ), it has been recently confirmed Figure 2. Shape-dependent parameter quantifying the magnitude of the size effect (α shape ) for various single element adopting a spherical morphology at the nanoscale [41,114,115].
experimentally that vibrations of nanoparticles are still ruled by continuum mechanics laws down to sizes of a few nanometers [44,45]. At frequencies higher than the Debye frequency, the 'lattice' is unable to 'see' the vibrations because the wavelength of the vibrations is smaller than the lattice parameter; therefore, the vibrations become independent from the lattice.
In conclusion, the universality of 1/D size-dependency has been confirmed by Vanithakumari et al. [46] and Xiong et al. [42] for most physico-chemical properties. Moreover, it is observed that low-temperature phenomena are described by a ½ scaling exponent while higher temperature phenomena are described by a unitary scaling exponent.

Melting temperature of NPs
The solid-liquid phase transition of nanoparticles has been investigated by the scientific community since 1850. Faraday was the first scientist emitting the idea that the melting temperature of nanoparticles should be smaller than the bulk melting temperature. A more detailed discussion concerning the most important theoretical models predicting the size-dependency of the melting temperature can be found in Refs [17,47,48]. Based on Equation (9), the size-dependent melting temperature is described by [49,50]: Where T m is the melting temperature of the nanoparticle having a size D, T m;1 is the bulk melting temperature, α shape is the parameter quantifying the magnitude of the size effect and D is the size of the nanoparticle. Almost all the theoretical models mentioned in Refs [47,48]. predict a size dependency as written in Equation (10), only the value of the α shape parameter may differ. The 1/D behavior of Equation (10) is illustrated in Figure 3 [17]. The 1/D behavior has been recently verified experimentally by heating gold nanoparticles having diameters below 5 nm in-situ a Transmission Electron Microscope (TEM) [51]. By using the fluctuations theory, the thermodynamic correction factor for a single component system is defined based on the fluctuating number of particles within the system [52]: where the brackets denote an ensemble average and N is the number of particles in the system. Equation (11) can be rewritten in a similar way as Equation (10) [53,54]: where C is a constant related to the surface effect that do not depend on the size D i.e. C is a function of α shape . Therefore, the thermal properties of a small system differs from those of a large system but in a predictable way, 1/D [52,55,56].

Energy bandgap of NPs
The energy bandgap, E g , of a semiconductor can also be predicted by using thermodynamic concepts. Indeed, the energy bandgap can be described as the variation of the Gibbs free energy between the conduction and valence energy bands, namely, E g ¼ ΔH cv À TΔS cv , where ΔH cv and ΔS cv are the variation of enthalpy and entropy between the conduction and valence energy bands, respectively [57]. At the nanoscale, the energy bandgap varies with the size of the semiconductor and it increases when the size is reduced. From Equation (9), it is known that any fermionic (S = 1/2) property, ξ, scales like Δ= 1 ¼ α shape D. Applying the aforementioned fermionic scaling law, Δ= 1 ¼ α shape D, to the energy bandgap and melting temperature, produces ΔT m T m;1 ¼ ΔE g E g;1 where T m (E g ) and T m,∞ (E g,∞ ) represent the sizedependent and bulk melting temperature (energy bandgap) of the material, respectively. Consequently, the size-dependent bandgap energy, E g , can be calculated by employing the following scaling law, E g E g;1 ¼ 2 À T m T m;1 [ [58][59][60][61]. The exact same scaling law was obtained by Li et al. [62] when expressing the size and temperature-dependency of the electrical conductivity as an Arrhenius equation: where σ 0 denotes a pre-exponential constant, Q D ð Þ is the size-dependent activation energy for electrical migration within a nanoparticle, k B is Boltzmann′s constant and T is temperature. For intrinsic semiconductors (non-doped), the size-dependent activation energy is related to the sizedependent energy bandgap as Q D ð Þ ¼ E g D ð Þ 2 [63]. Therefore, the following egality is obtained: At the melting temperature, T m , the electrical conductivity is assumed to be size-independent meaning that σ D; T m ð Þ¼σ 1; T m;1 À Á . Therefore, we get By combining Equations 13 and 14, we obtain the same scaling law, E g E g;1 ¼ 2 À T m T m;1 , as previously obtain when expressing the energy bandgap as E g ¼ ΔH cv À TΔS cv [64][65][66]. On Figure 4, the validity of the scaling law is illustrated on CdS nanoparticles. The exciton Bohr radius of  [116][117][118][119]. The following scaling law, E g E g;1 ¼ 1 þ α sphere D, has been used to predict the energy bandgap of CdS nanoparticles at the nanoscale with a parameter α sphere ¼ 0:38 nm [114].
CdS is 1.50 nm meaning that at 3 nm the energy bandgap starts being affected by quantum effects [67].

Phase diagrams of NPs
At the nanoscale, the range of properties is extended by tuning the size, shape of the system [68,69]; but it can also be extended by taking mixtures of elements to generate alloys [70]. Generally, there is an enhancement in specific properties upon alloying due to synergistic effects [71]. As illustrated in Figure 5, there is a very wide range of combinations and compositions possible for nano-alloys. Four types of mixing patterns (a, b, e, and f) can be identified in Figure 5: core-shell structure, Janus structure, random alloying and ordered alloying. Some variants are possible (c, d, g, h). The fingerprint of an alloy is its phase diagram. A binary phase diagram is a visual representation, at constant pressure, of phases at equilibrium between two elements for any combination of temperature and composition. Obtaining the phase diagrams for binary systems at the nanoscale has been a long-sought goal [72][73][74]. At the nanoscale, the experimental determination of these phase diagrams is almost impossible because it requires calorimetry measurements which are very challenging to perform because of the small mass involved [75][76][77]. In this area, theoretical modelling is very often the only tool that can be used to infer the thermodynamics of nano-alloys [78,79]. A list of the known binary phase diagrams at the nanoscale is indicated in Table 2. As it can be seen from Table 2, only 24 binary phase diagrams are known at the nanoscale for various nanoparticle shapes; so, there is still a huge amount of work to do in order to calculate the other 3136 possible binary phase diagrams. Indeed, around~80 chemical elements from the periodic table are stable; therefore, the number of 2-combinations among 80 elements gives us 3160 binary phase diagrams in total.
Most of the calculated phase diagrams have been obtained at the nanoscale by using nano-thermodynamics (or also called nano-calphad [73]). The Gibbs free energy (GFE) of the alloy within a single phase is given by [80,81]: where x i is the composition in element i, G 0 i is the GFE of the pure element i, R is the molar gas constant, and T is the temperature. Ω v ij is the interaction energy for each pair of elements ij, and ν is a parameter equal to 0 for regular solutions and 1 for sub-regular ones. In Equation (15), the first two terms represent the Gibbs free energy of ideal mixing while the last term represents the excess Gibbs energy of mixing. The Gibbs energy of the alloy within a two-phase mixture is given by: where x S and x L are the composition of the solid and liquid phase, respectively. The one-phase region is directly determined by the curve having the lowest Gibbs free energy while the two-phase region is simply determined by the equality between the Gibbs free energy of mixing in the liquid and solid states. For each temperature T between the melting points of the elements, the compositions that minimize the total GFE are determined. Therefore, at equilibrium, when the solid and liquid phases coexist (G S = G L ), the solutions of Equation (16) form the solidus and liquidus curves in the binary phase diagram. The success of the theory has been confirmed in Figures 6 and 7 where the experimental data points perfectly match the theoretical predictions for the particular Au-Cu phase diagram. Furthermore, Figures 6 and 7 confirm the fact that there exist two types of size effects as mentioned in section 5. Indeed, the solid-liquid phase transition undergoes a stronger size effect (= 1 ¼1 À α shape D) compared to a solid-solid phase transition (= 1 ¼ ). The reason is that a solid-liquid phase transition involves bond breaking, consequently involving electrons (fermions) while a solid-solid phase transition involves different vibrational characteristics between the two solid phases, consequently involving phonons (bosons). Therefore, Figures 6 and 7 illustrate the fermionic and bosonic nature of phase transitions. Another effect that may play a significant role at the nanoscale is the stress within the nanoparticle. Indeed, the curvature of the nanoparticle increases as its size decreases; consequently, initiating higher strain and then higher stress within the nanoparticle. This is effect becomes nonnegligible for very small nanoparticles (~2-3 nm in size) i.e. clusters [82][83][84]. The large curvature of small nanoparticles is responsible of the strain which gives a volume contribution into the excess Gibbs energy of nanoparticles. Moreover, strain-stress effects are more important in binary and multi-elements compounds, because of the size mismatch between different atomic species, consequently affecting not only non-crystalline but also crystalline structures.

Conclusions
Nanothermodynamics is a theory developed by T.L. Hill in the early 1960s for small equilibrium systems and it has become one of the major formalisms for quantitative treatment of equilibrium nanoscale materials. The key concept in the theory is a difference between differential and integral forms of many nonextensive thermodynamic quantities due to the smallness of a system. Consequently, power laws naturally emerge as general features describing those complex nanoparticles due to their non-extensivity. Another important consequence of Hill's theory is the sensitivity of nanoparticles to their   . Therefore, by going from the macro-world to the nano-world, there is a transition from exponential laws describing a continuous material to power laws describing a discrete material. The appearance of those power laws is the result of strong correlations between the physico-chemical properties with the size, shape and environment of the nanoparticle. Finally, let's remember Einstein's quote about thermodynamics: 'A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore, the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will never be overthrown' [85].