Thermodynamics of the Nanoparticle Consolidation

Thermodynamic functions have been derived that describe the processes of nanoparticle consolidation in solid–mobile phase twoand three-phase dispersed systems. An expression for the shrinkage pressure in a two-phase dispersed system has been deduced, which allows one to calculate stresses generating in the bulk of heterophase composite materials in the course of the nanoparticle consolidation. On the strength of these thermodynamic functions criteria have been suggested that allow one to predict the structure of a nanocomposite material.


Introduction
The phenomenology, kinetics, and the mechanism of nanoparticle sintering in nanocomposite materials are sufficiently well understood in material science.However, there are only a limited number of papers reporting the study of the thermodynamic state of nanoparticles [1][2][3][4].The author has not found publications reporting the studies of the nanoparticle thermodynamic consolidation in dispersed systems.
The aim of the present work is to derive thermodynamic functions that describe the nanoparticle consolidation and, based on the resultant relations, to justify some special features of the structure formation of nanocomposite materials.Nanoparticles with a crystalline structure have been the object of the investigation.

Experimental procedure
Typically nanoparticles are produced by two methods.The first of them may be termed "violent".It is realized by the force applied from the external power sources of energy, e.g., mechanical power in the intensive crushing and abrading of coarse particles, electric powder in electric discharge size reduction, etc.As a result of the absorption of the applied energy, nanoparticles have a heavily distorted structure and a great deviation from the equilibrium state in terms of thermodynamics.According to the second method, which is arbitrarily called "natural", nanoparticles are produced by growing (crystallization) from the matrix medium.These nanoparticles have a structure, which is close to their equilibrium state for the chosen production conditions.In this paper we will discuss nanoparticles produced by the second method.
For a correct choice of the thermodynamic study method, the most important problem is the determination of the minimum size of a nanoparticle.The author of [5] indicates fairly well that the size, whereby the symmetry elements inherent in this crystal start to disappear, should be considered the minimum size of nanocrystals.Such sizes are 0.5 nm for a body centered cubic structure and 0.6 nm for a face centered cubic structure [5].In thermodynamics terms, a critical size r cr of a new phase nucleus should be taken as the minimum size of a nanoparticle.All particles of radius r < r cr are unstable and disappear with time, dissolving in the mother phase.Particles of radius r > r cr are stable and able to grow.It is important in the thermodynamic studies that the state of a nanoparticle can be described in full measure by the macroparameters like temperature and pressure, while the effect of fluctuations on the parameters of the nanoparticle state should be negligible.
One of the structure features of a nanoparticle is a heavily curved closed surface that encloses the particle and inside the particle the volume of the surface layer is commensurable with the volume of the particle nucleus, which has properties of a macrophase.The surface layer of nanoparticles has a specific structure and properties, because of which technologists tend to produce particles, whose volume consists mainly of the surface layer volume.
For thermodynamic studies of the processes proceeding with the participation of nanoparticles Hill's method [6] is used.The author applied the thermodynamic method to describe the state not of an individual particle but of a great assembly of n particles.Each particle contains m i molecules of the ith sort.If we assume that the number of molecules in the system under study remains constant, then the energy of the assembly of particles changes with a change in the particle size r and number n.For such a system, the fundamental equation of state [6] can be written where u is the internal energy; η is the entropy, T is the temperature, p is the external pressure, v is the particle volume, μ is the chemical potential.
The u and η magnitudes belong to one particle, Λ is the new potential, which indicates the variation of the assembly of particles as the particle sizes change.The Λ potential is shown to be equal to the work of the small object formation [7] where s is the particle area, γ is the averaged surface tension.
The averaged surface tension is found from the expression , where s j is the area of the jth crystal face, , γ ∑ = s s j j is the surface tension of this face.Hereafter we will use the averaged values of the surface tension only and omit the upper index.
By integrating Eq. ( 1) we obtain the expression for the whole system U = Tη -PV + ∑m i μ i + Λn.For other thermodynamic functions we have F = -PV + ∑m i μ I + Λn, G = ∑m i μ I + Λn, where F is the free energy, G is the Gibbs free energy.The Hill method is used to describe the nanoparticle consolidation process in this paper.

Thermodynamics of consolidation of nanoparticles in a two-phase system
Let us consider a system consisting of n nanoparticles, which are denoted phase α, and dispersion medium ε (fig.1).Any mobile medium (a gas, a vapor, a liquid) may be dispersion medium ε.In the initial state (see Fig. 1a) nanoparticles are separated by the ε phase and concentrated in region II of the system.The dispersion medium is present in excess and occupies regions I and II.This allows us to impart properties of an unlimited phase to the medium.Regions I and II are separated by diaphragm III that is permeable for the ε phase.As a result of the consolidation of the α particles the system passed to the final state (see Fig. 1b), in which region II consists of particles α solely, phase ε is concentrated in region I, and diaphragm III shifted to a new position.We impose the following restrictions on the system: In the initial state of the system the Gibbs free energy for phase α is described by the following expression , where (′) indicates that the specified values refer to the initial state of the system, k is the number of the system components.Using relation (2) and taking into account that sn = S αε , we have According to [8], the chemical potential of a particle component i under the equilibrium conditions is not equal to the chemical potential of a component i in the dispersion medium, their interrelation is described by where μ r is the chemical potential of a component of a particle of radius r, μ ∞ is the chemical potential of a component of a particle of radius r → ∞, v m is the molar volume.
Taking into account of this expression, the Gibbs free energy of the whole system in the initial state has the form In a similar way, we derive equation for the system in the final state The change of the Gibbs free energy in the system transition from the initial to final state is .
Considering that the relation between interfacial areas S αε = 2gS αα , where g is the coefficient that allows for the variation of a particle geometric shape, and we obtain For some consolidation conditions Eq. ( 5) can be simplified.For example, in vacuum sintering of carbides, nitrides, and oxides of IV, V and VI groups transition metals of the Periodic system, refractory metals and some other substances, the partial pressures of their components range from 10 -6 to 10 -20 Pa.This means, that under the above conditions the concentrations of these components in the dispersion medium can be neglected.Then ' , , ' .Considering the above and equalities (3), we obtain )' ( )' ( For simplicity of the paper the final state of the system shown in Fig. 1b is idealized.In the real system, particles α in the initial state may have contacts and in the final state they form equilibrium dihedral angles.Under these conditions in the system in the initial state there exists the S αα contact surface.Because of the consolidation of particles α the contact surface area increases.For these conditions Eq. ( 6) takes the form The variation of the system free energy F is described by the relation In this relation the PΔV member allows for the work, which the system does against the external pressure P as the system volume changes.In the special case that the dispersion medium is a liquid ΔV ≈ 0, ΔG and ΔF values coincide, and the expression for the variation of the system free energy F is It follows from Eq. ( 7) that the consolidation of nanoparticles proceeds spontaneously at γ αα < γ αε .This condition is always fulfilled provided dispersion medium ε is gaseous.In a system where medium ε is a liquid, a reverse relation γ αα > γ αε is possible, e.g., in the WC-Co, WC-Ni, TiC-Ni systems.Under these conditions the spontaneous consolidation of nanoparticles is impossible in terms of thermodynamics and a reverse processdeconsolidation of a polycrystalline body into an assembly of particles takes place, i.e. spontaneous formation of a nanodispersed system.The deconsolidation process can be studied in the reverse transition of the system (see the fig. 1) from the final to the initial state.A variation of the process free energy F is described by the relation Equation ( 8) defines a thermodynamic potential of the system under study, therefore the system pressure may be found by the relation P = -(∂F/∂V) T, m .We will call this pressure "the shrinkage pressure P sh ".To understand the physical meaning of the shrinkage pressure, consider the process of shrinkage in the context of the work done by the system.In the system transition into the final state, diaphragm III has shifted, and the work A = P sh ΔV sh was done as a result.Here P sh is the pressure in the II region, ΔV sh is the change in the volume of the II region.At the same time the variation of the system free energy F equals the work done under isochoric and isothermal conditions.Thus, A = -ΔF.After corresponding substitutions and transformations we obtained the relation All particles can be conditionally divided into two types: small objects and macroobjects.Particles, whose chemical potentials, μ r depend on their sizes, belong to small objects, while particles, whose chemical potentials are not markedly affected by the particle size, belong to macroobjects.
To describe the thermodynamic state of particles of the second type, the Gibbs method may be used and to describe particles of the first type, Hill's method are applied.Nanoparticles belong to small objects.The paper published earlier [9] gives the following values (obtained by the Gibbs method) of the free energy F and shrinkage pressure variations in the consolidation of particles with macrophase properties: A comparison of these expressions with corresponding Eqs. ( 8) and ( 9) for nanoparticles allows a conclusion that the additional item 8) and ( 9) is due to the effect of size of a small object on the component chemical potentials.Because of this, for the nanoparticle consolidation, the variation of the system free energy ΔF and the shrinkage pressure P sh may be thought of as the sum of two items ΔF = ΔF ∞ + ΔF r , P sh = P sh ∞ +P sh r , where ΔF ∞ and P sh ∞ refer to macroobjects and ΔF r , P sh r account for the effect of small objects.The P sh value may be calculated by the procedure [9].The calculation shows that if a macroparticle, say, of size 10 μm, is placed in region II (see the fig. 1) then in the shrinkage of nanoparticles of size 10 nm the macroparticle is subjected to reduction by a pressure of 0.18 MPa (the data are given for a copper powder with a porosity of 20% at a temperature of 1279 K).Around a particle of size 10 μm in region II a field of stresses is formed, which can bring about the residual porosity and crack initiation.

Thermodynamics of consolidation of nanoparticles in a three-phase system
Let us consider a system consisting of n α particles of phase α, n β particles of phase β, and mobile phase ε.A mobile phase can be either a gas or a vapor or a liquid.The system has region II filled with particles of phases α and β and phase ε and region I filled with phase ε only.Regions I and II are separated by diaphragm III permeable for phase ε (fig.2).In the initial state of the system (fig.2, a) the particles in region II are separated by phase ε.In the finite state (fig.2, b) a consolidation process of particles of phases α and β occurred and particles formed two mutually penetrating skeletons.In this case, α-α and β-β contacts between particles α and β appeared, respectively, as well as α-β interphase contacts.As a result of the particle consolidation, the volume of region II decreased by the ΔV sh value, the diaphragm moved to a new position, and the volume of region I increased by the ΔV sh value.We impose the following limitations on the system:  Let us consider particle consolidation under constant pressure and temperature.Under these conditions, the function of the system state is the Gibbs free energy G.The Gibbs free energy variation (ΔG) in the system transition from the initial to the finite state can be described as the sum of the Gibbs free energy variations of two-phase subsystems.
Eq. ( 10) defines the variations of the Gibbs free energy during the formations of contacts by particles: particles α form only αα contacts (term 1), particles β form only ββ contacts (term 2), particles α form αβ contacts (term 3), particles β form βα contacts (term 4) and term 5 defines the Gibbs free energy variation in the dispersion medium ε.Using Eq.( 5), for each term we obtain . 1 1 If particles of the β phase are distributed evenly in the system volume and do not form ββcontacts, then Eq. (10) has no (10b) term.Depending on the system peculiarities, Eqs.(10а)-(10e) can be simplified.For example, in sintering W and Al 2 O 3 nanoparticles the component concentrations in the ε phase are very small, besides tungsten and aluminumoxide are not mutually soluble.Taking into account these peculiarities of the W -Al 2 O 3 system Eqs.(10а)-(10e) look like: In Eqs.(10c), (10d) and (11c), (11d) we have S αβ = S βα , γ αβ = γ βα .We will analyze some peculiarities of forming nanostructures in three-phase compositions by the example of the above-mentioned simplified model, which is described by Eqs.(11а)-(11d).It follows from Eq. (11a) that αα contacts are spontaneously formed at γ αα < γ αε and ββ contacts form spontaneously at γ ββ < γ βε .Contacts between particles α and β will be formed at γ αε > γ αβ and γ βε > γ βα .If some of the inequalities are not met, proper contact surfaces do not form.If components of α and β particles are soluble in a dispersion medium, then nanoparticle consolidation is described by Eqs.(10a)-(10e).
Nevertheless, it should be noted that the γ αε , γ βε , γ αα , γ ββ , γ αβ .surface energies strongly affect nanoparticle consolidation.Thus, by adding surface-active substances into a system, one can affect the consolidation of nanoparticles.

Conclusions
1. Thermodynamic functions describing the consolidation process in a two-and a three-phase system consisting of nanoparticles and a mobile phase have been derived.The mobile phase is gas, vapor or liquid.2. The notion of the shrinkage pressure, which generates in the system during the particle consolidation, has been introduced, and the equation for the shrinkage pressure has been derived.The equation allows both for interphase and surface tensions and for variations of the areas of contact and interphase surfaces.3. Criteria that make it possible to predict the structure of heterophase composite materials have been suggested.

Fig. 1
Fig.1Model of the nanoparticle consolidation in a solid -mobile phase two-phase system: the initial (a) and final (b) states of the system; α -nanoparticles, ε -dispersion medium.