Thermodynamics of the Consolidation of Nanoparticles and a

The thermodynamic study of the particle consolidation process in a system consisting of nanoparticles, inclusions of macroparticles, and mobile phase (gas, vapor, liquid) has been conducted. The thermodynamic functions describing this process have been derived. The conditions have been established, under which the process of consolidation proceeds to the end; the conditions, under which the process does not take place in terms of thermodynamics; and the conditions, under which only certain phases consolidate. It has been shown that in this system there are diffusion flows of the substance from nanoparticles to the macrophase. The conditions have been defined, under which a nanoparticle or a group of nanoparticles can be in an equilibrium state and maintain the size and shape arbitrarily long.


Introduction
Of all variety of heterophase materials the compositions, in which one of phases is included into the matrix consisting of microparticles of other phases as macroinclusions, have found wide applications.The progress in the field of nanopowder production allows us to initiate the development of new heterophase materials, which consist of a matrix formed of nanoparticles, and macroparticles of another phase.In these composite materials in the course of a sintering the consolidation of the nanoparticles, the interaction of nanoparticles with the macroparticles, and the intensive growth of the macroparticles takes place.In this connection at the first stage we shall study thermodynamics of macroparticle growth and dissolution nanoparticles.The analysis of the thermodynamics of the growth of large particles and disappearance of small particles in a nanodispersed system will make it possible to determine the conditions, under which this process proceeds spontaneously or does not take place, and to find the ways of the process inhibition in terms of thermodynamics.At the second stage we shall study consolidation of nanoparticles and their interaction with a macrophase.

The thermodynamics of large particle growth in nanodispersed composite materials
The studies were conducted on the model of a system, which contains n nanoparticles α and dispersion medium ε (Fig. 1).Any mobile phase (vapour, gas, liquid) can be a dispersion medium.The system contains a macroparticle β with properties of a macrophase.Nanoparticles and the macroparticle have the same composition.The number of nanoparticles n in the system is not limited and it should be such that the whole system is a macroobject, to which methods of thermodynamic analysis are applicable.Fig. 1.Model of a system consisting of nanoparticles α, macrophase β, and dispersion medium ε, S is the interface.
In the initial state of the system particles are of the equilibrium shape.Particles of the equilibrium shape but of different sizes are characterized by an unequal equilibrium concentration of the component in the liquid phase or by a partial pressure in the gas phase.The ratio of the equilibrium concentration of the particle component of radius r (C r ) to the equilibrium concentration of the component over a flat surface (C ∞ ), i.e. with r → ∞, is described by the known Gibbs-Thomson equation , where γ is the surface tension, v m is the component molar volume, R is the gas constant, T is the temperature.
It follows from this equality that in the neighbourhood of a small crystal, the component concentration is higher than that over the surface of macrophase.In [1] the Gibbs-Thomson equation was expressed in terms of chemical potentials: where μ r , μ ∞ are the chemical potentials of a component in a particle of radius r and in an unlimited volume of the phase, respectively, Ω is the volume of an atom or a molecule.From the Gibbs-Thomson equation and expression (1) it follows that the difference in concentration between the components in the neighbourhood of dispersed particle and macrophase makes the difference in chemical potential between the components, which causes a diffusion flow of the substance from a nanoparticle to the macrophase.In the course S α β ε of the component diffusion the substance of nanoparticles is transferred and deposited onto the surface of the macrophase.Because of this the volume of nanoparticles decreases and the macrophase volume increases.We assume that in the system initial state nanoparticles had the size r′ and in the final state the size was r′′.In this case, r′ > r′′.As a result of the mass transfer of the substance the volume of macroparticle increased.
The following limitations are imposed on the system: ) where m is the quantity of the ith component, indices α, β, ε, αε, βε indicate that the values they define refer to the corresponding phase or interface.
Limitation (2,b) testifies that the system does not exchange masses with the environment and consists of independent components, and the components can be redistributed between the phases.Besides, we assume that number n of nanoparticles in the system remains unchanged, all nanoparticles have the same radius r.The redistribution of a substance between nanoparticles and macrophase occurs at the constant temperature and pressure.Under these conditions the system characteristic function is the Gibbs energy G.For the thermodynamic studies of the process of the dissolution of small and growth of large (in our system a macrophase) particles, we use the Hill method [2].Here the thermodynamics were used to describe the state not of an isolated particle, but of a great group consisting of n particles.Each particle contains m i molecules of the ith component.If we assume that the number of components in the system under consideration remains constant, then the energy of the group of particles changes with the variation of the particle size r and the number of them n.For this system the fundamental equation of state can be written [3]: where u is the internal energy, η is the entropy, P is the pressure, v is the particle volume.The u and η magnitudes refer to the same particle, Λ is the new potential, which shows how the energy of the particle group changes with variation of the particle sizes.The author of [3] showed that potential Λ equalled the work of the small object formation: where γ is the surface tension, s is the particle surface area.By integrating Eq. (3), we obtain equation for the whole system U = Tη -PV + Σm i μ i + Λn.
For other thermodynamic functions we have F = -PV + Σm i μ i + Λn; G = Σm i μ i + Λn, where F is the Helmholtz free energy, G is the Gibbs energy.
The Gibbs energy, G, of the system in the initial (′) and in the final (′′) state consists of the sum of the Gibbs energies of each phase and interface S.
where the S index shows that the value denoted by it refers to interface S.
For nanoparticles we have . Taking into account Eq. ( 4) and the equality ns αε = S αε , where S αε is the surface area of all nanoparticles n in the system, we got Using Eq. ( 1), we converted the term ∑ Here μ i ∞ is the chemical potential of the ith component of the macrophase.Because the compositions of nanoparticles and macrophase are equal, μ i ∞ can be assigned to macrophase.
In expression With allowance made for the aforesaid for a nanoparticle, we have For the macrophase, dispersion medium, which has the macrophase properties as well, and surface S, we have respectively Thus, for a system in the initial state the Gibbs energy is of the form After the above transformations we have got the following expression for the final state of the system: In Eqs. ( 5) and (6) chemical potentials referred to macrophases, therefore, under the conditions of the equilibrium between the macrophases μ i The V′/r′ ratio is worthwhile to represent in terms of the surface area according to the V/r = gS relation, where g is the coefficient that allows for the geometry of particles.Substituting this expression into (7) and (8) gives: In the course of the mass transfer, surface tensions γ′′ s and γ′ s can be taken as equal.This permits us to neglect the last term in Eq. ( 9).In the second and third terms the chemical potentials are referred to macrophases β and ε (see the Fig. 1).We assume that macrophases β and ε are in the chemical equilibrium, i.e.
With allowance made for limitation (2,b), one can take . Thus, to change the Gibbs energy, we obtained Obviously S′ αε > S′′ αε .According to [3], as the radius r decreases, the surface tension of a nanoparticle also decreases and at r → 0, γ → 0. This means that γ′′ αε < γ′ αε , and inequality (10) is always true.Thus, to change inequality (10) in the system under study by methods of thermodynamics is impossible and in the presence of the nanoparticles dissolve in a dispersion medium and large particles grow.The mass transfer of a substance from nanoparticles to the macrophase has some peculiarities.During the mass transfer nanoparticles decrease spontaneously in size to the total disappearance.In this case, the diffusion flows lead to an increase in the gradient of components and, accordingly, an increase in the substance flows to the macrophase.The macrophase destabilizes the nanodispersed system.A nanoparticle can be in equilibrium with the medium, have an equilibrium shape, and live arbitrarily long, if it is placed into a limited volume (cell).In this case, there is no the exchange masses with the medium surrounding the cell, the cell volume and temperature are constant.Under these conditions a nanoparticle takes an equilibrium shape and is in equilibrium with the medium surrounding it.If in the result of fluctuations a nanoparticle partial dissolution occurs, solution surrounding the nanoparticle becomes supersaturated and the substance precipitation from the solution onto a particle takes place, and the system returns to the equilibrium.A group of nanoparticles of the same size that is in the dispersion medium is also unstable.In such a system due to fluctuations, a gradient of the chemical potential of any component appears, which brings about a diffusion flow of the substance that results in the appearance of the higher gradient of the chemical potential, increase and initiation of new flows of the substance, and in the intensive growth of some particles at the expense of others, i.e. condition (10) is met.It should be noted that the real systems are polydisperse, therefore, in their volume there always are gradients of chemical potentials, and in the course of the nanoparticle consolidation larger particles grow through the dissolution of smaller ones.
It follows from the above that for the stabilization of a nanodispersed system, each nanoparticle should be placed into a capsule obstructing the exchange components between particles.These capsules can be made by adsorbed or chemically bonded atoms of a substance on the particle surface that block the substance transfer into the medium surrounding the particle.Another way is in the development of a dispersion medium, through which the transfer of nanoparticle components is impossible.In this case, the dispersion medium acts directly as the envelopment.

Thermodynamics of the consolidation of nanoparticles and their interaction with a macrophase
The thermodynamic investigation was conducted on the model of a system consisting of n particles of phase α, particles of macrophase β, and mobile phase ε.The system has region I filled with phases α, β, and ε and region II filled with phase ε only.In the initial state of the system (Fig. 2, a) nanoparticles α and macroparticles β are separated by phase ε.In the final state (Fig. 2, b), a particle consolidation process is completed.As a result of the particle consolidation, interfacial surfaces α -ε and β -ε were replaced by contact surfaces α -α and α -β, respectively, and mobile phase ε was forced from region I to region II, and in region I a compact body was formed.
The following limitations are imposed on the system: where r α and r β are the particle sizes of phases α and β, respectively, indices α, β, ε, αα, αβ, αε, βε indicate that the values they define refer to the corresponding phase or interface.
In the system under study the sizes of particles of phases α and β do not change, but the geometry of them can be changed according to the consolidation conditions (limitation (11,c)).Limitation (11,c) allows us to consider an idealized system.In real systems, in the course of consolidation the sizes of nanoparticles change.First we consider the thermodynamics of the process when condition (11,c) is met and then when the process occurs under the conditions, in which the particle size is allowed to change.The particle consolidation process is conducted under isobaric and isothermal conditions.The characteristic function describing the process under the above conditions is the Gibbs energy (G).It should be noted that in the system under consideration particles of phase α have the nanophase properties and phase β particles have the properties of a macrophase.Because of this, to describe nanoparticles of phase α, we use the Hill method of thermodynamic study [2] and to describe macroparticles of phase β, the Gibbs method is used [4].
A variation of the Gibbs energy ΔG in the system transition from the initial into the final state may be shown as the sum of variations of the ΔG α value during the consolidation of nanoparticles α, ΔG β value caused by the replacement of interfaces α -β with β -α in particles β, and ΔG ε value describing the variation of the Gibbs energy of phase ε, ΔG = ΔG α + ΔG β + ΔG ε .
At the first stage we will define ΔG α , the variation of the Gibbs energy in the nanoparticle consolidation.In the initial state of the system the Gibbs energy of phase α is described by the expression , where the sign ′ indicates that the marked values refer to the initial state of the system, k is the number of the system components, n is the number of particles in the system.
Using relation ( 4) and taking into account that sn = S αε , we have Taking into account Eg. ( 1), the Gibbs energy of the system in the initial state of the nanoparticle α consolidation is defined by A change in the Gibbs energy in the particle consolidation may be represented in the following way: The last two terms of this expression allow for the variation of the Gibbs energy of phase ε.Taking into account that S αε =2g α S αα , where g α is the coefficient allowing for the variation of a particle geometry in the system passing to the final state, we have, For particle β in the initial state, Taking into account Eq. ( 1), we obtained In the final state the Gibbs energy of particle β is described by We assume that in Eqs. ( 12) and ( 13) S βε = g β S βα .Here g β takes into account the variation of geometry of particles β.
In the final form the relation describing the variation of the Gibbs energy of phase β becomes . The variation of the Gibbs energy of the system when passed from the initial to the final state (see the Fig. 2) is described by For simplicity of the presentation the final state of the system shown in Fig. 2 b is idealized.In the real systems, particles α in the initial state may contact each other and particle β and in the final state particles α may form dihedral angles.Under these conditions in the initial state there exist the S′ αα and S ′ αβ contact surfaces.As a result of the consolidation of particles α, the area of contact surfaces S′ αα and S′ αβ increases.With allowance made for the aforesaid, in Eq. ( 14) the S αα and S αβ magnitudes should be replaced by Δ S αα and Δ S αβ .
When a particular composition is considered, Eq. ( 14) may be simplified taking into account the peculiarities of the process under study.For instance, in the solid-phase sintering of nanoparticles produced from carbides, nitrides, or oxides of transition metals of the groups IV, V, and VI of the periodic table of elements and in the liquid-phase sintering of W, Mo, and Cr refractory metals in copper, silver, tin, aluminium oxide, etc. melts, the solubilities of nanoparticle components in a gas atmosphere and liquid phase are negligible.
In the system transition from the initial into the final state a change in the concentration of the ith component of phase ε may be neglected, then (μ ε ′ ~ μ ε ′′), and allowing for limitation (11,b) we have Taking into account that , we transform Eq. ( 15) to the form: The analysis of Eq. ( 16) shows that surface energies on the contact and interfacial surfaces play an important role in the nanoparticle interactions with each other and with phase β.To assess the effect of surface energies on the processes under study, we take the following assumptions: V α ′ = V α ′′ and V β ′ = V β ′′ = V β .In the real processes V α ′ ≠ V α ′′ and V β ′ ≠ V β ′′, therefore, the conclusions made should be applied with allowances made for the taken assumptions.If we take that in the initial state particles were of the spherical shape, then .With allowance made for these equalities Eq. ( 16) is transformed to It follows from Eq. ( 17) that in the system under study the following relations among the surface energies are possible: (3 + 4g α )γ αα < 10g α γ αε , (18,a) (3 + 4g α )γ αα > 10g α γ αε , (18,b) (9 + 2g β )γ αβ < 11g β γ βε , (18,c) (9 + 2g β )γ αβ > 11g β γ βε , (18,d) If inequality (18,a) is fulfilled, the consolidation of nanoparticles in the system under study occurs spontaneously, with inequality (18,b) the consolidation of nanoparticles does not occur, with inequality (18,c) contacts α -β form, i.e. nanoparticles intergrow with macroinclusion β, while with inequality (18,d) contacts α -β do not form and the inclusion is mechanically squeezed by the nanoparticles.If conditions (18,a) and (18,c) are met, a compact composite body is formed, and if the surface energy relations are subject to conditions (18,b) and (18,d), the nanoparticle consolidation process does not occur, but contacts α -β form, i.e. inclusion β becomes overgrown with nanoparticles.
The variation of the surface energy value, which is attained by introducing corresponding surface-active substances, is one of the methods to control the processes in the system under study.Surface-active substances, which decrease the γ αα and γ αβ values, facilitate the formation of contacts α -α and α -β and compact composite body, while surface-active substances that decrease the γ αε and γ βε values are undesirable because they can prevent contacts α -α and α -β from being formed.
Let us consider the consolidation of nanoparticles under the conditions, when the change of particle size takes place.Under these conditions a variation of the Gibbs energy ΔG in the system transition from the initial into the final state may be shown as the sum of variations of the ΔG, according to Eq.9 and Eq.14.Conclusions 1.A system consisting of a group of nanoparticles and a macrophase is unstable in terms of thermodynamics.In such a system there always are gradients of chemical potentials and diffusion flows of the substance from nanoparticles to the macrophase, which are induced by the chemical potentials.A system consisting of nanoparticles is stable under the conditions that exclude the exchange substances between nanoparticles, and between nanoparticles and the macrophase.2. The process of the consolidation in a three-phase system consisting of nanoparticles of phase α, macroparticles of phase β, and mobile phase ε has been studied.An equation for the variation of thermodynamic potential of this system in the consolidation process of particle α and β has been derived.The conditions have been established, under which (1) the particle consolidation proceeds to completion to form contact surfaces α -α and β -α, (2) the process proceeds partially to form only α -α or β -α contact surfaces, and (3) the consolidation does not occur.

Fig. 2 .
Fig. 2. System consisting of nanoparticles α, inclusion of macrophase β, dispersion medium ε; region I is filled by phases α, β, and ε, and region II is filled by phase ε; the system is in the initial (a) and in the final state (b).