On the Application of Laplace Pressure in the Science of Sintering

An equation of the Laplace pressure derived using the Gibbs thermodynamic method have been discussed and the correct applications of the equation have been substantiated. It has been shown that the expression is applicable only to macrovolumes for the description of surfaces with a constant curvature, but not to the description of nanodisperced systems and surfaces with variable curvature. The expression of the Laplace pressure applicable to a crystal and the cavity limited by a surface of any geometrical shape have been derived.


Introduction
The Laplace pressure value is defined as the product of the surface tension by interfacial area curvature.In spite of the very simple expression, its interpretation and application in the literature is ambiguous.In this expression in parallel with the surface tension the free surface energy is used [1,2].It is applied to nano-and macrodispersed systems to describe the moving forces of the particle consolidation [2 -4].In this case the notion of the "effective Laplace pressure" is introduced, which requires a clear physical interpretation.Because of this the necessity has arisen to describe the derivation of the equation of the Laplace pressure based on the philosophy of thermodynamics and taking into account the results obtained to substantiate the area of its correct application.

Thermodynamic study
To solve this problem, have been used a method of virtual changes of the system parameters when the system is in equilibrium state.Let us consider a system consisting of phases 1 and 2 and phase 2 is inside phase 1 and is restricted by arbitrary closed surface S 12 .Phases 1 and 2 may be solid, liquid, or gaseous and set up various combinations on the condition that there is interfacial surface S 12 .The following limitations are imposed on the system: where T is the temperature, ϑ is the entropy, V is the volume, m is the amount of the component, indices 1, 2, and 12 indicate that the values they define refer to the corresponding phase or interface.
To simplify the description it is assumed that the system is one-component.Limitation (1) indicates that there are no temperature gradients in the system.Limitations (2) and (3) are responsible for the conditions of the system interaction with the environment.The constancy of the entropy points to the thermal insulation of the system, while the constancy of the volume points to the mechanical insulation.Limitation (4) indicates that the system does not exchange mass with the environment.Under these conditions the thermodynamic potential of the system is internal energy U.For the system under consideration we have where P is the pressure, μ is the chemical potential, γ is the surface tension, S is the interfacial area; indices v and s show that the values they indicate belong to the volumetric and surface phases, respectively.
In the state of equilibrium the system internal energy is the lowest and under any virtual variations in the region near the equilibrium is zero, i.e.
For the system considered it is assumed that in the equilibrium state the chemical potentials of phases and interfacial area S 12 are the same, i.e.
It follows from Eq. ( 9) with allowance made for limitation (1) that for macrophases under virtual changes of interfacial area δS 12 pressures P 1 = const, P 2 = const, and γ 12 is also const.Only under these conditions and limitation (1) equality (9) is true.With allowance made for the above Eq.( 8) becomes The first three terms of Eq. (10) define the condition of the system thermal equilibrium, the next three define the condition of the mechanical equilibrium, and the last term define the condition of the system chemical equilibrium.These conditions are independent and can be written as follows: Limitation ( 2) and the constant temperatures in the bulk of the system guarantee the implementation of condition (11a), limitations (4) and (9) ensure the implementation of condition (11c).Let us transform condition (11b) in the following way.According to limitation (3), δV 1 = -δV 2 , therefore, from (11b) we found

S
, where K is the surface curvature, R 1 and R 2 are the main radii of the curvature at the given point.Taking into account this dependence we have Eq. ( 12) is a known dependence of the Laplace pressure on the surface tension and interfacial surface curvature.It may be derived for a multicomponent system as well.In this case an additional limitation should be imposed, namely, the system should consist of independent components.The Eq. ( 12) is true only if the limitations and assumption used in its derivation are allowed for.

Results and discussion
According to our thermodynamic study, the surface tension is used in Eq. ( 12) and not the free surface energy.In deriving Eq. ( 12) the Gibbs thermodynamic method was used, which was developed for macroobjects, therefore, the expression Eq. ( 12) is inapplicable to the nanosized objects.In the thermodynamics of nanosystems a new independent parameter appears, namely, nanoparticle size r [5], on which the chemical potential, surface tension, and thermodynamic functions depend.Because of this limitation (9) cannot be extended to nanodispersed systems, it is true for macroobjects only, hence, the applications of Eq. ( 12) is restricted to macrosystems only.
There is a contradiction in the expression of the Laplace pressure.Eq. ( 12) describes the pressure at a point of a curved surface, which contradicts the physical essence of the mere notion of the pressure, which is an integral characteristic and has the meaning only as applied to a certain surface area or to the bulk of the body.Because of this Eq.( 12) may correctly be used only to describe a surface with constant curvature K, i.e. spherical, cylindrical, or plane.To apply expression (12) correctly, it is necessary for presenting as follows , where K = ( 0, 1, 2).For a plane surface the Laplace pressure equals zero, that's why sometimes it is interpreted as an excessive pressure that is generated by a curved surface.Based on the above, the use of Eq. ( 12) for arbitrary curved surface (K ≠ const) is incorrect.Laplace pressure is the excessive pressure of the phase created by the closed surface, and for its writing it is necessary to use a symbol L P Δ .
Let's define the Laplace pressure in bodies of various geometrical shapes (fig.1).

A B
The cavities are filled by vapor.A cavity (fig.1a) is limited by a spherical surface.The spherical surface has a constant curvature, therefore, the pressure created by this surface can be defined by Eq. ( 12).According to Eq. ( 12), the Laplace pressure in the A area is A L P Δ = 0, and in the B area is 1b).Such a situation is impossible in a cavity filled by vapor.It is obvious that Eq.( 12) is unsuitable for determination of the Laplace pressure in a cavity (fig.1b) and in a crystal (fig.1c).To solve this problem, we use the results reported in [6].The author [6] has proved that pressure of crystal ,P r , is described by the following expression where Р m is the medium pressure, γ i, is the surface tension of face s i of a crystal, l i is the height of a pyramid constructed using face s i as the basis.
, where v is the crystal volume, s A is the mechanical work of the formation of a whole crystal surface, s A = i i s γ Σ .For a practical use it is necessary to find the average value of a surface tension , where s is the surface of a crystal.Taking into account these expressions, we have Eq. ( 13) can be applicable for closed cavities of any geometrical shape and for crystals.For the description of an equilibrium state of dispersed systems it is necessary to use Eq. ( 13), which is devoid of the contradictions inherent in the Eq. ( 12).
In the literature [2,3,4,7] the Laplace pressure is used to describe the motive force of the particle consolidation process.On the strength of the above contradiction in using Eq.(12) to describe the pressure in a cavity, formed by several particles, depending on a geometrical shape at different regions of the cavity various combinations of the Laplace pressure can be obtained.In order to apply the Laplace pressure for cavities of an arbitrary geometrical shape, the author [4, p. 39] introduce the so-called "effective Laplace pressure", which is described by expression where F s is the free surface energy, V is the volume, σ is the specific free surface energy, r 0 is the average radius of powder particles, Θ is the porosity.
During the particle consolidation the Helmholtz free energy decreases, therefore,

.
It seems likely that Eq. (14) should be presented as follows: