Thermodynamic Interpretation of a Dihedral Angle in Composite Materials

The equilibrium between solid and 1iquid phases in sintered composite materials has been studied. It is shown that closed surfaces, which bound dispersed phases, influence the mechanical equilibrium between these phases. An expression is derived for a dihedral angle in composite materials, which includes values of surface tensions at the phase interfaces as well as parameters of a composite equilibrium structure (phase composition, particle contiguity and coefficients of a particle geometry).


Introduction
Liquid phase-sintered composite materials, which consist of high-melting particles and a metal melt, exhibit highly developed solid-solid and solid-liquid interfaces.To evaluate the effect of surface energies on the material structure development, the values of dihedral angles formed by the intersection of the solid-solid interface and two solid-liquid interfaces are used [1][2][3][4][5].Based on the mechanical equilibrium of forces that generate in these surfaces, Smith [1] suggested the dihedral angle as follows: , 2 2 cos sl ss γ γ ϕ = (1) where γ ss and γ sl are the surface tensions at the solid-solid and solid-liquid interfaces, respectively.Expression (1) is valid for isotropic unbounded solid and liquid phases.
In composite materials, high-melting particles are a dispersed phase, which is bounded with closed surfaces.The closed bounding surfaces are known to have an essentia1 effect on the conditions of phase equilibrium [6,7].
In connection with this the question arises of the validity of the applicability of expression (1) to composite materials.The aim of the present work is to derive an expression for the dihedral angle formed by particles in composite materials.

Theoretical model
Let us assume that in a system (Fig. l) solid and liquid phases are in thermodynamic equilibrium under the conditions: , const sl ss l s where η is the entropy, V is the volume, m is the mass, i is the number of components, and s, l, ss, sl indices show that the as-indexed quantities belong to solid, liquid phases and solidsolid, solid-liquid interfaces, respectively.Condition (2a) suggests system entropy constancy, condition (2b) shows the volume constancy and condition (2c) demonstrates that processes related with the disappearance or appearance of new components do not occur and the system does not exchange mass with the environment.According to these conditions, the internal energy is a thermodynamic potential of the system.For the analyzed system, the internal energy U is given by where T is the temperature, P is the pressure, µ is the chemical potential, A is the phase interface area, V and A indices show that the as-indexed quantities belong to bulk phases or interfaces, respectively.In equilibrium, the internal energy of the system is minimum, thus for any virtual changes of energy, the work of the system equals zero, i.e. δU=T s δη s +T l δη l + T ss δη ss +T sl δη sl -P s δV s -P l δV l +γ ss δA ss +γ sl δA sl + The first four terms of the expression describe the condition of system thermal equilibrium, the next four terms describe the condition of phase mechanical equilibrium, and the last four terms describe the condition of chemical equilibrium.All these conditions are independent of each other, therefore we can write as follows: T s δη s + T l δη l + T ss δη ss + T sl δη sl = 0, (3a) -P s δV s -P l δV l + γ ss δA ss + γ sl δA sl = 0, Limitation (2a) and temperature constancy in the system volume ensures the fulfillment of condition (3a).Using (2c) we find from (3c) that the equality of chemical potentials of the components in all phases ensures chemical equilibrium in the system.
We rearrange condition (3b) taking into account that according to (2b) δV s = -δV l , and based on geometrical positions of A ss and A sl surfaces (Fig. 1), it follows that δA sl = -2δA ss cosϕ/2.
In [6,7] it was proved that for phases, which are bounded with closed surfaces, P s ≠ P l .Shcherbakov [6] has shown that the pressure of a dispersed phase is given by , 3 where a is the surface area of a particle, v is the particle volume and γ is the particle faceaveraged surface tension.
The γ value is given by , a a a sl sl ss ss where a=a ss +a sl .
For spherical particles, expression ( 5) is transformed into the Laplace equation.Contrary to the Laplace equation, expression (5) is applicable for particles of any shape.
Fig. 2 Model of the solid-liquid system: 1-particle, 2-cavity filled with liquid, 3-unit cell.Thus, in expression (4), when applied to composite materials: (P s -P l )δV l ≠ 0. It should be noted that any changes in the liquid content of a composite (∆V l ) would cause changes in the contact interface area of particles (∆A ss ).Let us derive this relationship.To do this, we consider a solid-liquid dispersed system, which consists of n s -number of equidimensional particles forming a skeleton as well as of uniformly arranged cavities filled with liquid (Fig. 2).The particles and cavities form a regular structure, which may be divided into nnumber of equal cells in such a manner that each cell would accommodate one cavity filled with liquid v l and some dihedral angles.The number of cavities filled with liquid relates to the number of particles in the system through the following relation: n = kn s , where k is the coefficient.The volume of the 1iquid phase in each cell is v l =V l /n.The liquid content of a cell and its interfacial area a sl can be written as v l =k v R 3 , a sl =k a R 2 , where R is the size of a cavity filled with liquid, k v and k a are the coefficients which allow for different cell geometries.When adding liquid of volume (∆v l ) into the cavity, the latter changes its volume and area ∆a sl =2k a R∆R+R 2 ∆k a , (8) In the above expressions, the second terms account for changes of the cavity geometry.If one assumes that when adding liquid of volume ∆v l , the cavity geometry does not change, then from ( 7) and ( 8) we can write Then, by multiplying the left-hand and right-hand sides of the above expression into n-number of cells we have for the whole system The cavity size can be written as is valid, where v is the particle volume, k s is the factor accounting for the particle geometry, and r is the particle size.In view of the last expression, we have , 1 where u is the volume content of the liquid phase in the system given in fractions.
In the above expression, it is worthwhile to introduce the specific surface of particles instead of r, according to equality v a k r 1 1 = . By substituting (10) in ( 9) and replacing ∆A ss by ∆A sl according to relationship, ∆A sl = -2 ∆A ss cosϕ/2, we have , 1 In view of ( 5) and (11), condition (4) takes the form: .
In sintered composite bodies high-melting particles can form a spatial configuration with equilibrium dihedral angles only in the case of a specific amount of the liquid phase u ϕ [3,8].If the liquid phase content of a composite body is u<u ϕ , then the body volume (V=V s +V l ) will be inadequate to construct a high-melting skeleton, in which partic1es form equilibrium dihedral angles.Thus, u ϕ is the minimum of the liquid phase content such that the particles form equilibrium dihedral angles in the composition.Thus, u ϕ is the constant for a given composition and the ratio (1-u ϕ ) 1/3 /u ϕ 1/3 = λ is its certain parameter λ.In expression (12), the liquid phase content u can be replaced by u ϕ and taken outside the variation sign.
Having made this transformation we find from (12) that , 0

Results and discussion
When deriving expressions (13) and ( 14) we assumed that particles of the dispersed phase are isotropic.Because of this expressions (13) and ( 14) are applicable to phases in which anisotropy of surface properties is poorly pronounced.To allow for the anisotropy of particle surface properties, the surface tension of particles is calculated from γ =Σa j γ j /Σa j where j is the number of faces on a particle.
Analysis of (1) and (13) shows that in dispersed systems particles form equilibrium dihedral angles whose values are higher than those of dihedral angles formed by unbounded phases.It should be also noted that expression (14) includes parameters of the composition structure when it is in a equilibrium state, namely contiguity, volume of the liquid phase u ϕ and, indirectly, in terms of k s, k v , k a , coefficients of the geometry of particles and their coordination number.

Conclusion
Closed surfaces, which bound dispersed phases, have a pronounced effect on mechanical equilibrium of solid and liquid phases.This effect is manifested in higher values of equilibrium dihedral angles formed by particles in dispersed systems as compared with dihedral angles formed by unbounded phases.
In composite materials the value of a dihedral angle is defined by surface tension at the solid-solid and solid-liquid interfaces as well as by parameters of the structure of the composition (phase composition, particle geometry and particle contiguity), which are featured by the composition as the system reaches the equilibrium state.

Fig. 1
Fig. 1 Schematic diagram showing the equilibrium state of solid and liquid phases at their point of contact.
As v l = V l /n and n = kn s , we have 3 .