Experimental sound velocities,
In this chapter, we study an interfacial phenomenon between liquid metals and ceramic substrates. Therefore, investigation of these phenomena is of great importance not only in technological applications but also in fundamental understanding of physical behavior of the adhesion between two different materials as far as their electrical structures and physiochemical properties are concerned. Moreover, adhesion energy is interpreted thermodynamically by the interfacial interactions and the nature of bonding between liquid metal and ceramic material. The adhesion energy in metal/ceramic systems is determined by using an electro-acoustical model based on the propagation of the acoustic wave in the interface and strongly depends on the electric properties of combination.
- Liquid metal
- Gap energy
- Acoustic parameters
Metalized ceramics by liquid metal have a crucial uses in several modern technological applications such as solar cell [1, 2, 3, 4] electrical devices [5, 6, 7] and Micro Electro Mechanical Systems (MEMS) [8, 9, 10]. Recently, these systems are used as the conductive wiring of microelectronic circuits; there has been considerable interest in the characterization of the structure and properties of liquid metal/ceramic interface .
However, the coating of ceramic surfaces can affect most of the properties of the interface. Therefore, the investigation of interfacial phenomena between metals and ceramic substrates is of great importance not only in technological applications but also in fundamental understanding of physical behavior of the adhesion between two different materials as far as their electrical structures and physiochemical properties are concerned. In fact, at the interface of a metal/ceramic system, adhesion occurs when the atoms or molecules of the two contacting surfaces approach each other so closely that attractive forces between approaching atoms (or molecules) bond them together. The strength of the bond depends on the size of the atoms, the distance between them, and the presence or absence of contaminant matter on the surface . Hence, the strength or wea1kness of bonds is the key factor to determine the interface stability: good adhesion, welded adhesion, perfect bonding, weak bonding smooth interface, etc. The metal/ceramic contact is characterized by the adhesion energy,
Adhesion energy represent in generally the sum of all interfacial interactions between two surfaces :
At the interface zone, the surface acoustic wave (SAW) propagation which depends on elastic properties of solid substrates is greatly affected: the response would be different depending on the weakness or strength of bonds due to impedance mismatching . Hence, in this context, we investigate the dependences of adhesion energy on acoustic parameters, in particular SAW velocities, for many metal/ceramic systems.
The objective of the electro-acoustic model  is the investigation of interfacial adhesion in liquid metal/ceramic systems subjected to non-reactive wetting in order to eliminate the non-equilibrium contribution Wnon-equil of adhesion work during a chemical reaction at the interface. A wide range of non-reactive liquid metals were used in this proposed model.
2. Choice of liquid metals
At the room temperature, most metals have a crystalline phase; the most widely used are iron, aluminum and copper. They are often present in oxide form (sodium oxide, magnesium oxide…), some metals are present in the non-oxidized state (precious metals: platinum, gold) or in the form of alloys. Metal alloys are in general the combination of two or more metals as in the case of brasses (alloys of copper and zinc); but they can also contain non-metallic elements (i.e. iron-carbon alloy). Metals and their alloys are usually very good conductors of heat and electricity; they are most often hard, rigid and plastically deformable. It should be noted that a large number of metals have a very high melting point, since they have relatively weak mechanical properties and are most often characterized by a wettability, a low thermal and electrical conductivity (as in the case of copper and gold). Therefore, the use of metals in metallized ceramic structures requires a fusion process in order to liquefy or melt these metals. For this, the role of metallization is to make the ceramic wettable by the liquid metal.
Several liquid metals parameters used in this investigation are listed in Table 1; sound velocities at melting temperatures are tabulated by Blairs , surface tension values are proposed by Keene , Liquid densities are taken by Crawley  and by Blairs . Whiles the elastic constants, solid densities and Rayleigh velocity are obtained from Briggs .
3. Relationship between the properties of metals in solid and liquid states
Analytical study has been proposed to express the relation between experimental sound velocities of liquid metals at the melting temperature,
Relationship between these parameters can be quantified by the following equation:
One can see also a clear tendency between the liquid metals densities,
The relationship that expresses this tendency can take the following form:
A close comparative between one of very important properties of liquid metal, which is the surface tension, σm, and Young’s moduli, E, values shows a linier dependence between these parameters, as can be seen in Figure 3.
To quantify the relationship between elastic moduli and surface tension, a simple plot was made; a linear correlation is defined, that it can be written as:
4. Determination of adhesion energy in liquid metal/ceramic systems
Very recently, an electro-acoustical model  has been proposed to estimate and interpreted the work of adhesion of non-reactive liquid metal/ceramic substrates systems in terms of the Rayleigh velocity of acoustic wave propagation in surface of all types of corresponding ceramic substrates, VRC.
In this model, several metals are considered (Au, Cu, Sn, Ga and Ag) on a great number of ceramic substrates (AlN, Al2O3, BN, CoO, Er2O3, Ho2O3, Lu2O3, MgO, NiO, SiC, SiO2, TiC, TiO, TiO2, Ti2O3, Y2O3, Yb2O3, ZnO and Zr2O3). The characteristics of all ceramic materials: energy gap,
The variations of work of adhesion on Rayleigh velocity for different ceramic substrate, VRC, in contacting with different non-reactive metals (Au, Cu, Sn, Ga and Ag) are investigated. In this study, some published data on wok of adhesion for different metals/ceramics systems are considered [12, 13, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].
In the first time liquid gold/ceramic combinations are taken the obtained results are illustrated in Figure 4.
In order to generalize the above observations obtained with liquid Gold/ceramic systems and to put into evidence the results reproducibility, several other nonreactive metals deposed in different ceramic substrates are considered, i.e., (Cu, Sn, Ga and Ag):
The obtained results are illustrated in Figure 5 in terms of work of adhesion as a function of ceramic Rayleigh velocities in contacting with several non-reactive metals. All the curves show the same behavior: the work of adhesion increases linearly with increasing VRC. However, two sets of linear dependences are distinguished that are regrouped according to the band gap energy of the ceramic substrate, as discussed below.
The dependence of Wad on VRC (Au) is quantified via curve fitting, (lines in Figures 4 and 5). We distinguish two parallel dependences for gold/ceramic substrate systems: for higher energy values (upper curve) the linear variation is found to be of the form:
Whereas, for small energy values (lower curve), the linear dependence is found to be of the form:
Moreover, it should be noted that the same behavior of two parallel lines is obtained for all metal/ceramic systems. Therefore, all curves have a same slop not only for small gap materials but also for large gap ceramics; the general expression takes the form:
where the subscript, (Me), represents any given investigated nonreactive liquid metal (Ag, Au, Cu, Ga and Sn), C and Ć are characteristic constants for each metal/ceramic combination.
The exact corresponding values of characteristic constants C (for small gap ceramic materials) and Ć (for large gap ceramic materials) of several liquid metal/ceramic systems are giving in the Table 3.
The similar dependence (with the same slope equal to 0.07VRC) is indicative of the existence of the same mechanism responsible for this behavior. However the existence of two parallel dependences for every system is due to the energy band structure of the ceramic materials in particular the energy gap (Table 1). A close analysis of Figure 5 and the Eg column clearly shows that the upper set of curves corresponds to solid ceramic materials with small energy gaps (Eg ≤ 3 eV), whereas the lower ensemble of curves represents ceramic materials with large energy gaps (Eg > 3 eV).
In fact, solid materials with small band gaps behave as conductors (Eg → 0) or semiconductors (Eg ≤ 3 eV). In this case, it was reported  that the high adhesion energy values of same metal/ceramic systems are associated with high electron density of metals and low band gap energy of solids ceramics. The interfacial adhesion between a metal and a ceramic crystal is assured by the electron transfer , it is interesting to define an interfacial propriety represents the minimum energy needed for appearance of a limit number of interfacial bonds responsible for generating of the adhesion between the metal and the ceramic, this energy is caused by Van der Waals interaction, WVDW. The intensity of the electron transfer at small band gap solid ceramic is increased because of its wealth by the free charges inside and the chemical equilibrium contribution Wchem-equil taking place.
For large band gaps, there will be practically a small number of free charges inside in the ceramic crystal. As a result, the chemical equilibrium contribution Wchem-equil, to the adhesion energy is negligible. Consequently, the adhesion energy is approximately resulted by from the Van der Waals interaction .
The Van der Waals contribution of adhesion energy rested constant and proportional with Rayleigh velocity of ceramic materials whether it is the band gap energy, for the first time it is determined exactly as follows:
The determinate WVDW energy values for different metal/ceramic systems depend directly on the choice of various parameters appearing in Eq. (3). For example, Mc Donald and Eberhart  calculated WVDW values equal to 500 ± 150 mJ/m2 for different metal/alumina systems, that in our model and for the same system we have found WVDW values equal to 396 mJ/m2. While Naidich  found WVDW values of 350 150 mJ/m2 for metal/oxide ceramic systems, this confirms the compatibility between our proposed model and other model of WVDW estimation.
For small gap ceramic materials, the characteristic constant C of Eq. (10) represents Wchem-equil contribution, this energy is relatively important compared to WVDW energy, it represents another interfacial property responsible for putting the stability and the perfection to the interface between metal and ceramic. The good convergence in Wchem-equil values for a given metal/small gap ceramics could be explained by the fact that for (Eg < 3 eV), here will be a big density of inside in the ceramic crystal and consequently height electron transfer.
In this work, an analytical approach  is adopted to express the relation between experimental sound velocities of liquid metals, c, at the melting temperature and determinate Rayleigh velocity of these metals at solid state, VRM, by SAM program. Hence, VRM is expressed in terms of c, as we recently reported .
The variations of chemical equilibrium energy on normalized Rayleigh velocity, (VRM/z) for different bulk metals in contacting with several small band gap ceramic materials are investigated, where z is number of coordination’s of each metal atom. In this investigation, we consider Eq. (10) to determine Wchem-equil and some published wok on adhesion energy for different metals/ceramics systems [12, 13, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. The obtained results are presented below.
The dependence of Wchem-equil on (VRM/z) is quantified via curve fitting, (line in Figure 6). We distinguish dependence for liquid metal/small band gap ceramic substrate systems: the linear variation is found to be of the form:
So, the chemical equilibrium contribution of adhesion energy in metal/ceramic system is related directly to the Rayleigh velocity of metals.
For large gap ceramic materials, the discrepancy in Ć values for a given metal/large gap ceramics could be explained by the fact that for (Eg > 3 eV), here will be a smaller density of inside in the ceramic crystal (practically no free charges inside) and consequently the electron transfer at metal/ceramic interfaces cannot be important . As a result, the characteristic constant Ć values are negligible compared to WVDW energy and/or especially to Wchem-equil energy.
Therefore, the general expression of adhesion energy takes the form:
a. For small gap ceramic materials:
b. For large gap ceramic materials:
The importance of the deuced relation lies in its applicability to all investigated metal/ceramic systems. It could be extended, through familiar relations, to other acoustic parameters. Similar results for longitudinal and transverse velocities were obtained. Moreover, preliminary results for elastic constants (Young’s modulus and shear modulus) are very satisfying.
In this work, an interfacial phenomenon between liquid metals and ceramic substrates has been investigated. Moreover, same liquid metal characteristics (sound velocity propagation in liquid metal, liquid density and surface tension) were predicted by the metal characteristics in solid state (Rayleigh velocity, solid density and Young’s modulus). Adhesion energy terms in metals/ceramic systems were determined by using an electro-acoustic model. It was shown that the adhesion energy increases linearly with Rayleigh velocity of ceramic substrates for all types of ceramics. Van der Waals term of adhesion energy was deduced only depends on Rayleigh velocities of ceramic. On the other hand, the chemical equilibrium term was deduced strongly depends on the energy gap of the ceramics materials: it was higher for small band gap ceramic materials and depends on Rayleigh velocities of metals, for the opposite case it was deduced negligible. These universal relations that could be extended to other acoustic parameters are applicable to all metal/ceramic combinations.