II. évfolyam 3. szám 2001. július
Volume 2  N^{o }3  July 2001
[HUN]  Magyar cikk
[ENG]  English article

Prediction of Wettability between
Liquid Metals and Covalent Ceramics
L.Zoltai
LIMOS R&D, Department of Physical Chemistry,
Faculty of Materials and Metallurgical Engineering, University of Miskolc,
3515 Hungary, Miskolc, Egyetemvaros
Abstract
Contact angle values have been estimated by the model developed by us earlier for 48 * 6 = 288 liquid metal / covalent ceramic
systems. 130 systems can be taken as ‘nonreactive’, i.e. the calculated value in this paper is expected to be an equilibrium
value, being independent of contact time. In 158 systems some chemical reaction or mutual solubility in the system is
expected, and so the calculated values will be valid in reality only during short contact
times. The calculated contact angle values vary within a relatively small
interval, ranging from 143^{o} for the Li/B_{4}C system to 170^{o} for the
Pd/b
SiC system. For the given liquid metal, the contact angle increases gradually in the following
row: B_{4}C à
Si_{3}N_{4} à
b
BN à
AlN à
a
BN à
b
SiC. The range of contact angle for one given metal decreases with the increasing average value of the contact
angle. For Li, with a highest polarising effect the contact angle varies in the interval of 10^{o}, while for Pd this interval is only 4^{o}. The effect of temperature has been found negligible in all systems
(changing in the interval between +1^{o} and –3^{o} while the temperature changes by 100 K). It has been
concluded, that due to high surface tension of liquid metals, in absence of chemical interaction and mutual solubility between them and covalent
ceramics, liquid metals never wet the surface of covalent ceramics better than 143^{o}.
Keywords:
wettability, contact angle, adhesion energy, model calculation, liquid
metals, B_{4}C,Si_{3}N_{4}, b
BN, AlN, a
BN, b
SiC. 
1. Introduction
Metals and ceramics interact with each other during innumerable engineering problems and manufacturing processes such as foundry
processes, composite production etc., thus the knowledge of adhesion energy and contact angle between these phases can be very
useful. A relatively large experimental databank on the contact angle between liquid metals and ceramics is available now in the literature [15]. In addition to
that, in recent years our research group have completed a large number of experiments on determination of the contact angle in different
metal/ceramic systems, including measurements on a covalently bonded Si_{3}N_{4} ceramic [67]. Based on literature data and our own
measurements, a complete adhesion theory was developed by us on the liquid metal – covalent ceramic systems [8].
The goal of the present paper is to apply the model developed in [8] to calculate the contact angle values in 48 * 6 = 288 liquid
metal/covalent ceramic systems. It should be noted, however, that some of the possible combinations are ‘reactive
systems’, meaning that chemical reaction or mutual solubility can take place between the
phases. Therefore, in reactive systems our calculated results will be only ‘initial’
values, corresponding to short contact times between the phases (see [2, 9]). In ‘nonreactive’ systems our calculated results will refer to the equilibrium contact angle
values, being valid independent of the duration of contact between the two
phases.
2. Short summary of the model [8]
The model is based on the LondonvanderWaals
equation, as between covalent ceramics and liquid metals no other interaction can be
envisioned. For the interaction of a pure metal with a multicomponent ceramics A_{x}B_{y}, summation should be performed over the tow components of the
ceramics. The final equation for the adhesion energy between the liquid metal and nonreactive covalent ceramic phase was derived
as:
(1)
where W  adhesion energy (J/m^{2}),
a
 polarizability (m^{3}),
I  first ionisation potential
(J/mol),
R  radius of atom (m),
indexes Me, C and i refer to liquid metal, ceramics and to the i^{th} component of the
ceramic, respectively,
c_{Mec}  the number of bonds per unit interface area
(mol/m^{2}):
(2)
where N_{Av}  is the Avogadro number (6.02 10^{23}mol^{1}),
V_{m}  is the molar volume (m^{3}/mol),
f  is the correction factor, being the function of the volume and surface occupancy
factors.
The contact angle can be calculated from the YoungDupré
equation:
(3)
where s
_{Melg} is the surface tension of the liquid metal (J/m^{2}).
3. Selection of Physical Constants [8]
In order to apply Eq.(1) for the estimation of the adhesion energy in the liquid metal – covalent ceramic
systems, first parameters used in Eqs. (13) should be defined. The molar volume of liquid metals is taken from the compilation [10]. The molar volume of ceramics is taken from [21]. The packing factor f_{Me} for liquid metals is taken equal 1.06 [20]. For the hexagonal and cubic ceramics f_{c} is estimated as 1.09.
The polarizability of the atom at the surface of the liquid metal was estimated as the 1/6^{th} part of the polarizability of the gaseous atom:
(4)
Data of polarizability of gaseous atoms are taken from [1314]. As follows from [13], the polarizability of covalent gaseous molecules can be approximately calculated additively from their atomic
polarizabilities. Therefore Eq.(4) will also be applied to the components of the covalent
ceramic.
The ionization potential of a metallic atom in the surface of the liquid metal can be approximately calculated
as:
(5)
where D
_{v}H_{Me} is the vaporization enthalpy of the liquid metal at its melting point
(T_{m}) [17],
R = 8.314 J/molK – the gas constant,
I_{Me,g} – the first ionization energies of the gas atoms are taken from [12],
coefficient 5/6 refers to the fact that the cohesion energy of the surface atom is
approximately the 5/6^{th} part of the cohesion energy of the bulk atom.
The ionization potential of different components of the ceramic should be estimated in different
ways, depending on, whether they are solid or gaseous as a pure phase under standard
conditions. As an example, in case of the Si_{3}N_{4} ceramics, silicon is
solid, while nitrogen is twoatomic gas in standard state. Therefore, for the two components of Si_{3}N_{4} the ionization energies should be estimated in slightly different
ways:
(6.a)
(6.b)
where D
_{s}H_{Si} is the sublimation enthalpy of Si [17],
D
_{f}H_{Si3N4} is the enthalpy of formation of Si_{3}N_{4} [17],
D
_{d}H_{N2} is the enthalpy of dissociation of 1 mol of N_{2}(g) into 2 moles of
N(g) [17].
The radii for liquid metals are taken from the table for atomic
(metallic) radii (coordination: 12) from [18]. The radii of the components of the ceramic is estimated from the smallest distance between the components in the
compound, taking into account their radii in pure state. As an example, the smallest distance between Si and N in the Si_{3}N_{4} compound is 1.71 10^{10} m [11]. Using the metallic radius of Si (1.34 10^{10} m [18]) and orbital radius of gaseous nitrogen (0.521 10^{10} m [18]), the radii of the components of the ceramics were taken
as: R_{Si} = 1.23 10^{10} m, R_{N} = 0.478 10^{10} m.
In Tables 12 the initial physical data are summarised for the metals and
ceramics, respectively.
Table 1. Physical parameters of metals needed for calculations
Me 
Tm [
17]

Vm at Tm[
10]

a
[
10]

f [
20]

a(gas) [
13]

I(gas) [
12]

D_{v}H at Tm[
17]

R[
18]

s_{lg} at Tm[
20]

dslg/
dT[
20]

K 
e6, m3/mol 
10^{4 }K^{1} 

e30, m3 
kJ/mol 
kJ/mol 
e10, m 
mJ/m2 
mJ/m2K 
Li 
453.69 
13.4 
1.9 
1.06 
24.3 
513.3 
155.3 
1.55 
470 
0.2 
Na 
370.98 
24.8 
2.54 
1.06 
23.6 
495.8 
104.1 
1.89 
190 
0.13 
K 
336.35 
47.1 
2.9 
1.06 
43.4 
418.8 
86.3 
2.36 
100 
0.08 
Rb 
312.65 
57.7 
3 
1.06 
47.3 
403 
78.6 
2.48 
86 
0.07 
Cs 
301.55 
72.2 
3.1 
1.06 
59.6 
375.7 
74.4 
2.67 
69 
0.06 
Be 
1560 
5.33 
0.71 
1.06 
5.6 
899.4 
309.1 
1.13 
1300 
0.31 
Mg 
922 
15.3 
1.6 
1.06 
10.6 
737.7 
132.6 
1.6 
590 
0.19 
Ca 
1112 
29.5 
1.6 
1.06 
23.9 
589.7 
159.1 
1.97 
370 
0.12 
Sr 
1050 
37 
1.1 
1.06 
27.6 
549.5 
146.3 
2.15 
300 
0.11 
Ba 
1002 
41.4 
0.81 
1.06 
39.7 
502.8 
160.9 
2.21 
280 
0.11 
Al 
933.45 
11.3 
1.5 
1.06 
6.8 
577.4 
314.2 
1.43 
950 
0.26 
Ga 
302.8 
11.4 
0.92 
1.06 
8.12 
578.8 
266.2 
1.39 
715 
0.088 
In 
429.76 
16.3 
0.97 
1.06 
9.6 
558.3 
238.2 
1.66 
565 
0.09 
Tl 
577 
18 
1.15 
1.06 
7.55 
589.3 
174.2 
1.71 
460 
0.1 
Si 
1685 
11.1 
1.4 
1.06 
5.38 
786.5 
393.2 
1.34 
830 
0.084 
Ge 
1210.4 
13.2 
0.89 
1.06 
6.07 
762.1 
339.1 
1.39 
610 
0.11 
Sn 
505.06 
17 
0.87 
1.06 
7.7 
708.2 
292.7 
1.58 
550 
0.076 
Pb 
600.6 
19.42 
1.24 
1.06 
6.8 
715.9 
188.1 
1.75 
455 
0.085 
Sb 
904 
18.8 
1.3 
1.06 
6.6 
833.6 
240.6 
1.61 
380 
0.07 
Bi 
544.52 
20.8 
1.17 
1.06 
7.4 
703.3 
196.6 
1.82 
385 
0.077 
Cu 
1358 
7.94 
1 
1.06 
6.7 
745.4 
316.9 
1.28 
1350 
0.3 
Ag 
1233.95 
11.6 
0.98 
1.06 
7.88 
731 
265.8 
1.44 
920 
0.24 
Au 
1337.58 
11.3 
0.835 
1.06 
6.14 
890.1 
348.2 
1.44 
1140 
0.23 
Zn 
692.65 
9.94 
1.5 
1.06 
6.3 
906 
120.5 
1.36 
820 
0.27 
Cd 
594 
14 
1.5 
1.06 
7.2 
867.6 
103.6 
1.56 
630 
0.2 
Hg 
234.29 
14.65 
1.77 
1.06 
5.4 
1007 
61.83 
1.6 
500 
0.19 
La 
1193 
23.3 
0.4 
1.06 
31.1 
538.1 
419 
1.87 
750 
0.1 
Ti 
1939 
11.6 
0.56 
1.06 
14.6 
658 
441.2 
1.46 
1600 
0.35 
Zr 
2125 
15.4 
0.54 
1.06 
17.9 
660 
579.6 
1.6 
1500 
0.29 
Hf 
2500 
14.9 
0.53 
1.06 
16.2 
642 
575.7 
1.59 
1500 
0.24 
V 
2175 
9.5 
0.6 
1.06 
12.4 
650 
457.5 
1.34 
1900 
0.29 
Nb 
2740 
11.9 
0.51 
1.06 
15.7 
664 
685.2 
1.45 
2100 
0.21 
Ta 
3287 
12.1 
0.46 
1.06 
13.1 
761 
748.3 
1.46 
2200 
0.19 
Cr 
2130 
8.27 
1.1 
1.06 
11.6 
652.7 
353.6 
1.27 
1700 
0.35 
Mo 
2897 
10.3 
0.53 
1.06 
12.8 
685 
585.9 
1.39 
2300 
0.32 
W 
3680 
10.5 
0.45 
1.06 
11.1 
770 
808.9 
1.4 
2500 
0.25 
Mn 
1517 
9.54 
1.6 
1.06 
9.4 
717.4 
246.7 
1.3 
1200 
0.38 
Re 
3453 
9.96 
0.44 
1.06 
9.7 
760 
706.7 
1.37 
2800 
0.3 
Fe 
1809 
7.94 
1.3 
1.06 
8.4 
759.3 
378 
1.26 
1900 
0.42 
Ru 
2523 
9.27 
 
1.06 
9.6 
711 
614.5 
1.33 
2300 
0.31 
Os 
3300 
9.46 
 
1.06 
8.5 
840 
747.5 
1.35 
2500 
0.25 
Co 
1768 
7.6 
1.4 
1.06 
7.5 
760 
396.6 
1.25 
1900 
0.4 
Rh 
2233 
9.27 
 
1.06 
8.6 
720 
517.4 
1.34 
2000 
0.34 
Ir 
2716 
9.61 
 
1.06 
7.6 
880 
625.7 
1.36 
2300 
0.32 
Ni 
1726 
7.43 
1.51 
1.06 
6.8 
736.7 
399.9 
1.24 
1800 
0.44 
Pd 
1825 
10.14 
1.17 
1.06 
4.8 
805 
348.5 
1.37 
1500 
0.26 
Pt 
2045 
10.31 
1.52 
1.06 
6.5 
870 
533.3 
1.38 
1750 
0.36 
Table 2. Physical parameters of ceramics needed for calculations
Ceramics 
Si_{3}N_{4} 
b
SiC 
B_{4}C 
AlN 
a
BN 
b
BN 
Structure 
hex 
cub 
trig 
hex 
hex 
cub 
Vm, 293 K [
21]

e6, m3/mol 
44 
12.5 
21.95 
12.53 
10.84 
7.2 
b [
21]

e6, 1/K 
8.25 
14.1 
13.5 
15 
3.3 
 
a
(gas, A) [
13]

e30,m3 
5.38 
5.38 
3.03 
6.8 
3.03 
3.03 
a
(gas, B) [
13]

e30,m3 
1.1 
1.76 
1.76 
1.1 
1.1 
1.1 
Dist(AB) [
11]

e10, m 
1.715 
2.2 
1.63 
1.885 
1.58 
1.567 
R(A, met) [
18]

e10, m 
1.34 
1.34 
0.91 
1.43 
0.91 
0.91 
R(B, orb) [
18]

e10, m 
0.521 
0.62 
0.62 
0.521 
0.521 
0.521 
I(A, gas) [
12]

kJ/mol 
786.5 
786.5 
800.6 
577.4 
800.6 
800.6 
I(B, gas) [
12]

kJ/mol 
1402.3 
1086.2 
1086.2 
1402.3 
1402.3 
1402.3 
DsH_{A}298K [
17]

kJ/mol 
450 
450 
560 
329.7 
560 
560 
DfH(AxBy)298K [
17]

kJ/mol 
744.8 
73.22 
71.13 
317.98 
254.39 
254.39 
DdH(B2)298K [
17]

kJ/mol 
945.4 
0 
0 
945.4 
945.4 
945.4 
Results of calculations
The results of calculations are summarized in Table 3 for 48 * 6 = 288 systems. In the first column the symbol of the metallic element, while in following six double columns the calculated values are given for the 6 different ceramics. In each double column first the calculated contact angle is given. After that, the category of this calculated result is marked by ‘eq.’ or ‘init.’. These two categories have been defined in this paper as follows:

‘eq.’: meaning that the calculated value is an equilibrium contact angle, taking place independent of the contact time in the ceramic/liquid metal system; this category was chosen, when the liquid metal has no chemical interaction (with negative standard change of Gibbs energy of reaction) with any of the components of the metal [17], and also they are not mutually soluble (or only to a smallest extent) [15, 16]. There are 130 ‘equilibrium’ values given in Table 3.

‘init.’: meaning the calculated value is just the initial contact angle, being valid within short contact times between the ceramic and the liquid metal; this category was chosen when chemical interaction [17] or significant mutual solubility [1516] between the phases is expected, and thus the contact angle is expected to decrease with contact time. The equilibrium contact angle cannot be calculated by the equations presented in this paper. There are 158 ‘initial’ values given in Table 3.
As one can see from Table 3., the contact angle values vary within a relatively small interval, ranging from 143^{o} for the Li/B_{4}C system to 170^{o} for the Pd/b
SiC system. For the given liquid metal, the contact angle increases gradually in the following row: B_{4}C à
Si_{3}N_{4} à
b
BN à
AlN à
a
BN à
b
SiC. The range of contact angle for one given metal decreases with the increasing average value of the contact angle. For Li, with a highest polarising effect the contact angle varies in the interval of 10^{o}, while for Pd this interval is only 4^{o}.
Finally we can conclude, that due to high surface tension of liquid metals, in absence of chemical interaction and mutual solubility between them and covalent ceramics, liquid metals never wet the surface of covalent ceramics better than 143^{o}. This conclusion is in perfect agreement with our measured data [67], and also with the large number of experimental data obtained by other groups [15]. Although, it should be mentioned that there is a significant number of papers showing contact angle as low as 120^{o} in nonreactive liquid metal / covalent ceramic phases. We presume that those experimental data probably refer to ceramics or liquid metals of oxidised surface. As oxides have ionic character, the new type of interaction, the ioninduceddipole interaction becomes possible, ensuring higher adhesion energy and lower contact angle in the system (see [19, 9]).
Conclusions
Contact angle values have been estimated by the model developed by us earlier for nonreactive liquid metal / covalent ceramic systems. Among the 288 systems studied, 130 systems can be taken as ‘nonreactive’, i.e. the calculated value in this paper is expected to be an equilibrium value, being independent of contact time. In 158 systems some chemical reaction or mutual solubility in the system is expected, and so the calculated values will be valid in reality only during short contact times.
The calculated contact angle values vary within a relatively small interval, ranging from 143^{o} for the Li/B_{4}C system to 170^{o} for the Pd/b
SiC system. For the given liquid metal, the contact angle increases gradually in the following row: B_{4}C à
Si_{3}N_{4} à
b
BN à
AlN à
a
BN à
b
SiC. The range of contact angle for one given metal decreases with the increasing average value of the contact angle. For Li, with a highest polarising effect the contact angle varies in the interval of 10^{o}, while for Pd this interval is only 4^{o}.
The effect of temperature has been found negligible in all systems (changing in the interval between +1^{o} and –3^{o} while the temperature changes by 100 K).
Finally it is concluded, that due to high surface tension of liquid metals, in absence of chemical interaction and mutual solubility between them and covalent ceramics, liquid metals never wet the surface of covalent ceramics better than 143^{o}.
Table 3. Results of calculations (for the meaning of ‘eq.’ and ‘init’ see the text above)
Me 
Si_{3}N_{4} 
b
SiC 
B_{4}C 
AlN 
a
BN 
b
BN 
Q

Type 
Q

Type 
Q

Type 
Q

Type 
Q

Type 
Q

Type 
Li 
145 
Init. 
153 
init. 
143 
eq. 
148 
init. 
148 
eq. 
146 
eq. 
Na 
149 
eq. 
155 
eq. 
146 
eq. 
151 
eq. 
152 
eq. 
149 
eq. 
K 
150 
eq. 
155 
eq. 
146 
eq. 
152 
eq. 
153 
eq. 
151 
eq. 
Rb 
152 
eq. 
156 
eq. 
148 
eq. 
153 
eq. 
154 
eq. 
152 
eq. 
Cs 
152 
eq. 
155 
eq. 
148 
eq. 
153 
eq. 
154 
eq. 
152 
eq. 
Be 
153 
Init. 
161 
init. 
154 
eq. 
156 
init. 
156 
eq. 
154 
eq. 
Mg 
159 
Init. 
164 
init. 
158 
eq. 
161 
init. 
161 
eq. 
160 
eq. 
Ca 
159 
Init. 
163 
init. 
157 
eq. 
160 
init. 
161 
eq. 
159 
eq. 
Sr 
161 
Init. 
164 
init. 
158 
eq. 
162 
init. 
162 
eq. 
161 
eq. 
Ba 
158 
Init. 
162 
init. 
156 
eq. 
159 
init. 
160 
eq. 
158 
eq. 
Al 
163 
Init. 
167 
init. 
162 
eq. 
164 
init. 
165 
init. 
163 
init. 
Ga 
157 
Init. 
163 
eq. 
157 
eq. 
160 
init. 
160 
init. 
158 
init. 
In 
162 
eq. 
166 
eq. 
161 
eq. 
164 
eq. 
164 
eq. 
163 
eq. 
Tl 
164 
eq. 
167 
eq. 
163 
eq. 
165 
eq. 
166 
eq. 
164 
eq. 
Si 
159 
Init. 
165 
init. 
159 
init. 
162 
init. 
162 
init. 
160 
init. 
Ge 
157 
init. 
163 
init. 
157 
eq. 
160 
init. 
160 
eq. 
158 
eq. 
Sn 
161 
eq. 
165 
eq. 
160 
eq. 
163 
eq. 
163 
eq. 
161 
eq. 
Pb 
165 
eq. 
168 
eq. 
164 
eq. 
166 
eq. 
166 
eq. 
165 
eq. 
Sb 
159 
eq. 
164 
eq. 
158 
eq. 
161 
init. 
161 
eq. 
160 
eq. 
Bi 
165 
eq. 
168 
eq. 
163 
eq. 
166 
eq. 
166 
eq. 
165 
eq. 
Cu 
160 
Init. 
165 
init. 
160 
init. 
162 
init. 
162 
init. 
161 
init. 
Ag 
161 
Init. 
165 
init. 
160 
eq. 
163 
init. 
163 
eq. 
161 
eq. 
Au 
164 
Init. 
168 
init. 
163 
eq. 
165 
init. 
165 
eq. 
164 
eq. 
Zn 
159 
eq. 
164 
eq. 
158 
eq. 
161 
init 
161 
eq. 
159 
eq. 
Cd 
162 
eq. 
166 
eq. 
161 
eq. 
163 
eq. 
163 
eq. 
162 
eq. 
Hg 
163 
eq. 
167 
eq. 
162 
eq. 
164 
eq. 
164 
eq. 
163 
eq. 
La 
159 
eq. 
163 
init. 
157 
init. 
160 
init. 
161 
eq. 
159 
eq. 
Ti 
160 
Init. 
165 
init. 
160 
init. 
162 
init. 
162 
init. 
161 
init. 
Zr 
162 
Init. 
166 
init. 
161 
init. 
163 
init. 
163 
init. 
162 
init. 
Hf 
162 
Init. 
166 
init. 
162 
init. 
164 
init. 
164 
init. 
163 
init. 
V 
159 
Init. 
165 
init. 
159 
init. 
162 
init. 
162 
init. 
160 
init. 
Nb 
161 
Init. 
166 
init. 
161 
init. 
163 
init. 
163 
init. 
161 
init. 
Ta 
163 
Init. 
167 
init. 
163 
init. 
165 
init. 
165 
init. 
163 
init. 
Cr 
157 
Init. 
163 
init. 
157 
init. 
159 
init. 
159 
init. 
158 
init. 
Mo 
162 
Init. 
167 
init. 
162 
init. 
164 
init. 
164 
init. 
162 
init. 
W 
163 
eq. 
168 
init. 
163 
init. 
165 
init. 
165 
init. 
164 
init. 
Mn 
157 
Init. 
163 
init. 
157 
init. 
159 
init. 
159 
init. 
158 
init. 
Re 
165 
Init. 
169 
init. 
165 
init. 
166 
eq. 
166 
init. 
165 
init. 
Fe 
160 
Init. 
165 
init. 
160 
init. 
162 
init. 
162 
init. 
161 
init. 
Ru 
162 
Init. 
167 
init. 
162 
init. 
164 
eq. 
164 
init. 
163 
init. 
Os 
164 
Init. 
168 
init. 
164 
eq. 
166 
eq. 
166 
eq. 
164 
eq. 
Co 
161 
Init. 
166 
init. 
161 
init. 
163 
init. 
163 
init. 
161 
init. 
Rh 
163 
eq. 
167 
eq. 
163 
eq. 
164 
eq. 
165 
eq. 
163 
eq. 
Ir 
164 
eq. 
169 
eq. 
165 
eq. 
166 
eq. 
166 
eq. 
165 
eq. 
Ni 
161 
Init. 
166 
init. 
161 
init. 
163 
init. 
163 
init. 
161 
init. 
Pd 
166 
Init. 
170 
init. 
166 
init. 
168 
init. 
168 
init. 
167 
init. 
Pt 
164 
Init. 
169 
init. 
164 
init. 
166 
init. 
166 
init. 
165 
init. 
References

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pp)

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