The interphase energy between a solid and melts has a significant effect on the transport of matter across the electrodeelectrolyte phases boundary. In metallurgy, the dependence of wetting on the droplet size obtained by the electrolysis of melts of liquid metals, the destruction of the lining of electrolysis baths, the appearance of an anode effect, etc. is known.
However, despite the significant role played by the interfacial energy at the solidliquid interface (melt) in various fields of science and technology, so far there is no direct method for measuring this quantity. Therefore, unfortunately, we have to resort to semiempirical methods for its determination. Moreover, it is seldom in the literature that in the same work a complex measurement of all the necessary quantities is carried out simultaneously to find the interfacial energy at the solidmelt (liquid) boundary of another substance.
To write this article, we were able to find in the literature the necessary quantities for calculating the interfacial energy of solid nickel at the boundary with the chlorides of alkali and alkalineearth metal chlorides in various environments [7].
Method of calculations
In the present paper we used thermodynamical formulas obtained by us earlier to determine the temperature coefficient of the nickel surface energy Δσ_{SV}/ΔT [4]:
σ_{SL} = σ_{LV} (2–3cosθ + cos^{3}θ)/4)^{1/3}; (1)
σ_{SV} = σ_{LV}[((2–3cosθ + cos^{3}θ)/4)^{1/3}+ cosθ]. (2)
In the formulas (1) and (2) Δσ_{SL} – the interphase energy of the solidbody boundary – is the intrinsic melt, σ_{LV} is surface energy of the vapor melt, σ_{SV} is surface energy of the solid, is the proper vapor, and θ is the contact angle formed by the proper melt on the surface of the solid.
From the known values of σ_{LV} and θ, formulas (1) and (2) make it possible to determine σ_{SV} at the melting temperature of the metal.
First, from the literature we find one experimental value of σ_{SV} of nickel, measured at a temperature different from the melting point, then we calculate the σ_{SV} of nickel by the formula (2). The difference between the surface energy calculated and measured by the values of Δσ_{SV}, related to the temperature difference at which the surface energy of nickel is determined, made it possible to determine Δσ_{SV}/ΔT.
Due to the fact that the nickel wetting angles with alkali and alkalineearth metals were measured at temperatures much lower than the melting point of nickel, the surface energy of solid nickel was recalculated by us for the temperature at which θ1 was measured.
As it is mentioned above, in the work [4] we developed a thermodynamic theory and obtained formulas that enabled us to calculate the surface energy of a solid body with its vapor σ_{SV} and the interfacial energy of a solid – its own melt σ_{SL}. from the known values of the surface energy of the melt, σ_{LV}, and the wetting angle θ at the equilibrium temperature of a onecomponent threephase system.
In [2], a table of relative values of σ_{SV}/σ_{LV} and σ_{SL}/σ_{LV} for all possible changes in the angle θ from 0 ° to 180 ° was compiled.
Direct measurement of the edge angles of solid metals by intrinsic melts with high melting points is a difficult task. With this circumstance, apparently, the absence in the literature of the experimental data θ1 is connected. Earlier, we calculated the edge angles of solid metals with our own melts, including for nickel, which turned out to be equal to θ1 = 21 ° [3]. It was found that the calculated values of σSL for singlecomponent systems correspond to the criteria for wetting lowenergy surfaces by lowtemperature liquids under equilibrium conditions. As is well known, the interphase energy in equilibrium systems depends on the difference and structure of the contacting phases and decreases when their properties approach. For example, for two phases of different polarity, the interphase energy will be the lower the lower the polarity difference of these phases (P.A. Rebinder’s polarity rule).
Eremenko V.N. expressed the principle according to which similar wetting is similar, the meaning of which, in the opinion of some researchers, is that the interphase energy between such substances is low (V.N. Eremenko’s principle of similarity).
Indeed, studies, including ours, show that for the onecomponent systems and for dissimilar lowenergy surfaces, when they are wetted with lowtemperature liquids, the “likesosolike” principle is valid.
For the contact of highenergy surfaces with lowtemperature liquids, good wetting does not guarantee a strong decrease in the interfacial energy. At the same time, in systems where simple contact is the only source of interfacial energy reduction, the establishment of a connection between a solid and a wetting liquid can be disturbed by the presence of contamination films that interfere with perfect contact. Such a metalfilmmelt system remains stable, since there is no tendency to transition and to the exchange of atoms between phases, since their chemical potentials are equal.
A different situation is observed in interacting nonequilibrium systems. For such systems, in addition to a purely contact process, the interfacial energy can decrease as a result of the interfacial interactionthe diffusion of one of the components of the wetting liquid from the surface layer into the volume, the dissolution of the metal in the liquid, etc., i.e. from the flow of irreversible processes of chemical interaction at the interphase boundary [5].
The authors, whose values of the boundary angles are used by us, for their calculations note that the measured wetting angles on nickel in the chlorine atmosphere may not correspond to the thermodynamic equilibrium state of their surface, since during the experiment a continuous process of their oxidation with chlorine occurred to form chlorides that did not accumulate on the their surfaces, but dissolved in salt melts, diffusing from the surface layer to the volume. In calculations such angles were used by us, considering them as contact angles, as is customary in terminology.
As the surface energy of molten nickel at the crystallization temperature, when calculating the interfacial energy and the surface energy of solid nickel at the melting temperature σ_{SV}, we used the averaged value σ_{LV} = 1770 mJ/m2 taken from [6]. Substituting the last value of σ_{LV} together with the boundary angle given above, using the indicated table, we calculate σ_{SL} and σ_{SV} of nickel at the melting point.
σ_{SL}(Ni) = 261 mJ/m2, σ_{SV} = 1914 mJ/ m2.
The value of σ_{SL} of nickel obtained by us is in agreement with the experimental data given in the review paper [8], found by the method of supercooling small drops.
The calculated σ_{SV} of nickel also agrees with the experimental value of σ_{SV}, measured by the zerocreep compensation method [1]. The value of the surface energy of solid nickel σ_{SV}(Ni) = 1940 mJ/m2 obtained by the authors at a temperature T = 1670 K, which is below the melting point by 58 K.
From the two found values of σ_{SV} of nickel, we calculate the temperature coefficient of the surface energy of solid nickel
Δσ_{SV}/ΔT = (19141940)/(17281670) = = 26/58 = – 0.4483 mJ/m2. (3)
In the literature, instead of Δσ/ΔТ, dσ/dТ is usually written, which, in our opinion, is not correct.
The values of the temperature coefficients are small, but finite. Their recording in a differential form a priori means that they are infinitesimal. Therefore, the differentiation of the surface energy with respect to temperature must be done by replacing the dots of the differential d, the Δ (delta) and the final result in terms of Δ.
In a onecomponent condensed system, as a rule, the temperature coefficient should be negative, i.e. increasing the temperature reduces the surface energy.
In double and multicomponent liquids and solids positive, negative and even zero temperature coefficients (socalled temperature buffering) are possible.
Table shows the angles of wetting of solid nickel by molten salts and their interfacial characteristics at different temperatures.
From Table it follows that as the temperature increases, σ_{LV} and WA of all compositions of salt melts decrease linearly. The σ_{SV} and σ_{SL} are also decreased in a given temperature range. Salt melts well wet the nickel, but a sharp decrease in the interfacial energy is not observed.
In the absence of a similarity between the wettable solid and the wetting liquid, good wetting is possible only under the condition σ_{SV} > σ_{LV}. In this case, θ < π/2, and in the case σ_{SV} < σ_{LV}, θ > π/2.
In calculating the phasetophase characteristics, we used the surface energy values of the molten chlorides of alkali and alkalineearth metals, measured by the maximum pressure method in a gas bubble (argon or chlorine) [7]. The authors presented the results of the temperature dependence of sLV as an empirical equation with the help of which we gave the values of σ_{LV} to the temperatures at which the angles of salt melts on solid nickel were measured. The edge angles presented in Table are rounded to the whole. The calculated values of interphase characteristics are also rounded up to integers of measure unit.
Calculations of the adhesion of salt melts to nickel were carried out by two formulas
WA = σ_{LV} + σ_{SV} – σ_{SL}, (4)
WA = σ_{LV}×(1 + cos θ), (5)
for selftesting and to avoid possible errors
It should be noted that the ratio (4), called the Dupre equation,expresses the fact that the change in the free energy of the system during its transition from one state to another as a result of the reversible isothermal process is equal to the difference in the free energy of the system in these two states. It directly follows that the increase in the binding energy of a liquid and a solid causes a decrease in the interfacial energy between the solid and the liguid. In particularlу, the latter position is confirmed by our calculations.
On the same basis described above, the energy release during the reaction between the solid and the liquid can be equated with the work of adhesion. This position was also taken into account in the calculations of interfacial energy in systems in which the contact angles were measured under the atmosphere of chlorine.
The angle of wetting of solid nickel by molten salts and their interphase energies at different temperatures
Molten salt 
Environment 
T, K 
σ_{lv}, mJ/m2 
θ, ° 
σ_{sv}, mJ/m2 
σ_{sl}, mJ/m2 
WA, mJ/m2 
LiCl 
Ar 
932 
138 
12 
2271 
2136 
273 
953 
136 
11 
2261 
2128 
269 

983 
134 
10 
2248 
2116 
266 

1022 
130 
10 
2230 
2102 
258 

Cl2 
929 
135 
10 
2272 
2139 
268 

968 
131 
9 
2255 
2126 
260 

LiClKCl (0,580,42) 
Ar 
704 
139 
10 
2373 
2236 
276 
752 
135 
9 
2352 
2219 
268 

805 
130 
9 
2328 
2200 
258 

849 
127 
8 
2308 
2182 
253 

926 
120 
7 
2274 
2155 
239 

1031 
112 
7 
2226 
2115 
223 

Cl2 
703 
136 
10 
2374 
2240 
270 

773 
130 
9 
2342 
2214 
258 

863 
122 
8 
2302 
2181 
243 

1013 
109 
7 
2234 
2126 
217 

LiClKCl (0,620,38) 
Ar 
798 
162 
13 
2331 
2173 
320 
861 
157 
14 
2303 
2151 
309 

921 
153 
12 
2276 
2126 
303 

972 
149 
13 
2253 
2108 
294 

1036 
145 
– 
2224 
– 
– 

Cl2 
800 
159 
13 
2330 
2175 
314 

853 
156 
12 
2306 
2153 
309 

1041 
142 
11 
2222 
2083 
281 
As an example, we will perform a procedure for calculating the surface energy of solid nickel at a temperature at which the contact angle formed by the molten salt of LiCl on the surface of solid nickel is measured.
According to the data of the authors of [7], this angle at 932 K is equal to θ = 12 °. Using formula (1) and the value of σ_{SV} of nickel at the melting temperature found above, we obtain
σ_{SV} = 1914 + 0, 4483 (1728 – 932) = 2271 mJ/m2 (6)
(recall that with decreasing temperature, σ_{SV} increases). Similarly, at a temperature of 932 K, it is necessary to bring the surface energy of the LiCl melt. For this purpose, we use the empirical equation obtained by the authors of [7].
σ_{LV} = σ_{0} – σ_{T}, (7)
where σ_{0} = 215 mJ/m2, β = 0, 0829 mJ/ (m^{2}×K) in argon and σ_{0} = 212 mJ/m2, β = 0,0836 mJ/(m^{2}×K) in the atmosphere of chlorine.
Substituting these values in (7), respectively, we obtain:
σ_{LV} = 138; 135 mJ/m2.
Now we find the interfacial energy at the interface of the LiCl – nickel melt.
σ_{SL} = σ_{SV} – σ_{LV}×cos θ = 2271 – 138 ×0.9781 = 2136 mJ/m2in argon.
Similarly, we find σ_{SL} in a chlorine atmosphere
σ_{SL} = σ_{SV} – σ_{LV}×cos θ = 2272 – 135 ×0,9848 = 2139 mJ/m2.
Calculations for other salt systems are carried out according to the same scheme.
In conclusion, we note that the calculation method presented here is applicable to the evaluation of interphase characteristics and other solidliquid systems.
Conclusions
1. The interphase energy σ_{SL} and the surface energy of solid nickel σ_{SV} at the melting point are determined.
2. The surface energy of solid nickel is reduced to the temperatures at which the edge angles of salt melts on nickel are measured.
3. Interfacial energies and work of adhesion of molten chlorides of alkali and alkalineearth metals to nickel were calculated.
4. It is shown that, in the absence of similarity between the wettable solid and the wetting liquid, a low value of the contact angle is not a prerequisite for a sharp decrease in the interfacial energy at the solidmelt interface of another substance.