The mechanism of formation of surface micro- and nanostructures in the AlCoCrFeNi high-entropy alloy during electron-beam treatment формирования поверхностных микро- и наноструктур в высокоэнтропийном AlCoCrFeNi сплаве

The paper is devoted to a study of the formation of submicron and nanosized cellular crystallization structures on the surface of a high-entropy AlCoCrFeNi alloy irradiated by high current electron beams with the energy density varying from 10 to 30 J / cm 2 and a pulse time of 200 μs. The study revealed that the combination of thermal, evaporation-capillary and thermoelectric instabilities induces the formation of submicro- and nanodimensional cellular structures similarly to high-entropy alloys. The proposed dispersion equation was analyzed to detect the conditions for the generation of this instability. The importance of the evaporation process was investigated by finding a solution to the heat problem with phase transformations. The temperature distribution over time calculated at different distances from the surface of the alloy samples demonstrated that the surface temperature is lower than the evaporation temperature for the energy density E s < 30 J / cm 2 , therefore, the term of evaporation in the dispersion equation was ignored for these values of the energy density. The analysis of the dispersion equation showed that for E s = 30 J / cm 2 , the wavelength λ m with the maximal growth rate of perturbations on the melt surface gains a value in the submicro-and nano-range, provided that the thermoelectric coefficient equals to ~4 –10 V / K, and the pressure of evaporation is ~10 5 Pa. If we exclude thermoelectric effects, these values λ m are observed for the pressure of evaporation ~10 11 Pa. The wavelength λ m was revealed to decrease according to a power law as the beam energy density increases.


Introduction
In recent years, the effect of concentrated energy flows, e. g. low energy high current electron beams on high-entropy alloys with different chemical formulae, has been the subject of many research works [1][2][3][4][5]. These works have demonstrated that multiple irradiations of CrFeCoNiMo and CoCrFeNiMo 0.2 alloys by an electron beam with an energy density of 4 J / cm 2 and a pulse time of 1.5 μs induces the formation of a fusion zone. Moreover, its thickness expands with the increasing number of pulses [1,2]. The structure of this zone comprises crystals with an average size of 109 nm. Several authors [3 -5] revealed that highentropy alloys (HEAs) synthesized in selective electronbeam melting are characterized by high mechanical properties, which are possible due to the formation of micro-and nanodimensional structures and phases. One of the most probable mechanisms responsible for their forming is thought to be various hydrodynamic instabilities, e. g. the Mullins-Sekerka instability [6], thermocapillary instability [7,8] and the Kelvin-Helmholtz instability [9,10]. An assumption was made [11,12] that micro-and nanostructure phase states develop in multicomponent alloys owing to the evolving combination of thermal, concentration-evaporation and thermoelectric instabilities. This study investigates the effect of an electron beam with the energy density from 10 to 30 J / cm 2  from 225.83 to 618.87 nm was formed on the treated surface. The dimensional distribution of cells displays one maximum in the range from 300 to 350 nm.
The cross-section of the samples ( Fig. 1) was analyzed, and a conclusion was drawn that the modified layer is a product of the electron-beam treatment. The thickness of this layer depends on the height of the crystallization columns and increases with the growing energy density of an electron beam.

Results and Discussion
As discussed above, the formation of micro-and nanodimensional structure and phase states on the surface during electron-beam treatment is induced by the combination of thermal-capillary, concentration-capillary and thermoelectric instabilities. We assume that this instability arises when an electron beam is applied to high-entropy alloys. The significant role of thermoelectric convection for the formation of a cellular crystallization structure is traced in the thickness of the molten layer ~1-20 μm. As stated in previous studies [13,14], thermoelectric effects fail to be neutralized totally by the thermocapillary convection if a concentrated flow of energy is supplied from above. When determining the thermoelectric coefficient, we proceed from the assumption that the convective flow, intensifying the thermoelectric effect, is of significant importance for the transfer of charge in the liquid state [12].
The values of specific heat of fusion and evaporation in the first approximation are determined according to the mixture rule: , where a i -volume percentage of an i-component in the alloy, x i -specific heat of fusion (specific heat of evaporation) for an i-component of the alloy. The temperature gradient is found as in [12]: G 0 = ((q − q out )/κ), where q = E s / t 0 -energy density of an electron beam, E s -surface energy density оf an electron beam, q out -evaporation-conditioned surface energy density, κ -thermal conductivity of a liquid metal. The role of evaporation for the power density [15] is calculated as follows: The role of evaporation in the formation of nanostructures was calculated according to the previously proposed heat model [16], which considers this phase transformation. Fig. 2 provides the data on the temperature distribution over time at various depths from the irradiated surface at 30 J / cm 2 .
It can be apparent from the data in this figure that the temperature on the irradiated surface is equal to the temperature of evaporation for this value of the energy density, whereas no evaporation is observed for any other values of the energy density below 30 J / cm 2 . Therefore, for E s < 30 J / cm 2 we consider ω p = 0.
Further, it is necessary to find the minimum wavelength, which initiates the combined instability. For this purpose, the solution (3) is determined in z i C 2 1 r  Z As a result, certain corrections for the frequency of capillary waves are obtained, which depend on thermal, evaporation-capillary and thermoelectric effects.
Turning to (4) from z to ω, it is obtained: Thermophysical constant variables and parameters of the electron beam were substituted, and it was concluded that the instability arises at λ > 203 μm for E s = 30 J / cm 2 and without evaporation thermoelectric effects. The numerical solution of Eq. (3) shows that the thermocapillary instability arises at λ > 210 μm. Given the evaporation pressure equals 10 5 Pa, the instability begins with a wavelength of λ = 43 μm; numerical calculations confirm this. The instability for wavelengths ranging from 220 to 300 nm, which are equated to the sizes of crystallization cells under the experimental study, was initiated at an evaporation pressure of 4.5 •10 11 Pa and 2 •10 11 Pa.
The numerical solution of the dispersion Eq. (3) showed that the maximum value of the perturbation growth rate under these conditions falls on the wavelengths of 380 and 530 nm (Fig. 3 a, curve 1 and 2). Considering thermoelectric effects, the wavelength attributed to the origination of instability is 225 nm (Fig. 3 a, curve 3) for an evaporation pressure of 10 5 Pa and a thermoelectric coefficient of 4.3 V / K; this outcome is in line with experimental data.
The maximum growth rate is recorded for a wavelength of 430 nm; it differs slightly from the SEM data of the cellular structure. As mentioned above, the presented results are obtained in the low-frequency approximation by finding a solution to Eq. (3). We turn to the discussion of numerical solutions to full dispersion Eq. (1). This equation is a cumbersome algebraic equation of the 16 th degree with respect to z 1 , so we omit it here. The study considers only such roots of Eq. (1), which meet the condition Re(ω) > 0 and Re(z 1 ) > 0, Re(z 2 ) > 0. As numerical solution shows, the equation has two roots. In both cases, the instability arises at λ > 220 nm, and the maximum growth rate is registered for a wavelength of 430 nm (Fig. 3 b, curve 1); this is in line with the experimental data and solutions of Eq. (3).
This fact allows one to conclude that the low-frequency approximation is adequate (3). If evaporation is not taken into account, the wavelength at which the maximum growth rate is observed is the same as at an evaporative pressure of ~10 11 Pa. This allows us to conclude that the thermoelectric instability prevails over the evaporative-capillary instability in this range of vapour recoil pressures. The most noticeable effect of evaporation becomes at an evaporative pressure of ~10 10 Pa (Fig. 3 b, curve 2). The maximum growth rate, in this case, will be observed at a value of λ m = 450 nm. Fig. 4 demonstrates the correlation between the wavelength with the maximum growth rate and the energy density varying from 10 to 30 J / cm 2 . This figure shows that the growing energy density E s brings about a decrease in the wavelength according to the power-law λ m = 583.16 · E s −2.125 (correlation coefficient 0.999); therefore, the most likely expected sizes of the cellular crystallization structure will be reduced. This correlation allows optimizing the processing conditions of high-entropy alloys by low energy high current electron beams.  Still, it necessitates the research into the role of the concentration of alloying elements for the surface tension of the alloy under study and the determination of the diffusion coefficients in these alloys.

Conclusion
The study establishes that thermoelectric phenomena significantly affect the initial stage of thermocapillary flow instability in a high-entropy alloy melt irradiated by electron beams. The critical outcome of the study is that evaporationcapillary instability is possible only for the energy density of 30 J / cm 2 and higher. The correlations between the growth rate of perturbations at the plasma-melt boundary and the wavelengths display one maximum only, which is recorded in the submicro-and nanometer range for the thermoelectric coefficient γ ≥ 4 V / K. Due to the obtained results, the mechanism responsible for forming surface micro-and nanostructures represents the combination of thermal, evaporation-capillary and thermoelectric instabilities. They can be relevant for determining the optimal conditions of electron-beam processing high-entropy alloys to form micro-and nanostructures. The further development of the model can be related to the consideration of concentrationcapillary effects.