The paper reports on an undulating topography initiating on the interface “coating/base material” under heterogeneous plasma flows to be generated by an explosion of yttrium powder on the aluminum-silicon base. We assumed that an undulating topography on the interface resulted from a combination of Rayleigh–Taylor and Kelvin–Helmholtz instabilities. A flow of an incompressible viscous two-dimensional fluid was considered in the field of bulk forces. The first layer made up of titanium or silumin is thought to be static, and the second one is accelerated perpendicular to the base material plane. A range of transversal velocities in the second layer was determined. Navier–Stocks equation and boundary conditions were stated for each layer. In a system Ti-Y Rayleigh–Taylor instability dominates at a transversal velocity of below 10m/s, changing into Kelvin–Helmholtz instability at velocities above 10m/s. In a system Al-Si-Y Kelvin–Helmholtz is dominant at velocities above 50m/s due to lower density of the base material in comparison with titanium and a high acceleration of yttrium powder. The study highlights importance of the transversal velocity in yttrium layer for reasoning of undulating pattern formation on the interface “coating/base material” and distribution of yttrium particles in depth of the modified layer.

To date, hard and wear resistant composite coatings have been widely used for wear protection of products surfaces [1]. A number of methods have been developed so far to deposit coatings onto surfaces of essential elements, e.g. electric arc and electro-slag facing [2,3], heterogeneous plasma flows generated by electric explosion of conductors [4], methods of chemical and physic deposition from a gaseous phase [5,6] and sol-gel method [7]. Coatings deposited using heterogeneous plasma flows generated by electric explosion of conductors take precedence over other methods since this technology makes possible a relatively quick (∼100μs) production of wear resistant coatings. The detachment of a coating from the base material cannot be ignored in operation. This defect might be caused by mechanical stresses arising at a contact load on the interface coating/base material due to the discrepancy between their elasticity moduli [8,9]. The surface of the interface is one of key factors [10,11], to cause redistribution of stress concentrators, keeping stable functional properties of a coating without long stripes of localized plasticity in the matrix. To form a developed surface topography of the interface coating/base material we need data on treatment conditions by heterogeneous plasma flows, which further its formation. Therefore, research focused on mechanisms of forming surface topography in conditions above is of high relevance. Experimental studies [12,13] suggest this topography has an undulating pattern similar to one registered when analyzing Rayleigh–Taylor instability. Its essence is that the interface of two media is instable provided that a medium with a higher density is accelerated to the normal of the surface with a low density. In line with this theory researchers [14,15] applied a hydrodynamic approach to modeling of accelerated coating and base material by incompressible viscous fluids. As a result, they stated and analyzed a dispersion equation for low harmonic disturbances of the interface [14]. The assessment of this equation found a wavelength with the maximal increment ∼10μm in systems Mo-Cu, Cu-Mo [14] and ∼1μm in systems Ti-Zr and Ti-Nb [15]. The latter study demonstrated a disagreement with experimental data, and authors [15] suggested the only maximum of increment vs. wavelength to be a result of a viscous-potential approximation and assumption of ultimately thick layers, whereas a second maximum in the long waves range might be registered in experimental conditions. Another study argued that an undulating topography of the interface is caused by Kelvin–Helmholtz instability of two viscous fluids [16]. In accordance with mutual penetration of two fluids [17,18] a dispersion equation was stated in [16], and its analysis showed a critical wavelength to be ∼10μm, being proved by experimental data. To sum up, we assume formation of an undulating topography of the interface coating/base material – is a consequence of Rayleigh–Taylor and Kelvin–Helmholtz instabilities combined together. Therefore, this study aims at exploring the combination of these two instabilities via stating and analyzing a dispersion equation for short-wave disturbances. A combination of Rayleigh–Taylor and Kelvin–Helmholtz instabilities was discussed in Refs. [19–23] for the case of large-scaled vortexes formation in the magnetopause of the Earth's magnetosphere. These studies suggest that shear flows do not have a clear impact on Rayleigh–Taylor instability. A linear stability theory [19,20] points at an increasing growth speed of Rayleigh–Taylor instability for any velocity of shear flow of layers, furthermore, this augmentation is monotonous. In the framework of this theory studies [21,22] highlight that a shear flow in a medium with higher density along a medium with a lower density causes a drop of its growth speed. Outcomes of numerical analysis also prove this fact [23]. Studies [19–23] report on data with relation to gases and plasma. On the interface “fluid/fluid”, “gas/fluid” there is a more difficult situation because of phase-to-phase surface tension, which displaces instabilities of the interface into the long-wave range, being also a key factor in stabilizing the interface surface.

2Problem statementFig. 1 gives an electron-microscopic image of surface layers in titanium [24] and aluminum [25] alloys treated in electric explosion of yttrium powder.

Electron-microscopic images of surface layers treated by a heterogeneous plasma flow [24,25]: (a) Ti-Y; (b) Al-Si-Y.

From the data in Fig. 1 it is apparent that a system Ti-Y comprises three layers with different morphology and dimensions of structural elements (Fig. 1a). The surface layer has the coarsest structure (I in Fig. 1a); a layer adjacent to the layer of thermal transformation (III in Fig. 1a) is more dispersive. The surface topography of the interface between layers II and III has a developed wave-shaped character, which results probably from a combination of Rayleigh and Kelvin–Helmholtz instabilities. Alongside with alteration of the surface topography this instability also causes a non-uniform distribution of alloying elements. Data obtained from X-ray micro-spectral analysis [26] show that yttrium is distributed in the least uniform way, with its concentration varying 36–2.4wt.%. In a system Al-Si-Y the interface between layers I and II (Fig. 1b) is also undulating, here yttrium pattern is also non-uniform [27]. The concentration of yttrium in layer I is 2.50wt.%, and in layers II and III – 0.14 and 0.39wt.%, respectively.

We use a method of finite elements to examine a combination of instabilities. We consider stability in a plane steady-state flow of incompressible two-dimensional fluid in the field of bulk forces. The direction of x-axis is selected along the interface of the layers, y-axis is perpendicular to x and directed towards the second layer (Fig. 2). The first layer (−l<x<l, −h1<y<η(x,t)) is made up of a fluid with a viscosity ν1 and density ρ1. The second layer (−l<x<l, η(x,t)<y<h2) is a fluid with a viscosity ν2 and density ρ2, moving at a constant velocity U0 to be directed along x-axis, with acceleration a, directed along y-axis.

Navier–Stocks equations are stated for layers (Eq. (1)):

where u→ – velocity vector, p– pressure, a→ – acceleration, ρ – density, μ – dynamic viscosity, n=1, 2 – layers. A system (1) was solved numerically using the method of finite elements. The evolution of surface interface was examined with the help of Phase field method [28,29]. Cahn–Hilliard equations are applied to the study on the dynamics of two-phase flow. This method estimates a scalar function φ in the entire area to be calculated (Eq. (2)):where u→ – velocity vector of a fluid, χ – fluidity parameter, w – energy density of the mixture, ɛ – parameter to determine the thickness of an intermediate layer (half of a cell size). A fluidity parameter was accepted χ=1mskg−1 in numerical computations. A density of mixture energy and thickness of the intermediate layer are associated with a surface tension coefficient by the relation (Eq. (3)):Start conditions were set as follows. As stated above, a horizontal velocity component for the upper layer of a melt (Fig. 2) is U0, and a vertical component is written as a periodic disturbance along y axis with an amplitude V0 (Eq. (4)):

Boundary conditions are presented in Table 1. In experiments to assess a disturbance the amplitude of velocity was assumed to be 1m/s. Material characteristics and parameters of external impact are listed in Table 2.

Physical properties of a melt and impact.

Designation | Description | Value | ||
---|---|---|---|---|

Base material | Coating material | |||

Ti | Al-Si | Y | ||

ρ | Density | 4120kg/m3 | 2700kg/m3 | 4470kg/m3 |

μ | Dynamic viscosity factor of metal melt | 3.71×10−3Pas | 1.2×10−3Pas | 1.83×10−3Pas |

σ0 | Phase-to-phase surface tension coefficient | 0.63N/m | 0.31N/m | – |

U0 | Horizontal velocity component of yttrium flow | – | – | 0m/s,5m/s,10m/s,50m/s |

ay | Acceleration | – | – | −6×109m/s2 |

A shear velocity is thought to be a key parameter to describe the formation of nanostructures within the model based on Kelvin–Helmholtz instability. To estimate a shear velocity on the contact interface we calculate a dynamic pressure to be developed by plasma jet on the surface of a base material (Fig. 3). Experimentally determined maximal dynamic pressure PD of a flow correlates well with physical data on the flow structure. An equation results from the mechanism stating a plasma jet impact on the surface (Eq. (5)):

where indexes Pl, G – are for plasma and gas parameters, respectively; ρ0 – air density at atmospheric pressure; γ – air adiabatic curve; vCB – velocity of a contact boundary. Start conditions of the experiment: ρ0=1.29kg/m3, γ=1.2; vCB=7000m/s. As determined [30] a radial distribution of pressure is stated as follows (Eq. (6)):We find a steady-state flow of a viscous fluid in area 1 (Fig. 3) using Navier–Stocks equation stated as follows (Eq. (7)):

A pressure is set on the surface of a layer (Eq. (6)). A no-slip condition is set on a boundary η (Eq. (8)):

For the purpose of calculations based on the finite element method we used a package COMSOL Multiphysics. Fig. 4 provides outcomes of calculations. Technological characteristics of units to generate heterogeneous plasma flows depended on the battery capacity (C, F), voltage of a charge (U0, kV) and weight of a foil to be exploded (m, mg). It is clear from Fig. 4 that a shear velocity on the inner layer edge ranges to 53m/s for a system Ti-Y and 65m/s for a system Al-Si-Y (Fig. 4).

Therefore, a horizontal velocity component of yttrium flow is to be in a range 0–55m/s to estimate the flow of materials.

3Results and discussionFirst, we consider formation of an undulating topography on the interface “modified layer/base material” for the case of yttrium flow onto titanium base. Fig. 5 demonstrates the distribution of yttrium and titanium densities in different points of time for a horizontal component of flow velocity 0m/s. As shown in Fig. 5 in a time point of 0.6μs the interface is “mushroom-shaped” (Fig. 5a), confirming the development of Rayleigh–Taylor instability. In points 1.0–1.4μs this disturbance tends to grow, afterwards a “mushroom leg” fractures (Fig. 5b–d). This destruction is caused by the initiation and development of Kelvin–Helmholtz instability to arise due to tangential discontinuity of a vertical velocity component on the edge of a “mushroom leg”.

In points of time t>1.4μs we registered intense remixing of yttrium and titanium (Fig. 5e and f), furthermore, vortexes penetrate at a depth of ≈80μm, that is the reason for yttrium being at depths exceeding a diffusion one. The flow of molten materials is different provided that a transversal velocity is taken into consideration. Fig. 4 shows a density pattern of a molten substance at a transversal velocity of U0=5m/s. This figure demonstrates that in a point of time 0.8μs the interface surface is also “mushroom-shaped”, however, it is slightly displaced from the vertical axis (Fig. 6a). At t=1μs a leg of the “mushroom” is undulating as in the case of no transversal velocity (Fig. 6b), splitting then into “drops” (Fig. 6c and d). These “drops” are subject to Rayleigh–Taylor instability, which causes their refinement, as a consequence, layer III (Fig. 1a) is more dispersive than layers I and II. In points t>1.4μs, as in the case above, processes of remixing dominate, but a share of areas with a density ρ>4300kg/m3 increases (Fig. 6e and f).

At a transversal velocity component 10 and 50m/s Kelvin–Helmholtz instability is a prevailing one; that is indicated by vortexes shape in points before 1μs (Figs. 7a and 8a). At t>1μs vortexes split and “drops” are formed, interestingly, this process is more intensive at U0=50m/s than at 5 and 10m/s (Fig. 8b and f). Dimensions of drops vary 1.25–7μm at 10m/s and 1.28–5.3μm at 50m/s. Alongside with vortex splitting into drops we observed grouping of small drops into big ones (Figs. 7e and 8e). A configuration of the interface (Figs. 7f and 8f) is similar to the interface of zones II and III (Fig. 1a). So, we can conclude that the consideration of a transversal velocity provides a reasonable explanation of undulating topography on the interface “coating/base material”.

In a system Al-Si-Y we observed nearly the same situation; the only difference was a slightly faster combination of Rayleigh–Taylor and Kelvin–Helmholtz instabilities than in a system Ti-Y, since silumin has a lower density in comparison with titanium and quicker acceleration of electric explosion products. Another distinction between these systems is a domination of Kelvin–Helmholtz instability over that of Rayleigh–Taylor at velocities above 50m/s. Therefore, we can find an explanation for a less developed undulating topography in this case (Fig. 1b). On the other hand, a close inspection of Figs. 9–11 shows that a mixing process is intensive in a ∼30μm thick layer, being the reason for a non-uniform distribution of yttrium in the modified layer (Fig. 12).

It is found that a combination of Rayleigh–Taylor and Kelvin–Helmholtz instabilities is a key factor in formation of an undulating topography on the interface coating/base material. The study demonstrates a change of instabilities in a system Ti-Y and in a system Al-Si-Y at 10 and 50m/s, respectively. Developed vortexes penetrate 30–50μm from the treated surface; that is much deeper than diffusion penetration of yttrium and other alloying elements. This fact gives account of yttrium atoms at these depths.

Conflicts of interestThe authors declare no conflict of interest.

This work was supported by the President grant for State Support to young researches MK-118.2019.2 and state order of Ministry of Science and Higher Education No. 3.1283.2017/4.6.

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