Interplay of viscosity and surface tension for ripple formation by laser melting

A model for ripple formation on liquid surfaces exposed to an external laser or particle beam and a variable ground is developed. The external incident beam is hereby mechanically coupled to the liquid surface due to surface roughness. Starting from the Navier-Stokes equation, the coupled equations for the velocity potential and the surface height are derived in a shallow-water approximation with special attention to viscosity. The resulting equations obey conservation laws for volume and momentum where characteristic potentials for gravitation and surface tension are identified analogously to conservative forces. The approximate solutions are discussed in the context of ripple formation in laser-materials processing involving melting of a surface by a laser beam. Linear stability analysis provides the formation of a damped wave modified by an interplay between the external beam, the viscosity, and the surface tension. The limit of small viscosity leads to damped gravitational and the limit of high viscosity to capillary waves. The resulting wavelengths are in the order of the ripples occurring in laser welding experiments, hinting at the involvement of hydrodynamic processes in their origin. By discussing the response of the system to external periodic excitations with the help of Floquet multipliers, we show that the ripple formation could be triggered by a a periodically modulated external beam, e.g., appropriate repetition rates of an incident laser beam. The weak nonlinear stability analysis provides ranges where hexagonal or stripe structures can appear. The orientation of stripe structures and ripples are shown to be dependent on the incident angle of the laser or particle beam where a minimal angle is reported. Numerical simulations confirm the findings and allow us to describe the influence of variable grounds.

Figure 1

Laser-induced periodic structures: (a) Weld sim on steel, welding with cw fiber laser, wavelength $\lambda =1.03\mu \text{m}$; (b) laser cut of a 6-mm-thick steel with a cw diode laser $\lambda =0.8-1.0\mu \text{m}$, nitrogen atmosphere; (c) $3\times 3\mathrm{mm}$ area on steel engraved with a pulsed fiber laser, repetition rate $f=200\text{kHz}$, $\lambda =1.06\mu \text{m}$; (d) LIPSS left after femtosecond laser, repetition rate $f=1\text{kHz}$, $\lambda =0.8\mu \text{m}$.

Figure 2

Liquid layer with free surface $h(x,y,t)$ on a solid substrate exposed to an external laser or particle beam, which creates a surface current due to a gradient of the surface. The vectors $\mathbf{n}$ and $\mathbf{t}$ describe the normal and tangent directions to the free surface. The bottom geometry $f(x,y,t)$ may be space and time dependent.

Figure 3

Sketch of the form of liquid pot due to laser melting sweep. The dimensions are $l_1=l_{y-}\approx l_{y+}\approx l_{x+}\approx l_z\approx 0.1-10\mathrm{mm}$ and $l_{x-}\approx 3-10\times l_1$.

Figure 4

The wavelength dependence of $b\left(k\right)$ and $a\left(k\right)$ (above) determining the growth rate Eq. (50), which allows us to discuss four different regions (below) according to $a\gtrless 0$ and $b\gtrless 0$. Here $k_0^2=-G/(2DH+\mathrm{\Gamma })$ and all indicated relations are holding for both scalings of $\tau$ in Eqs. (17) or (18).

Figure 5

The areas of stable (yellow) oscillating (green) and unstable (red, pink) regions according to Fig. 4 in dependence on the dimensionless wavelength and external current with parameters of Eqs. (16). Right side is a zoom of left figure. The data for Fe and Au differ visibly only for the range $a<0,b<0$.

Figure 6

The real (solid) and imaginary (dashed) part of the growth rate $\lambda =-i\omega$ for a horizontal cut of Fig. 5 corresponding to unstable (left) and stable (right) behaviors. The oscillating range is indicated by filling.

Figure 7

Time evolution of the one-dimensional fluid interface $h(x,t)$ for $G=1,\mathrm{\Gamma }=1.2,H=0.05$. The initial condition of all simulations is shown in (a) and corresponds to a resting fluid with an elevation in the center. (b), (c) Two snapshots of the time evolution of the interface without external current. (d), (e) Two snapshots of the onset of unstable behavior for $D=-0.2$ according to Eqs. (53), and in (f) one snapshot of stable oscillating behavior $D=0.2$ according to Eqs. (55) are selected. The corresponding wave lengths $\mathrm{\Lambda }$ are given above.

Figure 8

The real (solid) and imaginary (dashed) parts of the wavelength Eq. (59) in dependence on the external current for three different frequencies and the parameters of Eqs. (16).

Figure 9

Above: Stable (damped) case $D\ge 0$. Below: Unstable case $D<0$ of the two wavelengths Eq. (59) in dependence on the frequency for liquid Au with parameters of Eqs. (17). The corresponding real parts (above) and corresponding imaginary parts (below).

Figure 10

Comparison of Au and Fe for the corresponding larger wavelength of Fig. 9.

Figure 11

The contours of Floquet multipliers showing the alternating border between stable (blue) and unstable (yellow) behaviors for an external frequency in dependence on the wave vector and two different external beam amplitudes. The white area indicates the range which is marginally stable with ${\nu }_i=0$. The uppermost left parabola limits the range of oscillatory instabilities. The parameters for Au according to Eqs. (16) are chosen independent of characteristic time $\tau$.

Figure 12

Sketch of incident plane $x,z$ (black) given by the incoming beam $J_0$ under incident angle $\mathrm{\Theta }$ together with the surface $x,y$ (red) and the geometry of amplitude analysis used for hexagonal structures where $\varphi$ is the angle to the beam-$x$ axes at the surface.

Figure 13

The angle of stripe orientation $\varphi$ in the surface as in Fig. 12 as function of the incident angle $\theta$ of incoming beam for the collision model Eqs. (A7) (dashed) and the surface impingement model Eqs. (A13) (solid).

Figure 14

The range of possible stable stripes as function of the viscosity and the angle of stripe orientation with respect to the plane of incoming beam for four different amplitudes of the collisional model. The area with $\text{Re}\lambda <0$ is indicated by gray and the white area represents oscillating behavior. We choose $G+\mathrm{\Gamma }=0.1$ which parameter almost does not influence the result. The amplitudes and viscosity parameter are scaled in terms of external current.

Figure 15

Three time steps of the two-dimensional set Eq. (31) for $G=1$, $\mathrm{\Gamma }=0.5$, $H=5$ with an initially elevated surface. Left side: With external current $D_x=D_y=-2sin\omega t$ and $\omega =10$ corresponding to the unstable ripple formation of the upper left figure in Fig. 14 and $\varphi ={45}^{\text{o}}$, right side: without external current. The wavelengths of linear response are given above for comparison.

Figure 16

Time steps of the two-dimensional set Eqs. (31) at 1/10 of the initial elevation of Fig. 15 for $G=1$, $\mathrm{\Gamma }=0.5$, $H=2$ with external current $D_x=-6sin\omega t,D_y=0$, and $\omega =10$ without (left) and with (middle, right) tilted ground. The right column has been chosen with $D_x=-6.5$ being in the unstable regime for untilted bottom. The wavelengths of linear response are given above for comparison.

Figure 17

The geometry of incoming sphere with velocity $v_0$ from a current $I_{\mathrm{at}}$ colliding a sphere in the material under impact parameter $b$. Due to surface roughness and deformation of the surface by external beam, the surface becomes tilted from $x$ to $x^{\prime }$ direction, blocking half of impact parameter.

Figure 18

The geometry of incoming impingement on the surface $J_{0}cos\theta$ split to the surface parallel contribution where $b=tan\theta$.