Mechanisms proposed for rippling
Productivity in different welding techniques can be improved by increasing welding speed and current. This strategy, however, is limited by the appearance of surface defects such as rippling, humping, undercutting, etc . Weld ripples exhibit rather regular, arc-shaped topographic features on a solidified surface, for example, as shown in Figure 1 in EBW of Al 6061 . The ripples slightly elevate above the surface. Figure 2 shows a rippling structure on silicon surface irradiated by a p-polarized laser beam, provided by Pedraza et al. . Notice tiny little “fingers” in lower rim of fringes and asymmetry in fringe profile taken in downward direction.
On the other hand, more complicated humping shows an irregular and unpredictable surface contour consisting of a series of swelled protuberance, as can be seen in Figures 3(a)-(d). Morphologies of humped welds are quite complicated, which were roughly categorized into the gouging region and beaded cylinder morphologies . Typical gouging region morphology defects in GTAW at high currents and high travel speeds are shown in Figure 3(a) . The front of the weld pool exhibits a large depression known as the gouging region. Open, unfilled dry spots in between the humped beads can also be seen. In some cases, two small channels appear at the walls of the gouging region. The weld bead having parallel grooves at the side is the undercutting defect.
Figure 3(b) shows a parallel humping or a split bead, separated by an empty channel . Figure 3(c) shows beaded cylinder morphology defects, which are quite different from the gouging region morphology . The beaded cylinder morphology includes beadlike protuberances that sit above the workpiece surface and are connected by a narrow central channel. In some cases of disconnected protuberances, traces of a central channel can still be seen. It is interesting to find that the gouged region and bead cylinder morphology are inverse phenomenon. Different morphologies therefore can be revealed by simply interchanging the liquid and gas phases. Figure 3(d) shows humping in EBW of Al 5083 . Figure 4 also shows the a “star-like” or finger structure develops as well as the number of rays increases with laser power in etching of regular holes in Mo films with an laser beam immersed in atmosphere . Surface patterns of weld beads therefore are very complicated.
Even though formation of the defects have been extensively proposed in the past, more systematical understanding of pattern formations is still limited. The aims of this work are to provide rigorous and pictorial interpretation based on thermal science concepts, and clarify and propose some physics involved in the mechanisms of weld bead formation. Clear physical concepts associated with quantitative scale analysis are important and beneficial for predicting, controlling and avoiding the occurrence of surface defects .
2. Mechanisms of surface patterns
Different mechanisms of rippling have been proposed and summarized in Table 1.
|Solidification rate fluctuations||Cheever and Howden , D'annessa |
|Power source effects||Garland and Davies , Ecer et al. [ 11]|
|Thermocapillary instability||Fujimura et al. [ 12]|
|Kelvin-Helmholtz instability||Ang et al. |
|Rayleigh-Taylor instability||Bennett et al. , Lugomer |
|Instability due to evaporation||Emel’yanov et al. |
|Morphological instability||Weizman et al. , Style and Wettlaufer [ 18]|
|Thermocapillary edge flow||Anthony and Cline , Wei et al. [20, 21]|
|Laser interactions||Birnbaum , Siegman and Fauchet [ 23]|
On the other hand, different mechanisms responsible for humping are presented in Table 2.
|Rayleigh’s capillary instability||Bradstreet , Gratzke et al. , Albright and Chiang |
|Kelvin-Helmholtz instability||Kumar and DebRoy , Tytkin et al. |
|Hydraulic jump||Shimada and Hoshinouchi |
|Thermocapillary edge flow||Wei |
|Kelvin-Helmholtz instability||Dynamic pressure difference|
|Rayleigh-Taylor instability||Density difference|
|Rayleigh capillary instability||Capillary pressure difference|
|Morphological instability||Solute supersaturation|
|Thermocapillary instability||Surface tension gradient|
|Evaporation instability||Evaporation pressure difference|
|Hydraulic jump||Hydraulic pressure difference|
|Gravitational-electromagnetic instability||Interactions between gravitational and electromagnetic forces|
Surface roughness including rippling, gouging, humping, fingers, etc., therefore can be affected by mechanisms shown in Table 3. Understanding their physical meanings is described as follows.
3. Thermal science analysis of surface patterns
Rippling or humping is determined by the formation of capillary wave on the free surface. Under surface tension, the pressure differences created at a curve interface support deformation of the interface, as sketched in figure 5.
Mathematically speaking, it is governed by Young-Laplace equation
where, are liquid and gas pressures, , the surface tension, and two principal radii of a surface, respectively. Equation (1) can be simply derived from a normal stress balance on an interface . Any physical or chemical variables affecting pressures at the interface are responsible for different surface roughness patterns. The onset and mechanisms of instability are determined from perturbed deformation governed by equation (1) by substituting perturbed liquid and gas pressures. In this work, free surface instability can also be relevantly revealed from the concept of mass conservation, as shown in figures 6(a) and (b). Provided that velocity profiles maintain the same, the free surface is flat. However, the free surface deforms downward, if the mass of the outflow is greater than that of the incoming flow. Since surface deformation is enhanced, the free surface is suffered from instability.
The factors affecting surface patterns listed in Table 3 are rigorously described as follows:
4. Kelvin-Helmholtz instability
KH instability arises due to difference in velocities between gas and liquid. KH instability is the simplest and widely encountered instability, derived from Young-Laplace equation, where liquid and gas pressures are determined from conservation of mechanical energy (namely, the Bernoulli’s equation) with specified constant liquid and gas velocities and far away the location considered. Bernoulli’s equation indicates that the lower the velocity is, the higher the pressure is. Provided that deformation is toward the gas, the decrease in perturbed velocity results in an increase in perturbed pressure in the liquid, as illustrated in Figure 7. Opposite phenomenon occurs in gas phase. This results in a further increase in deformation under the action of surface tension. Wavelength for surface deformation can be scaled from equation (1)  and given by
Wavelength of surface deformation due to KH instability therefore reduces if the difference in velocities between gas and liquid increases. Provided that gas velocity is 10 m/s, the length for roughness in liquid metals is around 10within a relevant range of rippling spacing.
5. Rayleigh-Taylor instability
RT instability occurs when a heavier liquid overlies a lighter liquid. The pressure involved is hydrostatic pressure, which is a function of gravitational acceleration and the depth location considered. As illustrated in figure 8, the difference in hydrostatic pressures of liquid and gas across a surface with a deformation simply yields
Provided that deformation is toward the lighter fluid, a positive perturbed pressure deviated from the base state results. A negative perturbed pressure simultaneously occurs with deformation toward the heavier liquid. Deformation further increases. At later times, initial perturbations grow into spikes of heavier fluid “falling” into lighter fluid and bubble of the lighter fluid “rising” into the heavier fluid. Hence, RT instability has also been stated to occur when the pressure and density gradients are in opposite directions, or a lighter fluid pushes or accelerates a heavier fluid. Equation (4) indicates that an increase in surface tension or decrease in difference in densities across a free surface increases wavelength of ripples.
Acceleration in equation (4) may not be the earth’s gravity. RT instability can also occur at an interface separating fluids through which a blast wave has been transmitted from a heavier to a lighter fluid. This instability is Richtmyer-Meshkov instability, often called impulsive or shock-induced RT instability. In high intensity beam welding, the produced shock waves propagate with discontinuities of densities, pressures and velocities in different magnitudes and directions across the free surface. Provided that the interface subject to an oblique shock, it will give rise to complicated instabilities or patterns at the interface. The normal component of the shock generates RM instability, and the parallel component generates KH instability. If a normal acceleration is also present, RT instability occurs.
6. Rayleigh’s capillary instability
Rayleigh’s capillary instability is a crucial factor to understand a bulged or gouging region of surface roughness. The gouged region exhibits an inverse feature of the bulged region. Rayleigh’s capillary instability can be revealed from figure 9. Radius is at a location B near the minimum radius. Provided that wavelength of surface deformation is long, curvature >> at location A, as illustrated in figure 9(a). Since liquid pressure is primarily balanced by capillary pressure due to curvature, a greater liquid pressure is induced by a smaller radius of the cylinder. The positive difference in the pressures thus induces a perturbed flow from locations A to B. The system therefore is unstable and breakups the cylinder into droplets. However, figure 9(b) shows that for deformation with small wavelength, curvature >>at location A. The perturbed pressures at location A therefore decrease and it can be smaller than perturbed pressure at location B, leading to the flow from points B to A, and stabilizing the system.
Since the difference in the perturbed pressures at two locations affects instability, the minimum wavelength for onset of instability is the balance between two components of capillary pressure. The critical wavelength thus is of the same order of the radius of the liquid cylinder . Stability of a bead should depend on the boundary conditions at its contact lines on the surface. Gau et al.  experimentally found that cylindrical segments for water on hydrophilic stripes with the apparent contact angle less than did not break up into droplets, as would be expected. It displayed long-wavelength instability where all excess fluid gathered into a single bulge on a hydrophilic stripe. Speth and Lauga  theoretically confirmed the most unstable wavenumber for the instability —the one which was observed in an experimental setting — decreases to zero when the apparent fluid contact angle reached. The creation of bulges in the experiment corresponded with a zero-wavenumber capillary instability.
7. Morphological instability
Morphological instability is a consequence of thermal and metallurgical processes. Surface morphology becomes unstable by decreasing surface tension and increasing constitutional supercooling (namely, >0) [34,35], where m is the negative slope of the liquidus line in the phase diagram of a dilute solution, and are negative concentration and positive temperature gradients ahead of the solidification front, respectively. The decreased solute concentration increases liquidus temperature near the solidification front, as illustrated in figures 10(a) and (b), respectively, and revealed from the phase diagram of figure 10(c).
In view of the rapid drop of solute concentration ahead of the freezing front, the actual temperature may be below the liquidus temperature and result in constitutional supercooling or morphological instability, as shown in figure 11. Mathematically speaking, morphological instability can be simply found by considering interfacial temperature governed by Gibbs-Thomson equation 
where, , are, respectively, interfacial temperature, melting temperature and latent heat for solid-liquid transition. The variations of temperature and concentration at the interface subject to a forward deformation can be expressed by their spatial gradients, . Scaling equation (5) therefore leads to
8. Instability due to evaporation
It is well-known that a semi-infinite liquid heated below is susceptible to evaporative instability, as illustrated in figure 12(a). As surface deformation is closer to the bottom surface than the flat surface, temperature at the trough is greater than the equilibrium temperature, meaning that higher evaporation takes place at the trough than the base state. On the other hand, a deformation away from the bottom surface leads to lower evaporation rate than that at the base state. Liquid pressure therefore increases from the surface trough to crest. The induced flow from the trough to crest thus enhances surface deformation, leading to evaporative instability (referring to figure 6(b)). However, in welding and manufacturing processes, workpieces are irradiated by incident flux at the top surfaces. A liquid heated from above, which is the case opposite to previous figure 12(a), is stable, as illustrated in figure 12(b). In this work, it is found that evaporation may also induce instability subject to base temperature decreasing in radial directions, as illustrated in figure 12(c) . For a typical welding process the isothermal field is in a roughly spherical shape. Provided that surface deforms toward the bottom, temperature at the crest, which is close to the pool edge, can be lower than that at the trough. The decrease in pressures therefore pushes the liquid from the trough to crest and gives rise to instability.
9. Thermocapillary instability
All pure liquid metals and alloys containing minor surface active solutes such as O, S, Se, Te, et al. have negative surface tension coefficient (< 0). As illustrated in figure 13, thermocapillary force balanced by viscous shear stress at the interface is given by
A negative surface tension coefficient induces an outward surface flow, provided that surface temperature decreases in the outward direction. Consider an interface to be displaced toward the hot surface at the bottom, as illustrated in figure 14(a). Temperature at the trough thus is hotter than other points on the deformed surface. This results in an outward lateral flow from the trough to crest along the interface. To conserve mass, the perturbed liquid flows downwards and further deforms the interface with speed w (referring to figure 6(b)). The system thus is unstable. The well-accepted interpretation is incorrect. Rather than perturbed downward flow, it is usually interpreted as that instability results from incessant and amplified upward flow accompanying with enhanced energy at the trough due to increased thermocapillary force from the trough to crest .
In this work, it is proposed that a free surface heated from above and a negative surface tension coefficient may still cause instability, as illustrated in figure 14(b) . Provided that the surface is strongly deformed toward the bottom, significant heat conduction is transport from the interior to free surface. Enhanced downward thermocapillary surface flow therefore is required to balance horizontal conduction, and results in further deformation. This work also proposes that thermocapillary instability takes place near the edge of the molten pool, where > 0 in the presence of surface active solutes in the case similar to the mechanism (see figure 14(a)) provided by Pearson , as illustrated in figure 14(c). This is another reason responsible for serious roughness encountered in alloys having surface active solutes .
10. Thermocapillary edge flow
Deformation of the free surface near the solidification front is responsible for rippling or humping. As illustrated in figure 15, an increase in liquid pressure due to a decreased surface flow from the central to rear edge of the pool give rise to deformation of the free surface near the solidification front. Subtracting Young-Laplace equation at two locations which are away and near the edge of the pool surface, and introducing the pressure difference between two locations from Bernoulli’s equation,, the amplitude of ripples can be found to be [1,20]
where the loss coefficient is introduced to account for the energy loss near the pool edge, a, and are, respectively, roughness amplitude, thermal diffusivity and distance of the rear edge measured from the pool center. Surface speed can be scaled and related to thermocapillary force [37,38]. Based on a typical surface velocity of 1 m/s, pool width of 1 mm and loss coefficient of 0.99, rippling amplitude predicted from equation (8) is around a reasonable value of 10.
11. Hydraulic jump
Studying hydraulic jump usually assumes liquid pressure to be hydrostatic pressure. Hydraulic jump occurs if the pressure gradient becomes increasingly adverse as the flow proceeds downstream. As sketched in figure 16(a), an increased liquid height (>) increases hydrostatic force (= pressure x height) against the downward flow, and decreases downward velocity to satisfy conservation of momentum and mass. However, liquid height at the downstream location can also be less than that at the upstream location, as illustrated in figure 16(b). Hydrostatic force decreases whereas velocity increases. Provided that the Froude number, > 1, hydraulic jump occurs because of /> 1 . As Froude number < 1, the height ratio /< 1. Surface tension can also play a role in hydraulic jump , as observed the occurrence of polygonal patterns from breaking axial symmetry of a circular hydraulic jump.
Fingers can be accompanied with hydraulic jump (see figure 4). This is because pressure and density gradients are in opposite directions in the course of splashing, leading to RT instability. Allen  was the first to propose that the splashing of a droplet impact on a surface is an example of RT instability, caused by a rapidly decelerating interface. Replacing gravitational acceleration by in equation (4) and substituting from geometrical consideration lead to
Provided that time is short, equation (9) indicates that radius of the ring-shaped spread is less than wavelength of fingers. The ring-shaped spread thus is free from fingers. On the other hand, larger time results in the ring-shaped spread covered with n fingers.
13. Gravitational-electromagnetic instability
In plasma, Rayleigh-Taylor instability can occur because the magnetic field acts as a light fluid supporting a heavy fluid (plasma), . This is because the ion drift velocity U0 is in the direction of g B0. If a ripple in plasma develops on the interface as the result of random thermal fluctuations, the drift velocity U0 will cause the ripple to grow, as sketched in figure 17. Gravitational acceleration g is in the downward direction. The drift of ions causes a charge to build up on the sides of the ripple, and an electric field develops which changes sign as one goes from crest to trough in the perturbation. The drift E′ B0 is thus always upward in those regions where the surface has moved upward, and downward where it has moved downward. The ripple therefore grows as a result of these properly phased E′ B0 drifts.
Physical interpretation of bead defects is important and beneficial for controlling quality of welding joint. It involves inter-discipline among different sciences of thermal physics, aerodynamics, electromagnetism, optics and metallurgy, morphology, pattern selection, instabilities, and contact line dynamics. Phase transitions between liquid and gas, and solid and liquid are also included. Spaces and amplitudes of rippling and humping can be effectively revealed from scaling of a force balance between perturbed liquid and gas pressures and surface tension. Any factor which can induce pressure differences or influence surface tension is responsible for specific surface patterns. This study provides a general, relevant and rigorous interpretation of physical mechanisms involved in surface roughness.
The author is grateful for Mr. Sheng-You Tsai drawing the pictures