Y Pan
email: y.pan@m2i.nl

I M Richardson
email: i.m.richardson@tudelft.nl

Keyhole behaviour during laser welding of zinc coated steel

Keywords: Laser welding, zinc coated steel, keyhole dynamics, rear wall fluctuation, keyhole elongation

The production of consistent, high quality laser welds on zinc coated steels for the automotive industry remains a challenge. A simple overlap joint geometry is desirable in these applications but has been shown to be extremely detrimental to laser welding because the zinc vapour formed at the interface between the two sheets expands into the keyhole and disrupts fluid flow in the melt pool, which often leads to metal ejection. In the present work, laser welding on sheets with various coating thicknesses has been performed and it is observed that the sheets with thick coatings (~20 µm) show surprisingly good weldability. High speed video camera visualizations of the keyhole provide insight into the keyhole dynamics during the process. It appears that the dynamic pressure of zinc vapour can effectively elongate the keyhole and the process can reach a stable state when an elongated keyhole is continuously present. A simple analytical model has been developed to describe the influence of zinc vapour on keyhole elongation.

Introduction

Laser welding of zinc coated steel sheets in an overlap joint configuration is commonly used in the automotive industry. A major problem that arises when welding these materials is associated with the vaporization temperature of zinc (906°C), which is much lower than the melting temperature of steel (1530 °C). In an overlap configuration, the zinc vapour produced between the sheets during welding will vent through the keyhole, particularly when no gap is present between the overlapping sheets. This causes unstable fluid flow and molten metal is often ejected from the pool, resulting in the formation of pores and severe undercut.

One of the first methods introduced and still in use today to overcome this problem is to set a gap between the sheets prior to the welding, usually in the range of 0.1 to 0.2 mm, which provides a channel for the escape of the zinc vapour. [START_REF] Graham | Nd-Yag laser welding of coated sheets steel[END_REF][START_REF] Akhter | Laser welding of zinc coated steel[END_REF][START_REF] Bley | An alternative approach for the cost-efficient laser welding of zinc-coated sheet metal[END_REF][START_REF] Xie | Galvanized steel joined with lasers[END_REF] However, this solution can be undesirable because of the difficulty associated with maintaining such gaps in industrial configurations. Another possibility involves enlarging the keyhole in order to facilitate the escape of zinc vapour, by using elongated or dual focus laser spots. [START_REF] Xie | Galvanized steel joined with lasers[END_REF][START_REF] Forrest | Fundamental study of dual beam laser welding of Zinc Coated Steel Sheets in Lap Joint Configuration with Zero-Gap[END_REF] This method can minimise the destabilising influence of zinc vapour on the melt pool; however, it requires careful control of the laser power distribution. Reduced power intensity on the material surface and hence reduced productivity limits the attractiveness of this method.

Previously, it was reported that welding of zinc coated sheets without any special arrangements can be successfully performed under certain circumstances. [START_REF] Pieters | Laser welding of zinc coated steel in overlap configuration with zero gap[END_REF][START_REF] Fabbro | Study of CW Nd-Yag laser welding of Zn-coated steel sheets[END_REF][START_REF] Dasgupta | Laser welding of zinc coated steel: an alternative to resistance spot welding[END_REF] In the present work, the weldability of materials with two different coating thicknesses has been studied. High speed camera visualization techniques have been employed to explore dynamic changes in keyhole dimensions. It is found that while some sheets with a 7 µm zinc coating thickness are difficult to weld with zero gap, others with a 20 µm zinc coating thickness, which might be expected to exacerbate process instability, show good weldability. This surprising result is associated with keyhole shape and dynamic variations. An analysis of keyhole evolution is presented for both stable and unstable cases and possible reasons for the observed behaviour are considered. A simple analytical model is suggested relating operating parameters (welding speed) and material properties (coating thickness, surface tension and density) to describe the influence of zinc vapour on keyhole geometry.

Background

The stability of a weld pool is governed by a number of factors. Surface tension plays a significant role in retaining the weld pool, and is a complex function of temperature, [START_REF] Sahoo | Surface tension of binary metal -surface active solute systems under conditions relevant to welding metallurgy[END_REF] pool geometry and surface chemistry; [START_REF] Wang | Modeling of the effects of surface-active elements on flow patterns and weld penetration[END_REF] all of which may vary with time and position. Temperature is determined by the balance of heat input and heat losses and shows strong spatial variation in fusion welding processes. During laser welding the surface tension distribution is dependent on the laser beam power and the power density, which influence pool size and surface temperature distribution as well as pool chemistry (through vaporisation).

High power density laser beams for example can create local evaporation, and the reaction forces generated at the pool surface give rise to keyhole opening. The dynamic behaviour of the keyhole is decisive for the welding process. Keyhole fluctuations, particularly parallel to the welding direction, can stimulate weld pool oscillations. Conversely, large oscillation amplitudes of the melt pool may lead to a collapse of the keyhole and result in weld seam defects such as porosity and undercut. [START_REF] Matsunawa | Dynamics of keyhole and molten pool in laser welding[END_REF][START_REF] Klein | Forced oscillation of the keyhole in penetration laser beam welding[END_REF] To keep the keyhole open, the energy balance and the pressure balance at the keyhole wall must be satisfied. [START_REF] Andrews | Hydrodynamic limit to penetration of a material by a high-power beam[END_REF] The metal vapour inside the keyhole acts against the surface tension to keep the keyhole open. [START_REF] Andrews | Hydrodynamic limit to penetration of a material by a high-power beam[END_REF][START_REF] Duley | laser Welding' 83-85[END_REF] Inside the melt pool during a laser welding process, different types of melt flow can take place, driven by buoyancy, surface tension gradients, surface drag due to impinging gas flows or vaporisation. [START_REF] Semak | The role of recoil pressure in energy balance during laser materials processing[END_REF][START_REF] Dowden | The Theory of Laser Materials Processing[END_REF] The relative importance of the driving forces varies with operating conditions. Marangoni convection driven by surface tension gradients is dominant when welding at relatively low speeds, [START_REF] Kaul | Effect of active flux addition on laser welding of austenitic stainless steel[END_REF][START_REF] Mills | Marangoni effect in welding[END_REF] however, it has been reported to be less dominant with increasing welding speed [START_REF] Lampa | An analytical thermodynamic model of laser welding[END_REF][START_REF] Fabbro | Experimental study of the dynamical coupling between the induced vapour plume and the melt pool for Nd-Yag CW laser welding[END_REF] or when a side gas jet is applied. [START_REF] Fabbro | Experimental study of the dynamical coupling between the induced vapour plume and the melt pool for Nd-Yag CW laser welding[END_REF][START_REF] Kamimuki | Prevention of welding defect by side gas jet flow and its monitoring method in continues wave Nd:YAG laser welding[END_REF] Another major feature of the melt flow includes liquid transport around the keyhole. [START_REF] Dowden | The Theory of Laser Materials Processing[END_REF][START_REF] Beck | Aspects of keyhole/melt interaction in high speed laser welding[END_REF] The driving force for this motion is the evaporation recoil pressure at the front of the keyhole, controlled by the input laser power, power density and welding speed. [START_REF] Dowden | The Theory of Laser Materials Processing[END_REF] At lower welding speeds the melt pool between the keyhole and the melting isotherm is wide, resulting in a less constrained, low speed flow. At high welding speeds, a much higher liquid metal flow speed is generated because the volume of metal melted per unit time increases and the melt pool between the leading edge of the keyhole and the melting isotherm becomes dense. [START_REF] Kaplan | Modellrechnung des laser-Tiefschweißens[END_REF][START_REF] Lampa | Fluid flow and resolidification in deep penetration laser welding[END_REF] This melt flows around the keyhole, collides with the slowly moving weld pool and is forced upwards and downwards, forming a weld bead shape with a central peak and undercut at either side. [START_REF] Dowden | The Theory of Laser Materials Processing[END_REF] Fluid flow has a substantial influence on energy transport within a liquid weld pool, and changes in flow pattern may substantially alter the shape of the pool. [START_REF] Mazumder | Mathematical modelling of laser materials processing[END_REF] Changes in laser power, power density or pool size will have some influence on vaporisation and the reaction forces generated at the weld pool surface.

When welding zinc coated sheets in an overlap configuration, the influence that zinc vapour exerts on weld pool fluid flow and energy transport is strongly dependent on the flow geometry. During bead on plate welding or overlap welding with gaps of 0.1mm between the sheets (such that zinc vapour can escape), zinc vapour has only a small influence on weld pool size and shape. Conversely, when stake welding with no gap between the sheets, vapour escape through the melt pool can lead to unacceptable instability. [START_REF] Graham | Nd-Yag laser welding of coated sheets steel[END_REF][START_REF] Akhter | Laser welding of zinc coated steel[END_REF][START_REF] Bley | An alternative approach for the cost-efficient laser welding of zinc-coated sheet metal[END_REF][START_REF] Xie | Galvanized steel joined with lasers[END_REF] During laser welding the amount of zinc vaporised per unit time is the product of the mean coating thickness, the welding speed and the width of the region where the surface temperature exceeds the vaporisation temperature of the zinc layer. This in turn is influenced by the welding speed, the beam power and the surface power density as well as the material thickness and thermophysical properties. The way in which the vapour impinges on the weld pool and the subsequent influence on both process stability and weld pool geometry is considered in the present work.

Experimental Conditions

The present study was carried out using a Nd:YAG (wavelength 1064 nm) laser with a nominal power of 3 kW. The focal distance was 150 mm, projecting the laser beam to a spot of 450 µm diameter. The laser beam was focused on the top surface of the sheet with the beam perpendicular to the surface. No shielding or backing gases were used in the tests. All welds reported were made at a beam power of 3 kW and welding speeds were varied in the range of 25 mm s -1 to 55 mm s -1 .

Experiments were performed on 0.8 mm thick DX54D hot dip galvanized (GI) zinc coated sheet steels with nominal zinc coating thicknesses of 7 and 20 µm. The chemical composition of the base steel is given in table 1. After welding, the specimens were checked visually and cross-sections were made. Visual weld quality was quantified by a length fraction, defined by the length of the weld seam without any pores or severe undercut divided by the total length of the seam. The average length fractions were determined for 3 welds of 100 mm length per parameter set. The dynamic behaviour of the keyhole was filmed with a Phantom v5.0 CMOS camera at a sampling rate of 30,000 fps. The camera was placed coaxially with the laser beam. The laser focusing lens provided an image of the top surface of the keyhole and melt pool, directed toward the camera by means of a dichroic mirror mounted inside the laser head, as shown in Figure 1. Since the keyhole and vapour plume are well illuminated by process radiation, no external illumination was necessary. Temperature measurements were performed with standard 0.13 mm diameter K-type thermocouples to determine the position of the zinc boiling isotherm at the interface between the two sheets. The locations of the thermocouples are shown in schematically in Figure 2.

Thermocouples were attached on the cleaned surface along a line perpendicular to the welding direction. Small pieces of ceramic were used to isolate the individual thermocouple wires. The exact positions were measured after welding to determine the position where the maximum temperature approached the zinc vaporisation temperature of 1180 K.

Figure 2: Temperature measurements at the interface between the two sheets.

Results

Prior to welding, the coated sheets were sectioned to examine the original coating thickness and thickness variations. Measurements were performed at intervals of 1mm along a 100 mm sample length. Results show an average thickness of 21.4 µm for the 20 µm nominal thick coating thickness, with a standard deviation of ±1.6 µm, and an average thickness of 8.7 µm for the 7 µm nominal coating thickness with a standard deviation of ±2.2 µm (Figure 3). The weld quality, quantified by the length fraction, is shown in Figure 4. As expected the presence of the zinc coating reduces the weld quality. It is notable that better quality welds were obtained for the 20 µm coating thickness GI sheets, while the welds on steels with a 7µm coating thickness showed a significant number of defects. It is also found that better quality is obtained at lower welding speeds. Typical transverse cross-sections of welds are shown in Figure 5. It can be seen that weld quality is relatively good, with uniform zinc layers adjacent to the weld at the steel interface for the 20 µm coating thickness (Figure 5a), which is in agreement with previously published results. [START_REF] Pieters | Laser welding of zinc coated steel in an overlap configuration[END_REF] In contrast, a poor quality weld is found in the case of the 7 µm coating thickness, as shown in Figure 5b. Large pores are found in the fusion zone, some of which are filled with zinc, indicating that these pores connect with the sheet interface and that liquid zinc can flow during weld solidification. It is also found that there is only a very irregular appearance of zinc adjacent to the weld at the interface. According to the visual inspection and element mapping, no zinc is found in the weld zone; [START_REF] Pieters | Zinc transport phenomena in laser welding of coated sheet steel in overlap configuration[END_REF] this indicates that zinc diffusion into the weld pool is negligibly small and that almost all the zinc that is vaporized escapes through the keyhole. Weld end craters were sectioned longitudinally to examine the zinc distribution in front of the keyhole and results are shown in Figure 6. In Figure 6a, it can be seen that the liquid steel on the keyhole front wall is completely penetrated by the zinc vapour, and liquid zinc has flown through the resulting channel during solidification. In contrast, in Figure 6b a separation region appears at the fusion boundary without penetrating into the keyhole. In this case liquid zinc is found to flow between the solid steel surface and the prior liquid steel melt-pool.

Based on coaxial views of the keyhole opening for zinc coated and non-coated steels (Figure 7), the keyhole geometry can be approximately reconstructed with respect to the inclination of the keyhole front wall and the shape of the keyhole top aperture. The grey scale indicates a combination of both laser and radiative intensities. The bright zone on the images indicates the material directly exposed to the laser beam. The dark area in the centre of keyhole indicates that the steel sheets have been fully penetrated. Here the laser beam travels through the keyhole without any laser -material interaction. It is found that the keyhole widths are virtually constant, being defined by the incident focal spot dimensions. In contrast, the keyhole length increases when welding zinc coated sheets, and a channel for zinc vapour evacuation forms. This corresponds to the dark line in the middle of the keyhole front wall, which indicates that the zinc vapour penetrates the liquid film at this location and escapes into the keyhole. This elongation effect is readily apparent in the case of the 20 µm zinc coating thickness (Figure 7, images on the right).

During welding of bare steels, a characteristically sloping rear keyhole wall is expected as a consequence of laser beam reflection from the front wall. [START_REF] Fabbro | Study of CW Nd-Yag laser welding of Zn-coated steel sheets[END_REF] This is evident in the photographs in the left column of Figure 7. However, relatively straight rear walls are observed when welding zinc coated steels.

Discussion

Zinc vapour evacuation through the keyhole front wall

In the numerical simulation presented by Geiger et al. [START_REF] Geiger | A 3D transient model of keyhole and melt pool dynamics in laser beam welding applied to the joining of zinc coated sheets[END_REF] it was shown that as the gap at the interface between two uncoated sheets welded in an overlap configuration increases to 0.05 0.1 mm, the liquid metal on the keyhole front wall tends to separate; i.e., a channel opens connecting the interface gap to the keyhole. It is reasonable to expect that this will occur at smaller gaps when a pressure gradient exists between the interface and the keyhole, caused for example by the evolution of zinc vapour.

It is shown in Figure 3 that the thickness of the 20 µm coating is more uniform than that of the 7 µm coating. A more consistent supply of zinc vapour into the keyhole can be expected when welding the sheets with the thicker zinc coating. With a thick coating, as shown in Figure 5a and 5b, a more consistent zinc vapour evacuation channel will open, partly because of the constant zinc vapour pressure but also because a larger gap already exists after the vaporisation of the 40 µm zinc coating at the interface. The visible presence of zinc vapour evacuation channels, as shown in Figure 7, was analyzed to give the percentage of images showing the channel. For the 20 µm coating thickness, 94.6% show the presence of a channel at a welding speed of 25 mm s -1 and 92% at 45 mm s -1 , compared with 63.9% and 61.0% for the 7 µm coating thickness.

Keyhole elongation behaviour during welding

The keyholes with modified shapes shown in Figure 7 are similar to those reported by Fabbro et al. [START_REF] Fabbro | Study of CW Nd-Yag laser welding of Zn-coated steel sheets[END_REF] and Schimidt, [START_REF] Schmidt | Analysis of Yag laser lap-welding of zinc coated steel sheets[END_REF] which show that the keyhole shape is strongly influenced by the dynamic pressure of the zinc vapour. In order to develop a statistical evaluation of the keyhole geometry, a continuous video sequence of 10,240 images (representing a time period of 0.34s) for steady state welding was analyzed for each parameter combination. The sequence was divided into 10 bins (groups) of consecutive images, each of them comprising 1,024 images. Average keyhole lengths were calculated in each bin and the evolution of keyhole lengths is shown in Figure 8. When welding at 25 mm s -1 , the keyhole is elongated to about 0.93 mm (20 µm coating thickness) from 0.56 mm (bare steel). This effect is more significant when welding at 45 mm s -1 , ranging from 0.58 mm for bare steel to 1.34 mm for the steel with 20 µm thick zinc coating. It was reported that the inclination angle of the keyhole front wall increases with increasing welding speed, resulting in reduced direct transmission of the beam through the keyhole and increased beam absorption, due both to an increased exposure area and multiple reflections within the keyhole. [START_REF] Fabbro | Experimental study of the dynamical coupling between the induced vapour plume and the melt pool for Nd-Yag CW laser welding[END_REF][START_REF] Fabbro | Study of keyhole behaviour for full penetration Nd-YAG CW laser welding[END_REF][START_REF] Krasnoperov | Weld pool geometry during keyhole laser welding of thin steel sheets[END_REF] Fabbro et al. [START_REF] Fabbro | Study of keyhole behaviour for full penetration Nd-YAG CW laser welding[END_REF] report that for a 1 mm thick steel sheet, increasing welding speed above 7 m min -1 (117 mm s -1 ) until no direct beam transmission through the keyhole occurs, results in an elongated keyhole due to the plume emitted from the keyhole front wall and the influence of laser beam reflections. However, the range of welding speeds examined here (25 -55 mm s -1 ) is much lower than the speed reported to lead to an elongated keyhole and this behaviour does not explain the longer keyhole obtained when welding zinc coated sheets.

Observation of constant keyhole widths for all of the welding cases studied (as shown in Figure 7) suggests that approximately uniform zinc vaporization isotherms will arise ahead of the keyhole, independent of the coating thickness. This indicates that a higher mass flow rate is required to evacuate the zinc vapour in the case of the thicker coating. Hence a higher vapour dynamic pressure at a given position in the keyhole can be expected, which explains the longer keyhole obtained with the 20 µm coating. For a given coating thickness, it is observed that when the welding speed increases from 25 to 45 mm s -1 (a factor of 1.8), the cross-sectional area decreases. For the 20 µm coating thickness for example, the area decreases by a factor of 1.1. Since parallel sided weld profiles are obtained (as shown in Figure 5), the width of the weld waist also decreases by a factor of 1.1. Assuming that the zinc in the weld area is vaporized and leaves through the keyhole, the volume of the vapour expanding into the keyhole per unit time increases by a factor of 1.6 when increasing welding speed from 25 to 45 mm s -1 . Thus a higher mass flow rate is produced when welding at the higher welding speed, leading to a higher vapour pressure at a given position in the keyhole. This explains the longer keyholes obtained when welding at higher speeds.

The standard deviations of measurements, shown as error bars in Figure 8, indicate the amplitude of the keyhole fluctuation. In contrast to the very stable keyhole behaviour obtained when welding bare steels, both coated materials show strong fluctuations as a result of weld pool eruption and induced melt pool oscillations. The mean fluctuation magnitude for both coated materials is shown in Figure 9. It is evident that the sheets with 20 µm zinc coating thickness show slightly smaller oscillations than those with 7µm zinc coating thickness, independent of welding speed. It is also observed that the process is more stable when welding at the low speed. This is understandable because there is more time available for heat conduction and a larger zinc fusion zone can be produced at a lower welding speed. Such a zone is helpful because it reduces the effect of any sporadic change in the coating characteristics and consequently provides a more consistent supply of zinc vapour into the keyhole.

Although large fluctuation amplitudes are observed for both 20 and 7 µm coating thickness materials, it is found that the data distributions in a 0.034 s time period are quite different. The examples given in Figure 10 reveal two modes of keyhole fluctuation. In the case of the 20 µm coating thickness (Figure 10a), minimal fluctuations are observed and the keyhole mostly oscillates in a narrow range with occasional large excursions. This contrasts with the case of the thinner coating (Figure 10b), where the keyhole continuously oscillates over a large amplitude.

Analytical model of keyhole elongation

In order to evacuate the vapour produced when welding steels with thicker coatings, a higher zinc vapour flow velocity will arise, and higher dynamic pressure therefore develops. The influence of zinc vapour evolution on keyhole elongation may be explained from an analysis of the pressure balance at the keyhole wall. For a stationary keyhole with a high aspect ratio, the following pressure balance is given by kroos et al. [START_REF] Kroos | Towards a self-consistent model of the keyhole in penetration keyhole beam welding[END_REF][START_REF] Kroos | Dynamic behaviour of the keyhole in laser welding[END_REF]

v i d s p p p p p γ + + = + , (1)
where p v is the vapour recoil pressure, p i is the radiation pressure, p d is the hydrodynamic pressure due to a melt flow, p s is the hydrostatic pressure and p is the pressure due to surface tension. The right-hand terms try to close the keyhole whilst the left-hand terms have the opposite effect.

Calculations can be found in the literature to estimate the values of the above terms. [START_REF] Graham | Nd-Yag laser welding of coated sheets steel[END_REF][START_REF] Duley | laser Welding' 83-85[END_REF][START_REF] Aalderink | Keyhole shapes during laser welding of thin metal sheets[END_REF] It appears that p i and p s are small compared with other terms and for the purpose of the present discussion, can be neglected. The hydrodynamic pressure p d is ( )

2 2 max min 1 2 d p v v ρ = - , (2)
where is the liquid density and v max and v min are liquid flow velocities around the keyhole. The minimum velocity v min is assumed to be the welding speed v and the maximum velocity v max is calculated to be between 2 and 4 times the welding speed for welding speeds of 25 mm s -1 to 100 mm s -1 . [START_REF] Beck | Aspects of keyhole/melt interaction in high speed laser welding[END_REF][START_REF] Kaplan | Modellrechnung des laser-Tiefschweißens[END_REF][START_REF] Lampa | Fluid flow and resolidification in deep penetration laser welding[END_REF] Hence this term is also negligibly small (<0.5 kPa) at low to moderate welding speeds. For a cylindrical keyhole we can therefore write

v p p γ ≈ (3)
as a good first order description of the pressure balance. The surface tension pressure term is

p r γ γ = , ( 4
)
where is the surface tension coefficient = 1.6 N m -1 (typical for the range of mean pool surface temperatures experienced) [START_REF] Sahoo | Surface tension of binary metal -surface active solute systems under conditions relevant to welding metallurgy[END_REF][START_REF] Wang | Modeling of the effects of surface-active elements on flow patterns and weld penetration[END_REF][START_REF] Aalderink | Keyhole shapes during laser welding of thin metal sheets[END_REF] and r is the average keyhole radius determined from camera images to be ~0.26 mm. The surface tension pressure p is therefore of the order 6.2 kPa.

For the elongated keyhole, the keyhole rear wall is not significantly exposed to the laser beam due to the longer distance from the keyhole front wall, which explains the darker rear wall observed when welding zinc coated steels (see Figure 7). Based on geometrical considerations, radiation due to reflection from the keyhole front wall onto the rear wall may also be expected to be small, thus, there is little laser induced vaporization on the rear wall and the effect of p v is minor. The rear wall shape is mainly controlled by the surface tension of liquid metal, the hydrodynamic pressure due to the flow of molten steel and the zinc vapour pressure exerted on the rear keyhole wall. The zinc vapour impinges upon the rear keyhole wall restricting its forward motion. The resultant pressure p z plays an important role in maintaining an elongated keyhole. Fabbro et al. [START_REF] Fabbro | Study of CW Nd-Yag laser welding of Zn-coated steel sheets[END_REF] employed a simplified description of a free jet emitted from a nozzle, to estimate the pressure p z exerted on the keyhole rear wall. [START_REF] Schlichting | Grenzchicht Theorie[END_REF][START_REF] Abramovich | The Theory of Turbulent Jets[END_REF] The flow on the axis of the jet is

2 0 0 0 , z X p p x X x = ⋅ ≥ , ( 5
)
where x is the distance along the axis of the jet from the exit of the nozzle (keyhole length here), p o is the reservoir pressure that feeds the jet and X 0 is a characteristic distance for the decrease of the velocity field of the emitted flow. An evaluation of X 0 for turbulent flow is given by the following empirical equation 7 c 0 3 Re 16

D X = , ( 6
)
where D is a characteristic dimension of the nozzle exit, estimated here to be the total coating thickness at the interface, and Re c is the critical Reynolds number characterising the laminar to turbulent transition for free jet flows. A typical value for Re c of 40 is given in the literature. [START_REF] Schlichting | Grenzchicht Theorie[END_REF][START_REF] Abramovich | The Theory of Turbulent Jets[END_REF] It has been reported by Fabbro et al. [START_REF] Fabbro | Experimental study of the dynamical coupling between the induced vapour plume and the melt pool for Nd-Yag CW laser welding[END_REF] and Kamimuki et al. [START_REF] Kamimuki | Prevention of welding defect by side gas jet flow and its monitoring method in continues wave Nd:YAG laser welding[END_REF] that a side gas jet applied from a leading position can lead to a rearward melt flow. It is therefore reasonable to expect a similar effect when a vapour flow is applied from the middle of the keyhole front wall. Dasgupta et al. [START_REF] Dasgupta | Physics of zinc vaporization and plasma absorption during CO 2 laser welding[END_REF] have modeled the velocity field of the melt during laser welding of galvanized steel by considering the effect of zinc as an additional pressure. They show that the presence of zinc vapour increases liquid metal velocity by 30% when welding sheets with a 7 µm coating thickness, with a laser power of 3 kW at a welding speed of 31 mm s -1 . From the high speed images shown in Figure 7, it is observed that the melt pool at the sides of the keyhole become narrower when welding zinc coated steels, thus flow with higher velocity is required to maintain the mass balance. Hence the hydrodynamic pressure term p d needs to be taken into account. The magnitude of the hydrodynamic pressure, estimated based on the experimental observation of keyhole side walls, is given in table 2, which confirms that p d increases with increasing coating thickness. The pressure balance equation for an elongated keyhole is therefore

d z p p p γ + = . (7)
The surface tension term p is assumed to be constant and takes the same value calculated for the case of the cylindrical keyhole. This assumption is reasonable based on the observation that the radii of keyhole walls do not show significant differences when welding zinc coated or non-coated steels (see Figure 7). With known hydrodynamic pressure p d , the zinc vapour pressure exerted on the keyhole rear wall p z can be evaluated; p z decreases with increasing welding speed and coating thickness as shown in table 2. The surface tension term p is a constant and the hydrodynamic pressure term p d increases with the increasing welding speed or coating thickness, a decreasing p z is therefore required to balance both sides of (7). In addition, p z varies inversely as the square of the distance from the exit x according to (5). Combining (2), ( 5), ( 6) and (7), and replacing v min by the welding speed v, results in the pressure balance ( )

2 2 2 ec max 0 3 1 2 16 DR v v p p x γ ρ - + ⋅ = . (8)
The maximum velocity v max is controlled by mass conservation. The metal melted at the keyhole front wall should be transported to the melt pool behind the keyhole. The mass of liquid steel melted per unit time is

( 2 ) 2 p m w b v t ρ = + ⋅ ⋅ ⋅ & , ( 9
)
where w is the keyhole width, b is the width of liquid metal at the keyhole side wall (Figure10) and t p is the sheet thickness. The values of w and b, were measured from recorded images with a pixel resolution of ± 0.015 mm, and are given in table 2. The mass flow rate given by ( 9) must be balanced by the melt that flows around the keyhole (along the keyhole side walls), which is of the order max 2

2 p v b t ρ ⋅ ⋅ ⋅ , thus max 2 2 w b v v b + = ⋅ . ( 10
)
The calculated v max are given in table 2. In the case of bare steel, the results obtained are in agreement with the published calculated values [START_REF] Beck | Aspects of keyhole/melt interaction in high speed laser welding[END_REF][START_REF] Kaplan | Modellrechnung des laser-Tiefschweißens[END_REF][START_REF] Lampa | Fluid flow and resolidification in deep penetration laser welding[END_REF] (i.e., 2 to 3 times the welding speed). It is reasonable therefore to use this expression to estimate the flow speed when welding zinc coated steels. It is shown in table 2 that when welding sheets with the thicker coating, a higher melt flow velocity on the keyhole side wall is obtained, leading to a higher hydrodynamic pressure as discussed.

A number of estimations for the reservoir pressure p o can be found in the literature, [START_REF] Akhter | Laser welding of zinc coated steel[END_REF][START_REF] Fabbro | Study of CW Nd-Yag laser welding of Zn-coated steel sheets[END_REF] ranging from the saturation pressure of zinc vapour at the steel melting temperature (~5 MPa) to the ferrostatic head (~50 Pa). The reservoir pressure depends on the mass flow rate of zinc vapour given by , ,

z n z v v s z s m A v A v ρ ρ = = & , ( 11
)
where A n is the nozzle (escape channel) area, A s the area of the zinc vaporisation front and subscripts z, v and s refer to zinc and vapour and solid states respectively. Equating the reservoir pressure to the (vapour) dynamic pressure yields

2 , 2 2 1 , 2 , 1 2 s z s o z v v z v n A p v v A ρ ρ ρ = = . ( 12
)
Taking the first order assumption that the density of zinc vapour is proportional to the reservoir pressure; i.e., z,v = p o • º z,v /p atm , then

0 1 p k v ≈ , where , 1 , 2 s z s atm o n z v A p k A ρ ρ = . ( 13
)
Here v is the welding speed. p atm is the atmospheric pressure (1 bar), A n is the area of the gap at the keyhole front wall for zinc vapour escape, A s is the area of the zinc vaporization front. is the density and subscripts z, v and s refer to zinc and vapour and solid states respectively, and º z,v is the vapour density at standard pressure, which can be calculated using Clausius-Clapeyron equation.

The values of the variables involved in the equation are given in table 3. A n and A s are given by,

2 2 2 n c s v c w A t A r t π π = ⋅ ⋅ = ⋅ ⋅ (14)
where w and r v are defined in Figure 11. The keyhole width w is determined from the high speed images while r v is determined by temperature measurements performed at the interface between two sheets, and t c is the coating thickness. The values of the variables involved in the estimate of k 1 are given in table 3. An accurate evaluation of k 1 requires accurate data of the thermal field surrounding the weld and measurements of both the keyhole and zinc vapour jet. The estimate shown in table 3 is approximate because the nozzle area A n is affected by the reservoir pressure. When the pressure is less than the surface tension of the melt, this nozzle disappears. Hence the actual nozzle area A n in the case of 7 µm coating thickness is smaller than the calculated value. This is also supported by the low appearance rate of the vapour escape channel when welding with a 7 µm coating. According to equation 13, a decrease in A n results in a larger k 1 and p 0 . In contrast, for the case of the 20 µm coating thickness, A n can be larger than that calculated in table 3 because the vapour pressure makes a conically shaped nozzle as shown in Figure 6, leading to a reduction of k 1 and p 0 . Welding speeds also influence k 1 by decreasing A s with increasing welding speed. All these factors can result in some deviations from the estimation given in the table.

For simplicity, k 1 is treated as a constant of proportionality and chosen to match the measured values at a welding speed of 25 mm s -1 . The experimentally derived values of k 1 and p 0 are given in table 4. It can be seen that the values of k 1 shown in table 4 are generally of the same order as the calculated values shown in table 3. In addition, the differences in k 1 for the different coating thicknesses are supported by examining the order of magnitude of the surface tension of the melt acting to close the channel on the keyhole front wall. This is calculated using (4), with a local surface tension coefficient of 1.4 N m -1 (at 3000 K) [START_REF] Sahoo | Surface tension of binary metal -surface active solute systems under conditions relevant to welding metallurgy[END_REF] and radius of corresponding coating thickness, resulting in 7 × 10 4 Pa in the case of the 20 µm coating thickness and 2 × 10 5 Pa in the case of 7 µm coating thickness. The higher surface tension in the case of thinner coating explains the higher reservoir pressure required to open the channel. In this case, the vapour evacuation channel appears less often, and elongation is not sustainable because the delivery of zinc vapour is not consistent due to the none uniform coating, as shown in Figure2, resulting in significant keyhole fluctuations.

( )

ec 1 2 2 max 3DR 1 16 p - - 2 k v x v v γ ρ = ⋅ . ( 16
)
The calculated keyhole lengths are compared with measured values in Figure 12. Predictions from this simple analytical model are in good agreement with experimental measurements and the model provides an explanation for the major trends observed, including the elongation of the keyhole due to zinc vapour jetting (Figure 7) and extension of keyhole length with increasing welding speed and coating thickness (Figure 8). With increasing the coating thickness, both the characteristic dimension of the nozzle exit D and hydrodynamic pressure p d increase. These are consistent with the higher appearance rate of the zinc evacuation channel and the thinner side walls observed in the case of the 20 µm coating thickness. However, a smaller reservoir pressure p o is obtained with the thicker coating, which can be explained by the different nozzle sizes and smaller surface tension of the melt film on the keyhole front wall. A combined effect of these factors leads to a longer keyhole as shown in Figure 12. With increasing welding speeds, at a given coating thickness, p 0 and p d increase according to ( 13) and ( 2), as more zinc vapour needs to escape through the keyhole per unit time through the same nozzle size and the melt velocity around the keyhole increases when welding at higher speeds.

Conclusions

We have shown that laser welding of steel sheets with a 20 µm zinc coating thickness, in an overlap condition and without the introduction of a gap to vent the zinc vapour, can produce welds of acceptable quality. This surprising behaviour is linked to the development of an elongated keyhole with minimal positional fluctuations on the rear keyhole wall. The relatively high stability of the weld pool is ascribed to the consistent delivery of zinc vapour into the keyhole.

High speed coaxial visualization has shown that a zinc vapour evacuation channel is present on the keyhole front wall almost continuously during stable welding conditions and corresponding keyhole rear wall fluctuations are relatively small. Conversely, for sheets with a 7 µm zinc coating thickness, the vapour is found to be inconsistently emitted into the keyhole and the keyhole rear wall is subject to persistent and severe fluctuations.

For both coating thicknesses, relatively stable keyhole behaviour is observed at low welding speed, which is believed to be a result of a larger zinc vaporisation zone. More consistent zinc vapour generation can be expected and this in turn contributes to process stability and improved weld quality.

A simple analytical model has been developed to calculate the keyhole length and predictions show good agreement with experimental results, indicating that the dynamic pressure of the zinc vapour is responsible for elongation of the keyhole. A longer keyhole is obtained when welding sheets with thicker coatings or when welding at higher speeds.

Acknowledgment

This research was carried out under the project number MC8.05211 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl), the former Netherlands Institute for Metals Research.

Figure 1 :

 1 Figure 1: Experimental arrangement for high speed video visualization.

Figure 3 :

 3 Figure 3: Coating thickness variation of two GI coated steels.

Figure 4 :

 4 Figure 4: Fraction of acceptable weld for weld beads on 20 µm and 7 µm zinc coating thickness GI sheets and non-coated steel. Photographs show the cap and the root of the seams.

Figure 5 :

 5 Figure 5: Traverse cross-sections of welds made on (a) 20 µm, (b) 7 µm coating thickness coated sheets, and (c) bare steel with a laser power of 3 kW and travel speed of 45 mm s -1 .

Figure 6 :

 6 Figure 6: Longitudinal cross-sections of weld end craters showing (a) zinc flow into the keyhole (etched with 2% saturated nitric acid solution for 4 seconds) and (b) zinc trapped behind the keyhole wall (not etched). Processing power and speeds are 3 kW and 45 mm s -1 .

Figure 7 :

 7 Figure 7: Welding of non-coated and zinc coated steels at welding speeds of (a) 45 and (b) 25 mm s -1 . The first rows in each case show coaxial views of the keyhole at the top surface and the second rows show schematic keyhole profiles (laser power: 3 kW).

Figure 8 :

 8 Figure 8: Evolution of average keyhole lengths for (a) bare steel, (b) 7 µm and (c) 20 µm coating thickness GI coated steels, at welding speeds of 25 and 45 mm s -1 .

Figure 9 :

 9 Figure 9: Mean keyhole fluctuation magnitude for coated sheets at welding speeds of 25 and 45 mm s -1 .

Figure 10 :

 10 Figure 10: Data distributions in a bin (34 ms) for (a) 20 µm and (b) 7 µm coating thickness GI coated sheets (laser power: 3 kW, welding speed is 45 mm s -1 ).

Figure 11 :

 11 Figure 11: Definition of parameters w, b and r v .

Figure 12 :

 12 Figure 12: A comparison between the experimental and calculated values of keyhole lengths for welds made on GI zinc coated steels with 20 µm and 7 µm coating thicknesses. The error bars in figure represent the standard deviation of 10,240 measurements of keyhole lengths made over a period of 0.34s.

Table 1 :

 1 Composition of the base steel in wt%.

	Element	Al	C	Cr	Cu	Mn	Ni	P	S	Si	Ti	Fe
	wt%	0.026 <0.001 0.018 0.016 0.104 0.053 0.012 0.005 0.001 0.030 Bal.

Table 2 :

 2 Measured keyhole and weld pool properties and flow velocity and derived pressures for different welding speeds.

	Welding Speed	Keyhole / Pool Property	Bare steel	7 µm coated	20 µm coated
		L (mm)	0,56(±0.07) 0,72(±0.18) 0,93(±0.18)
		w (mm)	0.53	0.48	0.51
	25 mm/s	b (mm) v max (mm/s)	0.23 53 (±4)	0.15 68(±7)	0.04 175(±18)
		p d (Pa)	8	14	107
		p z (Pa)	NA	6.14 × 10 3	6.05 × 10 3
		L (mm)	0,57(±0.06) 0,82(±0.16) 1,17(±0.20)
		w (mm)	0.51	0.52	0.53
	35 mm/s	b (mm) v max (mm/s)	0.21 77(±5)	0.07 138(±19)	0.03 338(±32)
		p d (Pa)	17	64	404
		p z (Pa)	NA	6.09 × 10 3	5.75 × 10 3
		L (mm)	0,58(±0.05) 0,91(±0.20) 1,34(±0.18)
		w (mm)	0,52	0.49	0.48
	45mm/s	b (mm) v max (mm/s)	0.19 106(±9)	0.08 180(±25)	0.04 332(±35)
		p d (Pa)	33	108	385
		p z (Pa)	NA	6.05 × 10 3	5.77 × 10 3
		L (mm)	0,59(±0.06) 0,96(±0.22) 1,42(±0.24)
		w (mm)	0,53	0.53	0.49
	55 mm/s	b (mm) v max (mm/s)	0.16 144(±8)	0.07 223(±15)	0.04 419(±39)
		p d (Pa)	63	167	617
		p z (Pa)	NA	5.99 × 10 3	5.54 × 10 3

Table 3 :

 3 Variables involved in the calculation of reservoir pressure equation 13.

			7 µm coated			20 µm coated
	Welding speed (mm s -1 )	25	35	45	55	25	35	45	55
	w (mm)	From table 2, it can be seen that w is insensitive to welding speed and coating thickness. Here an average of 0.51 mm is used.
	r v (mm)	1.16	1.01	0.94	0,83	0,99	0,93	0.89	0,81
	A s /A n	4.5	4.0	3.7	3.2	4.1	3.8	3.5	3.1
	z,s (kg m -3 )				7140
	º z,v (kg m -3 )				1.8
	k 1 ( N s m -3 ) x 10 6	5.4	4.7	4.4	3.9	4.6	4.3	4.2	3.8
	p 0 ( Pa) x 10 5	1.4	1.7	2.0	2.2	1.2	1.5	1.9	2.0

Table 4 :

 4 Experimentally derived values of k 1 and p 0 obtained by matching the measured data at 25 mm s -1 .

			7 µm coated			20 µm coated
	Welding speed (mm s -1 )	25	35	45	55	25	35	45	55
	k 1 ( N s m -3 ) x 10 6		11.5				2.3
	p 0 ( Pa) x 10 5	2.8	4.0	5.1	6.3	0.6	0.8	1.1	1.3


Guillaume Ovarlez

Kapil Krishan

Sylvie Cohen-Addad

Investigation of shear banding in three-dimensional foams

Keywords: 83, 80, Iz -Emulsions and foams 83, 10, Gr -Constitutive relations

We study the steady flow properties of different three-dimensional aqueous foams in a wide gap Couette geometry. From local velocity measurements through Magnetic Resonance Imaging techniques and from viscosity bifurcation experiments, we find that these foams do not exhibit any observable signature of shear banding. This contrasts with two previous results (Rodts et al., Europhys. Lett., 69 (2005) 636 and Da Cruz et al., Phys. Rev. E, 66 (2002) 051305); we discuss possible reasons for this dicrepancy. Moreover, the foams we studied undergo steady flow for shear rates well below the critical shear rate recently predicted (Denkov et al., Phys. Rev. Lett., 103 (2009) 118302). Local measurements of the constitutive law finally show that these foams behave as simple Herschel-Bulkley yield stress fluids.

Introduction -Materials such as dense suspensions, colloidal gels, concentrated emulsions, foams or granular materials, present a jammed structure [START_REF]Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales[END_REF]. This results in the existence of a yield stress τ y below which they cannot flow. It was recently shown that some of these materials exhibit shear banding while others do not [START_REF] Dennin | [END_REF][3][4]. Shear banding means that in some conditions (near τ y ), in a homogeneous shear stress field, there is coexistence of liquid (sheared) and solid (unsheared) regions [3,5,6]. At the interface between both regions, the material flows at a nonzero critical shear rate 1 γc . This implies that shear banding materials cannot flow steadily at a shear rate smaller than γc . If one shears such systems between two boundaries at a low macroscopic shear rate γmacro < γc , local velocity measurements [7] then reveal shear banding [3]: the material splits into a region flowing at γc and a non flowing region, whose relative extent ensures that the shear rate spatial average is equal to γmacro . In macroscopic rheometric experiments, under applied stress, this leads to the viscosity bifurcation phenomenon [8,9]. This implies that, in addition to their yield stress, shear banding materi- 1 The situation is different in heterogeneous stress fields where shear localization occurs with any yield stress fluid. The shear rate at the interface between the sheared and unsheared regions is then naught for simple, non shear banding, yield stress fluids, whereas it is non-zero for shear banding materials [3]. als are also characterized by a critical shear rate γc , thus defining a timescale which must have its origin at a microscopic scale. Therefore, two different classes of jammed systems should be considered depending on whether they exhibit shear banding or not. To understand the physical origin of this phenomenon, it is thus of high importance to identify which materials do exhibit shear banding and which do not.

Shear banding of jammed systems is observed mostly in colloidal gels [5,6,10], where aggregation due to attractive interactions and thermally activated structuration mechanisms is in competition with shear. It is also observed in suspensions of noncolloidal particles where it results from competition between sedimentation-induced contact formation (which can be seen as an attractive force) and shear-induced resuspension [11]. Some yield stress fluids seem not to exhibit shear banding, although experiments only provide upper bounds on γc : neutrally buoyant noncolloidal suspensions [11], Carbopol gels [12], dense emulsions [13]. In all these cases, if a critical shear rate exists, it should be smaller than 10 -2 s -1 . Hence, it is suggested that attractive, thixotropic systems tend to develop shear banding whereas repulsive and non-thermal systems do not [3,10,12].

In this context, the case of foams is rather puzzling. Disordered foams and dense emulsions are both soft jammed packings of two immiscible fluids with similar structures and mechanical behaviors [14]. Their solid-like properties arise from their density of interfacial energy, and steady flow is accomplished by shear-induced structural rearrangements. Therefore, one could a priori expect foams to behave like emulsions with respect to shear banding. Shear banding in emulsions was reported once by Bécu et al. [15] but this result was later contradicted by Ovarlez et al. [13] who found γc < 10 -2 s -1 for several emulsions. It is now believed that the shear bands observed in [15] are transient shear bands (as observed in Carbopol gels [16]), which disappear in steady state. Several cases of shear banding in 2D and 3D foams have been reported in the literature. Many observations of shear banding in 2D foams are now attributed to viscous damping at the glass boundary [17][18][19][20] when the bubbles are confined by one (or two) glass plate. The case of bubble rafts is still unclear. On the one hand, Gilbreth et al. [21] have reported a consistent set of shear banded velocity profiles characterized by a unique γc ; these profiles are claimed to be consistent with a single continuum model in a 10 to 20 bubbles-wide zone, although no stress measurement is reported. On the other hand, Katgert et al. [20] do not observe any signature of shear banding for rafts of equivalent bubble size to gap ratios. Moreover their velocity profiles are not consistent with a single constitutive law, but can be described by a non-local model, such as what is found in confined emulsions [22].

The only measurement of local bulk velocity profile we are aware of in steadily sheared 3D foams was performed by Rodts et al. [23] and revealed strong shear banding. Velocity profiles were measured by MRI in a wet foam (gas volume fraction φ = 92%) sheared in a Couette geometry. At low rotational velocity of the inner cylinder, flow localization was observed with a discontinuity of the shear rate at the interface between the sheared and the unsheared regions, thus providing a critical shear rate γc ≃ 5 s -1 . Consistently, macroscopic strain rate measurements by Da Cruz et al. [9] in a similar 3D foam sheared in a homogeneous stress field (cone and plate geometry) showed a viscosity bifurcation, characterized by a critical shear rate γc ≃ 10 s -1 .

To account for the observed shear banding in 3D foams and predict the value of γc , Denkov et al. have proposed a model based on the thinning dynamics of the films between bubbles [24]. Shear flow induces rearrangements thus renewing bubble contacts. New films thin due to the capillary pressure set by the curvature of the Plateau borders. As their thickness reaches the range of van der Waals interactions, the film may thin abruptly to a Newton black film (of a few nm thickness) providing a strong adhesion between the bubbles and locally jamming the bubble packing. A critical shear rate then results from the competition between the timescale of shear-induced rearrangements and that of the Newton black film forma-tion. It reads: γc = 1.9

T 3/7 A 4/7 H η d 15/7 (1 -φ) 0.3 (1)
with A H the Hamaker constant (A H ≃ 4 10 -20 J), T the surface tension, η the viscosity of the solution, and d the average bubble diameter. For a given φ, γc does not vary much with T and exhibits a marked dependency with η and d . For instance, the value of γc expected for a typical Gillette foam is 10 s -1 (with T mN/m, η= 1.9 mPas, φ=92% and d = 40 µm). This prediction (see Fig. 1c) seems to be in good agreement with MRI data [23] and with recent qualitative measurements [24].

Finally, these results tend to show that 3D foams are not simple yield stress fluids, in contrast with dense emulsions. However, whereas solely local measurements of bulk velocity profiles can unambiguously determine the existence of steady shear banding in 3D foams, only one such measurement exists in the literature [23]. In this letter, we present a set of new experiments in 3D foams, with different gas volume fractions, bubble size and interfacial rheological properties. Local velocity profiles measured using MRI show that, in contrast with [9,23], 3D foams do not exhibit any observable shear banding. Consistently, they do not exhibit viscosity bifurcation. They appear to behave as simple Herschel-Bulkley fluids, as previously found in emulsions [13].

Materials and methods -We study a foam based on a mixed surfactant aqueous solution, denoted SLES. It contains Sodium Lauryl-dioxyethylene Sulfate (Stepan Co., USA) with concentration 0.33% g/g, Cocoamidopropyl Betaine (Goldschmidt, Germany) with concentration 0.17% g/g and 60% g/g Glycerol (Fluka, anhydrous p.a. 99.5% GC). The chemicals are dissolved in water (millipore milli-Q). The surface tension and the viscosity of this solution at 21±1˚C are T = 31.0 mN/m and η = 10.8 mPa s [25]. Foam is produced by simultaneously flowing the SLES solution and pressurized nitrogen gas saturated with perfluorohexane vapor through a glass bead column [26]. By adjusting the liquid flow rate and gas pressure, we produce samples with different controlled gas volume fractions φ. Using videomicroscopy, we characterize the average d of the bubble diameter d distribution and its dimensionless standard deviation µ = d 2 -d 2 / d for each foam sample. We prepare three materials with same d = 73 µm (and µ comprised between 0.5 and 0.6), of different φ: (i) (88.4 ± 0.3)%, (ii) (92.5 ± 0.3)% , and (iii) (95.3 ± 0.3)%. In addition, we have a sample with d = 45 µm, µ = 0.7, and φ = (92.5 ± 0.3)%. We also use Gillette shaving cream (Normal Regular), denoted Gillette NR, as in previous studies [27] for the sake of comparison with [9], with φ = (92.0 ± 0.5)%, d = 41 µm and µ = 0.6. Its constitutive foaming solution has T = 28.6 mN/m and η = 1.9 mPa s [25]. For the sake of comparison with [23], we use another Gillette shaving cream (Haute Protection) denoted Gillette HP. As the rigidity of the liquid-gas in- data from [24] data from [23] this study

Critical shear rate (s terfaces plays a role on the behavior of steadily sheared foams [28], it is interesting to test the influence of this parameter on shear banding. This can be done from our experiments (at φ = 92% and d ≃ 40µm) since in contrast to SLES foam, Gillette NR foam has rigid liquid-gas interfaces [25,28]. The material's local behavior is studied in a wide-gap Couette geometry (inner cylinder radius R i =4.1 cm; outer cylinder radius R o =6 cm; height H=11 cm). The use of a wide gap likely prevents (non-local) finite size effects [13,20,22]. Sandpaper is glued to the walls to avoid slippage; there is no observable slip in the velocity profiles. The rheometer is inserted in a 0.5-T vertical MRI spectrometer (24/80 DBX by Bruker). In all experiments, the velocity Ω of the inner cylinder is controlled. We measure the local velocity in the flowing sample for various constant Ω ranging from 5 to 100 rpm; the torque exerted by the material on the inner cylinder is measured using a Bohlin C-VOR 200 rheometer. The orthoradial velocity profiles V θ (R) are obtained through MRI techniques as described in [29,30]. The MRI setup also allows measurement of the local water content [30]. In all experiments, we checked that drainage can be neglected (it led to absolute volume fraction variations of order 0.5% in the 4cm high measurement zone over the duration of the experiments). We also checked that there is no observable shear-induced radial migration of water, as in emulsions [13].

As a complement and to allow for a direct comparison of our data with previous ones [9], we have performed macroscopic viscosity bifurcation experiments. Such macroscopic experiments are convincing only if the stress field is homogeneous; we thus use a 4˚cone and plate geometry (as in [9]), with serrated surfaces of 30 mm radius; the stress heterogeneity is then of the order of 0.5%. Note however that the gap is small (the gap at the edge is 50 bubbles high); therefore, one cannot exclude finite size (non-local) effects [20,22]. Experiments consist in first applying a preshear at high shear rate (20 s -1 ) during 60 s, ensuring that the whole material flows initially, before applying a constant stress for 150 to 300 s. The macroscopic shear rate is then plotted vs. time to check for the existence of a steady flow.

Velocity profiles and shear banding -In this section, we focus on the stationary velocity profiles. In Fig. 1a, we plot the dimensionless velocity profiles V (R)/V (R i ) obtained with SLES foam (φ = 92%) for various rotational velocities Ω. All the studied foams exhibit similar behavior. We observe that the material is sheared only in a fraction of the gap at low Ω: V (R) vanishes (within the measurement uncertainty) at some radius R c (Ω) < R o . R c increases with Ω. Beyond a critical velocity (that depends on the foam and is of order 40 rpm in Fig. 1a), the whole sample is sheared.

To analyze these observations, one should first be reminded that the shear stress distribution τ (R) is heterogeneous: it decreases with increasing radius R and reads

τ (R) = τ (R i ) R 2
i /R 2 from stress balance equations. When shear extends over the whole gap, the velocity profiles differ from those of a Newtonian fluid. Their strong curvature is due to the stress heterogeneity (since τ (R i )/τ (R o ) = 2.1) and is typical of shear-thinning fluid: the shear rate decreases more rapidly within the gap (i.e. when τ decreases) than for a Newtonian fluid. The shear localization observed at low velocity is a feature of yield stress fluid flow in Couette geometry. At low Ω (or applied stress slightly above the material yield stress τ y ), the flow stops at a radius R c within the gap where the local shear stress τ (R) equals τ y (i.e. R c = R i τ (R i )/τ y ). The decrease of R c (Ω) as Ω is decreased then comes from the rate dependence of the constitutive law at the approach of τ y . Note the discrepancy between this observation and that of Katgert et al. 2D foams [20] where dimensionless ve-locity profiles superpose and extend over the whole gap at any low Ω. We will discuss this differences in the section devoted to the constitutive law measurements.

From the shear localized velocity profiles, we now analyze the possible shear banding behavior of the studied foams. Two cases should be considered. On the one hand, if the material is a shear banding material, the shear rate γ(R) should tend towards a critical shear rate γc = 0 as τ approaches τ y i.e. as R tends to R c . As γ(R) = V /R -dV /dR, the local velocity should tend to zero with a non-zero slope |dV /dR| = γc , independent of the velocity at the inner cylinder. Shear banding should then result in a discontinuity of the slope of the velocity profile at the interface between the sheared and the unsheared regions, as in a homogeneous stress field [3]. This is precisely as observed by Rodts et al. [23] (see Fig. 1b). For various Ω, they found a consistent set of shear banded velocity profiles with a slope γc ≈5 s -1 at the interface. On the other hand, if the material is not a shear banding material, γ(R) should tend continuously to zero as τ approaches τ y i.e. as R tends to R c . This would mean that dV /dR = 0 at the interface between the sheared and the unsheared regions i.e. the velocity profile should tend smoothly to zero.

In Fig. 1b, we plot shear localized velocity profiles obtained by Rodts et al. and by us corresponding to a same position of R c (here ≃ 5.4 cm). We focus on the shape of the velocity profiles at the interface between the sheared and the unsheared regions. From this plot, it is clear that, in contrast with the Rodts et al. data, all of our systems exhibit a smooth transition from flow to rest and thus seem to be not shear banding materials. Of course, due to finite experimental resolution, we can never be certain that no critical shear rate exists. We can only provide upper bounds on the critical shear rate -if any -as shown in Tab. 1. Note that the accuracy of the shear rate measurement depends on the spatial resolution and on the MRI signal to noise ratio, which depends itself on the foaming solution and on external factors. The best experimental conditions were met with the 92% SLES foam, and led to a rather low upper bound ≈ 0.04 s -1 . Viscosity bifurcation -It is striking that we find upper bounds much lower than the critical shear rates γc previously reported [9,23]. From macroscopic measurements, Da Cruz et al. evidenced a viscosity bifurcation in the Gillette NR foam [9], with γc ≈10 s -1 . To understand if viscosity bifurcation experiments can be reconciled with our local measurements, we have performed these experiments on some of our systems, including the Gillette NR foam of [9]. We focus on measurements obtained for applied stress close to the yield stress to estimate the lowest steady state shear rates than can be achieved. Fig. 2 shows that steady state flow can be achieved under controlled stress conditions at shear rate values as low as 0.1 s -1 for the two Gillette foams and 0.02 s -1 for the SLES foam. Further experiments would have to be performed to improve the estimate of the upper bound of γc from these indirect measurements. At this stage, the conclusion is that local and macroscopic measurements are consistent, and lead to the same result: the absence of observable shear banding in all foams we studied.

Foam

Analysis -We now question the possible origin of the differences between our results and the previous ones, obtained on the same systems (the Gillette foams of [9,23]) using the same experimental displays. First, Da Cruz et al. [9] report a 200 s -1 preshear; we observe that applying such high shear rate to the Gillette NR foam with the geometry used in [9] leads to expelling a large amount of the material from the gap by inertial forces (this happens for shear rates higher than 30 s -1 ); therefore the protocol was inappropriate, leading to incorrect rheological data. Then the yield stress reported is ≈180 Pa, a value much higher than values reported in the literature [14,27,31]. This would mean that the material on which the measurements were performed in [9] was not what it was supposed to be. The case of the Rodts et al. data [23] is less obvious. They report a 7 Pa yield stress, which is lower than data from the literature [14,27,31]. Such a low value may indicate a material that is wetter than it should be, yet it is unclear why this would lead to shear banding.

It was recently shown that some non-shear-banding materials may exhibit transient shear banding [16] when a low velocity is applied after a short resting period; this is not observed when the low velocity is applied just after a high velocity. In order to check that the observations of Rodts et al. were not due to such a transient effect, we have studied the role of the shear history. MRI velocity measurements were performed under various conditions, when going from high to low velocities, from low to high velocities, and when rotating directly after a long resting time: all procedures were found to provide the same velocity profiles, i.e. no shear banding (even transient) was observed. The occurrence of shear banding in [23] might be due to uncontrolled traces of impurities in the system, e.g. clay particles, which were also studied with the same equipment. It was indeed shown that tiny amounts of colloidal clay particles dispersed in the continuous phase of a simple emulsion could form bridges between droplets and lead to thixotropic effects and shear banding [32]; the same might happen with a foam. Finally, let us stress that we provide here a consistent set of data performed with five different foams, including the Gillette foams of [9,23], with two different experiments. The conclusion is the absence of observable shear banding in all the foams we studied.

We now compare our upper bounds on γc with the model of Denkov et al. [24] (Eq. 1). Fig. 1c shows these bounds vs. the bubble radius for SLES foam (φ = 92%) as well as the γc value reported in [23]. It also shows the data obtained by Denkov et al. with foams of the same surfactants as in SLES foam mixed with myristic acid. Note that these last data were obtained in a 8 to 15 bubbleswide gap, a case where finite size effects may be observed [20]. When the timescale of shear 1/ γ is lower than the duration of bubble rearrangements, there may be stress heterogeneities at the bubble scale, leading to shear banding in a thin layer [33], which is not bulk shear banding.

There is a strong discrepancy between the model and our data; the model overestimates the critical shear rate γc for all the studied foams, as shown in Tab. 1. This means that, while the proposed mechanism in [24] is certainly relevant in some cases, it may not apply to the foams made from ionic surfactant solutions without salt addition that are used in this study. Liquid films actually thin until the balance between capillary and disjoining pressures is reached, and form equilibrium common black films of thickness in the range 10-100 nm rather than very thin Newton black films. Moreover, the model assumes rigid interfaces, which sets the boundary conditions of the film thinning flow profile. However mobile interfaces, as in SLES foam, should lead to faster thinning, and thus to even higher critical shear rate than with rigid interfaces. This qualitative argument shows that the interfacial rigidity cannot account for the difference between the experimental upper bound of γc and Eq. 1. More experiments, in systems with strong attraction between the films, are needed to further test the model.

Nevertheless, as noted in the introduction, the absence of shear banding in foams is consistent with the basic mechanisms that tune shear banding in other complex fluids [3]. In particular, it is consistent with observations in dense emulsions [13].

Constitutive law -The constitutive laws accounting for the materials' velocity profiles can be built from our experimental data. From the torque measurements T and the stress balance equations, one gets the shear stress distribution in the gap τ (R) = T /(2πHR 2 ). The local shear rate γ(R) in the gap is inferred from the velocity profiles V (R) through γ(R) = V /R -dV /dR. Both measurements performed at a given radius R for a given rotational velocity Ω thus provide a data point of the local constitutive law. We checked that the materials remain homogeneous unpon shear, which allows us to combine the data measured at various radii. The local constitutive laws τ ( γ) obtained for two of the studied foams are plotted in Fig. 3a and3b.

We find that the local flows curves obtained from experiments performed at various Ω are consistent with a single -local -constitutive law for each foam. This contrasts with recent results in 2D sheared foams [20] where no single local constitutive law can account for flow profiles obtained at different Ω. The same difference -non-local and local constitutive laws -is observed with emulsions flows in microchannels [22] or in our wide gap Couette geometry [13]. As in [13,22], the main difference between our observations and those of Katgert et al. [20] may stand in the bubble size to gap ratio (i.e. nonlocal effects are finite size effects). The gap of our geometry is indeed about 500 bubbles wide, whereas the gaps used in [20] are 20 to 40 bubbles wide. Leaving these possible finite size effects apart, it can finally be concluded that foams, together with emulsions [13] and Carbopol gels [12], belong to the class of simple yield stress fluids [3].

The local constitutive laws we obtain are well fitted to Herschel-Bulkley laws, with exponents consistent with those of foams with rigid (n=0.27) or mobile (n=0.46) interfaces [28]. We also remark that the Herschel-Bulkley consistency η HB of the SLES foam is in fair agreement with the Denkov et al. prediction (see Eq. 5 of [28]). Note finally that, due to the limited range of low strain rates measurements, the yield stress for Gillette NR foam is probably underestimated Conclusion -Our measurements demonstrate that three-dimensional foams do not exhibit observable signatures of shear banding, in contrast with previous observations [9,23]. The question remains open whether steady shear banding may occur in extremely dry foams since it has been observed upon shear start-up [26]. Our results disagree with the recent model of Denkov et al. [24], which is shown here to overestimate the critical shear rate -if any -of the foams we studied. Further experiments with strong attraction between films would be needed to test the mechanism proposed in [24]. While nonlocal effects have been recently evidenced [20], we have finally shown that the constitutive law of foams measured locally in a wide gap geometry is that of simple Herschel-Bulkley yield stress fluids, as emulsions [13] and Carbopol gels [12]. Experiments at both small and large scales on a same system, as in of Goyon et al. [22], remain to be performed to test the non-local modelling of 3D foam flows. * * *

Fig. 1 :

 1 Fig.1: a) Dimensionless velocity profiles V (R)/V (Ri) for the steady flows of SLES foam (φ = 92%, d = 45 µm), at various rotational velocities ranging from 5 to 80 rpm; the solid line is the theoretical profile for a Newtonian fluid. b) Comparison between localized velocity profiles of the foams we studied and that of Rodts et al.[23] (see φ and d in legend). c) Critical shear rate vs. bubble radius for foams with φ = 92-93%: Denkov et al. data[24] (squares), Rodts et al. data[23] (star); the hatched bars show the intervals of possible critical shear rates -if any -for the SLES foams in this study; the lines are Eq. 1 with AH = 4 10 -20 J (dashed line: T = 22.3 mN/m, φ = 93%, η = 3.8 mPa s; solid line: T = 31.0 mN/m, φ = 92%, η = 10.8 mPa s).

Fig. 2 :

 2 Fig. 2: Viscosity bifurcation experiments: shear rate vs. time at constant controlled stress in a cone and plate geometry. Thick lines: SLES foam with φ = 92% and d = 45µm (in grey scale, from black to light grey: 65 Pa, 45 Pa, 30 Pa, 20 Pa). Thin lines: Gillette HP as in [23] (black: 60 Pa, grey: 35 Pa). Empty symbols: Gillette NR as in [9], with d = 30µm (squares: 60 Pa, circles: 50 Pa).

Fig. 3 :

 3 Fig. 3: Local constitutive laws measurements of a) Gillette NR foam ( d = 41 µm), and b) SLES foam (φ = 92%, d = 45 µm). Each symbol corresponds to local measurements performed at a same rotational velocity. The solid lines are Herschel-Bulkley fits to the data τ = τy + ηHB γn , with τy = 24Pa, ηHB = 25Pa.s n , n = 0.27 for the Gillette NR foam, τy = 32Pa, ηHB = 12Pa.s n , n = 0.46 for the SLES foam.

Table 1 :

 1 Upper bounds on the critical shear rate γc obtained from MRI measurements and predictions of Eq. 1 for all foams.

		φ	d	Upper bound	Model
				on γc	(Eq. 1)
	SLES	92% 45 µm	0.04 s -1	1.5 s -1
	SLES	88% 73 µm	0.2 s -1	0.45 s -1
	SLES	92% 73 µm	0.2 s -1	0.52 s -1
	SLES	95% 73 µm	0.3 s -1	0.60 s -1
	Gillette NR 92% 41 µm	0.2 s -1	9.6 s -1
	Gillette HP	-	-	0.6 s -1	-

We thank P. Coussot and S. Rodts for providing their data and for open discussions on their papers. We also thank R. Höhler and N. Denkov for fruitful discussions, and H. Sizun for crucial technical help. We gratefully acknowledge financial support from E.S.A. (MAP No. AO99-108: C14914/02/NL/SH).