Christophe Almarcha

Bruno Denet

Joel Quinard
email: quinard@irphe.univ-mrs.fr

Denet B Almarcha C Quinard

Premixed flames propagating freely in tubes

Keywords: Premixed flame speed, front dynamics, instabilities, nonlinearity, Darrieus-Landau instability

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Premixed flames propagating freely in tubes

Introduction

The role of the hydrodynamic flame instability for premixed turbulent burning in gas turbine, spark ignition engines or industrial burners has been widely discussed [START_REF] Clavin | Premixed Combustion and Gasdynamics[END_REF][START_REF] Bychkov | Dynamics and stability of premixed flames[END_REF]. This effect is caused by thermal expansion through the flame front and is the most essential factor in the flame instability, together with the diffusivethermal effect that stabilizes the short-wavelengths perturbations depending on the Lewis number of the deficient reactant. Now, there is a mounting evidence that the flame instability is of primary importance for turbulent burning, leading for example to an increase by a factor of 10 of the flame propagation velocity of free expanding turbulent spherical flames [START_REF] Yu | Self-similar propagation of a free turbulent flame in premixed gas mixtures[END_REF][START_REF] Bradley | Flame Acceleration Due to Flame-Induced Instabilities in Large-Scale Explosions[END_REF]. It was also suggested that this instability plays an important role in the rapid increase of U t /U L with pressure in high-pressure environments [START_REF] Kobayashi | Burning Velocity of Turbulent Premixed Flames in a High-pressure Environment[END_REF], where U t /U L is the ratio of the turbulent flame speed to the laminar flame speed. Even weakly wrinkled flames maintain values of U t /U L ~ 3-4 down to very low values of the turbulence level u'/U L [START_REF] Savarianandan | Burning velocity of premixed turbulent flames in the weakly wrinkled regime[END_REF]. These combustion instabilities are relatively well described at the onset of instability, at least for some pure fuels, and can be handled in some direct numerical simulations, but it would be a huge task to account for them in actual burners since the characteristic length scale of the flow exceeds the typical flame thickness by many orders of magnitude. The theoretical analysis of these flames has been successful: nonlinear model equations have been derived, first by Sivashinsky in 1977 [START_REF] Sivashinsky | Nonlinear analysis of hydrodynamic instability in laminar flames ; Part 1 : Derivation of basic equations[END_REF] and by Frankel in 1990 [START_REF] Frankel | An equation of surface dynamics modelling flame fronts as density discontinuities in potential flows[END_REF]. Simple solutions of the Sivashinsky equation have been obtained in 1985 by Thual Frisch and Henon [START_REF] Thual | Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts[END_REF], and building on this discovery G. Joulin has studied theoretically large scale flame fronts in a number of configurations, spherical flames, oblique flames or flames of overall parabolic shape [START_REF] Joulin | On model evolution equations for the whole surface of threedimensional expanding wrinkled premixed flames[END_REF][START_REF] Joulin | On the hydrodynamic stability of curved premixed flames[END_REF][START_REF] Joulin | Potential-flow models for channelled two-dimensional premixed flames around near-circular obstacles[END_REF][START_REF] Denet | Wrinkled flames and geometrical stretch[END_REF]. Numerical simulations of the Sivashinsky equation have been performed in these configurations, as they are hundred times faster than direct numerical simulations, permitting the study of very large flames [14][START_REF] Karlin | Asymptotic modelling of self-acceleration of spherical flames[END_REF][16][17]. The qualitative agreement of these simulations with experiments is excellent [START_REF] Searby | Comparison of experiments and a nonlinear model equation for spatially developing flame instability[END_REF]. The observed dynamics is extremely complicated, involving cells on the flame front that continuously appear and then merge. This phenomenon possibly leads to a selfsimilar propagation of the flame with a fractal structure observed on free expanding spherical flames [START_REF] Yu | Self-similar propagation of a free turbulent flame in premixed gas mixtures[END_REF][START_REF] Bradley | Flame Acceleration Due to Flame-Induced Instabilities in Large-Scale Explosions[END_REF] or in numerical simulations [START_REF] Cambray | On a Scaling Law for Coarsening Cells of Premixed Flames: an Approach to Fractalization[END_REF][START_REF] Mukaiyama | Fractal structures of hydrodynamically unstable and diffusive-thermally unstable flames[END_REF]. It has been proposed [START_REF] Savarianandan | Burning velocity of premixed turbulent flames in the weakly wrinkled regime[END_REF][START_REF] Cambray | On a Scaling Law for Coarsening Cells of Premixed Flames: an Approach to Fractalization[END_REF][START_REF] Mukaiyama | Fractal structures of hydrodynamically unstable and diffusive-thermally unstable flames[END_REF][START_REF] Akkerman | Velocity of weakly turbulent flames of finite thickness[END_REF][START_REF] Kadowaki | The unstable behavior of cellular premixed flames induced by intrinsic instability[END_REF] that weakly turbulent premixed flames also have a fractal character in other geometries.

All these approaches refer to a characteristic length scale generally defined as the cut-off length scale λ c due to the balance between Darrieus-Landau (DL) hydrodynamic instability and stabilizing effects of thermal diffusion. This characteristic length is the one and only parameter in theoretical models. It can be evaluated to 10-40 d L from linear theories [START_REF] Clavin | Weakly turbulent wrinkled flames in premixed gases[END_REF] where d L is the laminar flame thickness, but numerical solutions of these models can hardly be compared to experimental results generally obtained in turbulent flows [START_REF] Savarianandan | Burning velocity of premixed turbulent flames in the weakly wrinkled regime[END_REF][START_REF] Gülder | Inner cutoff scale of flame surface wrinkling in turbulent premixed flames[END_REF][START_REF] Renou | Local scalar flame properties of freely propagating premixed turbulent flames at various Lewis numbers[END_REF][START_REF] Bychkov | The nonlinear equation for curved flames applied to the problem of flames in cylindrical tubes[END_REF][START_REF] Liberman | Numerical studies of curved stationary flames in wide tubes[END_REF].

The best way for checking these models and the whole literature on the subject is to study premixed flames propagating freely in quiescent gases. The movement of a flame element will then only results from its own propagation at the normal flame speed U L and from its advection at the local gas velocity produced by hydrodynamical effects induced by the whole flame shape. A recent experiment showed that it was possible to observe an almost stationary downwards propagation of premixed methane-and propane-air flames in tubes of various diameters, thanks to a new type of acoustic damper that eliminates the thermo-acoustic instability which otherwise would produce large acceleration effects on the flame and change drastically the flame propagation [START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF]. The attention was paid to the velocity of propagation of axis-symmetrical flames to be compared to self-similar relations describing free flame propagation in tubes [START_REF] Bychkov | The nonlinear equation for curved flames applied to the problem of flames in cylindrical tubes[END_REF][START_REF] Liberman | Numerical studies of curved stationary flames in wide tubes[END_REF]. The normalized self-turbulent flame speed U n = U t /U L increased up to 2.5 while the tube diameter increases from 36 to 141 mm, and its variation with the equivalence ratio of the gas mixture was qualitatively in agreement with a self-similar relation using as cut-off wavelength the shortest unstable wavelength calculated from linear theories. The fractal exponent was close to 1/3. However, slanted flames were also observed with a much larger propagation speed, a result not expected from Sivashinsky simulations where all solutions have the same speed and for which the more slanted flame is less stable [START_REF] Denet | Stationary solutions and Neumann boundary conditions in the Sivashinsky equation[END_REF]. The present work aims at a better description of these two regimes of propagation by measuring simultaneously the instantaneous position of the leading flame bump and its radius of curvature. It will be shown that the flame speed is well correlated to the curvature at the leading bump. The characteristic radius of curvature at the leading bump are then related to the cut-off wavelengths and compared to the results of Michelson-Sivashinsky simulations.

Experimental setup and procedure

The propagation velocity of laminar cellular flames is measured in vertical Pyrex tubes, 1.5 m long, with internal diameters D 0 of 54 and 94 mm. Flames in tubes with smaller diameter would be affected by heatlosses while tubes with larger diameter can lead to secondary instabilities on the flame [START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF]. The equivalence ratio, φ, of the premixed gas is controlled via a PC-interface connected to mass-flow regulators. The flame propagation is recorded using a high-speed video camera viewing the reflected images of the flame on two lines of sight at right angle (Fig. 1). The acoustic damper at the bottom part of the burner consists of a small annular slit, of height h and length l, that dissipates acoustic energy by terminating the tube with a real (resistive) acoustic impedance equal to the characteristic acoustic impedance of free air (see ref. [START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF] for details). A lightweight refractory fabric is placed over the open end of the burner to prevent mixing with ambient air while permitting the exhaust of the premixed or burnt gas. The procedure is as follows: after each run, the air flow is opened and maintained until the tube walls have cooled to ambient temperature. The flow of combustible, methane or propane, is then adjusted to the desired equivalence ratio and maintained for a time corresponding at least to ten fillings of the tube to ensure a homogeneous mixture. It is then stopped by closing the valve at the bottom of the burner and a delay of at least one minute is allowed before ignition to ensure that premixed gases are at rest. The ignition of the gas mixture is then performed by way of the fast discharge of a capacitor in a 0.1 mm radius steel wire that can be placed either on the burner axis to favour axis-symmetric flame shapes, or close to the wall to obtain slanted flames. About five runs are performed in each configuration: φ = (0.8, 1, 1.3), tube diameter = (54, 94) mm and axial or lateral ignition at the top of the burner. The video is generally recorded

!"##$%&'($ $)'*+,'&&-, ./ )"0-," 12 ! " !"##$%&'($ $)'*+,'&&-, 34, 56-& !4,,',#
at a frame rate of 250 images per second. The video movies are post-processed using ImageJ 1.45r software to obtain the instantaneous position of the upstream tip of the flame and to calculate the radius of curvature in each orthogonal section of the flame (Fig. 2). The effective radius of curvature is then simply obtained by 2/R f = 1/R 1 + 1/R 2 The self-turbulent flame speed is obtained by a linear fit of the displacement of the upstream flame tip over a distance at least equal to 0.5 m.

Michelson-Sivashinsky simulation

In this section, we give a brief presentation of the Michelson-Sivashinsky equation (or Sivashinsky equation, depending on the authors) that will be compared to experimental results. This equation is a model equation [START_REF] Sivashinsky | Nonlinear analysis of hydrodynamic instability in laminar flames ; Part 1 : Derivation of basic equations[END_REF] which describes the non linear behaviour of a flame submitted to a Darrieus Landau instability. We will write this equation in a one-dimensional case. Let us imagine a two-dimensional premixed flame propagating towards fresh gases (at the bottom of the domain) in the y direction. The transverse coordinate will be denoted by x, and the following equation for the front position !(!, !) can be derived (the flame laminar velocity is equal to 1, the width of the domain to 2!)

φ t + (1/2)φ x 2 = I(φ) + νφ xx + u(x, t)
orthogonal section of the flame (Fig. 2). The effective radius of curvature is then simply obtained by

2 R eff = 1 R 1 + 1 R 2
The self turbulent flame speed is obtained by a linear fit of the displacement of the upstream flame tip over a distance at least equal to 0.

Results

The first interrogation was on the existence of slanted flames : do they evolve with a well defined velocity of propagation slightly larger than the symmetrical ones as suggested by theoretical approaches [START_REF] Bychkov | Dynamics and stability of premixed flames[END_REF][START_REF] Joulin | On the hydrodynamic stability of curved premixed flames[END_REF]14], or is there a continuum of solutions as those obtained from the Sivashinsky equation [START_REF] Gülder | Inner cutoff scale of flame surface wrinkling in turbulent premixed flames[END_REF]? The flame trajectories in the tubes appear relatively rectilinear with two distinct velocities of propagation well correlated to the symmetry of the flame (Fig. 3), even if the flame can transit from one mode of propagation to the other one during the experiment. Lateral ignition produces generally slanted flames which propagate faster than almost symmetrical flames obtained after axial ignition. The same trends are observed with propane-air flames (Fig. 4), but the situation is not always clear : rich propane flames are always slanted in the smaller tube, whereas lean and rich methane-air flames are almost symmetrical.

Figure 3 : Trajectories of flames in a tube as a function of initial conditions for ignition I(φ) is a pseudo differential operator (the Landau operator) corresponding to multiplication by |k| (the wave vector) in Fourier space. The first two terms on the right hand side of the equation correspond to the Darrieus Landau linear dispersion relation, u(x,t) is a forcing term. The only non-linear term (1/2)φ x 2 is geometrical and describes the normal propagation of the flame at the laminar velocity (taken to be 1 here). It is extremely simple to solve this equation numerically with Fourier pseudo spectral methods, one example will be given in the following section. Let us note that the only non-dimensional parameter of the equation is !, it controls the number of unstable modes in the domain. In this description what is important is the ratio domain width/ cut-off length scale and is equal to 1/!. We now summarize some properties of the Michelson Sivashinsky equation. It has been found in [START_REF] Thual | Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts[END_REF] that simple solutions of this equation exist, described by a certain number of poles in the complex plane. The typical number of poles of the solution (called optimal number of poles in [START_REF] Denet | Stationary solutions and Neumann boundary conditions in the Sivashinsky equation[END_REF]) increases when the number of unstable modes 1/! increases, so that as the ratio domain width/ cut-off length scale is increased, the solution becomes more and more complicated. Among the many solutions that have been found in [START_REF] Denet | Stationary solutions and Neumann boundary conditions in the Sivashinsky equation[END_REF], some solutions are linearly stable, see particularly figure 1 of [START_REF] Denet | Stationary solutions and Neumann boundary conditions in the Sivashinsky equation[END_REF], where slanted or almost symmetrical solutions (see the experimental results of the next section for similar solutions in the cylindrical tube) have been obtained. One interesting property of these solutions found in this article is that a moderate noise (the term u(x,t) of the equation or the residual turbulence in the experiment) causes jumps between the different stationary solutions. Naturally these results are obtained for the one-dimensional Michelson-Sivashinsky equation, much less is known (see however [START_REF] Bychkov | The nonlinear equation for curved flames applied to the problem of flames in cylindrical tubes[END_REF]) in the cylindrical case. We will see in the next section how the experimental results compare to these predictions.

Results

4.1: Flame speed

The first interrogation was on the existence of slanted flames: do they evolve with a well defined velocity of propagation slightly larger than the symmetrical ones as suggested by theoretical approaches [START_REF] Bychkov | Dynamics and stability of premixed flames[END_REF][START_REF] Liberman | Numerical studies of curved stationary flames in wide tubes[END_REF], or is there a large number of discrete solutions as those obtained from the Sivashinsky equation [START_REF] Denet | Stationary solutions and Neumann boundary conditions in the Sivashinsky equation[END_REF]? The flame trajectories in the tubes appear relatively rectilinear with two distinct velocities of propagation well correlated to the symmetry of the flame (Fig. 3), even if the flame can transit from one mode of propagation to the other one during the experiment. Lateral ignition produces generally slanted flames, which propagate faster than almost symmetrical flames obtained after axial ignition. The same trends are observed with propane-air flames (Fig. 4), but the situation is not always clear: rich propane-and lean methane-air flames are always slanted in the smaller tube, whereas rich methane-air flames are almost symmetrical. There are thus essentially two velocities of propagation of premixed flames in tubes depending on the main shape of the flame. There is no indication that quasi-axis-symmetric flames are less stable than slanted ones, since a slanted flame may also transit to a quasi-symmetric flame. However it will be seen later that even slow flames obtained after axial ignition are slightly asymmetric. The measured values are in good agreement with previous measurements in the same conditions [START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF], except that much faster slanted flames speed were obtained here for stoichiometric propane-air flames thanks to the new type of igniter. It is worth noticing that previous measurements were done by igniting the flame with a lighter without any control of the initial condition. Slanted flames propagate at a speed 50% larger than quasi-symmetric flames. The same trends were observed with propane-air flames (Fig. 4). A question that arises is why some flames adopt a slanted shape with a faster velocity of propagation? Looking at the simultaneous measurement of the instantaneous flame speed at the upstream tip and of the integrated luminous intensity of the flame supposed to be proportional to the whole flame surface (Fig. 5a),

4.2: Effects of curvature

These local conditions at the tip can be characterized by the effective radius of curvature calculated from two orthogonal views of the flame front. Whatever the dispersion of the instantaneous measurements (Fig. 6-a), it appears clearly that the slowest flames are obtained with an axial ignition and have a larger (negative) radius of curvature than slanted flames obtained with lateral ignition. Here the flame speed is normalized by the laminar flame speed measured by Bosshaart and De Goey [START_REF] Bosschaart | The laminar burning velocity of flames propagating in mixtures of hydrocarbons and air measured with the heat flux method[END_REF]. For the slanted flames, the centre of curvature may even be found outside the tube (Fig. 6-b) but this results from erroneous measurements when cell splitting occurs at the flame tip, leading to false evaluation of the radius of curvature with too large values, or when the very curved part of the flame close to the wall is not clearly seen on the images. It can be seen that even axial ignition doesn't lead to really axisymmetric flames since the centre of curvature is slightly off-axis (Fig. 6-b). The same trends are observed in lean and stoichiometric mixtures of propane and methane-air in both tubes (54 and 94 mm), but the situation is much more confusing in rich mixtures, in which case there is no clear distinction between both types of ignition (Fig. 7). Thus even if the mean velocity of premixed flames propagating freely in tubes has generally two distinct ranges of values depending on the global symmetry of the flame, a result not expected from the Michelson-Sivashinsky analyses, it appears that the instantaneous velocities of propagation could sustain the ideas developed in these analyses: there is more than two stationary solutions when the ratio tube radius/ cut-off length is large as is the case in rich propane-air mixtures, so small perturbations of the flame can induce jumps between a great number of discrete solutions with different velocities of propagation .

3: Discussion

A statistical analysis of these data (Fig. 8) gives access to some characteristic lengths of the flame such as the most probable value (noted here R*) or the minimal value (noted R min ). The mean value, generally larger than R*, is not used here as it is probably biased by erroneous measurements. Slanted flames have a much narrower distribution of radius of curvature and peak at a smaller value, with also a smaller minimal radius of curvature. In order to compare these results to numerical models, we have to evaluate the cut-off wavelength in each configuration. According to analytical calculations of the stability of premixed planar flames including expansion effects, gravity and preferential diffusion [START_REF] Clavin | Premixed Combustion and Gasdynamics[END_REF][START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF][START_REF] Quinard | Stability limits and critical size of structures in premixed flames[END_REF], and for downward propagating flames above the threshold of cellular instability, there is a band of unstable wavelengths limited by two neutral wave numbers:

where E is the expansion ratio E=ρ u /ρ b given by GASEQ [START_REF] Morley | Gaseq a chemical equilibrium program for windows[END_REF], Ma is the Markstein number, d L the laminar flame thickness taken equal to D th /U L with D th =0.2 cm 2 s -1 the thermal diffusivity in the unburnt gases, and g the acceleration of gravity. The subscripts u and b refer respectively to unburnt and burnt gases. Using the values of Ma determined by Davis et al. [START_REF] Davis | Markstein numbers in counterflow, methane-and propane-air flames: a computational study[END_REF] and following the same approach as in [START_REF] Quinard | Self-Turbulent Flame Speeds[END_REF], three characteristic lengths can be calculated:

• The largest unstable wavelength Λ max = 2π/k n -is in the range 0.4 to 0.9 m for methane and 0.2 to 0.6 m for propane. It is much larger than the burner diameter in these experiments so it can be disregarded.

• The smallest unstable wavelength λ c = 2π/k n + , characteristic of the cut-off wavelength used in the models.

• The most unstable wavelength Λ*~ 2 λ c.

-10 These values are reported in Table I and on the Figure 9. The differences between axial and lateral ignition were not systematically observed in the smaller tube, especially with lean methane and rich propane mixtures in which case the flame was generally slanted, and with rich methane-air flames that were almost symmetrical. In each configuration, the evolution of the characteristic radius of curvature at the flame tip, the most probable R* or the minimal value R min , reflects the evolution of the intrinsic characteristic wavelengths, the most amplified wavelength Λ* or the cut-off wavelength λ c . These values increase with equivalence ratio for methane-air flames while they decrease for propane-air flames. The striking result is that the minimal radius of curvature is almost equal to R min ~ Λ*~ 2λ c , while the most probable value R* is of the order of 3Λ* and even 4.5 Λ* with stoichiometric methane-air flames in the wider tube. Such results were already obtained either by determining at which size cells produced by the Darrieus-Landau instability are prone to secondary instabilities [START_REF] Akkerman | Velocity of weakly turbulent flames of finite thickness[END_REF][START_REF] Liberman | Numerical studies of curved stationary flames in wide tubes[END_REF], or in numerical modelling of the Sivashinsky equation [14,[START_REF] Cambray | Mean Evolution of Wrinkle Wavelengths in a Model of Weakly-Turbulent Premixed Flame[END_REF]. But to the authors' knowledge, no experimental measurements were done with flames propagating freely in a tube. In Sivashinsky simulation [START_REF] Cambray | Mean Evolution of Wrinkle Wavelengths in a Model of Weakly-Turbulent Premixed Flame[END_REF], an initially flat flame is submitted to noise perturbations and evolved with a characteristic time t LD ~ 0.5λ c /U L corresponding to the time of growth of the most unstable wavelength. In the early stage, t < 2 t LD , there is a linear development of a cellular structure with a wavelength of the order of the most unstable wavelength. Then, after a while that depends on the noise level u'/U L , the cells start to coalesce with a linear increase of wavelength up to a point where the stabilizing effects due to curvature, roughly proportional to EU L /R [14], are too weak to support noise perturbations. A stationary state is finally obtained where cells splitting compensates for cells coalescence that produces a limiting mean wavelength Λ ∞ that is thus function of the noise level. In the simulation [START_REF] Cambray | Mean Evolution of Wrinkle Wavelengths in a Model of Weakly-Turbulent Premixed Flame[END_REF], this occurs at a time t ∞ such that Λ ∞ ~ 10 λ c at t ∞ / t LD =30 when u'/U L = 1/1000, or Λ ∞ ~ 4 λ c at t ∞ / t LD =10 when u'/U L = 1/10.

-8 -6 -4 -2 0 1 1.5 2 2.5 3 3.5 R f cm U n ! k n ± = E "1 4EM a d L 1 ± 1 " 8M a d L g (E "1)U L 2 # $ % & ' (
The transit time in the burner 1.5 m long is generally much larger than this limiting time t ∞ , so a stationary propagation of the flame can be assessed. The fact that R min ~ 2λ c let us suppose that curvature effects with this radius of curvature are sufficient to overcome the hydrodynamic instability in the cases studied. Then R* can reach values as large as 4.5L* ~ 9λ c without occurrence of the secondary instability, confirming the smallness of upstream perturbations. But it has to be noticed that our measurements are made at the leading bump, where the velocity field induced by the whole shape of the flame in the upstream gases can be very different from the one simulated with a flat flame on average. This is particularly true for slanted flames, which are also probably affected by boundary layer effects, and for which the fully developed flame cells are likely to appear on the flame skirt rather than at the leading bump.

Nevertheless, apart from the differences in the velocity of propagation between slanted and axisymmetric flames, these measurements are consistent with the results of numerical modelling of the Sivashinsky equation: the minimal radius of curvature is almost equal to the most unstable wavelength calculated from the linear theory, and flames with a small radius of curvature at the leading bump have a narrower range of fluctuations of their radius of curvature, reflecting a narrower range of unstable wavelengths due to stabilizing effects of curvature.

U t /U L 1, 9 2,6 2,1 3,3 1,8 2,7 2,3 3,4
Methane-air, φ=1.

Conclusion

Experiments were performed to analyse premixed flames propagating freely in wide tubes, first to give a better description of the two regimes of propagation already observed, then to relate their characteristic length scales to the cut-off wavelength used in numerical modelling of premixed flames dynamics.

Slanted flames propagate at a speed 50% larger than quasi-symmetric flames, three to four times the normal flame speed. This can be relevant for determining the flame flashback conditions. The variations of flame speed at the leading bump precede the variations in the flame surface, indicating that the local condition at the flame tip is the critical phenomenon that controls the flame propagation. This is confirmed by the correlation between the local radius of curvature and the flame speed, slower flames having a larger radius of curvature. The dynamics of the flame resembles the one described by the Michelson-Sivashinsky equation: slanted flames with a small radius of curvature have a restricted range of fluctuations, reflecting the stabilizing effect of curvature; the minimal radius of curvature is of the order of the most unstable wavelength Λ*~ 2λ c and the most probable radius of curvature can be compared to the limiting value Λ ∞ calculated with nonlinear theories. Table I: Characteristic length scales and normalized self-turbulent flame speed measured in tubes of internal diameter 54 and 94 mm.

Figures captions :

Figure 1 :

 1 Figure 1: Experimental set-up

Figure 2

 2 Figure 2-a

Figure 2 :

 2 Figure 2: Image processing to determine the position of the flame tip and the radius of curvature in each orthogonal section (left and right images on each picture; middle image: direct view of the flame). Propane-air flame, φ=1.0, tube ø 94 mm. a) axial ignition; b) lateral ignition.

2 :

 2 Image processing to determine the position of the flame tip and the radius of curvature in each orthogonal section ( left and right images on each picture). Middle image : direct view of the flame. Propane-air flame, !=1.0, tube ø 94 mm. a) axial ignition ; b) lateral ignition.

Figure 3 8 Figure 3 -b Figure 3 :

 3833 Figure3-a

Figure 4 :

 4 Figure 4: Histogram of stoichiometric propane-air flame velocities in a tube ø=94 mm as a function of initial conditions for ignition. Hatched/ full shading: lateral/ axial ignition.

  cm/s it appears that the flame speed is in advance compared to the flame surface. The same behaviour is recovered with Michelson-Sivashinsky simulations (Fig.5b) where the total flame length fluctuations are delayed compared to the flame speed fluctuations at the leading tip. The local conditions at the tip are thus critical, controlling the flame speed at the tip that determines the length of the flame skirt and the flame surface.

Figure 5 Figure 5 :

 55 Figure5-a

Figure 6 :Figure 7 :

 67 Figure6-a

Figure 8 Figure 8 :

 88 Figure8-a

Figure 9 :

 9 Figure 9: Characteristic lengths of methane-air (full symbols, full lines) and propane-air flames (open symbols, dash-dotted lines) as a function of the tube diameter and of the symmetry of ignition. (•,○= Λ* ; ▼,▽=λ c ; ◼,◻= R* ; ◆,◇= R min ). Lines represent only a smoothed interpolation between the calculated or measured values.

Figure 1 :

 1 Figure 1: Experimental set-up

Figure 2 :

 2 Figure 2: Image processing to determine the position of the flame tip and the radius of curvature in each orthogonal section (left and right images on each picture; middle image : direct view of the flame). Propane-air flame, φ=1.0, tube ø 94 mm. a) axial ignition; b) lateral ignition.

Figure 3 :

 3 Figure 3: Trajectories of methane-air flames in a tube ø=94 mm as a function of initial conditions for ignition. Full lines: lateral ignition; dashed lines: axial ignition.

Figure 4 :

 4 Figure 4: Histogram of stoichiometric propane-air flame velocities in a tube ø=94 mm as a function of initial conditions for ignition. Hatched/ full shading: lateral/axial ignition.

Figure 5 :

 5 Figure 5: Time traces of the instantaneous flame speed at the tip (full line) and integrated luminosity (left image, dash-dotted line) or mean flame speed (right image, dash-dotted line). a) Methane-air flame, φ=1, tube Ø=54 mm; b) Michelson-Sivashinsky simulation with 10% Gaussian white noise in space and time.

Figure 6 :

 6 Figure 6: Effective radius of curvature (a) and radial position of the centre of curvature (b) as a function of the normalized flame speed U n = U t /U L . Stoichiometric propane-air flame, tube Ø=94 mm. ◯ = axial ignition, ◢ = lateral ignition.

Figure 7 :

 7 Figure 7: Effective radius of curvature as a function of the normalized flame speed U n = U t /U L . Rich propane-air flame φ = 1.4, tube Ø=94 mm. ◯ = axial ignition, ◢ = lateral ignition.

Figure 8 :

 8 Figure 8: Histograms of the radius of curvature at the flame tip in both ignition conditions. Stoichiometric propane-air flame, tube ø = 94 mm. a) axial ignition; b) lateral ignition.

Figure 9 :

 9 Figure 9: Characteristic lengths of methane-air (full symbols, full lines) and propane-air flames (open symbols, dash-dotted lines) as a function of the tube diameter and of the symmetry of ignition. (•,○= Λ* ; ▼,▽=λ c ; ◼,◻= R* ; ◆,◇= R min ). Lines represent only a smoothed interpolation between the calculated or measured values.

  The most probable radius of curvature can be very large, up to 9λ c . It increases with the tube diameter and is generally larger in the case of axial ignition. No simple relation with λ c can be obtained from this study, but the most probable value R* is comparable to the value of Λ ∞ calculated with non-linear theories.

			Methane-air, φ=0.8, U L =23.6 cm/s		Propane-air, φ=0.8, U L =29.8 cm/s
			Λ*=1.2 cm, λ c =0.65 cm			Λ*=1.5 cm, λ c =0.8 cm
		Tube Ø = 54 mm	Tube Ø = 94 mm	Tube Ø = 54 mm	Tube Ø = 94 mm
	Ignition	Axial	Lateral	Axial	Lateral	Axial	Lateral	Axial	Lateral
	R* (cm)		2,15	4,15	2,5	2,15	2,8	4	3,2
	R min (cm)		1,4	1,5	1,4	1,5	1,7	2	1,9
	U t /U L		2,5	1,9	3	1,5	2,5	2,3	3,1
			Methane-air, φ=1, U L =36.3 cm/s		Propane-air, φ=1, U L =42.1 cm/s
			Λ*=1 cm, λ c =0.5 cm			Λ*=1.05 cm, λ c =0.54 cm
		Tube Ø = 54 mm	Tube Ø = 94 mm	Tube Ø = 54 mm	Tube Ø = 94 mm
	Ignition	Axial	Lateral	Axial	Lateral	Axial	Lateral	Axial	Lateral
	R* (cm)	2,8	2,3	4,5	2,8	2,8	2,5	3,9	2,9
	R min (cm)	1,1	1	1,5	1,5	1,4	1	2,2	1,4

Table I :

 I Characteristic length scales and normalized self-turbulent flame speed measured in tubes of internal diameter 54 and 94 mm.

				3, U L =26.3 cm/s		Propane-air, φ=1.4, U L =22.6 cm/s
			Λ*=2.1 cm, λ c =1.2 cm			Λ*=0.8 cm, λ c =0.4 cm
		Tube Ø = 54 mm	Tube Ø = 94 mm	Tube Ø = 54 mm	Tube Ø = 94 mm
	Ignition	Axial	Lateral	Axial	Lateral	Axial	Lateral	Axial	Lateral
	R* (cm)	3		3,5	2,9		1,8	1,3	1,1
	R min (cm)	2		2,2	2,2		0,8	0,9	0,5
	U t /U L	1,5		1,4	2,1		2,6	1,1	2,5

Acknowledgments:

Recurrent funding from the French CNRS and Aix-Marseille University supported this work.


José Antonio

Reglero Ruiz

Michel Vincent

Jean-François Agassant

Tarik Sadik

Caroline Pillon

Christian Carrot

Polymer foaming with Chemical Blowing Agents: Experiment and modeling

An experimental and theoretical analysis of the polymer foaming process using different chemical blowing agents (CBA) was performed. A simple experiment was designed to analyze the foaming process of polypropylene (PP)/CO 2 system under different pressure conditions using several chemical foaming agents. The expansion ratio was measured both by direct observation and from optical measurements and image analysis, showing a good agreement. A single bubble simulation based on relevant Differential Scanning Calorimetry (DSC) and Thermo-Gravimetrical Analysis (TGA) experiments, assuming each CBA particles as a nucleation site and accounting for gas diffusion in the surrounding polymer matrix has been built. The sensitivity of the model to physical and processing parameters has been tested. The calculation results are compared to the experiments and open the route to a simplified method for evaluating the efficiency of chemical blowing agents.

-Introduction

The foaming process to produce microcellular thermoplastics has been widely analysed in the last decades. Microcellular plastics are generally formed by cell nucleation and growth of bubbles in the polymer matrix. Chemical blowing agents (CBA) or physical blowing agents (PBA) are used to introduce the gas that creates the cellular structure.

A typical polymer foaming process involves several steps: the dissolution under an elevated pressure of a gas blowing agent (PBA or CBA) in the molten polymer, the nucleation of a population of gas clusters in the supersaturated solution upon the release of pressure to the ambient pressure and finally the growth of nucleated bubbles in the polymer to their ultimate equilibrium size. The final foam density depends on the original gas loading, the gas fraction which remains dissolved in the polymer matrix when it solidifies, the gas losses to the environment, and the depressurization rate. The cell size and cell size distribution depend on the kinetics of nucleation, the bubble growth process following nucleation and the coalescence during expansion.

Different steps can be considered when processing such materials. First the polymer/gas solution formation, second the microcellular nucleation, and finally the cell growth and the resulting density reduction. In the first stage, the polymer/gas solution formation is accomplished by saturating a polymer under a high gas pressure, forming a single-phase supersaturated solution governed by the gas dissolution in the polymer matrix which is a function of pressure and, at a lesser extent, of temperature. Numerous studies have been carried out analysing the dependence of the solubility of gas in several polymers, mainly CO 2 in polypropylene [1,2,3].

In the second stage, it is necessary to submit the polymer/gas solution to a thermodynamic instability to nucleate microcells. This nucleation can be achieved by lowering the solubility of the solution through the temperature and the pressure of the system. Usually, a rapid pressure drop produces a high nucleation rate in the polymer matrix and in the ideal case this nucleation occurs instantaneously.

In plastic foaming, nucleation refers to the process of generating gas bubbles in a polymer melt through a reversible thermodynamic process. Creation of small bubbles always leads to a free energy increase and they are in an unstable equilibrium with the environment. In the classical nucleation theory, there is a critical nucleus, which defines the minimum radius for a bubble to growth. Nucleated bubbles which size is larger than the critical nucleus radius will survive, whereas those smaller will collapse. Cell nucleation can occur homogenously or heterogeneously. The heterogeneous nucleation is usually 100 to 1000 times more favourable than homogeneous nucleation. Several additives or organic charges can be employed as nucleation sites in polymer foaming processes [4,5].

Once the cells are nucleated, they continue to expand by diffusion of the dissolved gas from the polymer matrix into the bubbles. In this stage of the process, a deep knowledge of the physical parameters that govern the diffusion properties of gas-polymer systems is necessary [6,7,8,9]. The cells grow reducing the polymer density as the gas molecules diffuse into the nucleated cells. The rate at which the cells grow is limited by the diffusion rate and the rheology of the polymer/gas solution. The cell growth process is controlled also by the time allowed for the cells to grow before solidification, the temperature and pressure of the system, the presence of other bubbles, etc… [10].

As explained before, there are two main processes to produce microcellular polymeric foams, basically depending on the way to introduce the gas into the polymer matrix. In the physical foaming agents (PBA) process, the gases are introduced directly into the barrel of an extruder or of the plasticating unit of an injection molding machine, in the polymer matrix. This method, (Mucell® process), has been extensively analysed by Park and Tomasko [11,12,13]. Several types of PBAs such as CO 2 , N 2 [14], chlorofluorocarbons or argon have been used with different polymer resins such as polypropylene, polyethylene or polystyrene [15]. Although PBAs can be employed both in extrusion and injection processes, the foaming injection molding process using CO 2 or N 2 is commonly used. However, PBA foaming needs special equipment such as gas dosage unit and controller, gas injector, a specially designed screw and a high-pressure gas source.

The second group of foaming processes involves the use of chemical blowing agents (CBA) that can liberate gases under certain processing conditions either due to chemical reaction or thermal decomposition. Most CBAs produce nitrogen (N 2 ) or carbon dioxide (CO 2 ) after decomposition [16,17,18]. CBAs reactions can be endothermic or exothermic. Azodicarbonamide is the most representative exothermic CBA, commonly having a high gas yield, with decomposition temperatures between 170°C and 200°C [START_REF] Eaves | Handbook of Polymer Foams[END_REF]. Sodium bicarbonate and zinc bicarbonate are the most common endothermic blowing agents [START_REF] Villamizar | [END_REF]. The injection-moulded plastics parts foamed by CBAs have reduced density, reduction of sink marks and are easy to process, but the bubbles are larger than that obtained with PBA, leading in principle to lower mechanical properties.

Both foaming processes present different physical scenarios. In the case of PBA foaming process the formation of a single gas-polymer supersaturated phase depends only on the saturation pressure and temperature of the system, but as said before, the temperature and pressure evolutions in the barrel can be difficult to control, so it is not straight forward to determine the total quantity of gas diluted in the polymer just before the nucleation and expansion process. In an ideal scenario, the gas dissolution process only occurs in the screw/barrel system, whereas the nucleation and growth phenomena take place during or after mould filling. Both processes are completely independent and separated.

In the case of CBA foaming process, it can be considered in a first approximation that the decomposition reaction is complete in the screw-barrel system, and that all the gas is generated, diffused and solubilised into the polymer matrix. The solubilisation of the gas into the polymer depends of the pressure in the plasticization unit but also of the ratio between the plasticization time and the diffusion time. If the decomposition reaction is not completed in the plasticization unit, the physical system to analyse becomes more complicated. Part of the gas remains unreacted in the CBA particles, and the foaming process begins even when all the gas has not been totally created.

This paper is focused on CBAs foaming process. The physical evolution of the polymer-gas system depends strongly of the temperature, pressure and kinetics of the chemical reactions. A simple experiment has been designed to analyse the foaming expansion as a function of time of a polypropylene containing three types of CBA, in static conditions (no flow). The expansion ratio has been measured by direct observation and from optical measurements and image analysis. A single bubble simulation based on DSC and TGA experiments, assuming each CBA particle as a nucleation site and accounting for gas diffusion in the surrounding polymer matrix has been built. The sensitivity of the model to physical and processing parameters has been tested and the results are compared to the experiments.

-Experimental

-Materials

The polypropylene compound was a 12 % mineral filled elastomer modified polypropylene (SUMIKA PP) with a melt flow index of 65 g/10 min (ISO R1133), a Newtonian plateau viscosity of 500 Pas at 210°C, determined from rotational rheological measurements, and a density of 0.91 g/cm 3 . Three different endothermic chemical blowing agents Hydrocerol® (CLARIANT) referred as CBA-818, CBA-828 and CBA-848 were analysed. These foaming agents are PE-based compounds with reactive elements (typically citric acid, sodium bicarbonate or a mix of both components). Table 1 presents the different CBA's employed and the weight concentration (wt. % respect to PE matrix) of reactive elements. In the following, CBA refers to the compound and not to the reactive elements only.

The Hydrocerol® foaming agent has been extensively employed in the last few years [21][22][23][24][25]. These chemical agents, with decomposition temperatures between 160 °C and 210 °C, can be added directly into the hopper of an injection moulding machine in the form of pellets in proportions from 1 % to 4 wt. %. Thermal characterization was carried out to determine the polypropylene fusion temperature and the decomposition temperatures of each reaction of the reactive elements included in the CBA pellets. Results are presented in figure 1. A heating rate of 6°C/min was chosen to assure the same heating rate as in the experimental foaming expansions. Constant heating rate of 6°C/min, from ambient temperature to 250°C.

CBA

The peak between 160 °C and 170 °C corresponds to the melting of polypropylene. For the three CBAs, the small peak around 90 °C corresponds to the PE melting temperature. For the CBA-818 and CBA-848, the peaks between 150 °C and 175 °C correspond to sodium bicarbonate decomposition reaction. A peak between 190 °C and 220 °C caused by the decomposition of citric acid is visible for CBA-828 and CBA-818. Finally, a small peak about 240 °C takes place in the CBA-818, caused by the coupling reaction of the remaining sodium bicarbonate and citric acid. This third reaction is only important at higher heating rates (above 20 °C/min). For low heating rates, the importance of this reaction is negligible, as it can be seen in figure 1. All the decomposition reactions of the CBAs start after the polypropylene fusion, which assures that the gas obtained from the CBA can be diluted in the melted matrix.

TGA measurements were carried out to determine the quantity of gas released by the reactive elements in the CBA particles (figure 2). The relative weight loss refers to the original weight of the granule containing 30 wt. % of polyethylene and 70 wt. % of reactive elements. This loss is associated to the gas escaping the sample, assuming that the pressure conditions do not allow any gas dissolution in the polyethylene. The TGA curve of CBA-848 shows that the decomposition reaction of sodium bicarbonate begins at 150 °C, reaching a weight percentage of created gas around 24 % at the end of the reaction, at about 210 °C. In the case of CBA-828, the decomposition reaction of citric acid begins at 215 °C and ends around 300 °C, with a weight percentage of gas created about 35 %. Finally, the CBA-818 presents a first decomposition reaction which begins at 150 °C. The citric acid decomposition reaction is probably coupled with the sodium bicarbonate reaction, as it can be observed in the small change in the slope about 200 °C. The maximum quantity of gas generated at the end of the decomposition reactions is about 28 wt. % at 300 °C. In all the cases, the PE decomposition begins at 450 °C.

It is possible to obtain the evolution of the total quantity of moles of created gas from the stoichiometry of the decomposition reactions. The sodium bicarbonate decomposition can be described as follows (reaction 1):

2 NaHCO 3 -> Na 2 CO 3 + H 2 O + CO 2 (1)
The citric acid decomposition is (reactions 2 and 3):

C 6 H 8 O 7 -> C 6 H 6 O 6 + H 2 O (2) C 6 H 6 O 6 -> C 5 H 6 O 4 + CO 2 (3)
Finally, a coupling reaction occurs when the sodium bicarbonate and citric acid react together to produce CO 2 and H 2 O (reaction 4):

C 6 H 8 O 7 + 3NaHCO 3 ->Na 3 C 6 H 5 O 7 + 3H 2 O + 3CO 2 (4)
In CBA-848 and CBA-828 only reactions 1 and 2 occur, respectively. As our experiments were carried out at ́ < 20 °C/min, the coupling reaction (4) for CBA-818 is neglected. According to the composition given in Table 1, 1 g of CBA contains 0.7 g of reactive element. The gas escaping the sample is composed both of CO 2 (molar mass of 48 g) and H 2 0 (molar mass of 18 g). From the stoichiometry of the reactions it is then possible to obtain the evolution of the number of moles of gas per g of CBA generated during the decomposition reaction (figure 3). It is expected that after foaming and cooling of the samples, the H 2 0 remains in the samples as condensed water vapor.

-Foaming experiment

The objective is to analyze the foaming behavior of different CBA/PP samples, obtained by mixing the three CBAs with PP granules in the same proportions as in injected samples. These materials are foamed in a steel mold and the expansion ratio and cellular structure will be analyzed.

Sample preparation and characterization

The challenge is to mix the components without activating the chemical reactions. A solid sample was fabricated starting from PP and CBA powders. Powders were mixed at room temperature in a fixed proportion of 98 wt. % of polypropylene and 2 wt. % of CBA. Then, cylindrical samples of 9 mm height (h i ) and 20 mm diameter () were fabricated in a steel mold by compression under a pressure of 20 MPa at 60 °C for 30 min. A total number of six samples, three groups of two samples with the same CBA, were fabricated, with a volume V i of 2.83 cm 3 . The density of all the samples ( sample ) was about 0.87 g/cm 3 , with a densification value up to 99.9 %, calculated as:

( ) (5)
taking  PP = 0.91 gcm -3 as the density of the solid PP. The number average radius and the number average volume can be obtained assuming a spherical geometry of the CBA particles, which is far from reality (equations ( 6) and ( 7)):

́ ∑ ∑ (6) ́ ∑ ∑ (7)
where N represents the total number of particles (N = 1800).

The average value of the reactive agent particle radius is = 4.46 m and the average volume is 2.8010 -9 cm 3 . This calculation was performed in three different SEM micrographs, with a dispersion value of  10 %.

Assuming each CBA particle contains 70 wt. % of reactive agent and 30 wt. % of low-density PE, and using the mix law (with  reactive agent  2 g/cm 3 and  LDPE = 0.9 g/cm 3 ), the density of a CBA particle is 1.67 gcm -3 , which leads to an average mass of a reactive agent particle of 4.7010 -9 g. The total mass of reactive agents in the sample (mass of 2.5 g) is obtained knowing that each sample contains 2 wt. % of CBA, in which the reactive agents represent 70 wt. % (0.035 g). The total number of reactive agent particles (N p ) in the solid sample is obtained simply by the ratio between the total mass of reactive agent in the sample and the average mass of a reactive agent particle: 0.035/4.7010 -9 = 7.45 10 6 .

Using the same procedure, it is possible to obtain the minimum distance between each particle (l), from the previous binarized image (figure 5a). The resulting histogram is presented in figure 6, together with the expression to calculate the number average value ́.

́ ∑ ∑ ( 8
)
The determination of ́ from equation ( 8) gives a result of 74 m. Another way for calculating ́ is to assume a uniform distribution of the reactive agent particles in the solid sample using crystallographic considerations. Considering for example a primitive cubic (PC) Bravais lattice, the number of reactive agent particles in the cubic cell (N p cell ) is 1. The determination of the distance ́is related to the volume of the solid sample V i , the number of reactive agent particles in each cubic cell N P cell and the total number of reactive agent particles N P through the expression:

́ (9)
Similar calculations can be carried out for the body-centered cubic (BCC) and face-centered cubic (FCC) bravais lattices, in which the number of reactive particles in each cubic cell, is 2 and 4, respectively. Table 2 presents the results of the three calculations, taking N p = 7.4510 6 and V i = 2.83 cm 3 .

Bravais lattice N P cell

̅ (m) PC 1 76 FCC 2 84 BCC 4 87 Table 2

-Average distance between CBA particles from crystallographic calculations

The average initial distance between reactive agent particles is between 76 m and 87 m which is similar to the value obtained from the binarized image. This distance ́ will increase during the expansion process.

Foaming experiment under pressure

Figure 7 presents the scheme of the foaming expansion experiment designed to analyze the behavior of the polypropylene/gas systems. The solid sample, with a number of reactive agent particles N P , has an initial height h i and an initial volume V i , and it is placed in a reservoir under an external pressure P ext . The reservoir is made of steel, with a thickness of 5 mm and a height of 200 mm. The inner diameter is 20 mm. The external pressure is applied with a weight deposited on a circular steel cap, with a diameter a little bit lower than the inner cylinder diameter. The device is introduced in an oven at a temperature T during a time t. A type K thermocouple is introduced in the sample to monitor the evolution of the temperature. Another one is located inside the oven. At the end of the heating step, the device is removed from the oven and cooled down at ambient air. When cooling is finished the weight is removed. The foamed sample has a final height h f , a final volume V f , and a cellular structure with a cell density N c and an average cell radius R. The height increase during the foaming process is measured optically with a recording camera, with a precision of  0.5 mm. It is important to mention that the temperature will vary differently as a function of time at different points of the sample, due to heat conduction from the oven. This means also that the foaming develops heterogeneously inside the sample, starting in the sample edges and propagating towards the core.

Figure 7 -Scheme of the foaming expansion experiment

Two external pressures were tested to evaluate their influence on the expansion rate, gas diffusion and final pore radius: 0.25 MPa (samples 1-1, 2-1 and 3-1) and 0.5 MPa (samples 1-2, 2-2 and 3-2).

Results and discussion

Figure 8 shows the evolution of the temperature with time. The sample and oven temperatures are not equal until the last part of the experiment. The average heating ramp of the sample temperature is about 6°C/min, considering a total heating time of 32 min from ambient temperature to 210°C.

Expansion

The measurement of the height variation started when the sample temperature reached 180 °C, slightly above the melting point of the PP, which is located between 160 °C and 170 °C according to DSC data (see figure 1). Figure 9 presents the samples height evolution from the beginning of the expansion measurements (about 23 min after starting the experience, see figure 8).

Figure 9 -Evolution of the height of each sample during the expansion process

All samples show a remarkable volume expansion due to the gas creation and the foaming process. The expansion process takes about 4 minutes for CBA-818 and CBA-848, and 6 minutes for CBA-828, leading to expansion ratios around 200 % with P ext = 0.25 MPa and 140 % with P ext = 0.5 MPa. After that time, the sample height reduces because of cooling and thermal shrinkage.The samples foamed with the CBA-828 showed a higher expansion than samples foamed with the other two CBAs. This may be explained by the larger number of moles of gas generated by the chemical reactions, as shown on figure 3. At the end of the process, shrinkage is clearly seen, especially in samples foamed with CBA-828. No shrinkage was observed in the lateral direction. For CBA-828 at 0.25 MPa, height reduces from 22.1 mm at 200 s to 20.1 mm at 260 s, which corresponds to a volume reduction of 10 %. For the same CBA at 0.5 MPa, the height reduction begins at 170 s with a maximum value of 14.8 mm and a final value of 13.3 mm at the end of the experience, with a similar volume reduction value.

The final volume V f and the expansion ratio after cooling are presented in table 3, as well as the density , calculated from the ratio between the mass and the volume of the expanded sample.

Height (mm) Time (min)

CBA-1 CBA-2 CBA-3

P ext = 0.50 MPa
It is important to estimate the volume of gas retained in the sample to analyze possible gas diffusion outside the polymer. Assuming that the decomposition reaction is complete, the theoretical total volume of gas V gas generated can be calculated from perfect gas equation.

( ) (9)
P ext is the external pressure (MPa), T is the foaming temperature, which is taken as a constant value of 210 °C (483 K) in a first approximation, and  is the gas constant (8.31 Jmol -1 K -1 ). The value of n(t) total corresponds to the total number of moles of gas created in the solid sample for the different CBAs at the end of the reaction.

According to figure 3, the total number of moles of gas created per gram of CBA are 0.0092 mol/g in the case of the CBA-828 and CBA-848, and 0.0135 mol/g in the case of the CBA-818. The total number of moles of gas generated in the sample n(t) total can be calculated by multiplying the previous values and the mass of CBA particles in each sample, in our case 2 wt. % of 2.5 g, and assuming that only the 70 % of each CBA particle contains reactive elements.

The volume of gas generated for each CBA at each external pressure, deduced from equation ( 9), is shown on table 4. This volume corresponds to the gas volume generated at the end of the expansion, before any cooling and thermal shrinkage of the polymer.

Table 4 -Theoretical total gas volume generated during the expansion processes for each sample, assuming that the decomposition reaction is complete

The fraction of gas in the foamed polymer can be also determined from image analysis. Several optical micrographs of the fracture surface of the expanded samples are presented in figure 10. The structure is homogenous throughout the whole sample volume, which indicates that the thermal gradients do not influence the final foamed structure, even when the heating ramp may be locally different in the sample. For a given CBA, increasing the pressure leads to a decrease of the bubble size, except for CBA-828, in which the bubble size remains almost the same. Figure 10 shows that CBA-828, containing citric acid as reactive element, induces a finer bubble size and higher expansion rate than the other CBAs. The addition of sodium bicarbonate (CBA-818) or CBA based only on sodium bicarbonate (CBA 848), do not add any remarkable improvement concerning the foaming behavior, in terms of reducing bubble size, as compared to CBA-828.

Sample

Using the ImageJ® software previously presented, it is possible to quantify the average bubble radius R bubble , the average distance between bubbles, (namely l bubble ), and the total volume of gas in each sample V gas . As an example, figure 11 presents the histogram of both bubble radius, assuming that each bubble can be considered as a sphere (figure 11a), and the distance between bubbles (figure 11b). This parameter is calculated for each bubble, taking the minimum distance between the edge of this bubble and the surroundings ones. Results presented have been calculated from test 1-1, but similar calculations have been performed for the other samples.

mm

Figure 11 -Resulting histograms of the SEM micrograph presented in figure 10a a ) Bubble radius b) Distance between bubbles

The average results are obtained from 10 micrographs of each sample with a maximum error of  5 %.The software accounts for the number of bubble in each image and the average radius. The total cell number N c in the sample was calculated using equation ( 10) [26], which accounts for the 3D extrapolation starting from a 2D image:

( ) (10)
 F represents the foam density, and  P the solid polymer density (0.91 g/cm 3 ). V f represents the final volume of the sample after shrinkage. The experimental gas volume can be also easily calculated from expansion measurements, following the next equation:

Both results (indirect method V gas indirect , from ImageJ® analysis and direct method V gas direct , from expansion measurements), together with the morphological determinations, are presented in table 5. They are in the same range, and the slight differences between both measurements can be due to specific errors associated to the software employed to analyze the optical micrographs. The last column of table 5, V gas remaining , represents the total theoretical volume of gas generated. It is assumed than after cooling and shrinkage H 2 O condensates into liquid water, so it is necessary to extract the volume of water vapor generated from the total volume of gas generated calculated in table 4. From the chemical reactions (1) (2) and (3) both citric acid and sodium bicarbonate decompose in one mole of H 2 O and one mole of CO 2 . The ratio between the molar masses of both components is 18/(18+48) = 0.27, indicating that the 27 % of the gas generated transforms into liquid water by condensation.

The proportion between the measured gas volume and the theoretical gas volume varies between 50 % and 75%. The order of magnitude of the thermal volume variation between 210 °C and room temperature for the polypropylene is around 20 %. This cannot explain the measured difference. An incomplete chemical reaction is unlikely according to the DSC and TGA measurements. It could be speculated that a part of the gas generated has not been nucleated and does not produce any expansion. It could remain dissolved in the polymer matrix or have diffused outside the sample during the expansion process. To test these hypotheses, the foamed samples were re-heated up to 210°C to analyze a possible second expansion produced by the remaining dissolved gas, with negative results. This indicates that the gas has probably diffused out of the samples. The gas diffusion outside the sample has been observed in HR-SEM micrographs, which shows the presence of micro channels in the sample surface (figure 12).

Figure 12 -Micro channel in the surface of the foamed sample

Another interesting parameter that can be analyzed from the foaming experiment is the coalescence. It is possible to define the ratio k between the final number of cells N c and the initial number of reactive agent particles N P . It lies between 0.2 % and 12 %, indicating that coalescence is a very important phenomenon that will be discussed lately.

-Theoretical approach

Many studies have been devoted to the development of numerical models for the bubble nucleation and growth in polymeric foaming process. In the classical work presented by Amon and Denson [27] a complete mathematical analysis of a bubble growth in a Newtonian matrix is presented. Bikard et al [28] and Bruchon [29] solved the same problem with a 3D finite element method which allows accounting for the simultaneous growing of multiple bubbles. Koopmans et al [30] introduced a 1 m viscoelastic multimode Maxwell behavior for the polymer matrix in a "bubble influence volume" surrounding the growing bubble. They also account for non-isothermal phenomena occurring at die exit in an extrusion process. Otsuki and Kanai [31] introduced a more realistic Phan-Tien Tanner viscoelastic constitutive equation which limits the dramatic increase of the elongational viscosity in the Maxwell model. Shafi et al [32,33] and Joshi et al [34] developed a homogeneous nucleation model that they coupled to the Newtonian Amon and Denson bubble growing model. Taki [35,36] compared these calculation results to experiments performed under several pressure release rates. Feng and Bertelo [37] investigated bubble nucleation in a viscoelastic polymer melt (Oldroyd B constitutive equation) containing nucleating agents.

All these works assume that the polymer is saturated with gas and that this gas diffuses from the matrix to the bubble. In our case, the diffusion process is in the opposite direction from the bubble, which is nucleated around the CBA particle, towards the polymer matrix. Recently Emami et al [38] analyzed the bubble nucleation in non-pressurized foaming CBA systems starting from solid materials, composed of PP and CBA powders as in our experiments. It was observed that the nucleation process proceeded in two distinct stages, namely primary and secondary nucleation. Primary nucleation occurred in the interstitial regions of the sintered plastic powder and the agglomerated blowing agent particles acted as nucleation sites, and secondary nucleation occurs in the polymer melt. The visual observations indicated that most of the first generation of bubbles endured the entire foaming process, whereas most of the bubbles generated during secondary nucleation disappeared over time. These results support the previous assumption which considers each reactive agent particle as a nucleation site, with no further nucleation phenomenon.

In the following, a kinetic model for a single bubble expansion in a Newtonian fluid coupled with the gas diffusion in the surrounding polymer matrix is proposed. The nucleation phenomenon will be simplified, assuming that each reactive agent particle can be considered as a nucleation site. The model has been applied to the experimental conditions presented in the previous section. The sensitivity of the model to several unknown parameters has been tested. Comparison between calculation and experiments will be discussed.

-Single bubble growth model

A schematic of the bubble growth model is shown in figure 13. A reactive agent particle creates n(t) created moles of gas by the chemical decomposition reaction deduced from the TGA curve. The bubble growth is governed by the competition between the gas which remains within the bubble and induces the growing mechanism (number of moles n(t)) and the moles of gas n(t) diffused which diffuses in the surrounding polymer matrix at the external pressure P ext . The gas concentration C(r,t) propagates concentrically in the surrounding polymer melt. The average concentration of gas at the bubble surface, C(r,t) r=R , is related to the gas pressure inside the bubble P gas through the solubility factor K.

Figure 13 -Single bubble growth model

The following assumptions are made:

1 -The bubble is spherically symmetric when it nucleates and remains so for the entire period of growth.

2 -The polymer matrix is Newtonian.

3 -The growth process is considered isothermal. Latent heat of reaction is neglected.

4 -Inertia effects are neglected and the fluid is assumed to be incompressible, which is reasonable in the pressure range studied.

5 -The gas released by the chemical reactions follows the ideal gas law.

6-The matrix is considered as an infinite medium and one single bubble is considered. This condition will be commented later, in terms of the gas diffusion distances, bubble radius and sample dimensions.

The first step consists in the determination of the number of moles of created gas n(t) created . Three different CBA kinetics are shown in figure 3. Considering the experimental foaming results, two different CBAs, CBA-818 and CBA-848, present similar results in terms of expansion rate and morphology. This indicates that the coupling reaction between citric acid and sodium bicarbonate has not occurred at the foaming temperature of the experience (210 °C), as indicated in the DSC curves figure 2. For this reason, two different numerical calculations results will be presented, first for the PP + CBA-828 samples (2-1 and 2-2) with the decomposition of citric acid, and then for the PP + CBA-848 samples (3-1 and 3-2) with the sodium bicarbonate decomposition reaction. Figure 14 presents the theoretical fitting of the experimental TGA data extrapolated to one single particle of reactive agent for both decomposition reactions, at a temperature of 210 °C. The low heating rate (about 6 °C/min according to the temperature measurements during the foaming experiment), permits, as a first approximation, the extrapolation of the non-isothermal results derived from the TGA to isothermal kinetics at 210 °C. The total reaction time was about 100 s, much lower than the foaming time employed during the expansion experiment, which assures that the chemical reactions are complete. Different kinetic models can be found in the literature, such as the

P gas n(t) R(t) C R(t+t)
Kamal and Sourour model [39]. In our case, the best fitting correlation was found using Boltzmann's exponential functions: (12) Fitting parameters, as well as the quality of the correlation are presented in table 6.

 (s) R 2 2
4.1710 -13 2.9510 -14 63.12 8.59 0.9991 3 3.9210 -13 2.1810 -14 57.87 9.20 0.9994 Table 6 -Fitting parameters of the number of moles of gas created during the decomposition reaction of the CBAs.

The ordinary differential equation for the bubble radius growth as a function of time writes [35]:

(13)
R is the bubble radius,  is the polymer viscosity and  is the surface tension. As explained before, the gas temperature T g is assumed to be constant (210°C) during the expansion process.

The variation of the number of moles of gas n(t) inside the bubble is derived from equation 14: The number of moles of gas which diffuse outside the bubble in the surrounding polymer matrix is obtained from the mass transfer of CBA at the gas-polymer interface and it can be expressed as follows:

(

D is the diffusion coefficient, and C is the gas concentration of the diffused gas. The concentration profile is given by a diffusion equation around the bubble which writes: (16) u is the velocity field related to the bubble expansion, in spherical coordinates. The ratio between the convection term and the diffusion term is a Peclet number given by (see Joshi et al [34]):

The diffusion coefficient value can be taken from the literature [35], with a typical value, for PP and PE / CO 2 systems, of D = 10 -8 m 2 /s. The calculation without diffusion will show that the bubble radius goes from 5.10 -6 to 2.10 -4 m and the growth speed of the bubble radius is around 10 -6 m/s. Therefore, convection can be neglected.

The numerical implementation requires defining boundary and initial conditions. Three physical parameters must be defined at time t=0. First, the average size of the reactive agent particles can be considered as the initial bubble radius R 0 , as a first approximation. This assumption is only an approximation, and the sensitivity of the calculations when changing the initial radius will be analyzed. The initial number of moles of gas n 0 can be directly calculated from the gas perfect law using the initial radius R 0 = ́= 4.46 m, at the two different external pressures 0.25 MPa and 0.5 MPa. The boundary condition for the gas concentration at the bubble surface C(R,t) is described by Henry's law:

(18)

-Numerical implementation

The numerical implementation of the previous equations is carried out by means of an incremental time marching approach during the decomposition reaction time (about 100 s, see figure 14). This approximation implies that the expansion process is limited to the decomposition reaction time, and that no further expansion occurs later. The variation of the number of moles of gas created is obtained by differentiating equation 12:

2 2 () ( , ) dC C C D T C u r t r dt t r r r r               𝑑 ( ) 𝑑 𝑢 𝑑 𝑑 = 4𝜋 2 𝜕𝐶 𝜕𝑟 𝑟= 𝐶( , ) = 3 4 𝑘 ( )ℜ 𝑔 𝜋 3 𝑣 𝑑 𝑢 ≈ 𝑢 (19)
250 iterations were employed to solve the equations. We adjust the time step t in order to obtain an incremental radius variation less than 1%: [35] Figure 15 presents the gas diffusion profiles outside the bubble for five different reaction times and both CBAs and an external pressure of 0.25 MPa. Results for the diffusion profiles taking an external pressure of 0.5 MPa were almost equivalent, which indicates that the slight variation in the external pressure does not affect significantly the gas diffusion. The quantity of gas which diffuses outside the bubble increases as the reaction develops. For reaction times close to 40 s, the gas diffusion penetration thickness is around 150 m for the CBA-828, and 300 m for the CBA-848, whereas for reaction times of 2 s, these distances are only 50 m and 80 m, respectively. The gas concentration at the gas-polymer interphase decreases strongly with time, even when the quantity of gas generated inside the bubble n(t) increases according to the TGA kinetics. The reason for this decrease is related to the dependence of C(R,t) with bubble radius in a proportion CR -3 (see equation 18). At high reaction times, the gas concentration at the interphase is negligible. The initial concentration values C(R,t) are higher for the CBA-828 due to the higher quantity of moles of gas generated by the citric acid decomposition.

-Results and discussion

Figure 16 compares the number of moles of gas created by the decomposition reaction, diffused and retained in the bubble during the expansion process for an external pressure of 0.5 MPa. These results show that, in the foaming conditions of figure 16, the diffusion process becomes noticeable from 20 s for both CBAs. From that time, the quantity of diffused gas increases, reaching a final value about 1.3510 -12 moles for the CBA-828 and a value about 710 -13 for the CBA-848. It can be seen also that at the end of the chemical reaction of CBA-828, the quantity of gas diffused is higher than the gas inside the bubble*. On the other hand, the number of moles of gas created by the decomposition reaction reaches a value between 410 -12 moles and 2.710 -12 moles, for the CBA-828 and CBA-848, respectively. These differences between both CBAs can be related to the kinetics of the decomposition reactions, which shows that the quantity of gas created by the CBA-848 is slightly higher than the quantity created by the CBA-828.

At the end of the reactions, the ratios between diffused and created gas are about 33 % and 28 % for CBA-828 and CBA-848 respectively. These values are in reasonable agreement with the experimental values found previously in table 5 (between 25% and 50% of diffused gas).

Figure 17 presents the predicted radius evolution for both CBA-828 and CBA-848. Two different situations are presented. In the first case, diffusion is not considered, and the number of moles of gas n(t) are taken directly from the TGA measurements. In the second case, the gas diffusion process is considered, and n(t) is obtained from the data of figure 16.

0,00E+00 The final bubble radius depends strongly on the external pressure and is obviously limited by the gas diffusion. For the CBA-828 (figure 17a), when the external pressure is 0.25 MPa, the bubble radius reaches 530 m when gas diffusion is neglected, and about 130 m when the diffusion process is considered. When the external pressure is increased up to 0.5 MPa, these values are 330 m and 90 m respectively. The theoretical values obtained (considering the diffusion process), are in qualitative agreement with the foaming experiments values (120 m and 85 m respectively, see table 5). A similar analysis can be performed for the CBA-848 (figure 17b). In this case, the theoretical values obtained considering the diffusion process are about 300 m and 180 m, for both external pressures of 0.25 MPa and 0.5 MPa. These results are also in qualitative agreement with the foaming experiments (350 m and 190 m respectively, see table 5).

Using the expansion R(t) value makes possible to compute the velocity value u at the bubble/polymer interface. The evolution of u for the CBA-828, considering only the results including the diffusion term, are presented in figure 18 (the results for the CBA-848 are similar). As it can be seen in figure 18, u values are around 10 -6 m/s which justifies neglecting the convection term in the diffusion equation (16) . The sensitivity of the model to control parameters, especially the initial radius R 0 , and physical parameters (the diffusion coefficient D and the viscosity ) on the bubble growth rate has been investigated (figure 19). The results are shown for CBA-848 for an external pressure of 0.25 MPa. In figure 19a, three different values of the initial radius R 0 were chosen (5 nm, 5 m and 100 m).

The lowest value corresponds to the typical critical radius value that can be found in the literature for homogeneous nucleation [4]. The final predicted bubble radius is only slightly influenced by the initial radius value (between 290 m and 320 m) but differences are obviously very important at intermediate time steps (till 30 s).

Varying the diffusion coefficient from 10 -8 m 2 /s to 10 -11 m 2 /s leads to a final bubble radius around 100 m and 600 m, respectively (figure 19b). Typical diffusivity values for a gas-polymer system place in the rage between 10 -8 -10 -10 m 2 /s. The chosen diffusivity value was taken directly from the literature [35], and corresponds to the diffusivity value of CO 2 into a PP matrix at 483 K. A more detailed analysis should include the diffusion process of water vapor into a PP matrix. It is possible to assume a value of the diffusion coefficient of water vapor slightly higher than the value for CO 2 due to the lower molar mass. However, the range of diffusion coefficients analyzed in figure 19b is expected to cover both CO 2 and water vapor diffusion process.

  

It can be observed that increasing viscosity  from 200 Pas to 1000 Pas induces a decrease of the final bubble radius from 600 m to less than 100 m (figure 19c). The viscosity was determined experimentally from shear rheological measurements and presents a Newtonian plateau around 500 Pa.s at the strain rates encountered during the foaming process. During the foaming process, the flow around a single bubble is purely elongational, which means that the chosen viscosity value determined by shear measurements may be significantly underestimated. This justifies testing higher viscosity values in the single bubble growth model, as seen in figure 19c.

The influence of the initial number of moles n 0 and the influence of the initial radius R 0 are correlated from the gas perfect law (n 0 R 0 3 ).

Finally, as the number of bubbles is less important than the number of CBA particles and as the final radius of the bubbles is more important than the mean initial distance between the particles, it is clear that bubble coalescence appears quite early in the process. Due to the important CBA particle size distribution (figure 5), one can believe that the biggest initial particles will induce the biggest bubbles in the first stage of their development (see figure 19a), which will coalesce with the small surrounding bubbles initiated by the smallest initial particles. Considering, for example, one bubble with a diameter of 100 m surrounded by eight bubbles with a diameter of 25 m (see figure 19a at t=5s), the final diameter of the central bubble will be only 108 m after coalescence. As a consequence, if 88 % of the bubbles disappear during the foaming process, (which corresponds to the scenario of one bubble surrounded by eight smaller bubbles), the final bubble diameter will be only enhanced by 8 %. This may explain the good agreement between the calculation result which does not account for coalescence and the experimental observation where coalescence takes place.

-Conclusion

A simple polymer expansion experiment has been designed to test the foaming behavior of different chemical blowing agents (CBA), submitted to different pressure conditions. The bubble size and bubble size distribution depend on the blowing agent and on the applied pressure and careful bubble size measurements obtained by Image Analysis correlate well with the global macroscopic expansion of the foamed sample.

A single bubble expansion model assuming nucleation on each CBA particle, accounting for the different chemical reactions and for the gas diffusion from the bubble to the surrounding polymer matrix agrees fairly well with the experimental results, despite the strong hypothesis. This allows us to build a simple method to estimate the capability of a CBA to develop a foamed structure and the resulting mechanical properties.

Further studies analyzing injection molding experiments with the same CBAs are in progress.

Figure 1 -

 1 Figure 1 -DSC curve of the chemical foaming agents and of the polypropylene compound.Constant heating rate of 6°C/min, from ambient temperature to 250°C.

Figure 2 -

 2 Figure 2 -TGA curve of the chemical foaming agents (heating ramp 10 °C/min from ambient temperature to 600 °C)

Figure 3 -

 3 Figure 3 -Quantity of gas created during the decomposition of the reactive elements in the CBA particles

Figure 4 - 3 Figure 4

 434 Figure 4 -SEM micrograph of the precursor foaming material

Figure 5 -

 5 Figure 5 -Calculation of the CBA particle average size a) Binarized image b) Resulting histogram

Figure 6 -

 6 Figure 6 -Resulting histogram of the calculation of distance between particles

Figure 8 -

 8 Figure 8 -Evolution of the height of each sample during the expansion process

Figure 10 -

 10 Figure 10 -Optical micrographs of the expanded samples a) Sample 1-1 b) Sample 1-2 c) Sample 2-1 d) Sample 2-2 e) Sample 3-1 f) Sample 3-2

Figure 14 -

 14 Figure 14 -Theoretical fitting of the number of moles created during the CBA decomposition a) CBA-828 b) CBA-848

Figure 15 -𝑑

 15 Figure 15 -Gas diffusion profiles obtained from the diffusion equation a) CBA-828 b) CBA-848

Figure 16 -

 16 Figure 16 -Evolution of the number of moles of gas during the foaming expansion of a single bubble a) CBA 828 b) CBA 848

Figure 17 -

 17 Figure 17 -Numerical prediction of the single bubble growth during the expansion process a) CBA 828 b) CBA 848

Figure 18 -

 18 Figure 18 -Bubble growth speed (u) during the expansion process (CBA-828)

Figure 19 -

 19 Figure 19 -Influence of several parameters in the single bubble growth (Test 3-1) a ) Initial radius R 0 b) Diffusion coefficient D c) Viscosity 

Table 1 -

 1 Chemical composition of the CBA particles

		Matrix	Reactive elements	Proportion
				(wt %)
	1	PE	Citric acid + sodium bicarbonate	35+35
	2	PE	Citric acid	70
	3	PE	Sodium bicarbonate	70

Table 3 -

 3 Final volume, expansion ratio and final density of foamed samples

	Sample	CBA	V f (cm 3 )	Expansion ratio	 (g/cm 3 )
	1-1	1	5.62	1.94	0.47
	1-2	1	3.98	1.38	0.65
	2-1	2	6.31	2.18	0.41
	2-2	2	4.17	1.44	0.63
	3-1	3	5.84	2.02	0.45
	3-2	3	3.92	1.36	0.66

Table 5 -

 5 Gas volume and morphological parameters in the expanded samples

	Test CBA Bubble count	R bubble (m)	V bubble (mm 3 )	L bubble (m)	 F (g/cm 3 )	N c (cm -3 )	V gas (cm 3 ) indirect	V gas (cm 3 ) direct	remaining (cm 3 ) V gas
	1-1	1	64	360	0.195	50	0.47	1.4310 4	2.89	2.73	3.77
	1-2	1	108	170	0.021	42	0.65	8.9710 4	1.19	1.09	1.88
	2-1	2	241	120	0.007	25	0.41	4.3010 5	3.11	3.42	5.53
	2-2	2	292	85	0.003	23	0.63	7.5810 5	1.35	1.28	2.76
	3-1	3	79	350	0.180	56	0.45	1.6510 4	2.90	2.95	3.77
	3-2	3	121	190	0.029	38	0.66	6.2410 4	1.19	1.03	1.88

Table 7

 7 presents the physical parameters and the initial conditions employed in the foaming simulation.

	Parameter	CBA 2	CBA 3	Units
	Initial number of moles n 0 (3.56 or 7.13) 10 -14 (3.56 or 7.13) 10 -14	mol
	Initial radius R 0	4.4610 -6	4.4610 -6	m
	Viscosity 	500	500	Pas
	Surface tension  *	0.020	0.020	Jm -2
	Diffusion coefficient D *	810 -9	810 -9	m 2 s -1
	Gas constant 	8.314	8.314	m 3 PaK -1 mol -1
	Gas temperature T g	483	483	K
	External pressure P ext	(2.5 or 5)10 5	(2.5 or 5)10 5	Pa
	Solubility parameter K*	1.1510 -4	1.1510 -4	molm -3 Pa -1

Table 7 -

 7 Physical parameters and initial conditions employed for the foaming simulation *Values taken from literature