Olivier Louisnard
email: louisnar@mines-albi.fr

Fabienne Espitalier

On the possibility of homogeneous nucleation of water droplets and/or ice crystals during the bounces of a cavitation bubble

Keywords: numbers: 47, 55, dd, 43, 35, Ei

Acoustic cavitation is known to trigger ice nucleation in supercooled water. Several competing and still debatable mechanisms have been proposed in the literature and are related to the pressure field in the vicinity of the bubble at the end of its collapse. Numerical simulations of the bubble dynamics show that during the bubble expansions in the bounces following the main collapse, the bubble core temperature reaches values far below 0 • C for time periods of about 500 ns. The water vapour present in the bubble during these time intervals explores the liquid and the solid region of the phase diagrams before going back to the vapour region. On the base of approximate nucleation kinetics calculations, we examine to what extent liquid droplets could nucleate homogenously in the bubble core during these excursions. We also discuss the possibility that the nucleated clusters reach the bubble wall and trigger ice nucleation in the surrounding liquid if the latter is supercooled.

I. INTRODUCTION

Gas bubbles submitted to a sound field undergo radial oscillations, a phenomenon known as acoustic cavitation. For enough large amplitude of the driving field, such bubbles undergo an explosive growth followed by a rapid collapse. The density energy in the compressed gas at the end of the collapse is large enough to break chemical bounds and produce light emission, known as sonoluminescence.

Cavitation bubbles are also known to trigger ice nucleation in supercooled water. In absence of ultrasound, nucleation is a stochastic process and can occur over a relatively wide range of supercooling temperatures. When subject to even short ultrasound bursts, supercooled water can freeze at lower supercooling level [1][2][3][4][5][6][7][8][9]. This ability to control the nucleation temperature has interesting and promising industrial applications. Despite various mechanisms have been proposed to explain this effect and several studies report experimental results, the mechanism of ice nucleation by a radially oscillating bubble remains unclear.

Two main available theories have been proposed [2,[10][11][12], and share a common feature: both attribute ice nucleation to a shift of the freezing point in the supercooled liquid phase, because of large pressure variations at the end of the bubble collapse. A large majority of species increase their freezing temperature T f as pressure increases. Normal ice (ice I) constitutes an exception since the solid phase is less dense than the liquid, so that a compression of supercooled water would therefore normally quench freezing. Hickling [10] argues that the very large pressures attained (typically several GPa) in the vicinity of the bubble allow the nucleation of ice V, VI and VII, which contrarily to ice I, have a positive dT f /dp slope.

Hunt & Jackson notices that the large pressure increase at the end of the collapse is followed by a very high negative transient pressure, which may increase supersaturation relative to ice I nucleation. Both theories are equally mentioned in studies of cavitation-enhanced water freezing. Due to the very short space-and timescales involved, a direct confirmation of one or the other mechanism appears hardly feasible, despite Hickling's statement has been favored by calculations of orders of magnitude [13].

In the present work, we examine the possibility of a third mechanism, which has been overlooked in past studies. The above-mentioned theories only examine the thermodynamic state of the surrounding water. However, in the curse of the volume oscillations of the bubble, water evaporates and condensates at the interphase, so that the bubble encloses a variable quantity of water vapor. Finite-rate mass diffusion of water vapor through noncondensable gas [14,15] and nonequilibrium evaporation/condensation at the bubble wall [14,16,17] are known to prevent water from condensing during the collapse. This water-trapping mechanism has been recognized to decrease the final collapse temperature [14,18,19], which plays a crucial role in sonoluminescence, and in particular explains the enhancement of single-bubble sonoluminescence (SBSL) in cold water [19][20][21][22].

However, poor attention has been paid to the fate of the bubble vapor content after the collapse. A first plausible reason for that is that sonoluminescence and sonochemistry studies are mainly concerned with the hot state of the bubble interior at the end of the collapse, so that the subsequent phases of the bubble dynamics are of little interest in this framework. A second reason is that a bubble is not granted to maintain a spherical shape and even to survive after its collapse in multi-bubble conditions. However, a single bubble in a levitation experiment is shape-stable for millions of cycles in single-bubble experiments [23,24], and bubbles stable for several cycles have been reported in some multi-bubbles experiments [START_REF] Mettin | Bubble and Particle Dynamics in Acoustic Fields: Modern Trends and Applications[END_REF].

This justifies the study of the water vapor content after the main collapse. Of special interest is the re-expanding phase of the bubble after the primary or secondary collapses. A naive examination of the radius-time curve around the collapse shows that the re-expansion also occurs on a timescale almost as fast as the collapse (see for example Fig. 2a). One may therefore expect that, as the collapse yields considerable heating of the bubble interior, the re-expansion might produce a large cooling of the bubble content, including water vapor. The goal of this work is the assess the latter point and to examine its implications on potential nucleation of liquid water droplets or ice crystals. From an historical point of view, this is an attempt to link the long known effect of cavitation on ice nucleation, to more recent theories modeling water and heat transport in an oscillating spherical bubble.

II. THE STATE OF WATER VAPOR DURING BUBBLE BOUNCES

To calculate the water state in the bubble, we use a simplified model based on thermal and mass diffusion layers popularized by studies on SBSL [15,[START_REF] Storey | [END_REF], and recently validated against more refined models [27]. In order to simplify the discussion, we neglect chemical dissociation of the bubble content. The bubble radial dynamics is described by a Keller equation [24,28,29]. We study the case of an air bubble in water, at ambient pressure p 0 = 101300 Pa, driven at 20 kHz. We first consider a bubble in water at ambient temperature T 0 = 298 K, with physical properties of water ρ l = 1000 kg/m 3 , µ l = 10 -3 Pa.s, σ = 0.0725 N.m -1

Figure 1 displays the variations of the radius (Fig. 1a) and core temperature (Fig. 1b) of an air bubble of ambient radius R 0 = 5 µm, driven by a 20 kHz sinusoidal field of amplitude p a = 130 kPa. The classical temperature peaks can be observed at the main and secondary collapses. Moreover, an interesting feature of the temperature curve can be observed on Figs 2a-b, which are zoom into Figs 1a-b around the bubble afterbounces: slightly after the main collapse and also after the secondary ones, the bubble core temperature falls down below 0 • C for a short time during the bubble re-expansion. Although this feature is visible in other works (see for example Fig. 1c in [30], Fig. 1d in [16], Fig. 4 in [31], Fig. 2 in [32]), to our knowledge, it has never been commented.

The physical origin of this feature shares some similarities with the adiabatic heating of the bubble core during the collapse: the expansion velocity is fast enough to partially inhibit heat conduction between the liquid and the bubble core, so that the expansion phases are almost adiabatic.

Moreover, owing to finite diffusion of vapor through air, there is excess water trapped in the bubble core during the bounces, which slowly condenses at the bubble wall from one bounce to the other [14,15]. Thus, when the temperature of the bubble core drops below 0 • C, the water vapor content is cooled down to freezing temperatures. This raises the question of whether this water vapor is thermodynamically stable against liquefaction or ice formation. In order to assess this issue, we first plot the path followed by the state of water in a temperature/water partial pressure phase diagram. Fig. 3 displays a subset of this path around the post-collapse bubble expansions materialized by red circle-marks (color online) on Fig. 2. It can be seen that the vapor state crosses several time the liquid-vapor boundary and even performs six excursions in the solid region, during time periods of hundreds of ns. This timescale, although very short compared to the acoustic period, is much larger than the collapse characteristic time. The partial pressure of water vapor ranges between 10 3 and 10 4 Pa, so that at this scale the liquid-solid boundary is almost the vertical line T = 0 • C. The numbers correspond to the order of appearance in the acoustic cycle, and the paths are described anti-clockwise. The second excursion yields the lowest temperature (-48.9 • C), as could also be seen in Fig. 2b.

The present simulation suggests therefore that even in a liquid at ambient temperature, the water vapor of the bubble becomes repeatedly metastable during the 1 around the collapses. The red (color online) circle marks on both curves materialize the times at which the bubble core is at a temperature below 0 • C bubble bounces, not only against liquefaction, but also against ice formation, for time lapses of about hundreds of ns. This opens the interesting issue of whether this metastable vapor has enough time to nucleate into droplets or ice crystals inside the bubble. Before going further, one should comment on the singularity of the above results, in view of the parameters used for the simulation yielding Figs. 123, which are typical of bubbles levitated in SBSL cells. Such bubbles are known to keep their spherical shapes for a large number of cycles, so that they indeed undergo bounces. The present result would thus suggest that inertial single bubbles commonly observed in levitation cells have their content ready for ice nucleation in the expansive part of their bounces, even at room temperature. Whether ice nuclei have indeed enough time to form in such bubbles or not has therefore to be clarified.

If the latter observation also held for a bubble surrounded by supercooled water, this result would provide a new explanation, albeit incomplete, of how cavitation can trigger so easily ice formation in supercooled water. In order to clarify this point, we reproduced the precedent simulation in supercooled water at temperature T ∞ L = -5 (blue thick solid lines). The phase boundaries are drawn in black thin dashed lines. The circles materialize the six negative peak temperatures visible in Fig. 2, numbered by increasing time. The triangles materialize the water vapor state slightly before the minimal temperature is attained, and are represented to illustrate the direction followed by the trajectories. The dashed lines are the boundaries between the vapor, liquid and solid states of water. The excursion of the trajectories in the solid region are, for the paths numbered from 1 to 6: 236 ns, 332 ns, 331 ns, 305 ns, 266 ns, 204 ns.

plane (T , p H2O

) is displayed as a solid line in Fig. 4. The path obtained for T ∞ L = 20 • C (as displayed in Fig. 3) is recalled in dashed line for comparison. Several differences can be observed. As expected, the bubble inner temperature is shifted toward low values since the surrounding liquid is colder. It is seen that the minimal temperatures attained during bounces are thus lower than for a bubble at ambient temperature (the minimum temperature attained reaches -67.5 • C in this case). Moreover, the water vapor pressure is also shifted toward lower values, since the bubble in colder water initially contains less vapor.

Thus, in the case of a bubble surrounded by supercooled water, the water vapor in the bubble is therefore not only metastable against liquefaction, but moreover enters in the temperature range below the so-called homogeneous freezing point of water at -42 • C, which has long been considered as the supercooling limit for liquid water [33], below which homogeneous freezing occurs. One might therefore be tempted to conclude that ice should nucleate, at least temporarily in the bubble during the excursions in the solid phase on Fig. 4. Whether freezing indeed occurs inside the bubble is the main question discussed in this paper. We first propose a short review of the abundant literature on water freezing, notably in the context of cloud microphysics, and extending in the more general frame of water physics.

T ∞ L = -5 • C; Dashed (blue) line: T ∞ L = 20
• C (same as Fig. 3). The phase boundaries are drawn in thin dashed (black) lines. The additional thin dashed line starting from the triple point is the equilibrium curve between supercooled water and vapor. There are 10 excursion loops in the solid whose respective durations are 908 ns, 663 ns, 531 ns, 455 ns, 420 ns, 417 ns, 413 ns, 425 ns, 445 ns, and 514 ns. The (red) solid line are the supersaturation ratio level curves from Eq. (1). The (green) dash-dotted line is an estimation of the vapor spinodal line (see appendix A)

III. REVIEW OF WATER FREEZING A. Cloud physics

The conditions in the bubble are similar to the ones encountered in clouds, and the related issue of the formation of liquid and/or ice in the latter. The state of water in clouds has been recently found to be crucial for climate change predictions, and motivated an active research in this field. Clouds are formed by expansion and cooling of humid air as it ascends, so that vapor becomes thermodynamically unstable and condensation into liquid droplets occurs [START_REF] Debenedetti | Metastable liquids: concepts and principles[END_REF][START_REF] Mason | Physics of clouds[END_REF]. This condensation is favored by the presence of soluble or insoluble aerosol particles called cloud condensation nuclei (CCN) [START_REF] Mason | Physics of clouds[END_REF]. Formation of ice can occur at moderate negative temperatures by several mechanisms involving specific aerosol particles (called ice forming nuclei, IFN) in low and middle troposphere clouds [START_REF] Cantrell | [END_REF]37]. However in high altitude clouds such as cirrus, the dearth of such IFNs prevents such mechanisms and supercooled water droplets can be found down to [38][39][40][41]. Below this temperature, the supercooled droplets in the cloud freeze homogeneously.

-40 • C

B. Water no man's land

These results are in close agreement with laboratory experiments, which have evidenced the so-called homogeneous freezing temperature T H below which supercooled droplets unavoidably undergo homogeneous freezing. According to classical nucleation theory, this temperature in fact slightly varies with the droplet size, since the number of nucleated embryos per unit time is proportional to the sample volume. At the time of Angell's early review on supercooled water [33], the lowest temperature ever reached for supercooled pure water was -42 • C in micronsized droplets [42].

However, it has been demonstrated that ultra-fast cooling of water allows to bypass homogeneous freezing, and to obtain amorphous ice with a glass transition at 136 K, which further crystallize into cubic ice (Ic) near 150 K upon heating [41]. There exists therefore a domain between 150 K and 231 K, termed as "no man's land " [43], where the existence and properties of supercooled water could not be assessed for long.

Entering the no man's land has been achieved first by the pioneering work of Bartell and co-workers [44,45], by expanding a mixture of carrier gas (neon) and water vapor through small supersonic nozzles. With cooling rates of 10 7 K/s, liquid water clusters of 74 Å (6000 molecules) could be observed and analyzed by electron diffraction spectroscopy. The clusters were found to freeze at temperatures as low as 200 K, and diffraction patterns showed that cubic ice (Ic) was nucleated. Similar recent experiments with slightly lower cooling rate (10 5 K/s) and argon as the carrier gas allowed to nucleate and freeze supercooled droplets between 202 K and 215 K [46]. The estimated homogeneous nucleation rates in supersonic nozzle experiment can reach about 10 30 m -3 s -1 , which is between 15 and 20 orders of magnitude larger than the nucleation rates observed slightly above the no man's land, where numerous results have been collected [45]. In spite if this large range of nucleation rate, models based on classical nucleation theory, and assuming nucleation of cubic ice (Ic), seem to yield rather good results, even if there remains uncertainties on the interfacial energy between liquid and ice (Ic) [46,47].

That deeply supercooled water freezes into cubic ice (Ic) rather than hexagonal ice (Ih) is supported by Bartell's electron diffraction spectra [44,45] and by the presence of cubic ice in clouds [48,49]. As noted by Murray and co-workers [37], this also agrees with Ostwald's rule of stages, which states that the metastable phase nucleate (in this case cubic ice) preferentially to the stable phase (hexagonal ice). Recent experiments, computer simulations, and careful reinterpretations of past studies, revealed that ice formed from supercooled water is in fact composed of randomly stacked layer of cubic and hexag-onal sequences [50]. As noted by Murray and co-workers [37], the precise phase of the ice critical nucleus remains unknown, and this constitutes an additional difficulty in the establishment of a definitive nucleation theory for supercooled droplets freezing, especially for the estimation of the liquid-solid interfacial energy, which has a huge influence on the nucleation rate. Despite the latter reservations, the freezing rate proposed in Ref. [47] seems to yield good agreement with a large set of experimental results, over a large range of supercooling, including the no man's land [46].

As a final remark on ice nucleation, there seems to have general evidence that direct homogeneous deposition of ice from vapor does not occur, neither in clouds, nor in supersonic expansions experiments [46]. Thus, liquid droplets would always nucleate prior to freezing, which, again, is a consequence of Ostwald's rule of stages. It sounds therefore reasonable to discard such a mechanism in the present case, and assume that the water vapor in the bubble cannot form ice nuclei without the prior formation of liquid droplets.

C. Relevance to the present problem

Following one of the paths visible on Fig. 4, it is seen that the water vapor in the bubble core can be supercooled to temperatures falling in the no man's land range. If direct ice deposition from vapor can be discarded, ice formation in the bubble core, if any, could only occur by primary condensation of vapor into droplets, which, further supercooled as the bubble expands, might undergo homogeneous freezing, possibly in the no man's land region. A reasonable theory for the latter process is available over a wide range of freezing temperatures [46,47]. The physics involved is therefore strikingly similar to the one encountered in supersonic nozzles phase transitions. In both cases, the fast cooling of the mixture of water vapor and carrier gas results from an almost adiabatic expansion. The latter owes to a supersonic flow in the nozzle divergent in one case, and by the bubble outward motion in the other.

There remains however the problem of the nucleation of droplets itself. Ultrasonic nozzle experiments [46] show that droplets first nucleate when supersaturation p H2O /p sat,liq (T ) reaches a critical value and further grow by condensation of the surrounding vapor. In doing so, heat is released in the carrier gas and increases its temperature, which in turn quenches liquid nucleation, yielding an almost monodisperse aerosol. Once water vapor has almost entirely been consumed, the subsequent expansion in the nozzle cools the droplets until they freeze homogeneously. During the growth phase, droplets are hotter than the surrounding mixture, as vapor condenses at their surface.

Models for this three-step process, droplet nucleation / droplet growth / droplet freezing, are now available and show good agreement with nozzle experiments [51][52][53][54].

Transposing such models to the present case is technically feasible, by supplementing the ODE set describing the bubble motion with the ODEs describing the droplets nucleation and growth. This would require however to reconsider the thermal model for the bubble interior and add a source term accounting for the heat released by vapor condensation. Embarking in such a procedure is out of the scope of the present paper.

Nevertheless, it is interesting to investigate at least the first step along a given bubble bounce, to assess whether a significant number of droplets can be nucleated, their sizes, and the fraction of the vapor available in the bubble core that is condensed into clusters. Before carrying out these nucleation calculations, we must make sure that spinodal liquefaction of water vapor in the bubble can be discarded in the present case. As can be seen on Fig. 4, the paths followed by water in the (T, p H2O ) plane remain sufficiently far from the estimated vapor spinodal line (dash-dotted line), whose estimation is deferred to appendix A. This ensures that condensation of liquid water in the bubble, if any, can only occur through homogeneous droplet nucleation.

IV. KINETICS OF DROPLETS NUCLEATION A. Nucleation model

The instantaneous supersaturation ratio related to the vapor-liquid transition is classically defined by: S(p H2O , T ) = p H2O p sat,liq (T ) (1) where p sat,liq (T ) is the equilibrium vapor pressure at temperature T . As we are dealing with potentially supercooled water, the correlation used for the latter must extend into the deeply supercooled regime, and we use the results of Ref. [55]. On the other hand, it should be emphasized that p H2O is not constant in our case, because the bubble continuously exchanges water with the surrounding liquid phase through its interface. This is why, contrarily to most studies, the supersaturation ratio does not depend on T only. The evolution of S on the second rebound is displayed on Fig. 5 in the two cases corresponding to Fig. 4: T ∞ L = -5 • C ( solid line, blue online) and T ∞ L = 20 • C (dashed line, red online). We also displayed the iso-S curves in the (T, p H2O ) plane in Fig. 4 (red online).

The classical nucleation theory (CNT) [START_REF] Kashchiev | Nucleation : Basic theory with applications[END_REF] assumes that nucleation occurs through progressive build-up of clusters of molecules, that are precursors of the new phase. A g-sized cluster can yield a (g+1)-sized cluster by attachment to a water molecule (termed as "monomer"), and conversely the latter can loose a monomer by the backward reaction: Under the so-called capillary approximation, the formation energy of a g-sized cluster can be written as

W (g) = kT Θg 2/3 -g ln S , Θ = aσ kT (3)
where a is the area of a monomer, σ is the interfacial free energy between the bulk phases, and Θ its dimensionless form. Assuming spherical clusters, a = (36πv 2 0 ) 1/3 , where v 0 is the molecular volume.

The energy of formation has a maximum W * , which is the nucleation barrier, for a cluster of critical size g * , defined by :

W * = 4 27 Θ 3 (ln S) 2 kT g * = 2 3 Θ ln S 3 , (4)
so that the nucleation rate is essentially the rate of production of critical clusters. Assuming that the cluster size g is a continuous variable, the kinetics of the reaction set (2) can be described by a standard balance equation:

∂f (g, t) ∂t = - ∂J(g, t) ∂g , (5)
where f (g, t) is the concentration in g-sized clusters and J(g, t) is the cluster flux along the cluster size axis (g). This flux can be shown to read:

J(g, t) = -β(g, t)f (g, t) ∂ ∂g f (g, t) f eq (g, t) , (6)
where β(g, t) is the attachment rate by collisions between monomers in concentrations f (1, t) and a g-sized cluster:

β(g, t) = f (1, t)ag 2/3 kT 2πm 1/2
with m mass of a water molecule. The quantity f eq (g, t)

is the so-called equilibrium concentration of clusters deduced from the law of mass action. The correct expression for the latter has been a matter of debate [START_REF] Kashchiev | Nucleation : Basic theory with applications[END_REF] since the initial work of Becker and Döring [START_REF] Becker | [END_REF]. We use the self-consistent expression proposed by Girshick and co-workers [58][59][60], which, contrarily to the classical formulation, has the advantage to be valid for g = 1:

f eq (g, t) = f eq (1, t) exp - W (g) -W (1) kT (7)
The set of equations ( 5)-( 6) is known as master equation of nucleation. It has no analytic solution in the general case of time-dependent supersaturation, as in the present problem. In particular, as far as we are aware, analytic treatment of the number of nuclei produced by a supersaturation pulse has been poorly explored. A noticeable exception can be found in the work of Trinkaus & Yoo [61] who used Green functions formalism to derive approximate analytic solutions of the master equation, in the case of an idealized nucleation barrier whose location has a parabolic time-dependence around a minimum g * (t). However, the analytic expressions proposed by these authors is restricted to the case where the system is still supersaturated when the supersaturation pulse has relaxed, which is not the case here. We must therefore revert to some approximation. The two solutions adopted are described hereafter.

We first note that the simpler problem of transient nucleation in response to a supersaturation step has been extensively studied [START_REF] Kashchiev | Nucleation : Basic theory with applications[END_REF][62][63][64]. In this case, the transient duration is of the order of the so-called nucleation time lag:

τ = δ 2 2β(g * ) , (8)
where

δ = - 1 2kT ∂ 2 W ∂g 2 -1/2 g=g * = 3g * 2/3 Θ -1/2 (9)
is the width of the nucleation barrier. The time-lag is physically the order of magnitude of the time required for the clusters to populate the subcritical region g < g * by fluctuations.

B. Quasi stationary nucleation rate

If supersaturation has been maintained constant for a duration sufficiently larger than the time-lag, nucleation can be considered stationary, and ∂f (g, t)/∂t = 0. Under some approximations, Eq. ( 5) can then be solved to obtain the steady-state cluster concentration [START_REF] Kashchiev | Nucleation : Basic theory with applications[END_REF]. The stationary nucleation rate J S is defined as the cluster flux through the critical size J(g * ) and is found to write:

J S = 2σ πm 1/2 p sat,liq (T ) kT 2 v 0 e Θ S × exp - 4 27 Θ 3 (ln S) 2 . ( 10
)
We calculated the latter quantity along the second bounce, in the conditions of Fig. 4, for a liquid at ambient temperature and for a supercooled liquid at T ∞ L = -5 • C (Fig. 6). In both cases, the nucleation rate increases sharply and reaches a maximum after about 100 ns after entering in the metastable liquid zone. The main difference is that both the nucleation rate and the temperature drop much more slowly in the bubble surrounded by the supercooled liquid. On the other hand, a counterintuitive result is that the maximum nucleation rate is lower for the bubble in the supercooled liquid. This is mainly due to the pre-factor (p sat /(kT )) 2 which is lower for the supercooled bubble because there is less vapor in the latter (see Fig. 4). As proposed by Kashchiev [56,61,65], the nucleation rate Eq. ( 10) can still be used if supersaturation evolves on a time scale much larger than the time-lag, a situation referred to as quasi-stationary nucleation. Assuming this assumption valid in our case, the number of critical nuclei formed at time t can be expressed as:

N QS (t) = t 0 J S S(s), T (s) V (s) ds, (11)
whereas the number of water molecules condensed into nuclei reads:

n QS (t) = t 0 J S S(s), T (s) V (s) g * (s) ds, (12)
In both integrals, the origin of time is chosen when the vapor becomes supersaturated. In the present case, calculated instantaneous values of the time lag τ are found to be larger than 1 µs, except near the supersaturation peak where it becomes of the order of 100 ns. It has therefore the same order of magnitude as the typical time scale of the supersaturation variations, so that the assumption of quasi-stationary nucleation may be not fully reliable here. For a given change of the supersaturation value, nucleation does not reach steady state, and the use of Eqs. (11) probably yields an overestimation of the number of nuclei formed.

C. Transient nucleation

The second method used is to revert to some treatment of transient nucleation in response to a supersaturation step. In the present case, supersaturation evolves rather as a dome-shaped pulse, and a turnaround must be used. Since the rise of S(t) is very abrupt, we replace it by a square pulse between t 0 and t F , defined as (see Fig. 7). The latter times are chosen such that J S (t 0 ) = J S (t F ) = αJ max

S

, where α is a free parameter. If we restrict our analysis to times lower than t F , all happens as if the system had undergone a supersaturation step at t 0 , and classical results of transient nucleation can be used up to time t F . The main drawback of the method is that the choice of times t 0 and t F is somewhat arbitrary, and the sensitivity of the results to this parameter will be examined a posteriori.

We then used the results of Shi, Seinfeld & Okuyama [63], who used a combination of a boundary-layer method and Laplace transform to solve the nucleation master equation ( 5) for a supersaturation step at t = 0. They obtained an analytic expression of the instantaneous cluster size distribution [70]:

f (g, t) f eq (g) = 1 2 erfc g -g * δ + exp - t τ + λ - 1 2 erfc g -g * δ + e λ , (13)
where

λ = g * -1/3 -1 + ln 3g * 1 -g * -1/3 δ . ( 14
)
The concentration of nuclei produced

F (t) = t 0 J(g * , s) ds
was also obtained in analytic form by the authors as:

F (t) = J S τ 2 E 1 (e 2(λ-t/τ ) ) -E 1 (e 2λ ) , (15)
where J S is the stationary nucleation rate given by Eq. ( 10), τ is the nucleation time lag from Eq. ( 8), and E 1 is the exponential integral.

In order to use these analytic results, we set the quantities g * , δ and τ to their values at t 0 and replace t by tt 0 , up to t Ft 0 . The total number of nuclei N (t) = tF t0 J(s)V (s)ds formed in the bubble was then approximated by:

N TRANS (t) ≃ F (t -t 0 )V m , t 0 < t < t F (16)
with F given by Eq. ( 15), and V m is the mean volume of the bubble during the equivalent supersaturation pulse between t 0 and t F . The number of water molecules condensed into critical nuclei reads similarly:

n TRANS (t) ≃ F (t -t 0 )V m g * , t 0 < t < t F (17)
Since we replaced the exact dome-shaped supersaturation pulse by a smaller square pulse, these estimations are expected to yield lower bounds of the number of nuclei produced and the number of molecules consumed, respectively.

D. Results

Figure 8 displays the number of nuclei produced, N TRANS (t) from Eq. ( 16) (thin solid lines) and N QS (t) from (11) (thick solid lines), in the conditions of Figs. 456. The curves for the bubble in ambient water T ∞ L = 20 • C end with round symbols (red line online), and the one in supercooled water T ∞ L = -5 • C with square symbols (blue online). The times t 0 and t F have been chosen on the criterion J S (t 0 ) = J S (t F ) = 1 2 J max

S

. The two estimations are expected to yield lower and upper boundaries, respectively, of the number of liquid nuclei formed in the bubble. The number of nuclei obtained at t F in the two cases are displayed as bold lines in Tab. I. As expected from the comparisons of the nucleation rates on Fig. 6, less nuclei are produced in the bubble surrounded by supercooled water. We can also make use of Eqs.( 12)-( 17) to calculate upper and lower bounds for the number of water molecules that have clusterized into nuclei, or more eloquent, the fraction x of those molecules relative to the initial number of water molecules in the bubble as the vapor becomes supersaturated (rightmost columns of Tab. I). It can be seen that during the second rebound of the bubble surrounded by liquid at ambient temperature, between 5 % and 10 % of the water vapor initially present can clusterize into droplets. This fraction drops down to between 0.35 % and 2.5 % in the case of a bubble in supercooled water.

Table I also displays results obtained for larger driving acoustic pressures. It can be seen that increasing the driving decreases the number of nuclei formed. This is due to the fact that larger driving amplitudes produce hotter collapses, so that the initial temperature at the beginning of the rebounds becomes larger. Thus, vapor supersaturation decreases and so does the nucleation rate.

Finally, we also varied the arbitrary factor α used to define the equivalent supersaturation step (see Fig. 7) in order to assess the sensitivity of the results to this quan- The calculation have been performed twice for each parameter set, using α = 1/2 and α = 1/3, respectively (see Fig. 7).

R0 pa T ∞ L α = Jcut Jmax N TRANS N QS x TRANS x QS µm kPa • C % % 5 130
The two bold lines correspond to the data displayed on Fig. 8 tity. Table I shows that the latter is reasonably weak.

A last useful information is the evolution of the cluster size distribution during the supersaturation pulse. The size distribution non-dimensionalized by the equilibrium distribution f (g, t)/f eq (g) calculated from Eq. ( 13), is displayed on Fig. 9a, for metastable water at T ∞ L = -5 • C in the same conditions as Fig. 5, for 10 equidistant times covering the supersaturation pulse (thin lines). The classical steady-state erfc function is also displayed in thick solid line and the critical size is materialized by the vertical dashed line. It can be clearly seen that the steadystate is not reached, which demonstrates that the latter assumption would be difficult to justify in the present problem.

Another interesting result is the order of magnitude of the largest cluster formed in the bubble during the supersaturation pulse. The number of clusters of each size formed in the bubble was calculated by: N (g, t) = f (g, t)V m where f was estimated from Eq. ( 13), and V m is the mean volume of the bubble during the equivalent supersaturation pulse. The result is displayed as thin lines on Fig. 9b (time increasing from bottom to top). The largest cluster formed is seen to reach only a dozen of monomers.

V. DISCUSSION

Using a classical reduced model of bubble dynamics including heat and mass transfer, we have shown that in typical inertial cavitation conditions at 20 kHz, the water vapor in an air bubble becomes strongly metastable during the bubble second rebound, both against liquefaction and freezing. Approximate nucleation calculations show that vapor-liquid phase transition indeed takes place and that a few percents of the initial water content clusterize during the supersaturation pulse. However, owing to the huge supersaturation level, the critical size is about 5 molecules and the largest cluster formed reaches roughly 10 molecules. It is therefore difficult to conclude to the formation of real droplets, in contrast with the experiments in supersonic expansion, where clusters of several thousands molecules are formed, and have additional time to freeze homogeneously. This is due to the typical time scales involved in the respective experiments: the full process in nozzles lasts for about tens of microseconds, whereas the metastable state of the vapor in the bubble lingers for less than 1 µs (see caption of Fig. 4). Since nucleation is a kinetic process, it cannot proceed efficiently over such a small timescale.

Homogeneous freezing of such small liquid clusters is therefore very unlikely and the scheme of ice crystals nucleating inside the bubble, as was initially suggested by Fig. 4 must be abandoned. However, the fate of the small clusters formed remains an interesting issue. As supersaturation decreases after the peak, one may expect these clusters to dissociate back to monomers (after about 600 ns in the case of Fig. 5). This short lifetime casts some doubts on their ability to redisperse in the surrounding liquid. We note however that the same issue can be raised about the redispersion of the radicals produced by water sonolysis during the bubble collapse, in the frame of sonochemistry. Indeed, such radicals are also short-lived species whose precise mechanism of redispersion in the liquid after the bubble collapse remains unclear. In spite of this lack of knowledge, the entrance of OH • radicals in the liquid is the mechanism commonly put forward to explain their contribution to chemical reactions in the liquid phase. Whether the liquid clusters formed in the bubble are actually able to hit the surrounding liquid pertains to the same questioning. If for the purpose of reasoning, one does not exclude such a mechanism, a highly supercooled liquid cluster coming into contact with the surrounding supercooled liquid might act as a pre-existing entity able to trigger freezing. All would happen as if some part of the surrounding supercooled liquid (typically at T ∞ L = -5 • C) underwent a sudden fluctuation down to minus tens of Celsius. Among the plausible contacting mechanisms, non-symmetrical bubble collapse or bubble coalescence may be invoked. Of course, we do not claim that all bubbles undergo such an event precisely when clusters have formed inside, but that such an event has a nonzero probability over billions of bubbles collapsing 20000 times per second. The whole process constitutes an alternative explanation of the ability of cavitation to trigger freezing in supercooled liquids.

Whatever the existence of such a process, the large cooling of the gas/vapor mixture during the bubble rebounds constitutes an interesting and, as far as the authors are aware, unreported feature of acoustic cavitation. More complete models of the bubble interior based on direct Navier-Stokes simulations [14] would be required to confirm and quantify more precisely the reached level of supercooling. If the answer is affirmative, the cavitation bubble would (again) constitute a rather uncommon physical object, whose content is able to cool down to temperatures seldom encountered for water vapor, after having heated up to thousands of Kelvins. 10: (Color online) Thin (black) lines: vapor pressure curve (solid) and vapor spinodal curve (dashed) calculated from hard sphere/Yukawa EOS (A1)-(A2). Thick (blue) solid line: vapor pressure of water used in this paper [55]. Thick (blue) dashed line: corrected vapor spinodal estimated from (A3). The (red) square symbols are the vapor spinodal data for water tabulated in Ref. [68]

FIG. 1 :

 1 FIG.1:a. Temporal evolution of the radius of a 5 µm air bubble in water driven by a 20 kHz field at pa = 130 kPa. b. Temporal evolution of the bubble core temperature.

50 FIG. 2 :

 502 FIG.2: Zoom of Fig.1around the collapses. The red (color online) circle marks on both curves materialize the times at which the bubble core is at a temperature below 0 • C

FIG. 3 :

 3 FIG.3:(Color online) Evolution of water conditions in the phase diagram of water (T, pH 2 O) in the conditions of Fig.1(blue thick solid lines). The phase boundaries are drawn in black thin dashed lines. The circles materialize the six negative peak temperatures visible in Fig.2, numbered by increasing time. The triangles materialize the water vapor state slightly before the minimal temperature is attained, and are represented to illustrate the direction followed by the trajectories. The dashed lines are the boundaries between the vapor, liquid and solid states of water. The excursion of the trajectories in the solid region are, for the paths numbered from 1 to 6: 236 ns, 332 ns, 331 ns, 305 ns, 266 ns, 204 ns.

FIG. 4 :

 4 FIG. 4: (Color online) Evolution of water conditions in the phase diagram of water (T, pH 2 O) for a 5 µm air bubble in water driven by a 20 kHz field at pa = 130 kPa. Solid (blue) line:T ∞ L = -5 • C; Dashed (blue) line: T ∞ L = 20• C (same as Fig.3). The phase boundaries are drawn in thin dashed (black) lines. The additional thin dashed line starting from the triple point is the equilibrium curve between supercooled water and vapor. There are 10 excursion loops in the solid whose respective durations are 908 ns, 663 ns, 531 ns, 455 ns, 420 ns, 417 ns, 413 ns, 425 ns, 445 ns, and 514 ns. The (red) solid line are the supersaturation ratio level curves from Eq. (1). The (green) dash-dotted line is an estimation of the vapor spinodal line (see appendix A)

FIG. 5 :

 5 FIG. 5: (Color online) Supersaturation ratio in a 5 µm air bubble in water driven by a 20 kHz field at pa = 130 kPa, along the second bounce. The conditions are the ones of Fig. 4. The origin of time is chosen when the water vapor state crosses the liquid-vapor equilibrium curve in each case. The markers on the curves indicate the bubble core temperature in • C.

FIG. 6 :

 6 FIG.6: (Color online) Stationary nucleation rate of droplets in a 5 µm air bubble in water driven by a 20 kHz field at pa = 130 kPa, along the second bounce, calculated from Eq.(10). Square symbols (blue online):T ∞ L = -5 • C; Round symbols (red online): T ∞ L = 20• C (same as Fig.3). The origin of time is chosen when the water vapor state crosses the liquid-vapor equilibrium curve in each case. The markers on the curves indicate the bubble core temperature in • C.

FFIG. 7 :

 7 FIG. 7:Choice of start-time t0 and end-time tF to apply the transient nucleation results of[63]. The values are chosen so that the stationnary nucleation rate JS lies in the rangeαJ max S ≤ JS ≤ J max S, where α is a free parameter of order 1 (typically α = 1/2, see text and Tab. I)

FIG. 8 :

 8 FIG. 8: (Color online) Number of nuclei produced during the supersaturation pulse; Thick solid lines: assuming quasistationary nucleation ; Thin solid lines: assuming transient nucleation. The conditions are the same as in fig. 6. Square symbols (blue online): T ∞ L = -5 • C; Round symbols (red online): T ∞ L = 20 • C (same as Fig.3). The supersaturation is recalled in dashed lines (right ordinate axis) for the two conditions.

FIG. 9

 9 FIG. 9: a. Thin lines: dimensionless cluster size distribution calculated from (13) at ten equidistant times covering the supersaturation pulse (time increasing from bottom to top). The thick line represents the number of clusters formed in steady state. b. Number of clusters formed in the bubble.

  FIG.10:(Color online) Thin (black) lines: vapor pressure curve (solid) and vapor spinodal curve (dashed) calculated from hard sphere/Yukawa EOS (A1)-(A2). Thick (blue) solid line: vapor pressure of water used in this paper[55]. Thick (blue) dashed line: corrected vapor spinodal estimated from (A3). The (red) square symbols are the vapor spinodal data for water tabulated in Ref.[68]

TABLE I :

 I Number of clusters created and number of water molecules consumed during the buble second rebound, for various bubble ambient radii and driving pressure amplitudes.

	-5	1/2	7.8 × 10 4 6.7 × 10 5 0.35 2.5
	5 140 -5	1/2	7.5 × 10 4 6.5 × 10 5	0.32 2.5
	5 150 -5	1/2	7.1 × 10 4 6.4 × 10 5	0.31 2.4
	5 130 20	1/2	5.7 × 10 6 1.2 × 10 7 5.6	10.
	5 140 20	1/2	4.9 × 10 6 1.1 × 10 7	4.7	9.3
	5 150 20	1/2	4.2 × 10 6 1.0 × 10 7	3.9	8.3
	5 130 -5	1/3	6.5 × 10 4 7.2 × 10 5	0.32 2.8
	5 140 -5	1/3	6.5 × 10 4 7.1 × 10 5	0.31 2.7
	5 150 -5	1/3	5.9 × 10 4 6.9 × 10 5	0.28 2.6
	5 130 20	1/3	5.0 × 10 6 1.3 × 10 7	5.5	11.
	5 140 20	1/3	4.5 × 10 6 1.2 × 10 7	4.7	9.9
	5 150 20	1/3	4.2 × 10 6 1.1 × 10 7	4.2	9.0

Appendix A: Calculation of the vapor spinodal curve

The vapor spinodal curve is generally poorly documented in thermodynamic database. We use therefore the approach of Kim and co-workers [66] who assumed that the ratio p spinodal (T /T C )/p sat (T /T C ) for the fluid of interest was the same as that obtained for an equation of state (EOS) describing a hard sphere fluid corrected by an attractive Yukawa potential [67]. We recall the main lines of calculations in this appendix.

The fluid pressure and chemical potential are given respectively by the EOS:

where ρ is the molecular density, η = πσ 3 ρ/6 is the packing fraction, σ is the hard sphere diameter, and α is the amplitude of the attractive potential. The corresponding dimensionless critical density ρ C and critical temperature T C are found to be:

The binodal curve is calculated by solving simultaneously

for the equilibrium densities ρ sat l and ρ sat v , and the liquid and vapor spinodal ρ spin l and ρ spin v densities are the roots of:

The equilibrium curve and the two spinodal curves in the (p, T ) plane are then obtained by applying the EOS (A1) to the values obtained. The equilibrium and vapor spinodal curves obtained are displayed as thick lines on Fig. 10, and the H 2 O equilibrium curve from Ref. [55] used in this paper in thin solid line. We then assume the real vapor spinodal curve to be:

where p sat is the vapor pressure calculated from Ref. [55]. Figure 10 displays the predicted binodal (thin solid line) and spinodal (thin dashed line) of the hard-sphere Yukawa equation of state. The binodal of Ref. [55] used throughout this paper is represented by a thick solid line and the spinodal deduced from Eq. (A3) by a thick dashed line. The latter results displays reasonable agreement with the theoretical prediction of Ref. [68] (square symbols).


Nicolas Le Moigne
email: nicolas.le-moigne@mines-ales.fr

Martial Sauceau
email: martial.sauceau@mines-albi.fr

Margot Chauvet

Jean-Charles Bénézet

Jacques Fages

Microcellular Foaming of (Nano)Biocomposites by Continuous Extrusion Assisted by Supercritical CO 2

In many industrial fields, the development of porous and light polymer composite structures is of great interest. Continuous extrusion assisted by supercritical carbon dioxide (sc-CO 2 ) is an emerging technology for the microcellular foaming of polymer and composite materials. Thanks to its specific physical properties, sc-CO 2 can act as a removable plasticizer and physical blowing agent during processing. Unlike chemical foaming processes, sc-CO 2 assisted extrusion avoids leaving residues in the final products. It also allows a rapid mixing and dissolution of sc-CO 2 in the polymer melt as compared to batch processes. In this chapter, the foaming of biopolymer matrices by sc-CO 2 assisted extrusion is reviewed and extended to the development of (nano)biocomposites foams loaded with micro/nano-sized (in)organic fillers.

Introduction

It is well known that a light cellular structure like foams may have several advantages compared to a massive solid of the same chemical nature. Beyond the advantages related to weight reduction and improved mechanical properties (flexibility, compressibility,…), several other properties, like insulation, sound absorption, cushioning can be of industrial interest. As an example, foams have undergone a large development in the sport industry for shoes, helmets,… Tridimensional porous structures may also offer new possibilities for re-colonization of a grafted implant by living tissue, the foam playing the role of a scaffold. Foamy structures may also act as a favorable medium to deliver progressively an active ingredient previously incorporated in the solid upon dissolution in a biological fluid (1).

It is noticeable that a lot of such porous structures are ubiquitous in nature with different roles and a wide diversity of pore size: wood, eggshells, bone tissue, corals, zeolites,… Humanity has understood for ages that mimicking nature is a mighty tool to develop new materials with specific end-user properties (2). Among them, thermoplastic foams have been tremendously developed since the mid-20 th century, starting with the Styrofoam® used in the "Liberty ships" of the US Navy and having since then given a huge number of different applications. However, due to the shortage of fossil resources, and the raise of environmental concerns, it has become evident in the last two or three decades that polymers from different origin must be used. The so-called biopolymers (bio-based, biodegradable and/ or biocompatible) have consequently elicited an increasing interest and an evergrowing number of scientific works have been devoted to them (3).

In this chapter, the foaming of biopolymer matrices by supercritical carbon dioxide (sc-CO 2 ) assisted extrusion is reviewed and extended to the development of (nano)biocomposites foams loaded with micro/nano-sized (in)organic fillers. In particular, the combined effects of the processing parameters and the use of fillers on the foam structure are discussed.

Foaming of Biopolymers

There are two main routes to produce biopolymer foams depending on the blowing agent used, which can be either chemical or physical. The first route uses chemical blowing agents, which are chemicals able to release a gas upon thermal decomposition, e.g. carbonate salts releasing carbon dioxide. These chemicals present however the drawback of leaving a solid residue which may be detrimental in some applications as in biomedical. This is why the alternative route of using gas injection under pressure has been gaining interest and is progressively replacing chemical blowing agents. Indeed, better cell morphologies can be obtained at reduced production costs (4).

Among these gases, carbon dioxide in supercritical phase (sc-CO 2 ) appears to be the most used since it does not present the classical disadvantages of other fluorinated gases known for their toxicity to the environment. CO 2 is cheap, chemically inert, non-toxic and readily soluble in most polymers with a strong plasticizing effect [START_REF] Mchugh | Supercritical Fluid Extraction: Principles and Practice[END_REF].

In this regard, extrusion-foaming process using sc-CO 2 injection in the barrel of a single screw extruder (Figure 1) exhibits major features among them: absence of organic solvents, single step process (intense and integrated system), continuous operation mode,… and therefore provides a relevant contribution towards a sustainable industrial production of foams. It clearly belongs to the modern approach of chemical engineering named "green process engineering" which combines process intensification, environment preservation, economic relevance and sustainable process management to produce greener products [START_REF] Charpentier | [END_REF].

In such a process, the die geometry and the operating temperatures are the main parameters influencing the final morphology and properties of the foams: cell density, type of porosity (open or close), pore size distribution, expansion ratio, etc. It is the thermodynamic instability in the gas/polymer binary system, induced by the pressure drop at the exit of the die, which allow the cell nucleation and growth phenomena to occur. The gas concentration upstream the die is also a major parameter to control. In a recent review, Chauvet et al. ( 3) described the applications of extrusion assisted by sc-CO 2 on the microcellular foaming of biopolymers as defined in Figure 2.

Chauvet et al. (7) have investigated the effect of the die temperature -in the range 100 to 160 °C -on the foam characteristics of poly-lactic acid (PLA) (PLE001, melt flow index MFI of 6 g / 10 min -210 °C, 2.16 kg -NaturePlast, France), a thermoplastic biopolyester matrix commonly used in packaging and biomedical applications. The experimental equipment is basically a single screw extruder equipped with a gas-injection system. A static mixer has been inserted between the screw and the die to enhance the mixing capacity of the extruder. The die has a cylindrical shape and a diameter of 2 mm. A detailed description of this equipment can be found in previous publications (8)(9)(10). Several authors have shown that the die temperature is the most influent parameters explaining the expansion ratio, and the resulting porosity of the foam (3). The expansion ratio presents a curve with a maximum as a function of die temperature (T 6 on Figure 3b). At high temperature, diffusivity of the gas is high, its solubility and the melt viscosity are low: as a consequence only poor expansion ratio can be obtained with large pores due to gas escape, bubble growth and coalescence. Lowering the die temperature results in higher melt strength and a skin effect on the periphery of the extrudate, this in turn allows keeping more gas in the foam. This leads to higher nucleation and porosity while limiting the growth and maintaining a low level of coalescence. However, too low processing temperatures (below a temperature threshold) end in a stronger stiffness of the extrudate resulting in lower expansion.

The CO 2 content in the metastable phase upstream the nozzle has obviously a large influence too. It must be reminded that CO 2 solubility varies with pressure and inversely with temperature (11). CO 2 solubility is quite high in most thermoplastics. Chauvet et al. (7) found that up to 10 wt% CO 2 could homogeneously be dissolved into melted PLA. Figure 3a shows that porosity up to 97 % could be obtained with only 6.4 wt% of CO 2 . When foaming a polymer, it is also of interest to consider the type of porosity. Depending of the targeted application, it might be useful to get rather an open porosity (e.g. for filtration purposes or for tissue regeneration) while, in other applications, getting a close porosity would be beneficial (e.g. for cushioning or sound absorption). Chauvet et al. (7) demonstrated that the threshold temperature (109 °C in this case) corresponds to a change in the kinetics of crystallization leading to a higher crystallinity of the polymer when manufactured at the lowest temperatures. Other authors have also elucidated the role of melt strength and crystallinity, which determine the porosity formation behavior (12). Another interesting feature lies in the fact that extrusion foaming performed at lower temperatures (below the threshold) exhibits not only a decreased total porosity but also a much lower open/closed porosity ratio leading to a final structure with smaller closed pores. This observation is evidenced in In conclusion, the extrusion-foaming of biopolymers using sc-CO 2 appears to be a practical, sustainable and promising tool to develop new challenging applications in various industrial sectors. Many efforts and achievements still need to be done for a deeper understanding of the multiple phenomena involved: gas sorption, cell nucleation and growth, coalescence of the bubbles, effect of micro-and nano-fillers to cite just a few. The process optimization also deserves more work for the elucidation of the exact role of the die and screw geometry, the pressure and temperature,... adapted to the different biopolymer matrices. Both experimental and modelling work will contribute to a better mastering of microcellular foam manufacture in the near future.

Foaming of Biocomposites

As reviewed by Nofar and Park (13) for the case of PLA based systems, microsized (in)organic particles can be used to control the morphology of the foams. They can indeed act as heterogeneous cell nucleation agents and increase the cell nucleation rate during foaming. According to the heterogeneous cell nucleation theory (14,15), the presence of stiffer phases within the polymer melt, i.e. hard particles or crystals, can promote cell nucleation through local pressure variations at the interface ( 16), hence greatly enhancing the cell density of the foam.

Literature on the continuous foaming by sc-CO 2 assisted extrusion of biocomposite systems, i.e. biopolymer matrices with (in)organic particles, is rather scarce.

Pilla et al. (17,18) investigated the effect of talc (grade-JetFil700C, average particle size of 1.5 µm, Luzenac) on the microcellular foaming by sc-CO 2 assisted single screw extrusion of PLA (Ingeo TM 3001D, MFI of 15 g / 10 min -190°C, 2.16 Kg -NatureWorks LLC, USA) and (non-) compatibilized PLA/poly (butylene adipate-co-terephthalate) (PBAT) blends (pure PBAT for the non-compatibilized blend, Ecoflex, melt viscosity of 2.5-4.5 ml / 10 min at 190°C, and Ecovio for the compatibilized blend, PLA/PBAT 45:55 wt%, melt viscosity < 2.5-4.5 ml / 10 min at 190°C, BASF). In overall, the use of talc (0.5 wt%) in the formulations allowed to decrease cell size and increase cell density (Figure 5) while greatly increasing the degree of crystallinity of PLA from 9 to 45 % (17). According to the die temperature ranging from 160 to 130 °C, minimum cell size of about 20 µm and maximum cell density of roughly 8.6×10 6 cells/cm 3 were reached for PLA/talc system. Non-compatibilized PLA/PBAT blends did not allow any improvement on cell morphology of the foam, cell size and density being rather similar or even worse in terms of cell density as compared to neat PLA with or without talc. In contrast, the compatibilized PLA/PBAT blend (Ecovio) allowed substantial improvement with a decrease of cell size down to 10 µm with 0.5 % talc and maximum cell density of ~2×10 7 cells/cm 3 at 150 °C with or without talc (Figure 5), i.e. two order of magnitude higher than neat PLA ~2×10 5 cells/cm 3 at same die temperature and processing conditions (18). Besides, volume expansion ratio and open cell content were decreased for PLA/PBAT blends and those containing talc with values not exceeding 1.4 and 40 % respectively, except for the non-compatibilized PLA/PBAT blend without talc which showed higher expansion ratio and open cell content up to 1.7 and 55 %, respectively. Pilla et al. ( 17) also used a chain extender CE (CESA-Extend OMAN698493 containing 30 % of multifunctional epoxy-based CE, Clariant) to counterbalance PLA degradation and reduction of molecular weight occurring during processing. The simultaneous addition of talc and CE led to denser and more uniform cell structure until a CE content of 1 wt%, with maximum cell density of 5.2×10 7 cells/cm 3 at 160 °C. Based on these results, these authors clearly evidenced that additive, i.e. talc and CE, and blending strategies to control the foam morphology of PLA are relevant, and can significantly modify the morphology of the foam with higher cell density and lower cell size and open cell content, especially in the case of compatibilized blends. This is attributed to higher melt strength and increased cell walls stability with talc particles and PLA crystals in formation within the melt that increase heterogeneous cell nucleation and limit cell coalescence. These phenomena are promoted for compatibilized blends showing good interfacial interactions between the biopolymer phases. However, the volume expansion ratio could not be substantially improved because of the too high strength of the polymer melt. The melt strength thus appear as a critical parameter for better foamability and should be tuned with care by the addition of particles or blending to ensure an improvement of foam morphology. (17,18), 2 wt% epoxy functionalized CE (Joncryl ADR4368-C, styrene-acrylic oligomer, molecular weight of 6800 g/mol and an epoxy equivalent weight of 285 g/mol, BASF, Germany) and 2 wt% talc (HTPultra5 L, median diameter of 0.65 μm, IMI FABI SpA, Italy) were added in the formulation to increase the melt strength of PLA and its crystallization kinetics, and hence improve its foamability. According to die temperature ranging from 130 to 116 °C, talc and CE allowed to reach ultimate porosity of 97.1 % when decreasing die temperature and for lower processing temperatures than neat PLA. The addition of 5 wt% of cellulose fibers (Arbocel BWW40, average fibre length and diameter of 200 and 20 μm, respectively, J. Rettenmaier & Söhne GmbH, Germany) or basalt fibers (Basaltex KVT 150tex13-I, linear density of 150 tex, filament diameter of 13 μm and initial fiber length of 10 mm) in the PLA/talc/CE formulation provided a wider processing window for PLA foaming with die temperature ranging from 135 to 110 °C. Due to the lower applicable temperature profile, greater crystallinity and increased melt strength were obtained. Furthermore, natural fibers substantially increased the melt viscosity and promoted the heterogeneous cell nucleation during foaming while enhancing the degree of crystallinity of PLA. In optimal processing conditions, microcellular PLA biocomposite foams with ultimate porosity of 96.1 and 97.4 % were obtained with cellulose fibers and basalt fibers, respectively. However, the heterogeneous fiber distribution within the melt and the weak fiber / PLA interfacial adhesion led to less uniform cell structure and increased open cell content compared to the PLA/talc/CE foam (Figure 6). The authors suggest that the weak interfacial adhesion induces local fiber-matrix debonding and micro-holes which favor gas loss and locally hinder the cell growing ability, hence leading to non-uniform cell size distribution. The compressive mechanical properties of the resulting foams were also measured. It was found that basalt fibers can substantially reinforce PLA foam and provide two times higher compression strength, 40 towards 20 kPa for neat PLA foam. However, the higher compression strength was achieved for the PLA/talc/CE foam, roughly 100 kPa. Cellulose fibers did not provide any mechanical reinforcement, due to poor interfacial adhesion, higher polydispersity and increased open-cell ratio.

Based on these results, the use of natural fibers appear as an interesting strategy to control foam morphology and reinforcement, providing that good interfacial adhesion with the matrix and homogeneous fiber distribution coherent with the foam cell size are achieved. Matuana et al. (19) studied the microcellular foaming of PLA/wood flour biocomposites by sc-CO 2 assisted extrusion. Wood flour / plastic composites are indeed a relevant industrial bio-based material, especially for decking and automotive applications. As reviewed by the authors, wood flour can greatly increase the viscosity of the polymer melt (20-22) which induces higher pressures in the extruder and higher resistance to cell growth during foaming. Furthermore, wood flour particles can act as heterogeneous nucleation agents, finer particle sizes leading to finer foam cells (20,23). Wood flour particles are thus potentially interesting bio-based fillers to control the nucleation rate and the morphology in foamed biocomposites. However, the thermal stability of wood flour is limited and too high processing temperatures are likely to release water vapor and volatiles during extrusion which could adversely affect the development and homogeneity of the cellular structure. For these reasons, Matuana et al. (19) developed a strategy to control the cell structure of PLA/wood flour biocomposites by the use of a low molecular weight rheology modifier, allowing to reduce the melt viscosity while reducing the processing temperatures and adding substantial amounts of wood flour in the formulation. PLA (Ingeo TM 2002D, D-lactide content of 4 %, MFI of 5-7 g / 10 min -210°C, 2.16 kg -NatureWorks LLC, USA) was mixed with 0.5 wt% talc (Mistron Vapor-R, median particle size of 2 μm, specific surface area of 13.4 m 2 /g, Luzenac Corp), various amount (10, 20 and 30 wt%) of wood flour (Pine flour, grade 12020, 120 mesh size, American Wood Fibers, Schofield, WI) and the rheology modifier (Epolene E-43, molecular weight 11 200 g/mol, maleic anhydride-modified polypropylene, Eastman Chemical Co.). The effects of wood flour and rheology modifier contents on the rheological behavior of the melt were investigated in order to identify suitable formulations for microcellular foaming of PLA/wood flour composites by sc-CO 2 assisted single screw extrusion. The increased melt viscosity induced by the resistance to flow of wood flour particles was counterbalanced by adding the rheology modifier in the formulations. Matching the melt flow index (MFI) of the melted biocomposites to that of neat PLA matrix was shown to be an efficient strategy to produce microcellular PLA/wood flour foams with uniform and homogeneous cellular structures similar to PLA foams (Figure 7). Cell density varying from 1.02 down to 0.26×10 9 cells/cm 3 and cell sizes varying from 8.9 up to 11.3 µm were achieved for PLA/wood flour 10 to 30 wt% foams, respectively (vs. 1.46×10 9 cells/cm 3 and 7.4 µm for neat PLA foams). The authors concluded that uniform microcellular morphologies were successfully obtained for wood flour concentrations up to 20 wt%.

In conclusion, the use of micro-sized (in)organic fillers such as talc or ligno-cellulosic fibers in biopolymer matrix appears as an interesting strategy to control cell morphology during continuous microcellular foaming by sc-CO 2 assisted extrusion. Increased melt strength and higher cell walls stability are obtained and allow limiting coalescence while heterogeneous cell nucleation brought by the particles promotes finer and homogeneous microcellular structure. However, the volume expansion ratio can be lowered by too high melt strength, which must thus be tuned with care so as to improve foamability and control foam morphology. It should be pointed out that the quality of the interfacial adhesion of the particles with the matrix as well as their dispersion and size distribution could play an important role on the final foam structure. (19). Copyright 2010 American Chemical Society.

Foaming of Nano-Biocomposites

Several studies reported on the preparation and foaming of synthetic polymer nanocomposites systems by continuous sc-CO 2 assisted extrusion. When well dispersed in the melt, nanoparticles allow achieving significant modifications of the rheological behavior of polymer melts at much lower filler content than micro-sized particles. Most of the works dealing with the foaming of nanocomposites by sc-CO 2 assisted extrusion use clays due to their high aspect ratio and platelet-shape structure upon exfoliation, in the polymer melt. Different processing methods were used: e.g. two-step extrusion process of polypropylene (PP) / clays systems (24-26), sc-CO 2 assisted single-screw extrusion of PP / clays systems (27), single-screw extrusion with prior preparation of exfoliated polystyrene (PS) / clays nanocomposites by in-situ polymerization (28) or the injection of sc-CO 2 / clays mixtures into the PP melt during extrusion (29). In most cases, the authors obtained an increased cell density and smaller cell size. Indeed, if well dispersed, the clays act as nucleating sites that promote heterogeneous cell nucleation, but they also greatly increase the melt strength of the polymer that limits in turn cell coalescence (13,30,31). Enhanced clays dispersion of the resulting exfoliated nanocomposite foams was also reported.

More recently, continuous sc-CO 2 assisted extrusion has been considered for the foaming of nano-biocomposites based on biopolymer matrices, in particular biopolyesters (e.g. PLA or PHAs), loaded with nano-sized fillers, especially clays. The rheological behavior of the systems which accounts for their foamability, and the morphology of the resulting foams in terms of cell density and size and expansion ratio has been analyzed in details.

Liu et al. (32) studied the foaming of PLA/clay systems. The authors firstly investigated the effects of the chain extender (CESA extend BL 10069 N, containing 10 wt % multi-functional epoxy-based CE, molecular weight < 3000 g/mol, average functionality > 4) and organo-modified clays (I.34TCN modified by a quaternary ammonium salt containing methyl tallow bis-2-hydroxyethyl, Nanomer Products Inc.) on the rheological behavior of PLA (Ingeo TM 2002D). It was found that the melt elasticity of PLA/CE systems was increased with low clay content. In contrast, too high amount of clays (> 3 wt%) resulted in a decrease of melt strength and elasticity due to clay agglomeration and extensive thermal degradation of PLA, i.e. reduction of its molecular weight. The thermal degradation of biopolyesters matrices (PLA, PHAs,…) in the presence of clays is a well-known phenomenon related to (i) the presence of Al Lewis and Bronsted acid sites in the inorganic layers of clays which catalyze the hydrolysis of ester linkages (33,34), and (ii) the clay organo-modifiers and their potential decomposition products that could also catalyze the degradation upon extrusion (35,36). Besides, Liu et al. (32) showed that the chain extending reaction was limited in the presence of clays due to interactions between CE and clays. Based on these observations, the rheological behavior of melted PLA, i.e. its melt strength and elasticity, and hence its foamability could be modulated by controlling the chain extender / clay ratio. The PLA/CE/clays formulations were then foamed by sc-CO 2 assisted extrusion. The introduction of low amounts of clays within the PLA/CE system promoted heterogeneous cell nucleation and higher cell density up to 1.1×10 7 cells/cm 3 with expansion ratio of 3.07 were obtained with 2 wt% of clays. In contrast, higher clay contents resulted in an increased cell size and a sharp decrease of the cell density (1.9×10 6 cells/cm 3 at 4 wt% of clays) and expansion ratio due to PLA degradation and extensive cell coalescence. The authors concluded that a homogeneous and finer cellular morphology could be obtained when controlling the chain extender / clay ratio.

The effect of organo-modified clays, Cloisite 30B (C30B, modified by a quaternary ammonium salt -tallow alkyl bis(2-hydroxyethyl) methylammonium chloride -Southern Clay Products, Inc., USA), on the foamability of PLA (Ingeo TM 2002D ) by sc-CO 2 assisted extrusion was also investigated by Keshtkar et al. (37). The authors prepared PLA based nanocomposites by counter-rotating twin screw extrusion with clay contents varying from 0.5 to 5 wt%. A masterbatch approach was used; preparing a PLA/5 wt% clays batch that was then diluted at the desired clay concentration. Good clay dispersion with intercalated and partially exfoliated structures was obtained as revealed by Transmission Electron Microscopy (TEM) and X-Ray Diffraction (XRD). The nanocomposites were then foamed with 5 to 9 wt% of sc-CO 2 in a tandem single-screw extruder system allowing a prior dispersion of sc-CO 2 within the PLA melt followed by an enhanced mixing and initial cooling of the mixture in the secondary extrusion zone. The structures of the resulting foamed extrudates were analyzed in terms of expansion ratio and cell density. In particular, the influence of the die temperature -and therefore of the die pressure -was investigated. The results clearly evidence that decreasing the die temperature and increasing the sc-CO 2 content lead to higher expansion ratio and finer cellular morphology with higher cell density, especially in the presence of clays (Figure 8). Indeed, clay addition had a great influence on the rheological behavior of the melt and increasing amounts of clays resulted in higher melt strength, more favorable to foaming with limited cell opening and coalescence. The authors also noticed that the incorporation of clays in PLA increased its extensional viscosity resulting in a better foamability with a more closed-cell structure due to higher cell walls stability. The effect of clay dispersion was analyzed by comparing the PLA/C30B system with a PLA/Cloisite 20A (C20A, modified by a quaternary ammonium salt -dimethyl, dihydrogenated tallow-Southern Clay Products, Inc., USA) system, the latter clay having poor dispersion ability in PLA and primarily leading to intercalated structures. It was found that the better dispersion of C30B clay resulted in much higher cell density and expansion ratio. Indeed, well dispersed clays have a strong nucleating role and promote higher cell density due to local variations in the elastic properties of the melt and consecutive stress distribution, that was assumed by Wang et al. (14) to be one of the main reason for the decreased energy barrier for cell nucleation in polymer/fillers systems. Accordingly, the maximum cell density and expansion ratio, i.e. 6.8×10 8 cells/cm 3 and 43 respectively, were reached at a die temperature of 115 °C, 9 wt% of sc-CO 2 and with 5 wt% of clays (Figure 9). Furthermore, a comprehensive study on the effect of clay, sc-CO 2 content and shear stress on the crystallization behavior of PLA was conducted as crystallization reveals to have a great influence on the final foam morphology. By means of differential scanning calorimetry and rotational rheometry experiments, the authors demonstrated that the crystallization kinetics of PLA was significantly enhanced by the presence of clays and the straininduced crystallization during processing. Additional experiments with various temperature profiles revealed that decreasing the operating temperatures allows increasing cell density and expansion ratio in relation with accelerated isothermal crystallization of PLA. Besides, increased degree of crystallinity of the final foams were well correlated with increased expansion ratio (up to 30 % crystallinity for maximum expansion ratio of 43), highlighting that the expansion of the foam at the exit of the die could also result in strain-induced crystallization. Based on these results, the authors assumed that promoting the nucleation of PLA crystals during processing should be responsible for enhanced cell density and expansion ratio due to heterogeneous cell nucleation around the crystals and improved melt strength and cell walls stability. 38) also reported that the dispersion of clays allows both homogeneous and heterogeneous nucleation to occur during the foaming process. Indeed, SEM observations at high magnification revealed that the membrane of larger cells contained a large number of fine nano-sized cells for the PLA/clay system. The authors assumed that a fraction of the cells were nucleated earlier due to rapid pressure drop in the nozzle leading to homogeneous nucleation, while another fraction was formed later due to the presence of clays in PLA, which provided sites for heterogeneous nucleation. As a consequence, cells nucleated earlier had a longer time for growth, resulting in cells of larger cell size (case of homogeneous nucleation) compared to cell nucleated later which had a shorter time for growth, leading to smaller cell size (case of heterogeneous nucleation). This mechanism resulted in a bimodal distribution of the cell size.

Jiang et al. (39) and Zhao et al. (40) reported that sc-CO 2 assisted extrusion could also contribute to the better dispersion of clays (Octadecyl amine-modified Nanocor I30P, USA for Jiang et al. (39); C30B for Zhao et al. (40)) in PLA (Ingeo TM 2002D). As revealed by XRD, the interlayer distance between the clays platelets was increased owing to the plasticizing ability of sc-CO 2 during processing that favors the diffusion of PLA macromolecules within the interlayer region of organo-modified clays, inducing their further exfoliation. Zhao et al. (40) assumed that nucleation and growth of tiny cells in between clay platelets was also a driving force for their dispersion and better exfoliation in PLA.

The foaming of nano-biocomposites by sc-CO 2 assisted extrusion has been explored with other biopolymer matrices. Le Moigne et al. (10) studied the foaming of poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) (HV content of 13 wt%, nucleated and plasticized with 10 wt% of a copolyester and a weight-average molecular weight of 600 000 g/mol, Biomer, Germany) / C30B clays systems. PHBV is particularly interesting for its high biodegradation rate (41) which makes it a material of choice in packaging and short-term uses. The authors pointed out that the prior preparation of a PHBV/clay masterbatch was a necessary step to obtained good clay dispersion and limit PHBV degradation during the extrusion foaming process. The masterbatch was prepared by melt intercalation by using a co-rotating twin-screw extruder with 10 to 20 wt% of clay. Masterbatches were then diluted to a clay content of 2.5 wt% before sc-CO 2 extrusion-foaming. In this study, the sc-CO 2 content had a great importance. The authors found a narrow processing window, sc-CO 2 mass fraction from 1.5 to 3.5 wt% and die temperature of 140 °C being the best compromise between a good foaming capacity and a limited clogging of the die. In these processing conditions, good clay dispersion allowed to favor cell nucleation and limit coalescence thanks to higher cell walls stability. At 1.5 wt% of sc-CO 2 , PHBV/clay nano-biocomposite foams with good homogeneity, and porosity higher than 45 % were obtained. Higher sc-CO 2 mass fraction allowed achieving cell density of 1.2×10 6 cells/cm 3 and cell size of 70 µm but the expansion of the foam and its overall porosity were drastically reduced due to rapid cooling and the increased stiffness of the frozen polymer layer at the periphery of the extrudates. In this case, increasing amounts of sc-CO 2 had no significant influence on the clay dispersion as revealed by XRD and intercalated structures remained predominant. The authors also pointed out that the high degree of crystallinity of PHBV hampers the diffusion of sc-CO 2 within the matrix and its expansion ratio, hence limiting the growth of the cells and the ultimate porosity of the foams.

In conclusion, nano-sized fillers such as clays can greatly contribute to obtain better cell morphologies during continuous microcellular foaming of biopolymer matrices by sc-CO 2 assisted extrusion. As compared to micro-sized fillers, they allow to increase the strength of the polymer melt at much lower concentration, i.e. maximum 5 wt%, while greatly promoting heterogeneous cell nucleation around the nanoparticles and the possibly induced nucleated crystals (42). This leads to finer and more homogeneous microcellular structure with high cell density reaching up to ~10 9 cells/cm 3 and small cell size of ten microns or less. Furthermore, the dispersion of nano-fillers within the cell walls could contribute to limit cell coalescence, especially for high aspect ratio particles such as nanoclays which have a platelet-shape. Indeed, Nofar (42) showed that nanoclay platelets can align along the cell walls of PLA foams due to biaxial stretching occurring during expansion (30) (Figure 10). This could enhance cell walls strength (30,42) and possibly limit or delay CO 2 diffusion due to an improved barrier effect (40), both phenomena contributing to inhibit cell rupture and coalescence during cell growth. (scale-bar: 100 nm). Reproduced with permission from ref. (42). Copyright 2016 Elsevier.

Conclusion

Foaming with supercritical CO 2 is a very active and trendy research area. In their recent review, Di Maio and Kiran (1) have underlined the stakes and the progress still to be made. Using a solvent-free and continuous process like extrusion-foaming is a major advantage over more classical batch processes. In this process, melt strength mastering, CO 2 dissolution kinetics and cell formation and growth are certainly the main challenges to be overcome. In this chapter, we have discussed several recent advances done on these three topics, some of them coming from a better mastering of the main operational parameters.

The use of biopolymers (whichever definition of this term is used) reveals very promising results towards more sustainable foams processing. The addition of micro/nano-fillers either organic or minerals influence greatly the foamability of biopolymers by increasing the melt strength and promoting heterogeneous cell nucleation. This might bring significant improvements in terms of the microstructural and physical properties of the final foams.

On a more fundamental facet, progress is expected in the development of more accurate methods for the measurement of CO 2 sorption in the polymer melt determining the level of the plasticization effect. The mechanisms of foam formation through the passage of the die deserve to be better understood: predictive modelling of the nucleation and growth of cells will probably bring new insights in the near future [START_REF] Chauvet | Proceedings of the 16th European Meeting on Supercritical Fluids[END_REF].

Figure 1 .

 1 Figure 1. Example of an experimental device used for the foaming of polymers by sc-CO 2 assisted single-screw extrusion. Courtesy of Jacques Fages, Centre RAPSODEE, IMT Mines Albi. Copyright 2018.

Figure 2 .

 2 Figure 2. The main biopolymers and their production routes. Reproduced with permission from ref. (3). Copyright 2017 Elsevier.

Figure 3 .

 3 Figure 3. PLA porosity (a) and expansion (b) obtained by extrusion foaming with sc-CO 2 . Reproduced with permission from ref. (7). Copyright 2017 John Wiley and Sons.

Figure 4 .

 4 The evolution of foam morphology is clearly going from (a) to (i) from large open pores to smaller closed pores.

Figure 4 .

 4 Figure 4. PLA foams. The first number is the die temperature (°C), the second is the CO 2 flow-rate (ml/min). Reproduced with permission from ref.(7). Copyright 2017 John Wiley and Sons.

Figure 5 .

 5 Figure 5. Variation of cell density with formulation and die temperature for PLA and PLA/PBAT systems with or without talc. Reproduced with permission from ref. (18). Copyright 2010 Elsevier.

Figure 6 .

 6 Figure 6. SEM micrographs of the cell morphologies obtained for the highly expanded (ε > 95 %) foams. PLA (a), PLA/talc/CE (b), PLA/talc/CE with 5 wt% cellulose fibers (c), PLA/talc/CE with 5 wt% basalt fibers (d). Reproduced with permission from ref. (8). Copyright 2014 eXPRESS Polymer Letters.

Figure 7 .

 7 Figure 7. Effect of wood flour content on cell morphology of foamed PLA samples: (a) 0 wt%, (b) 10 wt%, (c) 20 wt%, and (d) 30 wt% wood flour. Reproduced from ref.(19). Copyright 2010 American Chemical Society.

Figure 8 .

 8 Figure 8. Cell morphologies of the neat PLA and PLA/clay nanocomposites (PLACN) foams obtained at die temperatures of 115, 125 and 135 °C and 9 wt% of sc-CO 2 . Reproduced with permission from ref. (37). Copyright 2014 Elsevier.

Figure 9 .

 9 Figure 9. (a) Cell density and (b) expansion ratio of PLA and PLA/clay nanocomposites (PLACN) foams obtained at various die temperatures and 9 wt% of sc-CO 2 . Reproduced with permission from ref. (37). Copyright 2014 Elsevier.

Figure 10 .

 10 Figure 10. Transmission Electron Microscopy image of a cell wall in foamed PLA/1 wt% clays showing the alignment of nanoclay platelets along a cell-wall(scale-bar: 100 nm). Reproduced with permission from ref.(42). Copyright 2016 Elsevier.

Acknowledgments

C2MA is member of the European Polysaccharide Network of Excellence (EPNOE), http://www.epnoe.eu.