Jose Delgado Alves
email: jose.alves@fcm.unl.pt

António Marinho

Maria José Serra

Delgado Alves

José Delgado Alves

MD José Delgado Alves

Tocilizumab: is there life beyond anti-TNF blockade?

come

Introduction

The last 20 years have shown a remarkable change in overall perspectives on rheumatoid arthritis, from the point of view of both patients and doctors. In the 1990s, methotrexate (MTX) was shown to be the most effective therapy of all the disease modifying drugs (DMARD) [START_REF] Kay | Methotrexate: the gold standard without standardisation[END_REF]. By the end of the last decade of the twentieth century, anti-tumor necrosis factor alfa (anti-TNF-α) therapy raised the standards by achieving impressive results regarding symptom relief, improvement in quality of life and labour hours [START_REF] Hyrich | Benefit of anti-TNF therapy in rheumatoid arthritis patients with moderate disease activity[END_REF][START_REF] Augustsson | Patients with rheumatoid arthritis treated with tumour necrosis factor antagonists increase their participation in the workforce: potential for significant long-term indirect cost gains (data from a population-based registry)[END_REF] and by preventing the progression of joint destruction [START_REF] Van Der Heijde | Disease remission and sustained halting of radiographic progression with combination etanercept and methotrexate in patients with rheumatoid arthritis[END_REF][START_REF] Jamal | Adalimumab response in patients with early versus established rheumatoid arthritis: DE019 randomized controlled trial subanalysis[END_REF]. The power of these new drugs associated with a relatively low incidence of major adverse effects granted them a level of efficacy that allowed, for the first time in this disease, radiographic and clinical long term remission to be considered as a therapeutic target [START_REF] Van Der Kooij | Drug-free remission, functioning and radiographic damage after 4 years of response-driven treatment in patients with recent-onset rheumatoid arthritis[END_REF].

However, after the initial enthusiasm, long term data showed that much had still to be done in order to claim victory over rheumatoid arthritis. Monotherapy with anti-TNF-α is not significantly superior to Methotrexate alone and an association is still needed for optimal results [START_REF] Hochberg | Comparison of the efficacy of the tumour necrosis factor alpha blocking agents adalimumab, etanercept, and infliximab when added to methotrexate in patients with active rheumatoid arthritis[END_REF]. Furthermore, even with combination therapy, a significant proportion of patients still do not achieve satisfactory results [START_REF] Silman | Available therapeutic options following failure of a first anti-TNF agent[END_REF], and despite some reassuring data put forward by national registries, infections and cancer are still major causes for concern [START_REF] Dixon | Drug-specific risk of tuberculosis in patients with rheumatoid arthritis treated with anti-TNF therapy: Results from the British Society for Rheumatology Biologics Register (BSRBR)[END_REF][START_REF] Peña-Sagredo | Listeria monocytogenes infection in patients with rheumatic diseases on TNF-alpha antagonist therapy: the Spanish Study Group experience[END_REF][START_REF] Mariette | Lymphoma in patients treated with anti-TNF: results of the 3-year prospective French RATIO registry[END_REF].

Regardless of the undisputed importance of TNF-α as a pro-inflammatory cytokine, directly responsible for joint destruction and the perpetuation of inflammation in rheumatoid arthritis [START_REF] Moran | Human rheumatoid arthritis tissue production of IL-17A drives matrix and cartilage degradation: synergy with tumour necrosis factor-alpha, Oncostatin M and response to biologic therapies[END_REF], the existence of a significant group of patients who did not respond to anti-TNF-α treatment, led to the search for other potential targets, both molecular and cellular [START_REF] Okamoto | Molecular targets of rheumatoid arthritis[END_REF][START_REF] Kukar | Biological targets in the treatment of rheumatoid arthritis: a comprehensive review of current and in-development biological disease modifying anti-rheumatic drugs[END_REF][START_REF] Tansey | The TNF superfamily in 2009: new pathways, new indications, and new drugs[END_REF]. More than a decade after the generalization of anti-TNF-α therapy, (and despite the newcomers in this therapeutic group -e.g. golimumab and certolizumab), new challenges have been defined whilst a significant set of problems have still to be overcome.

Common questions on a daily practical basis include: management of patients who do not tolerate methotrexate or with disease refractory to anti-TNF-α treatment, management of extra-articular manifestations, treatment of patients who have a high risk of infection, or with previous or ongoing malignancies, and finally whether remission can indeed be achieved in a greater number of patients. B-cell depletion [START_REF] Quartuccio | Long-term effects of rituximab in rheumatoid arthritis: clinical, biologic, and pharmacogenetic aspects[END_REF] and blockage of CTLA4-associated lymphocyte co-stimulation [START_REF] Maxwell | Abatacept for rheumatoid arthritis[END_REF] have taken the lead Interleukin -6 (IL-6), a pivotal cytokine in the inflammatory response has become the most recent target to be "scrutinised" in clinical trials, through the inhibition of its soluble receptor by the monoclonal antibody tocilizumab (TCZ) [START_REF] Rose-John | The IL-6/sIL-6R complex as a novel target for therapeutic approaches[END_REF]. What news came from the first major clinical trials using this new drug? Should we expect more in comparison with what might be already consider as classic biologic therapy (i.e. anti-TNF-α)? Have the old problems been solved or at least are we getting better results? Is there life beyond anti-TNF-α blockade?

Managing patients without methotrexate

Anti-TNF-α treatment has still to show a clear detachment of its efficacy from the association with MTX. In fact, the clinical trials regarding the comparison of any of the anti-TNF-α drugs on monotherapy with MTX has failed to show an outstanding difference as far as clinical response is concerned, and in randomized controlled trials (RCTs) none has actually presented any benefit before the first 6 months. Nevertheless this data should be read very carefully, because in real life registers, with a large number of patients ineligible for RCTs, anti-TNF-α monotherapy showed a clear benefit over MTX [START_REF] Hochberg | Comparison of the efficacy of the tumour necrosis factor alpha blocking agents adalimumab, etanercept, and infliximab when added to methotrexate in patients with active rheumatoid arthritis[END_REF][START_REF] Kievit | The efficacy of anti-TNF in rheumatoid arthritis, a comparison between randomised controlled trials and clinical practice[END_REF][START_REF] Zink | rheumatoid arthritis in an observational cohort study: comparison of patients according to their eligibility for major randomized clinical trials[END_REF]. Initial studies with infliximab showed an increase of 38-46% of responders [START_REF] St Clair | Active-Controlled Study of Patients Receiving Infliximab for the Treatment of Rheumatoid Arthritis of Early Onset Study Group. Combination of infliximab and methotrexate therapy for early rheumatoid arthritis: a randomized, controlled trial[END_REF] and some studies suggest that the use of this monoclonal in monotherapy is associated with a relevant increase in acquired resistance to the drug due to the production of neutralising antibodies [START_REF] Ebert | Non-response to infliximab may be due to innate neutralizing anti-tumour necrosis factor-alpha antibodies[END_REF]. Only 46% of patients achieved an ACR 20 on standard-dose monotherapy with adalimumab during its major phase III trials [START_REF] Van De Putte | Efficacy and safety of adalimumab as monotherapy in patients with rheumatoid arthritis for whom previous disease modifying antirheumatic drug treatment has failed[END_REF] and etanercept reached a significant difference only after one 1 year of treatment [START_REF]Therapeutic effect of the combination of etanercept and methotrexate compared with each treatment alone in patients with rheumatoid arthritis: double-blind randomised controlled trial[END_REF]. This has justified a strong recommendation in most countries for these drugs not to be used in monotherapy, unless a clear intolerance or contra-indication to the use of methotrexate is present.

Tocilizumab has been tested in monotherapy in a major clinical trial: the AMBITION study where treatment with tocilizumab 8mg/kg every 4 weeks was compare with MTX (up to 20mg weekly). [START_REF] Jones | Comparison of tocilizumab monotherapy versus methotrexate monotherapy in patients with moderate to severe rheumatoid arthritis: the AMBITION study[END_REF] In the AMBITION study, patients naïve to MTX or anti-TNF agents, who were treated with tocilizumab for 24 weeks, showed a significant improvement in clinical parameters (ACR20, 50 and 70 and DAS28) at week 24 and a reduction to the normal range of hsCRP at week 12 and these were statistically different from the placebo group [START_REF] Jones | Comparison of tocilizumab monotherapy versus methotrexate monotherapy in patients with moderate to severe rheumatoid arthritis: the AMBITION study[END_REF]. A clear statistical difference between anti-TNF-α monotherapy and MTX had never been shown before for a period of time as early as 6 months.

Additional Therapy for DMARDs -resistant patients

In the large pivotal trials in Tocilizumab's recent history, three randomized controlled trials have made in MTX and non-biological DMARDs inadequate responders: the OPTION, TOWARD and LITHE studies. In the OPTION study, an expected result of a significant improvement of the clinical and biologic response at 24 weeks was found, in patients treated with TCZ + MTX (8 mg/kg) when compared with MTX alone. In this setting 27% of the patients achieved DAS remission and 79% good to moderate DAS responses [START_REF] Smolen | Effect of interleukin-6 receptor inhibition with tocilizumab in patients with rheumatoid arthritis (OPTION study): a double-blind, placebo-controlled, randomised trial[END_REF].

More recently, a structural joint damage study (LITHE) showed remission at 52 weeks in 47% of patients on TCZ 8mg/kg plus MTX, compared to 8% in a MTX monotherapy group. This is particularly impressive when one considers the fact that the mean disease duration in this cohort was 9 years [START_REF] Roy Fleischmann | LITHE: Tocilizumab Inhibits Radiographic Progression and Improves Physical Function in Rheumatoid Arthritis (RA) Patients (Pts) at 2 Yrs with Increasing Clinical Efficacy Over Time[END_REF]. Most of the previous studies looking at remission with anti-TNF-α treatment used study populations with a much shorter disease duration, which is known to be associated with a higher possibility of remission.

Refractoriness to anti-TNF treatment

Multiple studies have addressed the issue of non-responders (or poor-responders) to anti-TNF-α therapy. Overall, most studies show a benefit of some magnitude when a different anti-TNF-α is introduced after failure of a first one, regardless of which as been administered in the first treatment course, specially if there was a loss of efficacy rather than a primary absence of response [START_REF] Scrivo | Switching between TNF alpha antagonists in rheumatoid arthritis: personal experience and review of the literature[END_REF]. The lack of a clear advantage of any of the anti-TNF-α agents when used as a second course of treatment for non-responders, with special emphasis for primary non-responders (29), has been somewhat confirmed by the different national registries, which leads to the general impression that there is no advantage in switching anti-TNFs in this context.

The RADIATE study is a controlled study to compare efficacy of tocilizumab vs. placebo on a MTX background in patients with rheumatoid arthritis refractory to at least one anti-TNF-α biologic at 24 weeks. A total of 50%, 28.8% and 12.4% of the patients on TCZ (8mg/kg) achieved ACR20, 50 and 70 responses respectively compared to 10.1%, 3.8% and 1.3% of the placebo group (30). There were no differences regarding the type or the number of anti-TNF-α drugs previously used. These data are not dramatically different from other studies on anti-TNF-α drugs, but allows another option to be considered in non-responders.

Extra-articular manifestations

Despite being a systemic disease, the predominance of the articular involvement in rheumatoid arthritis has determined that the assessment of treatment response should take into account the changes (either clinical or radiologic) recorded in the joints.

Quality of life and overall patient satisfaction has only been considered as a main therapeutic target more recently. Anti-TNF-α treatment has provided a relevant improvement in both those variables. However, other aspects, such as fatigue or other organ involvement has very rarely been considered. IL-6, being one of the most important inflammatory cytokines, is directly associated with the systemic features of the disease. There are no data addressing the response to different treatments regarding rheumatoid nodules or interstitial lung disease, and, until now, even anaemia was not thoroughly considered in these patients. Fatigue and inflammatory-disease-related anaemia are important aspects of the overall burden of rheumatoid arthritis which are now being addressed by anti-IL6 treatment. In fact, improvement of red blood cell count in these patients has been consistently shown by TCZ in both the pivotal and follow-up studies (31). In instance the higher mortality rate among rheumatoid arthritis (RA) patients in comparison with the general population is largely attributable to cardiovascular (CV) disease, particularly coronary atherosclerosis, but also non-fatal myocardial infarction and heart failure. Biological agents such as anti-TNF-α agents delay and even reverse the progression of endothelial dysfunction and atherosclerosis.

Tocilizumab leads to changes in lipid profiles without increasing adverse vascular events, however is the dramatic effect on inflammation is long-term potential benefit on CV disease control (32).

Remission is the new target

Remission is now being presented as an important endpoint in clinical trials for rheumatoid arthritis. However, only one study has presented it as a primary endpoint:

COMET. In this study, two groups of approximately 270 patients each were randomised for a 2 year treatment course with etanercept and MTX vs. MTX alone. All the patients had been diagnosed with rheumatoid arthritis less than 2 years before. After the first Regarding IL-6 receptor blockade, the TOWARD study, involving 1220 patients already on DMARD treatment for RA, compared a group treated with tocilizumab 8mg/kg in association with any DMARD with a group treated with DMARD only. At 6 months, 30.2% of the patients in the tocilizumab group achieved clinical remission as assessed by DAS28<2.6, a strong result when compared to 3.4% of the placebo group.

The ACR70 response was 21% for the TCZ group vs. 3% for the placebo group) (36).

In the AMBITION trial, patients treated with tocilizumab on monotherapy were five time more likely to reach remission (DAS28 < 2.6) than patients on MTX at week 24.

(33.6% vs. 12.1%) [START_REF] Jones | Comparison of tocilizumab monotherapy versus methotrexate monotherapy in patients with moderate to severe rheumatoid arthritis: the AMBITION study[END_REF].

Interestingly, a similar result of 30.1% of remissions at 6 months was found when TCZ was used in patients who did not respond to previous treatments with anti-TNF-α (30), confirming a remarkable consistency in the induction of remission regardless of the context or background of where this drug is being used. One could remark that IL-6 blockage dramatically normalizes C-Reactive Protein (CRP) , overestimating the outstanding DAS remissions achieved by tocilizumab, however acute phase reactants have little contribution on composite indexes used in clinical practice. In manner of fact

Safety

Major safety concerns in the "biologic era" include overall infections and particularly serious infections, tuberculosis and malignancy. International registries have been monitoring these issues and it is now evident that there is an acceptable risk of infections when anti-TNF-α treatment is used, with the exception of a small number of patients who were subjected to high-dose anti-TNF-α therapy [START_REF] Leombruno | The safety of anti-tumour necrosis factor treatments in rheumatoid arthritis: meta and exposure-adjusted pooled analyses of serious adverse events[END_REF]. Data are not so clear as far as malignancies are concerned, but a large study of the first 6 years after the start of anti-TNF-α therapy in routine care showed no overall elevation of cancer risk and no increase with time of follow up, (with an advantage for the use of etanercept over the monoclonal antibodies for a smaller risk of lymphoma) [START_REF] Askling | Cancer risk in patients with rheumatoid arthritis treated with anti-tumor necrosis factor alpha therapies: does the risk change with the time since start of treatment?[END_REF].With regards to high risk cancer patients (patients with personal or strong family history of cancer), there is still no clear data available, and therefore a definitive limitation for the use of these drugs in patients with previous malignancies is the current standard of care [START_REF] Chapman | The role of anti-tumor necrosis factor receptor agents in cancer survivors: does the risk justify the benefit?[END_REF].

Regarding TCZ the first major set of data from the patients exposed to IL-6 receptor blockade, includes 4700 patients treated for almost 8000 patient-years [START_REF] Okada | Actemra (tocilizumab) for Rheumatoid Arthritis FDA Perspective[END_REF]. Both in the five pivotal studies and in the two long-term follow-up studies, the incidence of serious adverse events (SAE) was low (less than 16 SAE/100pts-yrs) and it did not increase with increasing duration of TCZ exposure [START_REF] Dixon | Serious infections following anti-tumor necrosis factor alpha therapy in patients with rheumatoid arthritis : lessons from interpreting data from observational studies[END_REF].

The rate of serious infections in RA patients treated with Tocilizumab inhibitors is 6 events/100 pts-yrs [START_REF] Listing | Infections in patients with rheumatoid arthritis treated with biologic agents[END_REF]. (In patients treated only with DMARDS, it is 2-3 events/100 pts-yrs ( 44)). Major infections included pneumonia and cellulitis (<1.4% and <0.8% of all the patients exposed in both the pivotal and follow-up studies). For most of the studies, patients were not requested to have a tuberculosis screening test or prophylaxis, in accordance with the relatively low relevance of IL6 in granuloma production. In all of the studies that comprise the TCZ program, only 9 cases of TB were identified, the majority of which presented more than 18 months after the start of the treatment, suggesting a de novo infection, rather than the reactivation of a latent one [START_REF] Van Vollenhoven Ronald | Long-Term Safety and Tolerability of Tocilizumab Treatment in Patients with Rheumatoid Arthritis and a Mean Treatment Duration of 2.4 Years[END_REF].

There was no difference between the expected incidence of malignancies (including non-melanoma skin cancers) and the frequency observed in RA patients, suggesting that TCZ does not appear to implicate an additional risk, as observed in populations with high risk of digestive malignancies [START_REF] Nishimoto | N Long-term safety and efficacy of tocilizumab, an anti-IL-6 receptor monoclonal antibody, in monotherapy, in patients with rheumatoid arthritis (the STREAM study): evidence of safety and efficacy in a 5-year extension study[END_REF]. Liver abnormalities are common, particularly in patients treated with the association of AST levels, and 15 to 20% had elevation of ALT levels. These elevated levels responded to reduction and/or interruption of TCZ treatment and there were no cases of hepatic insufficiency or liver failure reported [START_REF] Nishimoto | N Long-term safety and efficacy of tocilizumab, an anti-IL-6 receptor monoclonal antibody, in monotherapy, in patients with rheumatoid arthritis (the STREAM study): evidence of safety and efficacy in a 5-year extension study[END_REF].

Other major concerns during treatment with TCZ seem to be the reduction of neutrophils and the increase in cholesterol lipids, particularly LDL-cholesterol.

A reduction in the neutrophil count seems to be associated with a response to treatment.

It is not a real fall in the number of circulating neutrophils but a marginalisation of those cells in the blood vessel. In fact, there is no relationship between infections and the neutrophil count, and furthermore, the white cell response to an infection seen in these patients seems to be similar to the normal population.

A significant proportion of patients on TCZ show an increase in total cholesterol as well as in plasma LDL and HDL. Approximately 7% (in the long-term studies) required statin treatment and showed an appropriate response. There was no increase in cardiovascular adverse events in patients treated with TCZ, when compared to control treatments. Finally rare cases of gastrointestinal perforation have been described, and some of them were fatal, the relative risk is still not well characterized, but caution should be used in patients with previous diverticulitis or peptic ulcer disease (47).

Overall, the safety profile of this drug seems to be similar (although with a slightly different adverse event profile) to the other biologic treatments already in use, however some of the concerning safety problems have to be confirmed in long term studies, particularly hepatic toxicity and cardiovascular risk (48) [START_REF] Jones | Comparison of tocilizumab monotherapy versus methotrexate monotherapy in patients with moderate to severe rheumatoid arthritis: the AMBITION study[END_REF].

Final comments

Anti-TNF-α biologic treatment has started a new era in the management of rheumatoid arthritis. A remarkable improvement in the outcome of these patients regarding symptom control and quality of life, alongside the almost complete arrest of joint destruction, has raised the standards for treating this condition to a different level. However, some remaining problems have yet to be solved, opening a window of opportunity for new targets and different drugs. IL-6 blockade is one of the most recent alternatives. It has shown promising results in the main clinical trials which were confirmed in long-term follow-up studies and provided new opportunities for solving some of the issues that were not being particularly well addressed by Anti-TNF-α therapy. Naturally, the lack of head-to-head studies and the inexistence of registry-based data for this new drug make it difficult to fully endorse all the aspects that individual DAS scores, assuming the data of all extended reports of pivotal trials. Nevertheless its efficacy and security should also be proved in registries, as so far as been done with anti-TNF-α therapies.

Disclosures, funding and acknowledgements

There are no disclosures or funding to be reported by any of the authors.

  even good results regarding some of the above items, but much is still unknown.

  (as measured by DAS28 after one year) was achieved in 50% of the patients treated with etanercept and MTX vs. 28% in the MTX alone arm. Radiographic remission (defined by a non-progression in the Total Sharp Score) was achieved by 79.7% of the patients in the etanercept arm, compared with 58.7% of the MTX group (33).The BEST program, involving patients treated with infliximab, was a treatment strategy whereby patients were treated in 4 different groups: sequential monotherapy, step-up combination therapy, initial combination with prednisone and initial combination with infliximab. Again the target population was patients with disease duration of less than 2 years. After 5 years, 58% of the group treated with infliximab were off anti-TNF-α treatment and, of those, 19% were off any treatment (34).The PREMIER study comparing adalimumab plus MTX against MTX alone showed a total of 43 and 49% of patients achieving DAS28 remission at weeks 52 and 104, respectively. The MTX arm reached 21 and 25%. In this study disease duration could not be longer than 3 years. Interestingly the remission rate for the patients treated with adalimumab alone was very similar to the MTX group: 23 and 25% at 52 and 104 weeks (35).

  have been showing. However there is sufficient evidence to consider anti-IL6 receptor blockade as a valid therapy for the treatment of rheumatoid arthritis. Tocilizumab might have a promising significant advantage over anti-TNF-α treatment, particularly in the management of patients who are non-tolerant or in whom the use of methotrexate is contraindicated and in patients with significant extra-articular manifestations. Regarding the treat to target goals, tocilizumab is probably the biological who achieves the largest number of patients with sustained remission and low

  A, Ciurea A, Brulhart L, Möller B, Walker UA, Courvoisier D, Kyburz D, Dudler J, Gabay C; Arthritis. Which subgroup of patients with rheumatoid arthritis benefits from switching to rituximab versus alternative anti-tumour necrosis factor (TNF) agents after previous failure of an anti-TNF agent? Ann Rheum Dis. 2010 Feb;69(2):387-93. 30 -Emery P, Keystone E, Toy HP et al. IL-6 receptor inhibition with tocilizumab improves treatment outcomes in patients with rheumatoid arthritis refractory to anti-TNF biologics: results from a 24-week multicentre randomised placebo controlled trial. Ann Rheum Dis 2008 Nov;67(11):1516-23. 31 -Dayer JM, Choy E. Therapeutic targets in rheumatoid arthritis: the interleukin-6 receptor. Rheumatology (Oxford). 2010 Jan; 49(1): 15-24. 32 -Atzeni F, Turiel M, Caporali R, Cavagna L, Tomasoni L, Sitia S, Sarzi-Puttini P. The effect of pharmacological therapy on the cardiovascular system of patients with systemic rheumatic diseases. Autoimmun Rev. 2010 Oct; 9 (12): 835-9. 33 -Emery P, Breedveld FC, Hall S, Durez P, Chang DJ, Robertson D, Singh A, Pedersen RD, Koenig AS, Freundlich B. Comparison of methotrexate monotherapy with a combination of methotrexate and etanercept in active, early, moderate to severe rheumatoid arthritis (COMET): a randomised, double-blind, parallel treatment trial. Lancet 2008; 327: 375-82 34 -Goekoop-Ruiterman YPM, de Vries-Bouwstra JK, Allart CF, et al. Clinical and radiographic outcomes of four different treatment strategies in patients with early rheumatoid arthritis (the BeSt study): A randomized, controlled trial. Arthritis Rheum 2005; 52:3381-3390. 35 -Breedveld FC, Weisman MH, Kavanaugh AF, Cohen SB, Pavelka K, van Vollenhoven R, Sharp J, Perez JL, Spencer-Green GT. The PREMIER study: A multicenter, randomized, double-blind clinical trial of combination therapy with adalimumab plus methotrexate versus methotrexate alone or adalimumab alone in patients with early, aggressive rheumatoid arthritis who had not had previous methotrexate treatment. Arthritis Rheum. 2006; 54(1): 26-37. 36 -Genovese MC, McKay JD, Nasonov EL et al. Interleukin-6 receptor inhibition with tocilizumab reduces disease activity in rheumatoid arthritis with inadequate response to disease-modifying antirheumatic drugs. Arthritis & Rheumatism 2008; 58 (10): 2968-2980.

  Vollenhoven RF, Keystone E, Furie R, et al. Gastrointestinal safety in patients with rheumatoid arthritis (RA) treated with tocilizumab in the Roche clinical trials database. Presented at the 73 rd Annual Scientific Meeting of American College of Rheumatology (ACR), October 16-21, 2009, Philadelphia, Pennsylvania. 48 -Nishimoto N, Ito K, Takagi N. Safety and efficacy profiles of tocilizumab monotherapy in Japanese patients with rheumatoid arthritis: meta-analysis of six initial trials and five long-term extensions. Mod Rheumatol. 2010 Jun; 20(3): 222-32.

F

Biologic agent Target/Affinity Immune actions Half-life Administration

	Etanercept	TNF-α,	Decreases	4 days	Subcutaneous
	(humanized	lymphotoxin-α	circulating TNF-		injection of
	fusion protein)	(TNF-β).	α; partial		50mg once a
		Soluble TNF-α	blockade; no		week
			lysis of TNF-
			expressing cells
	Infliximab	TNF-α.	Monocyte and	9 days	Infusion of 3-
	(chimeric	Soluble and	T-cell apoptosis;		5mg/Kg at 0, 2,
	mAb g )	membrane TNF-α	lysis of TNF-		6 weeks and then
			expressing cells		every 6-8 weeks
	Adalimumab	TNF-α.	lLysis of TNF-	14 days Subcutaneous
	(humanized mAb)	Soluble and membrane TNF-α o r	expressing cells; possible effects monocytes and on apoptosis,		injection of 40mg every 2 week
	Tocilizumab (humanized mAb)	natural killer cells Competitive e P IL-6 receptor r blocking of IL-6 e receptor	5-10 days Infusion of 8 mg/Kg every 4 weeks
			R
			e
			v i e
				w
				O n l
					y

Table 1 .

 1 Comparison between tocilizumab and the most used anti-TNF agents in the context of autoimmune rheumatic diseases.


Ghalia Boukellal

Audrey Durin

Rudy Valette
email: rudy.valette@mines-paristech.fr

Jean-François Agassant

Helmut Münstedt

using conventional and planar elongation flow optical rheometers

Keywords: Planar elongational flows, Birefringence, Laser-Doppler velocimetry, Tube based constitutive equations

come

Evaluation of a tube-based constitutive equation using conventional and planar elongation flow optical rheometers

Introduction

Since the advent of the Doi and Edwards (1986) theory, tube based constitutive equations for polymer melts have been increasingly successful in predicting linear rheometric flows for both monodisperse and polydisperse linear entangled melts, together with providing a clear link between their macromolecular characteristics and model parameters.

Further constitutive models have then been introduced in order to correct the shortcomings of the initial theory, particularly in 'strong', nonlinear rheometric flows such as fast elongational flows (Bach et al. 2003). These advanced models include, for example (Ianniruberto and Marrucci 2001;Likhtman and Graham 2003;Marrucci and Ianniruberto 2003), chain stretch and convective constraint release.

The pertinence of advanced models to predict complex flows of monodisperse and polydisperse melts has been studied using various experimental approaches, coupled with numerical solution procedures. In these studies, different flow geometries such as contraction, contraction-expansion slit or cross-slot were investigated, for moderately 'strong' flows, as follows. First an a priori model parameter identification was performed. This was peformed using linear and nonlinear (mainly elongational) rheometry, coupled with a suitable optimization procedure to compute the values of the many model parameters. Then, 2D or 3D numerical simulations were performed and compared to experimental measurements using flow-induced birefringence (FIB) in recent studies (Lee et al. 2001;Collis et al. 2005;[START_REF] Valette | Matching time dependent pressure driven flows with a Rolie Poly numerical simulation[END_REF]Hassell et al. 2008;Hassell et al. 2009;[START_REF] Scelsi | Experimental observations and matching viscoelastic specific work predictions of flow-induced crystallization for molten polyethylene within two flow geometries[END_REF]Auhl et al. 2011) or both FIB and laser-Doppler velocimetry (LDV) in earlier ones [START_REF] Schoonen | A 3D numerical/experimental study on a stagnation flow of a polyisobutylene solution[END_REF][START_REF] Peters | On the performance of enhanced constitutive models for polymer melts in a cross-slot flow[END_REF]Agassant et al. 2002;[START_REF] Verbeeten | Viscoelastic analysis of complex polymer melt flows using the eXtended Pom-Pom model[END_REF][START_REF] Verbeeten | Numerical simulations of the planar contraction flow for a polyethylene melt using the XPP model[END_REF]Gough et al. 2008). Most of the cited studies have shown that a rather good agreement could be obtained.

However, apart for linear monodisperse melts in recent studies, a theoretically-based link (or missing link) between polymer melts characteristics and nonlinear model parameters was not discussed a priori. Moreover, in the recent studies, which involved the most advanced constitutive models, only FIB experimental patterns were compared to computed FIB patterns (Lee et al. 2001;Collis et al. 2005;[START_REF] Valette | Matching time dependent pressure driven flows with a Rolie Poly numerical simulation[END_REF]Hassell et al. 2008;Hassell et al. 2009;[START_REF] Scelsi | Experimental observations and matching viscoelastic specific work predictions of flow-induced crystallization for molten polyethylene within two flow geometries[END_REF]Auhl et al. 2011). Consequently it was not possible to check consistency between measured velocity and resulting FIB patterns.

In this study, comparison of experimental and computed FIB pattern was proposed using an alternative approach. In this approach, LDV measurements were performed to obtain velocity and velocity gradients on the symmetry axis of a smooth contraction flow and then used to compute FIB patterns from the tubebased model of Marrucci and Ianniruberto (2003). This approach was inspired by the work of [START_REF] Schuberth | Simultaneous measurements of velocity and stress distributions in polyisobutylenes using laser-Doppler velocimetry and flow induced birefringence[END_REF], who developped an 'optical elongational rheometer' for polymer fluids of low viscosity, following earlier studies of [START_REF] Quinzani | Birefringence and Laser-Doppler velocimetry (LDV) studies of viscoelastic flow through a planar contraction[END_REF] and [START_REF] Quinzani | Use of coupled birefringence and LDV studies of flow through a planar contraction to test constitutive equations for concentrated polymer solutions[END_REF]. Prior to this comparison, dynamic, capillary and elongational measurements at a wide range of temperatures and deformation rates were performed. A preliminary parameter identification is then proposed and discussed in terms of relevance to molecular theory. Computed stress field were then compared with the one experimentally measured using FIB. In order to reduce the discrepancy between experimental and numerical stress field some nonlinear parameters of the constitutive equation were modified. Consistency between these new parameters and preliminary rheometrical measurement was finally checked .

Polymer and Rheometry

In this paper a commercial grade Polystyrene (Styron TM 648 from the Dow Chemical Company) was used. Molecular characteristics of PS 648 are listed on table 1.

The complex modulus is measured (using a Rheometric Scientific ARES 600) at various temperatures. The WLF equation (parameters table 2) allows to obtain a master curve for the complex modulus (figure 1) and for the complex viscosity (figure 2).

Capillary measurements were obtained on a Gottfert Rheotester 1000 using several dies (1mm in diameter and various L/D ratios: 10, 20, 30) for Bagley corrections (Macosko 1994). Weissenberg-Rabinowitsch corrections (Macosko 1994) allowed to deduce the viscosity as a function of the true shear rate. Capillary measurements were superimposed with dynamic measurements (see figure 2), which were consistent with Cox and Merz observations (Macosko 1994). A Newtonian plateau is observed at low shear rates, then a classical power-law decrease of the viscosity, followed by a change of slope above 500 s -1 which could anticipate a second Newtonian plateau at higher shear rates.

Elongational viscosity measurements were performed on a Meissner (1972) RME rheometer at 180 o C for elon-gational rates ranking between 0.16s -1 and 1s -1 (figure 3). The transient elongational viscosity followed the linear regime for short times and the lowest rates. For longer times and higher rates, the elongational viscosity deviated from the linear regime and then reached a maximum before sample breakage. The plateau just before breakage could eventually be considered as a 'stationary elongational viscosity' (Bach et al. 2003).

Constitutive equation and preliminary identification

Previous rheometry data were used to fit a set of multimode Marrucci and Ianniruberto model (2003) 1986), that occur at different time scales. To overcome the shortcomings of previous models where orientation and stretch were decoupled (Ianniruberto and Marrucci 2001;[START_REF] Wapperom | Prediction of rheometrical and complex flows of entangled linear polymers using the double-convection reptation model with chain stretch[END_REF], this model was built using an 'undecoupled' (Marrucci and Ianniruberto 2003) equation for tube conformation tensor C:

Ċ = ∇vC + C∇v T - f Θ (C - I C 3 I) - 1 3θ r (f I C -1) I(1)
where ∇v and I are the velocity gradient and unit tensors, I C the trace of C, Θ and θ r the orientation and stretch relaxation times, and f is the finite extensibility factor. Orientation time Θ is defined as:

1 Θ = 1 θ + 1 θ r - 1 θ β (f I C -1) 1 + β (f I C -1) (2)
where θ is half of the reptation only disengagement time and β is a numerical CCR parameter of order 1 (Marrucci and Ianniruberto 2003), governing the transition of Θ from θ (slow flows) to θ r ('strong' flows). The finite extensibility factor f is defined as:

f = b -1 b -I C (3)
where b has the meaning of the square of the maximum stretch ratio. For polystyrene, Dhole et al. (2003) suggested that b = 66, whereas [START_REF] Vh | Modeling non-Gaussian extensibility effects in elongation of nearly monodisperse polystyrene melts[END_REF] proposed a maximum stretch ratio of 5, which implies b = 25. Finally the polymeric stress tensor is defined:

τ = 3Gf C - 1 3 I (4)
where G is the elastic modulus, which is around 2 × 10 5 P a for polystyrene (Dhole et al. 2003). For this study a 'crude' multimode extension of the Marrucci and Ianniruberto model for polystyrene was investigated. This model is able to capture qualitatively the main features which have been experimentally observed in the preceding section: Newtonian plateau, shear thinning effect as well as nonlinear behaviour in elongation. A 8 modes relaxation spectrum was first identified from the dynamic data (figure 1) using the non linear regression method early proposed by Baumgaertel and Winter (1989) (table 3).

For each mode, the three non linear parameters, β, θ r and b were then identified. For all set of parameters, it was found that the stationary shear viscosity curve was mainly sensitive to the parameter β, as expected (Ianniruberto and Marrucci 1996). The value b = 66, as prescribed by Dhole et al. (2009) for each mode was used as a starting parameter for the identification. Finally, a simplified form of classical scaling theories (Doi and Edwards 1986;Likhtman and McLeish 2002) was used to determine θ r :

θ ri = θ e θ ri θ e 2 3 (5)
where θ e would correspond to the relaxation time for a macromolecular chain with only one entanglement. This relaxation time was choosen of the order of the inverse frequency corresponding to the end of the rubberry plateau (10 -5 s on figure 1). Using these three non linear parameters (β = 1, b = 66, θ e = 10 -5 s) led to a reasonable agreement for the shear viscosity but not for the uniaxial elongational viscosity (figure 4). A better fit for stationary shear viscosity was obtained by decreasing the CCR parameter β to 0.92, strain hardening was obtained by increasing the exponent of equation ( 5) ( 2.44 3 instead of 2 3 ) and decreasing the finite extensibility parameter b to b i = (10,10,10,10,10,10,20,30) from the fastest to slowest mode (figure 5). These values are more consistent with the ones prescribed in [START_REF] Vh | Modeling non-Gaussian extensibility effects in elongation of nearly monodisperse polystyrene melts[END_REF]. It should be noted however that the more accurate fit was obtained using a non unique value of b for each mode. It should also be noticed that elongational results at low strain rates could exhibit measurement errors [START_REF] Vh | Modeling non-Gaussian extensibility effects in elongation of nearly monodisperse polystyrene melts[END_REF], which would explain the discrepancy at low elongational rates.

Velocity and stress field measurements

We used a single screw extruder Kaufman (screw diameter 40mm) coupled with a servo controlled gear pump to obtain a constant and controlled flow rate within a partially transparent die. This die was made of a semi circular converging geometry (Combeaud et al. 2004). The reservoir had a square section (14mm×14mm) and the final slit gap was 1mm thick and 20.75mm long (figure 6). Transparent Pyrex glass windows on each side of the die allowed to analyse carefully the flow field.

A laser doppler Dantec velocity measurement technology was used. In this study we focused on the velocity field in the mid plane of the die and on the symmetry axis. The precision of laser Doppler velocity measurements is higher on the symmetry axis than at the wall, and typical 1% precision was obtained for each point after 1 min sampling time (Hertel, 2008). A complete set (for all investigated flow rates) of velocity and birefringence measurement required a few hours and several kg of polymer. Figure 7 shows velocity plotted for several flow rates ranging from 0.05g/s to 0.7g/s. Velocity was approximated using a high order polynomial fit and then elongational rates could be computed as shown on figure 8. For the largest flow rate, the elongational rate reached values as high as 18s -1 .

The optical set-up for FIB measurements [START_REF] Robert | Flow birefringence study of the stick-slip instability during extrusion of high-density polyethylenes[END_REF][START_REF] Robert | Stick-slip flow of high density polyethylene in a transparent slit die investigated by laser Doppler velocimetry[END_REF]) used a monochromatic sodium source (wave length λ = 589nm). An initial polarizer filter, a first 1/4 wave plate, positioned at 45 o from the direction of polarization eliminated isoclinic fringes interfering with isochromatic fringes. On the other side of the transparent slit die, a second 1/4 wave plate compensated for modifications in total birefringence produced by the first plate. A final analyser filter enabled the observation of chromatic birefringence fringes in the flow field.

The transverse aspect ratio in the reservoir and the slit were, respectively, 1 and 14. While three-dimensional effects may arise in low aspect ratio regions (Wales, 1976), it was shown by Clemeur et al. ( 2004) that these effects may not have a great impact on the observed birefringence. Consequently, it is reasonable to consider that the principal stress difference may be related to the order k of the isochromatic extinction lines by:

σ 1 -σ 2 = kλ CW (6)
where C is the stress optical coefficient and W the die width. On the symmetry axis σ 1 = -p + τ xx and σ 2 = -p which means that we had directly access to the polymeric stress component in the flow direction.

The value of C = -4.0 10 -9 P a -1 at 180 o C was taken from Han and Drexler (1973) and Macosko (1994). For the conditions probed, a high number of fringes were observed, which were in part due to the high value of the stress optical coefficient (figure 9). Using a dedicated image analysis (which consisted in finding the set of local maxima of grey levels) allowed to detect precisely the position of each isochromatic line, and so the k value from which we obtained the stress component in the flow direction (figure 10). According to the stressoptical law, we observed elongational stress values as high as 10 6 P a. However, it was shown by [START_REF] Venerus | Stress and birefringence measurements during the uniaxial elongation of polystyrene melts[END_REF]

τ ′ = 3G C - 1 3 I (7)
5 New identification step

As the proposed convergent flow geometry induced non uniform and high planar elongational rates, it was used to test the pertinence of both the constitutive equation and parameters identified previously. The general principle of the identification procedure was based on using the flow field measurements on the symmetry axis and the computed planar elongational rate as input parameters in the Marrucci and Ianniruberto constitutive equation, together with the previously identified linear and nonlinear viscoelastic parameters. 'Corrected elongational FIB stress' distribution on the axis in the flow direction was then computed using equation 7 for each investigated flow rate, and compared to the one measured using flow birefringence. On figure 11 we observed that the agreement was qualitatively fair, but the computed corrected FIB stresses were overestimated at high elongational rates and underestimated at low elongational rates. Origins of the observed discrepancies could be two fold: first, FIB patterns integrated 3D effects especially in the reservoir, second, the parameters of the constitutive equation were identified in slow rheological flow conditions, with constant shear rates and elongational rates, which were different from those presented in figures 7 and 10.

We then revisited the identifying procedure accounting for these new field experiments; relaxation spectrum were obviously keep constant ; CCR parameter were slightly increased (0.95 instead of 0.92) ; finite extensibility coefficient b were modified (smaller values for the three fastest modes) and θ ei has been changed to 5 × 10 -4 for the three fastest modes. A line search method was used to minimize the quadratic distance between all measured and computed curves. The step size was restricted to 0.01 for the CCR parameter, 1 to the finite extensibility coefficient b and 0.01 to the exponent of equation 5. As a result of these changes, the final agreement for the 12 different flow rates was much improved especially at high rates (figure 12). The identified parameters are listed table 4.

The last step consisted in checking that the steady shear and elongational viscosity curves were still well predicted using these new parameters, which was found to be the case (figure 13).

Discussion and conclusion

This study was a first attempt to test the validity of constitutive equations and to identify their non linear parameters using both conventional and optical rheometry for polymer melts. It was shown that optical elongational rheometry could extend the conventional rheometry measurement window in two ways: it extends the range of investigated strain rates and it provides additional data in non uniform planar elongation. Moreover, it was shown that a simple multimode tube-based model (Marrucci and Ianniruberto 2003) could predict most of the experimental resuls using a set of theoreticaly addmissible set of parameters, apart for some finite extensibility parameters that needed to be set to unrealistically small values.

The next step would be to measure velocity and-FIB along the flow axis of the downstream channel and to check the relevance of the constitutive equation in that domain. In fact, most of the constitutive equations which give reasonable agreement for stress build up are less relevant for stress relaxation [START_REF] Wapperom | Numerical simulation of large amplitude oscillatory shear of a high-density polyethylene melt using the MSF model[END_REF].

In a second step the whole velocity and birefringence fields may be considered. On each streamline, the balance between elongational and simple shear would be different and this would represent a more severe test for the constitutive equations. It would be first necessary to check the validity of the experiments and especially the 3D effects generated by the shape factor of the upstream reservoir (Hertel et al. 2008[START_REF] Sirakov | Threedimensional numerical simulation of viscoelastic contraction flows using the Pom-Pom differential constitutive model[END_REF] and the boundary conditions along the two side glass windows. In a second step 3D viscoelastic computations need to be performed with selected constitutive equations at high rates

C 1 C 2 T 0 ( o K) ( o C) 5.02 122.4 180
θ i (s) G i (P a) β θ r i (s) b
7.63 10 -3 8.86 10 4 0.95 2.00 10 -3 5 1.08 10 -1 5.35 10 4 0.95 1.69 10 -2 5 6.72 10 -1 2.09 10 4 0.95 7.28 10 -2 5 1.35 1.08 10 5 0.95 1.27 10 -1 10 4.93 8.40 10 3 0.95 3.58 10 -1 10 1.70 10 1 1.71 10 3 0.95 9.65 10 -1 10 3.81 10 1 5.02 10 2 0.95 1.60 20 1.79 10 2 9.31 10 1 0.95 5.52 30 (10,10,10,10,10,10,20,30), θ e = 10 -5 s)

  (Dhole et al. 2009; Likhtman and Graham 2003; Wagner et al. 2005; Wagner et al. 2008).

  Baaijens F, Bastian H, Bernnat A, Bogaerds ACB, Coupez T, Debbaut B, Gavrus AL, Goublomme A, van Gurp M, Koopmans RJ, Laun HM, Lee K, Nouatin OH, Mackley MR, Peters GWM, Rekers G, Verbeeten WMH, Vergnes B, Wagner MH, Wassner E, Zoetelief WF (2002) The matching of experimental polymer processing flows to viscoelastic numerical simulation. International Polymer Processing 17:3-10 Auhl D, Hoyle DM, Hassell D, Lord TD, Mackley MR, Harlen OG, McLeish TCB (2011) Cross-slot extensional rheometry and the steady-state extensional response of long chain branched polymer melts. Journal of Rheology 55:875-900 Bach A, Rasmussen HK, Hassager O (2003) Extensional viscosity for polymer melts measured in the filament stretching rheometer. Journal of Rheology 47:429-441 Baumgaertel M, Winter HH (1989) Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheologica Acta 28:511-519 Clemeur N, Rutgers RPG, Debbaut B (2004) Numerical evaluation of three dimensional effects in planar flow birefringence. J. Non-Newtonian Fluid Mech 123 105-120 Collis MW, Lele AK, Mackley MR, Graham RS, Groves DJ, Likhtman AE, Nicholson TM, Harlen OG, McLeish TCB, Hutchings LR, Fernyhough CM, Young RN (2005) Constriction flows of monodisperse linear entangled polymers: Multiscale modeling and flow visualization. Journal of Rheology 49:501-522 Combeaud C, Demay Y, Vergnes B (2004) Experimental study of the volume defects in polystyrene extrusion. Journal of Non-Newtonian Fluid Mechanics 121:175-185 Dhole S, Leygue A, Bailly C, Keunings R (2009) A single segment differential tube model with interchain tube pressure effect. Journal of Non-Newtonian Fluid Mechanics 161:10-18 Doi M, Edwards SF (1986) The theory of polymer dynamics. Oxford University Press, New York Gough T, Spares R, Kelly AL, Brook SM, Coates PD (2008) Three-dimensional characterisation of full field stress and velocity fields for polyethylene melt through abrupt contraction. Plastics Rubber and Composites 37:158-165 Han CD and Drexler LH (1973) Studies of converging flows of viscoelastic polymeric melts. I. Stressbirefringent measurements in the entrance region of a sharp-edged slit die. J. Appl. Polymer Sci, 17:2329-2354 Hassell DG, Auhl D, McLeish TCB, Mackley MR (2008) The effect of viscoelasticity on stress fields within polyethylene melt flow for a cross-slot and contractionexpansion slit geometry. Rheologica Acta 47:821-834 Hassell DG, Mackley MR (2009) An experimental evaluation of the behaviour of mono and polydisperse polystyrenes in Cross-Slot flow. Rheologica Acta 48: 543-550 Hassell DG, Hoyle D, Auhl D, Harlen O, Mackley MR, McLeish TCB (2009) Effect of branching in crossslot flow: the formation of 'W cusps'. Rheologica Acta 48:551-561 Hertel D (2008) Flow of polyethlene melts within and into rectangular ducts investigated by laser Doppler velocimetry. PhD Thesis, Friedrich-Alexander-Universität, Erlangen-Nürnberg Hertel D, Valette R, Münstedt H (2008) Three dimensional entrance flow of a low-density polyethylene (LDPE) and a linear low-density polyethylene (LLDPE) into a slit die. Journal of Non-Newtonian Fluid Mechanics 153:82-94 Ianniruberto G, Marrucci G (1996) On compatibility of the Cox-Merz rule with the model of Doi and Edwards. Journal of Non-Newtonian Fluid Mechanics 65:241-246 Ianniruberto G, Marrucci G (2001) A simple constitutive equation for entangled polymers with chain stretch. Journal of Rheology 45:1305-1318 Lee K, Mackley MR, McLeish TCB, Nicholson TM, Harlen OG (2001) Experimental observation and numerical simulation of transient "stress fangs" within flowing molten polyethylene. Journal of Rheology 45:1261-1277 Likhtman AE, McLeish TCB (2002) Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35:6332-6343 Likhtman AE, Graham RS (2003) Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation. Journal of Non-Newtonian Fluid Mechanics 114:1-12 Luap C, Muller C, Schweizer T, Venerus DC (2005) Simultaneous stress and birefringence measurements during uniaxial elongation of polystyrene melts with narrow molecular weight distribution. Rheologica Acta 45:83-91 Macosko CW (1994) Rheology Principles, Measurements and Applications. VCH Publishers, New York Marrucci G, Ianniruberto G (2003) Flow-induced orientation and stretching of entangled polymers. Philosophical Transactions of the Royal Society of Lon-

Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 7 Fig. 8

 1234578 Fig. 1 Master curves of the complex modulus at 180 o C

Fig. 9 Fig. 10 Fig. 11

 91011 Fig. 9 Typical flow induced birefringence pattern (temperature 180 o C, flow rate 0.25g/s)

Fig. 12 Fig. 13

 1213 Fig.12Comparison of experimental and computed stress field along the symmetry axis using the new set of non linear parameters(table 4)

  and Luap et al. (2005) that linearity in the stress optical law failed beyond a certain stress value around 10 6 P a (which may vary depending on sample polydispersity). In[START_REF] Venerus | Stress and birefringence measurements during the uniaxial elongation of polystyrene melts[END_REF]), Luap et al. (2005) and Rolón-Garrido et al. (2006), departure from linearity was clearly explained by the role played by finite extensibility effects: at high elongational rates, stress increases faster than 'FIB stress'. Consequently, for model -experiments comparisons in the following, we computed 'corrected FIB stresses' from the Marrucci and Ianniruberto model by dropping the finite extensibility term f for each mode in equation 4. We then used a 'computed corrected FIB stress' τ ′ defined as:

Table 1

 1 Molecular characteristics of PS 648 C molecular mass M n molecular mass M w molecular mass M z temperature (10 3 kg.m -3 ) (10 3 kg.m -3 ) (kg.mol -1 )

	Density	Density	Number-averaged	Weight-averaged	Z-averaged	Glass transition
	at 23 o C	at 200 o (kg.mol -1 )	(kg.mol -1 )	( o C)
	1.047	0.973	136	296	460	100

Table 2

 2 Coefficients of the WLF equation for PS 648

Table 3

 3 Relaxation spectrum for PS 648 at 180 o C

	θ i (s)	G i (P a)
	7.63 10 -3 8.86 10 4
	1.08 10 -1 5.35 10 4
	6.72 10 -1 2.09 10 4
	1.35	1.08 10 5
	4.93	8.40 10 3
	1.70 10 1	1.71 10 3
	3.81 10 1	5.02 10 2
	1.79 10 2	9.31 10 1

Table 4

 4 Parameters identified at 180 o C for the Marrucci and Ianniruberto constitutive equation


Valérie Vidal

François Soubiran

Thibaut Divoux

Jean-Christophe Géminard

Degassing cascades in a shear-thinning viscoelastic fluid

Keywords: numbers: 05.45.-a, 47.57.-s, 83.80.Qr, 83.60.Rs

We report the experimental study of the degassing dynamics through a thin layer of shear-thinning viscoelastic fluid (CTAB/NaSal solution), when a constant air flow is imposed at its bottom. Over a large range of parameters, the air is periodically released through series of successive bubbles, hereafter named cascades. Each cascade is followed by a continuous degassing, lasting for several seconds, corresponding to an open channel crossing the fluid layer. The periodicity between two cascades does not depend on the injected flow-rate. Inside one cascade, the properties of the overpressure signal associated with the successive bubbles vary continuously. The pressure threshold above which the fluid starts flowing, fluid deformation and pressure drop due to degassing through the thin fluid layer can be simply described by a Maxwell model. We point out that monitoring the evolution inside the cascades provides a direct access to the characteristic relaxation time associated with the fluid rheology.

I. INTRODUCTION

Non-Newtonian fluids peculiar dynamics has focused a large attention over the last decades. Their ability to flow or break [1,2], to exhibit viscous or elastic behavior [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF], thixotropy effects [START_REF] Mewis | [END_REF][5][6] or to sustain loads without flowing [7,8] opened a wide range of applications, including cosmetics, food industry, environment and biology. Among them, micellar fluids differ from polymers by their internal dynamics: they continuously break and restructurate [9][10][START_REF] Gelbart | Micelles, Membranes, Microemulsions and Monolayers[END_REF], and their internal kinetics determines their length distribution -and, hence, their macroscopic properties [START_REF] Cates | [END_REF]13].

The behavior of bubbles rising through such fluids is complex, due, in particular, to the non-trivial coupling between the bubble and the fluid rheology [START_REF] Chhabra | Bubbles, Drops and Particles in non-Newtonian Fluids (Chemical industries series #113[END_REF]. This has prevented, up to now, a complete theoretical description of the system, and favored experimental studies. Due to the viscoelastic properties, the bubble shape is generally elongated [START_REF] Bird | Dynamics of Polymeric Liquids[END_REF][START_REF] Chhabra | Bubbles, Drops and Particles in non-Newtonian Fluids (Chemical industries series #113[END_REF][START_REF] Belmonte | [END_REF][16][17], its tail ends by a cusp [START_REF] Chhabra | Bubbles, Drops and Particles in non-Newtonian Fluids (Chemical industries series #113[END_REF][START_REF] Belmonte | [END_REF], and both its geometry and velocity oscillate during its rising through the fluid [START_REF] Belmonte | [END_REF]16,18]. In shear-thinning fluids, the local perturbation due to a rising bubble -or a falling sphere -creates a negative wake [19] and a corridor of reduced viscosity [20,21]. As a consequence, successive bubbles may interact one with the other [22][23][24][25], if the emission period is shorter than the time for the perturbation created by the bubble to vanish.

An extensive study of the degassing regimes when a constant air flux is imposed at the bottom of a complex, yield-stress fluid column (gel or immersed granular material) has revealed the existence of three different regimes [25][26][27]. On the one hand, at low flow-rate, bubbles rise independently one from the other. On the other hand, at high flow-rate, an open channel connects the bottom nozzle to the fluid surface; this channel develops instabilities, forming a 'bubble chain' [28]. Finally, at intermediate flow-rate, the system spontaneously oscillates between the two previous regimes, exhibiting a complex intermittent dynamics [25].

In this article, we extend these previous works to the case of a fluid without yield stress, and for a thin fluid layer, i.e., when the fluid height is smaller than the typical size of a bubble. We report the existence of a peculiar degassing regime in a thin layer of micellar fluid (CTAB/NaSal mixture). When a constant air flow is imposed at its bottom, in the bubbling regime, the air is released through periodic series of successive bubbles, hereafter named cascades. When the layer is thin enough, the cascades merely consists in successive opening and closing of the fluid layer above the air injection point. For convenience, these periodic degassing apertures will also be named bubbles in the following. We measure the overpressure at the base of the fluid column, at the injection point. The main characteristics of the cascades (periodicity, maximum overpressure) are analyzed in regards with the fluid rheology. We show that inside a cascade, the bubble properties vary continuously: the mean overpressure drops, whereas the emission time increases.

Q F δP V water h d FIG. 1. Experimental setup.
A constant air-flow is supplied at the base of a thin layer of micellar fluid by an air-flow controller F (flow-rate Q), via a chamber of volume V . h is of the order of a typical bubble diameter (a few millimeters to centimeters).

The transition between two regimes inside the cascades is directly linked to the fluid rheology and, in particular, provides a direct measurement of the viscoelastic characteristic time associated with the Maxwell fluid.

II. EXPERIMENTAL SETUP

The experimental cell consists of a cylinder made of plexiglass (diameter 74 mm, height 270 mm), filled with the fluid up to a height h (Fig. 1). Air is injected at constant flow-rate Q (from 0.17 to 1.72 mL/s) through an injection hole (diameter d = 2 mm) at the bottom of the fluid column. The air-injection system consists of a massflow controller (Bronkhorst, Mass-Stream series D-5111) connected to a chamber of volume V , from which the air flows at the column bottom. The volume of the chamber can be easily tuned by changing the water level into the chamber (see Fig. 1). The water itself makes it possible to inject humid air inside the fluid, thus avoiding any drying of the sample over the experimental time. A differential pressure sensor (223 BD-00010 AB, MKS Instruments) measures the variations of the overpressure δP inside the chamber, corresponding to the pressure variations at the bottom of the cell.

The fluid is a semi-dilute micellar system obtained by a mixing at equimolar concentration, inside pure water, sodium salicylate (NaSal, Sigma Aldrich) and hexadecyltrimethylammonium bromide (CTAB, Sigma Aldrich). The mixture of this two chemical components causes the formation of a network of giant entangled micelles, which break down and reform continuously [9]. On a macroscopic point of view, the fluid exhibits shearthinning, viscoelastic properties [29,30], which can be tuned by varying the fluid concentration c (from 0.03 to 0.5 mol.L -1 ). The rheology of these well-controlled mixtures is characterized by rheometer measurements (see Appendix A and B). The height h of the thin layer ranges from 5 to 35 mm. Unless specified, the results presented here are for a typical height of 5 mm and a concentration c = 0.1 mol.L -1 .

III. DEGASSING REGIMES

When varying the system parameters (c, V, Q, h), we observe three different degassing regimes through the thin layer: a bubbling regime, for which bubbles are emitted one after the other; an open channel regime, for which the system is able to sustain a stable channel connecting the injection nozzle at the base of the fluid column to the fluid free surface; and an intermittent regime, for which the system alternates spontaneously between the bubbling and the open channel regime -this latter pinching off intermittently. Similar degassing regimes have been reported in yield-stress fluids: gels [25] or in immersed granular media [26,27]. Typically, when increasing the injected air flow-rate Q, all other parameters being constant, the system goes from the bubbling, intermittent and finally open channel regime (Fig. 2). When increasing the fluid concentration, it becomes easier for the system to sustain an open channel, and it is necessary to go to higher volumes V and smaller flow rate Q to observe the bubbling regime (Fig. 2a andb) [START_REF]When the fluid concentration increases, the characteristic time associated with the Maxwell model increases[END_REF]. On the contrary, increasing the fluid column height makes it more difficult to open a channel, and shifts the regime boundaries toward smaller volumes and higher flow rates. Note that the precise boundary between two different regimes is difficult to determine, due to the finite acquisition time (the system may not have the time to switch from bubbles to an open channel, or vice-versa).

The formation of an open-channel can be qualitatively explained. Indeed, due to the fluid shear-thinning properties (Appendix A), when a bubble rises through the fluid, its wake is characterized by a local viscosity smaller than the surrounding fluid. If the flow rate is large enough, the following bubble will rise through a fluid with a smaller effective viscosity -and thus, will rise faster, and so on, until the system is able to sustain an open channel through the fluid column. Note that this channel does not resemble a cylinder, but rather a bubble chain, similar to previous observations in yield-stress and non yield-stress fluids [25,28].

IV. BUBBLE CASCADES

In a wide range of fluid concentration (from c = 0.04 to 0.14 mol/L), when the fluid layer height h is of the order of the size of a bubble (typically, h < 10 mm), we do not observe, in the bubbling regime, successive bubbles but rather periodic series of bubbles, hereafter named bubble cascades. These cascades are clearly observed in the overpressure signal δP recorded at the bottom of the fluid column (Fig. 3,top).

The pressure signal corresponding to one cascade exhibits successive rises and drops, each of them corresponding to the pressure increasing in the chamber, followed by a bubble emission (Fig. 3 middle, a andb). After each bubble cascade, the overpressure remains at δP ∼ 0 for a few seconds, before the next bubble cascade (Fig. 3 middle,c). During this time interval, the air escapes continuously through the thin fluid layer. The value of the overpressure is given by the charge loss of the air flow through the hole (roughly a few Pa, Fig. 3,middle,c). This hole then suddenly closes, and the next cascade starts. The sequence consisting of a bubble cascade followed by a hole opened through the fluid layer repeats periodically in time. The main goal of this work is to describe thoroughly the cascades properties.

The general properties of the cascades are reported in Figure 4. On the one hand, we observe that the time interval between each sequence (cascade + hole) is very stable and, over the range of parameters explored, does not depend neither on the injected flow rate Q, nor on the chamber volume V . The same observation is reported for the maximum overpressure reached inside each cascade (Fig. 4, inset). On the other hand, the number of bubbles emitted per cascade, n, depends linearly on the injected flow rate (Fig. 5a). The associated slope, dn/dQ, varies with the chamber volume V . For h = 5 mm and small volumes, we report a linear, decreasing relationship dn/dQ = -ζV , where ζ is a constant (Fig. 5b, black dots). For large values of V (> 105 mL) or larger height, however, dn/dQ is independent of the chamber volume. These results can be interpreted as follows.

For small h, the layer is of the order of a bubble height, and whenever a bubble is emitted, it pierces the layer. Via the open channel thus formed, the overpressurized air escapes the chamber. For small chamber volume V , the chamber reservoir quickly empties, and a dependence on V is clearly seen. When V reaches higher values, it acts as a pressure reservoir and dn/dQ remains roughly constant.

For larger h, the air is no longer able to pierce the fluid layer, and bubbles are emitted in the fluid, rise and burst without connecting anymore the injection nozzle to the fluid free surface. In this case, the overpressurized air trapped in the chamber cannot escape directly through the fluid, but via the successive bubbles. The bubble size is fixed by the nozzle size and the fluid rheological properties, and dn/dQ is independent of V .

The total volume of gas emitted during a cascade can be written V T ∼ Q∆t casc . As the number of bubbles emitted per cascade always depends linearly on Q, we can write n = αQ, where α = dn/dQ is a constant for a given series of experiment (c, h and V fixed). We thus find that the average gas volume emitted per bubble, v b = V T /n, is constant. For h > 5 mm, this constant does not depend anymore on V (Fig. 5b).

Inside a cascade, however, the bubble properties (maximum overpressure and emission duration) vary continuously (Fig. 3, top and middle). In the next two sections, we investigate these variations, and see which informations they bring on the system.

V. EVOLUTION INSIDE A CASCADE

During the release of a single bubble, the overpressure δP exhibits three different stages (Fig. 3, bottom). First, we observe a linear pressure increase (Fig. 3, bottom, region 1). The overpressure signal then departs from the linear tendency (Fig. 3, bottom, region 2) until the bubble is emitted (sudden pressure drop, Fig. 3, bottom, region 3). In this section, we describe each part of the pressure signal, and show that a simple Maxwell model can account for the different observations.

A. Linear pressure increase

When submitted to a sudden stress (pressure increase), at a short time scale, the CTAB/NaSal mixture does not flow, and the system is equivalent to a chamber of volume V continuously filled by a gas flow Q. The overpressure is given by

δP = P 0 V Qt (1)
where P 0 = 10 5 Pa denotes the atmospheric pressure. The experimental slope is consistent with this linear pressure increase (dashed gray line, Fig. 3 bottom, region 1), without any adjustable parameters. In the following section, we estimate the threshold pressure δP Y above which the fluid starts flowing when the pressure at its bottom increases.

B. Threshold pressure δPY

After a certain time t Y , associated with an overpressure δP Y (point Y , Fig. 3, bottom), the overpressure departs from its linear increase. At this point, the fluid starts flowing, and a bubble is nucleated and grows at the tip of the injection hole (Fig. 3, bottom, region 2). In order to estimate this threshold pressure, we describe the fluid with a Maxwell model, consisting of an ideal elastic spring attached to an ideal dashpot (Fig. 6). This simple model represents the viscoelastic behavior of energy storage and viscous loss, which can be quantified by the elastic (G ′ ) and viscous (G") modulus, respectively. These modulii can be considered constant as a function of the applied stress in a given frequency range (ω ∼ 1 Hz) representative of the frequency of bubbles rising through the fluid. The modulii are estimated from the plateau obtained from oscillation measurements (see Appendix B, Fig. 10) to G ′ ∼ 50 Pa and G ′′ ∼ 10 Pa.

The equation describing the opening of a hole of radius r due to a bubble rising in the fluid layer can be written:

ξρr d 2 r dt 2 = - 2γ r -αG ′ (r -r n ) + δP -ρg∆h (2)
where ξ is a constant. The first term in the right-hand side represents the closing force due to the fluid surface tension γ, where γ ≃ 40 mN/m. The second term describes the elastic force which tends to shift the spring back to its initial length, where α is a constant which can be approximated to the inverse of the nozzle radius 2/d. δP is the pressure inside the chamber (Fig. 1) -and, thus, inside the bubble which starts being generated at the injection nozzle. Finally, the last term quantifies the weight associated with the fluid layer height ∆h above the newborn bubble. At the limit where the fluid starts flowing, r ∼ d/2, ∆h = h and the threshold pressure, given by the condition d 2 r/dt 2 ≥ 0, can be written Before the emission of the first bubble in the cascade, r 0 = 0 and we can estimate δP Y ∼ 180 Pa, which is consistent with the pressure signal measured in the experiments (Fig. 3,top).

δP Y = 4γ d + ρgh + 2G ′ d d 2 -r n . (3)
As the bubble properties vary continuously inside the cascade, we investigate the evolution of δP Y as a function of the bubble number. We find that, in the main part of the cascade, δP Y is a linear, decaying function of the bubble number (Fig. 7). The corresponding slope, constant from one cascade to the other, increases with the fluid concentration (Fig. 7, inset). This linear relationship can be explained by a simple heuristic model, based on the key ingredients of the Maxwell model (Fig. 6).

We consider successive bubbles rising through or piercing the thin fluid layer. When the bubble rises during the characteristic time t b , r ∼ d/2 (Fig. 6) and we can write the differential equation describing the temporal evolution of r n :

G ′ d 2 -r n = η ṙn (4)
which gives, right after the bubble rise

r n (t b ) = r n-1 - d 2 e -t b /τ + d 2 (5)
where r n-1 is the initial condition from which r n evolves and τ = η/G ′ the characteristic time associated with the Maxwell fluid.

After the bubble rise, we suppose that the fluid layer closes almost immediately, due in particular to the surface tension. The fluid is hence at rest, and r n relaxes towards 0 during a time T -t b , where T is the average time between bubble emission. For sake of convenience, we consider here that T and t b are the same for each bubble. During this stage of relaxation, r n obeys the following equation:

τ ṙn + r n = 0 , (6)
which gives, after a time t = T -t b , r n = r n (t b )e -t/τ . We can therefore write the recurrence equation giving the displacement after the n th bubble:

r n = r n-1 e -T /τ + d 2 e -T /τ e -t b /τ -1 . (7)
By recurrence, and considering that r 0 = 0, we thus get the equation describing the displacement r n :

r n = d 2 e -t b /τ -1 e -T /τ -e -nT /τ 1 -e -T /τ . ( 8
)
For t b < τ and T < τ (both the characteristic times of bubble rising through the fluid layer and bubble formation and emission are smaller than the Maxwell time), the slope dr n /dn can be approximated to (d/2)(t b /τ ). From Eq. 3, we thus obtain the following expression for the threshold pressure:

δP Y ∼ 4γ d + ρgh + G ′ - 1 2 G ′ t b τ n (9)
The average time for a bubble emission (formation and growth, regions 2 and 3 in Fig. 3, bottom) inside a cascade, for c = 0.1 mol.L -1 , can be estimated to t b ∼ 0.2 s. The characteristic time τ = η/G ′ is obtained from rheological measurements (see Appendix) and can be estimated to τ ∼ 0.5 s, with η ∼ 25 Pa.s and G ′ ∼ 50 Pa.

We thus estimate a slope -(G ′ /2)(t b /τ ) of about -10, in agreement with the experimental results (see Fig. 7 for c = 0.1 mol.L -1 ).

C. Fluid deformation

Above the threshold overpressure δP Y , the fluid starts flowing and a bubble grows at the tip of the injection hole (Fig. 3, bottom, region 2). The overpressure then departs from its linear increase, as a consequence of the volume increase due to the bubble growth.

By deriving the general equation for an ideal gas, and denoting v b the volume of the bubble hence formed, we get

dP dt + P V dv b dt = RT V Q V mol (10)
Further integration leads to the general expression of the pressure variation in time:

P (t) = RT V Q V mol t - P V v b ( 11
)
where R = 8.314 J K -1 mol -1 is the ideal gas constant and V mol = 24.7 L is the molar volume of air at 25 • C.

Here we do not develop further the calculation, but note that the departure from the linear trend is linear with the bubble volume, which provides a rough estimation of this latter, from about v b ∼ 1 mL to 10 mL from the beginning to the end of the cascade.

D. Bubble emission

Finally, the bubble reaches the free surface -we remind here that the fluid layer is of the order of the bubble size -, the gas is suddenly released and the overpressure quickly drops (Fig. 3, bottom, region 3). In order to get an estimate of the characteristic time over which the pressure drops, we write, on the one hand, Bernoulli's equation to describe the air flowing from the chamber of volume V through the opening of diameter d, up to the surface of the fluid layer: 1/2ρv 2 = δP , which gives the flow velocity through the opening [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF]:

v = 2δP ρ . (12)
On the other hand, by considering the air as an ideal gas, we get δP/P ∼ v b /V , where V is the initial gas volume, equal to the chamber volume, and v b the volume variation corresponding to the volume of the bubble connected to the injection point. The typical time to empty the chamber can be written

τ * = δV /Q v , where Q v = π(d 2 /4)v
is the volumetric flow-rate through the hole. By using Eq. 12, we get the characteristic time for the pressure drop:

τ * = V P 1 πd 2 ρ a 32δP (13)
where ρ a = 1.2 kg.m -3 is the air density, and P = 10 5 Pa the atmospheric pressure. τ * is of the order of a few milliseconds, compatible with the measurements of the pressure drop (region 3 in Fig. 3, bottom). Note that the drastic pressure decrease due to the hole opening is followed by oscillations observed right after the bubble emission, which correspond to the elastic response of the fluid to the sudden stress imposed by the closing of the bubble walls, after the air release [13].

E. Open channel lifetime

we develop qualitative arguments to estimate the lifetime of the open channel. We note r * the hole radius for which the channel remains open at the end of a cascade, and r * n the associated value of the inner displacement from the Maxwell model developped above (Fig. 6). When the channel from the injection nozzle to the fluid free surface remains open, d 2 r/dt 2 = 0 and we can write, based on Eq. 2:

- 4γ d -G ′ d/2 -r * n d + δP -ρgh = 0 (14)
and Eq. 14 provides the value of r * n after the bubble in the cascade, while Eq. 15 gives the channel radius right after the pressure drop in the system. By substracting Eqs. 14 and 15 and neglecting the capillary forces when the channel if fully open, we find that the channel remains open when the overpressure δP ∼ G ′ . This rough approximation is consistent with the experimental value. Indeed, from Fig. 3, the channel remains open once the overpressure reaches a value close to 50 Pa, of the order of G ′ for an excitation frequency of the order of the frequency of bubbles rising through the fluid (ω ∼ 1 Hz, see Appendix B).

- 2γ r * -G ′ r * -r * n d -ρgh = 0 (15)
After the channel opening, we can write the balance between the main forces at stake, the weight and viscous dissipation, η ṙ/d ∼ ρgh/2. The characteristic lifetime of the open channel can therefore be roughly estimated to τ c ∼ 2η/ρgh, of the order of a second. Experimentally, τ c is of a few seconds. This discrepancy can be explained first, by the rough approximation for the radius dynamic equation; then, by the effective fluid viscosity which, for micelles, can be larger during elongational flowing than the viscosity measured under shear [START_REF] Prud | [END_REF]34].

VI. FROM CASCADES TO RHEOLOGY

The bubble emission characteristics vary continuously inside the cascade: through time, a bubble inside the cascade is emitted with a smaller pressure drop δP , and a longer emission time δt (Fig. 3). Note that for the heuristic model developped in section V B, we considered δt roughly constant and equal to T = δt . In this section, we investigate the variations of both δt and δP for the successive bubbles inside one cascade.

Figure 8 displays the mean value of the overpressure δP , over a bubble release, as a function of the time interval δt over which the same bubble is emitted, for a se-ries of bubble cascades. All data from different cascades, from the same experimental series, have been superimposed. Over different cascades, the data all collapse in the same curve, for a given set of parameters (c, h, Q, V ). The curves remain unchanged when both the volume V and the flow rate Q vary.

Two different regimes can be distinguished, at short and long time scales, respectively (Fig. 8, solid lines). The limit between both regimes defines a pressure and time threshold, respectively δP c and δt c . A statistical study over different volumes (56 ≤ V ≤ 161 mL) and flow rates (0.2 ≤ Q ≤ 1.2 mL/s) gives a constant pressure threshold δP c = 71.6 ± 5.2 Pa. By comparing the characteristic time δt c to the Maxwell characteristic time τ of the fluid (see Appendix), we get the direct relationship

δt c ∼ τ . ( 16
)
This relationship holds true for the three different fluid concentrations where bubble cascades are observed (Fig. 8, inset). Measuring the characteristic time linked with the bubble cascades therefore provides a direct signature on the fluid rheology.

VII. CONCLUSION

Injecting air through a thin layer of micellar fluid displays a wide range of dynamic behavior. We report the existence of a peculiar degassing regime, the bubble cascades regime, for which the air is released via successive bubbles which properties (maximum overpressure and emission duration) vary continuously through time. This regime is observed over a wide range of parameters (air flow-rate Q, chamber volume V and fluid height h). The cascades repeat periodically in time, separated by a few seconds during which a channel remains open between the injection nozzle and the fluid free surface, through which the air degass continuously. Measuring the overpressure at the injection point makes it possible to investigate the different stages: pressure increase in the chamber, bubble formation and fluid flow, bubble emission and pressure drop. We find that the cascade periodicity and maximum overpressure depend neither on Q nor V . The number of bubbles emitted per cascade depends linearly on the injection flow-rate Q. All the different steps of the overpressure evolution can be explained by a simple heuristic model, following the classical Maxwell description of a viscoelastic fluid.

Finally, we underline the interest of such simple experiment. In addition to reporting a new phenomenon (the bubble cascades), we point out that measuring the evolution of the overpressure inside the cascades provides a direct insight into the fluid viscoelastic characteristic time, linked with its rheology. Further work will concentrate on the microscopic behavior of the micelles. In particular, the time over which the rising bubble shears the fluid (typically, 0.2 s) suggests a partial alignment of the micelles (see Fig. 9, Appendix A). Visualizing the fluid by birefringence will make it possible to determine if the micelles are aligned through time by the shear flow generated by the successive bubbles, and how this alignment is linked with the local rheology. We also propose to investigate the effective fluid viscosity under elongation, which can be much larger than the shear viscosity, and could explain the discrepancy between the estimated and measured time for the open channel lifetime (section V E).

Appendix A: Flow curve

The rheology of the CTAB/NaSal mixtures is characterized by measurements on two different rheometers: C-VOR 150, Bohlin Instruments and AR1000, TA Instruments. All measurements are performed with a plateplate geometry. Sand paper is glued to the plates in order to prevent any sliding at the walls (typical rugosity of the order of 1 µm). The following results do not aim at a full rheological characterization of the samples used in our experiments. They only provide the general mechanical behavior of the different mixtures, and a support for the interpretation proposed in section VI. More detailled informations on the rheology of such systems can be found, for instance, in [9,10,[START_REF] Cates | [END_REF]13,29,30,[35][START_REF] Lerouge | Polymer Characterization: Rheology, Laser Interferometry, Electrooptics[END_REF][START_REF] Berret | [END_REF].

The semi-dilute solution behaves as a Newtonian fluid at low shear rate ( γ < γ1 = 2.1 s -1 ), with a viscosity plateau η ∼ 25 Pa, and exhibits non-Newtonian properties for higher shear rate (Fig. 9). The flow curve is classical for micellar fluids under shear [13,[START_REF] Lerouge | Polymer Characterization: Rheology, Laser Interferometry, Electrooptics[END_REF], with a plateau in σ vs. γ between γ1 = 2.1 s -1 and γ2 = 31.6 s -1 . The first transition in the flow curve ( γ1 ) provides an access to the characteristic time associated with the viscoelastic Maxwell model (Fig. 6), τ = 1/ γ1 ∼ 0.5 s. For γ < γ1 , the flow is homogeneous and isotropic; for γ1 < γ < γ2 , shear-bands appear and the flow is strongly inhomogeneous; for γ > γ2 , finally, the flow is homogeneous and nematic, with the micelles aligned in the shear direction [13]. As the typical time for a bubble to rise up the fluid layer is t b ∼ 0.2 s (section V B), we expect to be in the stress plateau and, therefore, to observe a partial alignment of the micelles (section VII). displayed in Fig. 10 provide another estimation of the Maxwell time:

τ = G ′ G ′′ ω (B1)
We find τ ∼ 0.5 s for c = 0.1 mol.L -1 , consistent with the estimation from the previous viscosity measurements (Appendix A).

The average plateau value strongly increases as a func-tion of the fluid concentration (Fig. 11). The values of τ for different fluid concentrations (Fig. 8 inset, section V B) have been estimated from the oscillatory measurements and Eq. B1. Note, finally, that another estimation of the characteristic time τ can be obtained by

τ = η G ′ (B2)
which, again, gives τ ∼ 0.5 s, in agreement with the above estimations.

FIG. 2 .

 2 FIG. 2. Phase diagram of the different degassing regimes depending on the parameters (V,Q) for c = 0.1 mol.L -1 (a) and c = 0.5 mol.L -1 (b) [(•) bubbles, ( ) intermittence, ( ) open channel]. Dashed lines are eyeleads. The gray zone correspond to the space of parameters where bubble cascades are observed [h = 5 mm].

FIG. 3 .

 3 FIG.3. Top: Pressure signal displaying the gas discharge via bubble cascades. δPmax indicates the maximum overpressure reached in the cascades. Middle: (a) The first bubbles in the cascade exhibit a linear pressure increase. The fluid starts flowing only when the bubble is about to be emitted (pressure jump). (b) The last bubbles in the cascade exhibit a curved pressure increase, characteristic of fluid deformation (flowing) and bubble growth. (c) Between two bubble cascades, an open channel connects the air nozzle to the fluid free surface. The overpressure is constant, almost equal to zero (see text). Bottom: Detail of the pressure signal for a bubble formation and emission inside a cascade: (1) linear pressure increase; the dashed gray line corresponds to the linear pressure increase in the fixed volume V , δP = (P0/V )Qt, without any ajustable parameters (see text);(2) bubble formation and growth; (3) the bubble pierces the free surface and the gas is released. Note the pressure oscillations subsequent to the bubble emission [c = 0.1 mol.L -1 , V = 147 mL, Q = 0.65 mL/s].

Q

  FIG. 4. Time interval ∆tcasc separating a sequence 'bubble cascade + hole', as a function of the air flow-rate Q. Inset : Maximum overpressure δPmax reached in the cascades (see Fig. 3). (Symbol, V [mL]): (•, 56), (•, 71), (△, 83), ( , 161). Dashed lines indicate the mean value of all the data.

FIG. 5 .

 5 FIG. 5. (a) Number of bubbles per cascade n as a function of the air flow rate Q. (Symbol, V [mL]): ( , 48), (•, 56), (•, 71), ( , 105), ( , 161). The gray lines correspond to the linear interpolation for each series of experiments (V fixed) [h = 5 mm]. (b) dn/dQ as a function of V . The series reported in (a) display a linear, decreasing relationship up to V = 105 mL. For larger fluid layer height, the slope is almost constant. (Symbol, h [mm]): (•,5); (•,8); (△,18); ( ,27); ( ,35). Inset: dn/dQ as a function of the fluid concentration [V = 147 mL, Q = 0.65 mL/s, h = 5 mm]. No apparent relationship is found.

FIG. 6 .

 6 FIG.6. Maxwell model for the CTAB/NaSal. This simple model consists in an ideal elastic spring (elastic modulus G ′ ) and a dashpot representing the viscous loss (viscosity η). r and rn indicate the displacements generated by the rising bubble and the dashpot shift after the n th bubble, respectively.

FIG. 7 .

 7 FIG. 7. Evolution of δPy as a function of the bubble number in the cascade. In the main part of the cascade, δPy is a linear decaying function of the bubble number n (Symbol, c [mol.L -1 ]): (•, 0.1), ( , 0.07) [V = 147 mL, Q = 0.65 mL/s].

FIG. 8 .

 8 FIG. 8. Mean pressure drop for a bubble vs. time interval for bubble emission. [V =147 mL, h=5 mm, c=0.1 mol.L -1 , Q=0.65 mL/s]. Inset: Variations of δtc with the fluid characteristic time τ , determined here as τ = G ′ /G"ω (see text). From left to right: c = 0.1, 0.07 and 0.05 mol.L -1 .

FIG. 9 .

 9 FIG. 9. The flow curve, shear stress σ vs shear rate γ obtained by increasing γ [c = 0.1 mol.L -1 ]. Inset: Viscosity as a function of shear rate. The arrow indicates the shearthinning behavior [C-VOR 150 Bohlin rheometer, plate-plate geometry, diameter 60 mm, gap 400 µm, waiting time 60 s per point].

FIG. 10 .FIG. 11 .

 1011 FIG. 10. Elastic (G ′ ) and viscous (G ′′ ) modulii as a function of the applied stress for the CTAB/NaSal mixture at c = 0.1 mol.L -1 [oscillation test ω = 1 Hz, AR1000 rheometer, plate-plate geometry].

Appendix B: Elastic and viscous modulii

Figure 10 displays the elastic (G ′ ) and viscous (G") moduli for the fluid c = 0.1 mol.L -1 , for an oscillation test (ω = 1 Hz, typically the period between two bubbles in our experiments). We observe that over a large range of applied stress, the elastic and viscous modulii are constant. The average plateau values of G ′ and G"