J Aubry

P Bydlowski

Energy Storage System Sizing for Smoothing Power Generation of Direct

Keywords: Energy Storage System (ESS), power smoothing, Direct Wave Energy Converter, Supercapacitor, Power Quality

This paper examines the sizing energy storage system (ESS) for energy converter as the SEAREV. The ESS is to insure a smoothed output power profile. First, the output set point power is considered constant and equal to the average during a sea state representative time ( hours). Two others causal method of smoothing will be studied in a second part. We introduce two state of charge (SOC) control strategies in order to maintain SOC between two limits and also two power quality criteria in order to quantify difference of the real output grid power with respect to the desired set point. The life cycle cost of the ESS is analyzed according energy rating in case of supercapacitor technology. The life expectancy is also studied in order to determine a possible replacement of the ESS its life.

Introduction

The minimization of power fluctuations is one of the keys for the development of direct electricity production from fluctuating renewable energies especially in the case of direct wave energy where the output power is far from smooth [ smoothing of direct wave energy converter ( power production due to the summation of the production of each single converter spatially dispersed in a farm has been studied in [START_REF] Molinas | Power Smoothing by Aggregation of Wave Energy Converters for Minimizing Electrical Energy Storage Requirements[END_REF][START_REF] Tissandier | Study of the smoothing effect on the power production in an array of SEAREV wave energy converters[END_REF]. This sm effect could be wisely combined with an individual (WEC scale) or global (farm scale) energy storage system (ESS) in order to improve the quality of the produced energy and also to provide system services This paper examines the sizing problem of an energy storage system (ESS) for a direct wave . The aim of this output power profile. power is considered produced power during a sea state representative time (about two Two others causal method of smoothing will We introduce two state tegies in order to between two limits and also two criteria in order to quantify the power with respect The life cycle economical according to its storage energy rating in case of supercapacitor technology. The life expectancy is also studied in order to determine a possible replacement of the ESS during Energy Storage System (ESS), power smoothing, , Supercapacitor, Power

The minimization of power fluctuations is one of the keys for the development of direct electricity production from fluctuating renewable energies [START_REF] Abbey | Supercapacitor Energy Storage for Wind Energy Applications[END_REF],

wave energy conversion where the output power is far from smooth [START_REF] Baker | Direct drive wave energy converters[END_REF]. The smoothing of direct wave energy converter (DWEC) power production due to the summation of the production of each single converter spatially dispersed ]. This smoothing effect could be wisely combined with an individual (WEC scale) or global (farm scale) energy storage system (ESS) in order to improve the quality of the produced energy and also to provide system services for the grid (furniture of reactive power, balancing, low voltage ride through capability…).

Here, the studied DWEC is the SEAREV SEAREV consists of a completely enclosed floating buoy with an embedded pendular wheel. Excitation forces of the swell on the buoy generate a re motion between the float and the wheel; this oscillating motion is damped to produce energy through a Power Take Off (PTO). For the PTO, we consider the case of direct-drive energy conversion (all in such a way that the power ca hydro-pneumatic storage as in the Pelamis, for example. Furthermore, the control strategy of the PTO (ie viscous damping torque with leveling of the recovered power) is not able to produce smoothed power without an important loss of pro Because of its high cycling capability, we supercapacitor (SC) technology for the ESS. Compared to other electricity storage technologies, SC satisfies better the WEC constraints [START_REF] Murray | Applications of Supercapacitor Energy Storage for a Wave energy Converter System[END_REF].

Hypotheses

In order to simplify, this study, we make a certain number of hypotheses in this work -The sea state is considered to be always the same.

The significant height and the pic period are respectively 3m,8s.

-The Energy Storage System is composed of supercapacitors modules. for the grid (furniture of reactive power, power system balancing, low voltage ride through capability…).

WEC is the SEAREV [START_REF] Babarit | Hydrodynamic optimisation and optimal control of a Wave Energy Control[END_REF][START_REF] Ruellan | Sizing methodology of a wave energy converter[END_REF]. The SEAREV consists of a completely enclosed floating buoy with an embedded pendular wheel. Excitation forces of the swell on the buoy generate a relative motion between the float and the wheel; this oscillating motion is damped to produce energy through a Power Take Off (PTO). For the PTO, we consider the case of drive energy conversion (all-electric chain) [START_REF] Aubry | Minimization of the kWh cost by optimization of an allelectric chain for the SEAREV Wave Energy Converter[END_REF], in such a way that the power cannot be smoothed by pneumatic storage as in the Pelamis, for example. Furthermore, the control strategy of the PTO (ie viscous damping torque with leveling of the recovered power) is not able to produce smoothed power without an important loss of productivity. Because of its high cycling capability, we will study supercapacitor (SC) technology for the ESS. Compared to other electricity storage technologies, SC satisfies ].

Principle of the SEAREV Wave Energy Converter.

In order to simplify, this study, we make a certain mber of hypotheses in this work:

The sea state is considered to be always the same. The significant height and the pic period are

The Energy Storage System is composed of supercapacitors modules. The nominal voltage of the ESS, corresponding to a State Of Charge (SOC) of 1, is 1200V. SOC is 0 when the voltage is 0V. The electrical characteristics (serial resistance and capacity) are constant.

-The only losses considered are those in supercapacitors. The losses in power electronics converters and the grid are not considered because they are second-order in this preliminary study.

First ESS requirement estimation

A first order estimation of the storage for an AWS farm has been studied in considered power production profile was si with a strong sea state, for a park with 30 AWS grid power profile was deduced from the production profile by a low-pass filtering. The energy ra ESS was the difference between the maximum and the minimum value of the time-integral of the difference between the production (input) power and the grid (output) power.

Figure 2: Energetic representation of

We can use this method in our case shows one example of production power profile for one SEAREV on a sea state Hs=3m, Tp=8s levelling control (of 1 MW). The average profile is equal to 270kW and it is considered as the output grid power. Then, the power in the ESS is equal to the difference of this two power profile As shown in Fig. 4, the difference between the maximum and the minimum value of the integral of the difference between the two power profiles is 45MJ The losses in power electronics are not considered order in this preliminary estimation storage energy rating for an AWS farm has been studied in [START_REF] Daratha | A firstorder energy storage requirements estimation for an Archimedes Wave Swing Park[END_REF]. The profile was simulated for a park with 30 AWS. The grid power profile was deduced from the production pass filtering. The energy rating of the the difference between the maximum and the integral of the difference between the production (input) power and the grid Energetic representation of the ESS. in our case. The Fig. 3 production power profile for one SEAREV on a sea state Hs=3m, Tp=8s and with power average of power is equal to 270kW and it is considered as the in the ESS is equal to the difference of this two power profiles.

ܲ ாௌௌ ሺݐሻ ൌ ܲ ௗ ሺݐሻ െ ܲ ௗ ܧ ாௌௌ ሺݐሻ െ ܧ ாௌௌ ሺݐ ൌ 0ሻ ൌ න ܲ ாௌௌ ௧

ௗ ሺݐሻ

(1)

න ܲ ாௌௌ ሺݐሻ݀ݐ (2)
of a production power profile for the 4, the difference between the maximum and the minimum value of the integral of the difference between the two power profiles is 45MJ (12.5kWh). It corresponds to the stored energy in a capacitance of 84F between 1200V and 600V (Note:ܧ ൌ ½ܥሺܸ ௫ ଶ െ ܸ ଶ ሻ).

Figure 4: Integral of the ESS power profile

The power production profile of the simulated on the basis of a profile. This profile is based on spectrum [START_REF] Babarit | Optimal latching control of a wave energy device in regular and irregular waves[END_REF]. It gives the amplitude of each frequency component while the phase of each component is randomly drawn uniformly between 0 and 2 a same sea state (here Hs=3m, T some temporal differences in the power production profile while the average produced relatively constant. Fig. 5 shows for 200 the sensitivity of the necessary to insure a constant grid power profile average produced power.

We want to point out the significa random draw of initial phases components, and then the one of the sea elevation and power production temporal profile at the same e content, on the sizing of the ESS. The minimum value is 5.7kWh (energy stored in 38F between 1200V and 600V) and the maximum is 18.3kWh (energy stored in 122F between 1200V and 600V).

SOC control strategies

In the previous paragraph, the energy rating of the ESS is determined to insure every time the grid power (here our power considered as constant equal to the power).

Another way to tackle this problem to introduce a State Of Charge (SOC) control Indeed, for a given value of storage may not be possible to insure every time the 45MJ International Conference on Ocean Energy, 6 October, Bilbao (12.5kWh). It corresponds to the stored energy in a capacitance of 84F between 1200V and 600V ሻ).

Integral of the ESS power profile ܲ ாௌௌ ሺݐሻ.

The power production profile of the Fig. 3 is particular sea elevation profile. This profile is based on the Pierson Moskovitz the amplitude of each frequency the phase of each component is randomly drawn uniformly between 0 and 2π.Then, on a same sea state (here Hs=3m, Tp=8s), we could have some temporal differences in the power production average produced power kept shows for 200 random draws, necessary energy rating of the ESS power profile equal to the the significant influence of the phases for each frequency one of the sea elevation and power production temporal profile at the same energy on the sizing of the ESS. The minimum value is 5.7kWh (energy stored in 38F between 1200V and 600V) and the maximum is 18.3kWh (energy stored in 122F between 1200V and 600V).

Sensitivity of the random draw of phases for a sea state on the minimum needed energy storage rating to insure a constant grid power.

SOC control strategies

In the previous paragraph, the energy rating of the every time the set point of ower set point is firstly considered as constant equal to the average produced Another way to tackle this problem of ESS sizing is a State Of Charge (SOC) control strategy. storage energy rating, it nsure every time the power set point ܲ ௦௧ especially when the SOC is equal to 0 (0V) or 1 (here 1200V). Then the grid power must dissent from the power set point according a SOC control strategy.

First SOC control strategy

Teleke [START_REF] Teleke | Control Methods for Energy Storage for Dispatching Intermittent Renewable Energy Sources[END_REF] has introduced a rule based control of the SOC. The idea of this strategy is to insure the set point during the maximum of time. We illustrate this strategy on Fig. 7 and Fig. 8. The grid power is kept constant equal to the set point during the maximum of time. But when SOC limitations are reached, the grid power becomes equal to the produced power and then its large fluctuations are not smoothed. In this example the production power profile is this one of Fig. 3. The energy rating of the ESS is 6.3kWh and the initial SOC is 65%.

ܲ ௗ ൌ ܲ ௦௧ ݂݅ ቐ ܥܱܵ ∈ ]0.3; 1[ ܥܱܵ = 1 & ܲ ௗ < ܲ ௦௧ ܥܱܵ = 0.3 & ܲ ௗ > ܲ ௦௧ ܲ ௗ = ܲ ௗ ݂݅ ൜ ܥܱܵ = 1 & ܲ ௗ > ܲ ௦௧ ܥܱܵ = 0.3 & ܲ ௗ < ܲ ௦௧

Power quality criteria

In order to quantify the difference of the grid power with respect to the set point, we introduce power quality criteria. The first one is equivalent to the one used by Teleke. We denote it Δ

௦ Δ ௦ = 1 ܶ න หܲ ௦௧ -ܲ ௗ ห݀ݐ ் (3)
The second that we will use is based on the rms value of the error between the power set point and the power fed to the grid. We denote it Δ ௦ :

Δ ௦ = ඨ 1 ܶ න ൫ܲ ௦௧ -ܲ ௗ ൯ ଶ ݐ݀ ் (4)
This second criterion penalizes more strongly, the important deviations from the set point. Values of these two criteria in the presented example are respectively 47kW and 143kW.

Normalization of these criteria

In the following, we will normalize these criteria with respect to the ones calculated without energy storage. Without storage, values of ߂ ௦ and ߂ ௦ would have been respectively 247kW and 310kW. This normalization allows quantifying the gain of use of an ESS on these criteria. For the example above, these two normalized criteria are respectively 19% and 46%.

Second SOC control strategy

We introduced another strategy for the control of the SOC. This strategy avoids so far as possible large and quick fluctuations of the grid power from the set point thanks to the direct relation between the grid power and the SOC. We illustrate this strategy on Fig. 10 and Fig. 11. For this second example, the two criteria are respectively 25% (61kW) and 35% (107kW) for the same energy rating of the ESS (6.3kWh). The power quality is worse according to Δ ௦ criterion but better by the second Δ ௦ . The calculation of the power quality criteria Δ ௦ and Δ ௦ must be done with care. As in the previous chapter, there is an important influence of the random draw of initial phases for each frequency components of the sea elevation profile. There is also an influence of the initial SOC. That is the reason why in the following, Δ ௦ and Δ ௦ will always be the average of 50 production power profile random draws and 11 initial states of charge.

Power quality criteria versus storage energy rating

Energy losses versus storage energy rating

Another interesting aspect is to calculate average losses due to the serial resistance of the supercapacitor ESS. We consider here that the ESS is composed of several BMOD0063 P125 Maxwell modules (63F 125V). The serial resistance of one module is 18mΩ [START_REF] Bmod | Module 63F 125V[END_REF]. The total serial resistance of the ESS is :

ܴ ௦ = 18. 10 ିଷ ܰ ௦௦ ܰ ( 5
)
Where ܰ ௦௦ is 10 because we consider 1200V as the maximum voltage of the ESS. ܰ depends on the desired storage energy rating. Fig. 14 represents the evolution of average losses with respect to the storage energy rating.

ܲ ாௌௌ ሺݐሻ = ܸ ாௌௌ ሺݐሻܫ ாௌௌ ሺݐሻ (6)

ܲ ௦௦ ሺݐሻ = ܴ ௦ ܫ ாௌௌ ሺݐሻ ଶ (7)
With an average produced power of 270kW, the efficiency on cycle of the supercapacitor ESS varies between 96% and 99% according to the storage energy rating (∝ ܰ ).

ߟ = 1 - 〈ܲ ௦௦௦ 〉 〈ܲ ௗ 〉 (8)

Cost analysis on life cycle with taking into account of energy losses

With an expected life of 20 years, the cost of these losses must be compared to the initial cost of the ESS. We consider an initial cost of the ESS of about 20k€/kWh (actual price for small serie production). The feed-in tariff of the produced energy is supposed to be fixed at 15c€/kWh (French value). Fig. 15 shows the cost on life cycle with taking into account of energy losses in storage device.

ܥ ሺ݇€ሻ = 20. ܧ ௧ ሺܹ݇ℎሻ + 15.10 ିହ . ܲ ௦௦ ሺܹ݇ሻ. 20ሺݕሻ.8760ሺℎሻ [START_REF] Daratha | A firstorder energy storage requirements estimation for an Archimedes Wave Swing Park[END_REF] On life cycle of 20 years, there is a minimum cost for about 5kWh of storage energy rating (corresponding to ܰ = 4ሻ. It means that all the solutions below this value (except 0kWh which has no cost) are not interesting because they exhibit a higher cost and higher power quality criteria (Δ ௦ and Δ ). It may be noted that the two strategies are equivalent in terms of economic cost on life cycle. Their differences are mainly on the two power quality criteria.

The optimal energy value is dependent of the feed-in tariff of the electrical energy and the initial cost (€/kWh) of the ESS.

Taking into account of aging effect

In the precedent section, the cost on life cycle has been calculated without taking account of possible replacements during the life duration of the ESS. In order to verify this assumption, we examine the aging effects on life cycle. We calculate this aging on the basis of a life expectancy law depending on temperature and voltage [START_REF] Diab | Comparison of the Different Circuits Used for Balancing the Voltage of Supercapacitors: Studying Performance and Lifetime of Supercapacitors[END_REF].

ܮ ௫ = ܥ ଵ exp ൬ ܷ ܥ ଶ + Δܶ + ܶ ܥ ଷ ൰ (10)
The constants C1, C2 and C3 are deduced from Fig. 16. ܥ ଵ = 1,8.10 ଵ ;ሻݏݎܽ݁ݕሺ ܥ ଶ = -0,1278 ܸ ିଵ ; ܥ ଷ = ܥ°55,31 ିଵ We assume that the thermal time constant of supercapacitor modules is such that the temperature rise is equal to average losses multiplied by the thermal resistance (the thermal capacitance is such that the thermal inertia of modules is assumed infinite). According to Maxwell's datasheet, the thermal resistance of a 63F 125V module is equal to .ܭ°230,0 ܹ ିଵ . The temperature rise is:

Δܶ = 0,032. < ܴ ௦ ܫ ாௌௌ ଶ > ܰ ௦௦ ܰ ሺ°ܭሻ (11)
A degradation factor ߛ can be used to compute the aging on an operation cycle. The variation of this factor over a cycle is

݀ߛ = න 1 ܮ ௫ ሺܷሺݐሻ, ܶሻ ݐ݀ ் (12)
In our case we calculate this degradation factor over one year and then apply this degradation on the capacitance and serial resistance according to the following equations:

ܴ ௦ = ሺ1 + ߛሻܴ ௦,௧ (13
) ܥ = ሺ1 -0,2ߛሻܥ ௧ (14) Every year, the degradation factor is the average of 50 production power profile (depending on the random draw of phases cf Sec. 4.3) and 11 initial states of charge. The process is iterated every year until the degradation factor becomes one or the life duration reaches 20 years.

Fig. 17 shows life expectancy calculated for the two SOC control strategies. Above ܰ = 3, the life expectancy exceeds 20 years. The ambient temperature ܶ is fixed at 30°C (Note that it plays a very important role). Fig. 18 shows the degradation factor after the end of life (or after 20 years). We can see that the second SOC control strategy deteriorate less the ESS than the first strategy.

The calculation of cost on life cycle made in the previous section is the valid when ܰ > 3. Below this value, we should have introduce the additional cost due to one or more replacements of the ESS. In this calculation, voltage has been limited to 1200V (2,5V per supercapacitor cells). If this voltage would have been limited to 1250V, the life expectancy would have been less of 20 years for most values of the storage energy rating (cf. Fig. 19) except for the second strategy which exhibits a life expectancy of 20 years for values of ܰ higher than 5.

Influence of the grid power set point

Up to there, the set point of the output power fed to the grid was the the average produced power. Actually, this set point cannot be known precisely even with a sea state forecast. Teleke has introduced in his simulations a prediction error of 10% around the real average power for the set point of each dispatch hour [START_REF] Teleke | Control Methods for Energy Storage for Dispatching Intermittent Renewable Energy Sources[END_REF]. In the following, we will see the influence, in terms of power quality criteria and economical life cycle cost, of a causal determination of the grid power set point (i.e. without prediction of the incoming power thanks to a sea state forecast).

Low-pass filtering

Firstly, the set point can be deduced from a low-pass filtering of the instantaneous production power profile. Fig. 20 illustrates this set point. In this example the time constant of the low pass filter is fixed at 200s and the initial SOC is 0.5 for a storage energy rating of 6.3kWh. The second SOC control strategy is used. In order to compare with the previous values of power quality criteria, we calculate in the following Δ ௦ and Δ according to the average power (in the formulas we replace ܲ ௦௧ by ܲ ). Then for this example, the criteria are respectively 30% and 34%.

Piecewise constant power

Another method to determine the set point for the grid power is to set a constant power during a certain period. This power set point is equal to the average production power during the preceding period. This method is causal. We illustrate this on Fig. 24. The period is equal to 314s. All others parameters are the same as the previous example.

Conclusion

In this paper we presented and reviewed methods for the sizing of an Energy Storage System with two SOC control strategy and two power quality criteria. It was applied to the smoothing of the power produced by a Direct Wave Energy Converter (as the SEAREV).

A cost analysis on life-cycle was conducted in order to find the best values of storage energy rating. Storage energy losses and ageing effect has been analysed on a life cycle of 20 years. According to the power quality criterion, one or the other of the strategies is the best.

We observe that the smoothing performances are relatively the same with or without short-term (~hour) prediction of the power production.

Further work can be done for a farm of wave energy converters. In this context, the smoothing due to the ESS can be wisely combine with the aggregating effect due to the summation of the production of each single DWEC spatially dispersed.

Another aspect that could be in-depth is the influence of the leveling of the produced power on the sizing of the ESS. This problem will be more important if the cost of power electronic converter is taken into account.

3

  rd International Conference on Ocean Energy 1 Energy Storage System Sizing for Smoothing Power Generation of Direct Wave Energy Converters , P. Bydlowski 1,2 , B. Multon 1 , H. Ben Ahmed 1 and B. Borgarino

Figure 1 :

 1 Figure 1: Principle of the SEAREV Wave Energy Co

Figure 3 :

 3 Figure 3: Example of a production power profile for the SEAREV.

3

  rd International Conference on Ocean Energy 2 the ESS, corresponding to a State Of Charge (SOC) of 1, is 1200V. SOC is 0 when the voltage The electrical characteristics (serial resistance and capacity) are assumed to be The only losses considered are those in

Figure 5 :

 5 Figure 5: Sensitivity of the random draw of phases for a same sea state on the minimum needed energy storage rating of the ESS to insure a constant grid power

Figure 6 :

 6 Figure 6: Rules and diagram of the first SOC control strategy.

Figure 7 :

 7 Figure 7: Example of the first SOC control strategy.

Figure 8 :

 8 Figure 8: SOC profile corresponding to the Fig. 7.

Figure 9 :

 9 Figure 9: Rules and diagram of the second SOC control strategy.

Figure 10 :

 10 Figure 10: Example of the second SOC control strategy.

Figure 11 :

 11 Figure 11: SOC profile corresponding to the Fig. 10.

Figure 12 :

 12 Figure 12: Δ ௦ vs energy rating of the ESS.

Figure 13 :

 13 Figure 13: Δ ௦ vs energy rating of the ESS.

Fig. 12

 12 Fig.12and 13 show the decrease of the two power quality criteria with respect to the ESS energy rating. According to the criterion, one of two strategies is

Figure 14 :

 14 Figure 14: Average losses vs energy rating of the ESS.

Figure 15 :

 15 Figure 15: Cost on life cycle of the ESS versus its energy rating.

Figure 16 :

 16 Figure 16: Lifetime of a supercapacitor versus voltage and temperature [13].

Figure 17 :

 17 Figure 17: Life expectancy versus energy rating of the ESS.

Figure 18 :

 18 Figure 18: Degradation factor after the end of life.

Figure 19 :

 19 Figure 19: Life expectancy versus energy rating of the ESS for a maximum voltage of 1250V.

Figure 20 :

 20 Figure 20: Example of low-pass filtering for the determination of the grid output power set point for a filter time constant of 200s.

Fig. 21 to

 21 Fig. 21 to Fig. 23 represents the evolution of the power quality criteria and losses versus the storage energy rating for different values of the filter time constant, only for the second SOC control strategy. For values of time constant greater than 500s, the performance of the ESS and his SOC control strategy are relatively similar.

Figure 21 :

 21 Figure 21: Evolution of Δ ௦ for different value of filter time constant and the second SOC control strategy.

Figure 22 :

 22 Figure 22: Evolution of Δ ௦ for different value of filter time constant and the second SOC control strategy.

Figure 23 :

 23 Figure 23: Evolution of average losses for different value of filter time constant and the second SOC control strategy.

Figure 24 :

 24 Figure 24: Example of piecewise average for the determination of the grid output power set point for a step period of 314s.

Fig. 25

 25 Fig. 25 and Fig. 26 represent the evolution of the power quality criteria versus the storage energy rating

Figure 25 :

 25 Figure 25: Evolution of Δ ௦ for different value of step period and the second SOC control strategy.

Figure 26 :

 26 Figure 26: Evolution of Δ ௦ for different value of step period and the second SOC control strategy.

System Sizing for Smoothing Power Generation

	Borgarino 2

The nominal voltage of International Conference on Ocean Energy, 6 October, Bilbao Energy Storage

SATIE, ENS Cachan Bretagne, CNRS, UEB av Robert Schuman, F-35170 Bruz judicael.aubry@bretagne.ens-cachan.fr

LMF, Ecole Centrale de Nantes, CNRS 1 rue de la Noë, F-44000 Nantes

Acknowledgements

This work was supported by the Bretagne regional council under Grant ARED SEAREV_2.

Authors would like to thank Jacques Courault (former engineer at AREVA T & D) for his advice on the quality criteria of energy.


Sébastien Manneville
email: sebastien.manneville@ens-lyon.fr

Annie Colin

Gilles Waton

François Schosseler
email: schossel@ics.u-strasbg.fr

Wall slip, shear banding, and instability in the flow of a triblock copolymer micellar solution

Keywords: numbers: 83.60.-a, 83.80.Qr, 47.50.+d, 43.58.+z Wormlike micelles, copolymer, wall slip, shear banding, elastic instability

come

Wall slip, shear banding, and instability in the flow of a triblock copolymer micellar solution

I. INTRODUCTION

Triblock copolymers PEO-PPO-PEO based on poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) have interesting self-assembly behavior in aqueous solutions due to their amphiphilic character [1][2][3]. PPO is soluble in water only at low concentrations and for molecular weights smaller than about 4000 [4]. As a homopolymer, PEO is insoluble in water at low and high temperatures. The binodal appears as a closed loop in the temperature-concentration phase diagram, and the miscibility region shrinks with increasing molecular weight [5][6][7]. Thus the hydrophilic/hydrophobic balance of these copolymers can be tuned by changing the PEO/PPO molar ratio and the temperature. Above the critical micelle concentration (cmc) and the critical micellar temperature (cmt), the copolymer chains aggregate and form micelles with a PPO core and a PEO corona [START_REF] Mortensen | for a recent review see[END_REF]. The cmc and the cmt both depend on the total molecular weight [1][2][3] and on the concentration of added components like, e.g., monovalent salts [9][10][11].

The structure of the micelles has been elucidated by us-ing scattering techniques [START_REF] Mortensen | for a recent review see[END_REF]12]. Above the cmt and the cmc, spherical micelles are obtained with the aggregation number being an increasing function of the temperature [13,14]. At low temperature many water molecules are still present in the PPO core but are progressively expelled as the temperature increases. Yet the radius of the core increases and there is a growing entropic contribution from the stretched PPO blocks to the free energy per micelle. Hence the aggregation number cannot increase indefinitely and a spheroidal or cylindrical shape can be expected [13,14] if the size of the PEO blocks is not too large compared to that of the PPO block [15]. This is basically the same packing parameter criterion as the one used for conventional surfactants [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF]. Clear experimental evidence for the growth of elongated micelles has been provided by scattering techniques for the commercial Pluronic copolymers P85 (M w = 4600 g mol -1 , PEO weight fraction ≈ 0.5) [START_REF] Mortensen | [END_REF][18][19] and P84 (M w = 4200 g mol -1 , PEO weight fraction ≈ 0.4) [20][21][22][23].

At low concentration of the copolymer micelles, the temperature-induced micellar growth has a huge influence on the viscoelastic properties. Typically, for a Pluronic P84 weight fraction of 0.04 in 2 M NaCl brine, the zero-shear viscosity increases by a factor 10 5 when the temperature increases from 30 • C to 40 • C [24]. Below 30 • C the solution contains only dilute spherical mi-celles with a negligible effect on the viscosity. As the aggregates grow into wormlike micelles, the solution enters the semidilute regime of living entangled polymers and the zero-shear viscosity jumps accordingly. Above about 41 • C, the solvent becomes progressively poorer for the PEO corona and the aggregates become more compact as the cloud point is approached [23].

The non-Newtonian behavior under steady shear is also strongly influenced by micellar growth. The steadystate flow curves (shear stress σ vs shear rate γ) measured in the concentric cylinder geometry display a different behavior below and above a transition temperature T c ≈ 37.7 • C at weight fraction 0.04 [24]. Below this temperature the shear stress is a monotonic increasing function of the imposed shear rate, while above T c the flow curve exhibits two increasing branches separated by a stress plateau [24]. The presence of this plateau is often interpreted as the hallmark of shear-banded flows, i.e., of the existence of a non-uniform velocity gradient across the gap of the rheometer. The occurrence and the stability of non-uniform shear flows have been studied theoretically for about twenty years . The existence of a plateau in the flow curve has been indeed observed in a number of experimental systems, mainly in conventional surfactant solutions . However only a few studies have been able to measure directly the velocity profile in the rheometer gap using NMR velocimetry [52,53,56,57,64], particle image velocimetry [55,68], heterodyne dynamic light scattering [60,70] or ultrasonic speckle velocimetry (USV) [65]. The recently developped USV technique allows one to measure velocity profiles with a temporal resolution of 0.02 to 2 s depending on the shear rate [71]. Thus the stability of inhomogeneous velocity profiles can be investigated and the transition to unstable flow regimes can be studied.

In Ref. [24] two of us used small-angle light scattering to study Pluronic P84 solutions in brine sheared in the concentric cylinder geometry. Strong indications for shear-induced demixing below T c and for shear banding above T c were found but no direct evidence from velocity measurements was provided. The aim of the present paper is to use the USV technique to simultaneously measure the rheology and velocity profiles of the same solutions in experimental conditions very close to those of Ref. [24]. We study the system at different temperatures, below and above the transition temperature T c (where a plateau appears in the flow curve). We are thus able to follow the evolution of the velocity profiles as a function of the applied shear rate and of the temperature, and to identify different flow regimes that turn out to be more complex than those usually observed in shearbanded flows.

The paper is organized as follows. Experimental details about the samples and about the rheological and velocimetry setups are given in Sect. II. Results obtained at two different temperatures are described in Sect. III. We show that (i) at T = 37.0 • C the bulk flow remains homogeneous and Newtonian-like, although significant wall slip is measured at the rotor that can be linked to an inflexion point in the flow curve, and (ii) at T = 39.4 • C the stress plateau in the flow curve is associated to stationary shear-banded states characterized by two high shear rate bands close to the walls and a very weakly sheared central band, together with large slip velocities at the rotor. In both cases, on the high shear branch of the flow curve, the flow becomes highly non-stationary and twodimensional or even possibly three-dimensional. These results are further discussed in Sect. IV where interpretations of wall slip, three-band structure, and instability are proposed in light of recent theoretical models and experimental results on shear banding.

II. EXPERIMENTAL SECTION

A. Sample preparation

We used without further purification the commercial Pluronic P84 supplied by BASF. The total molecular weight is 4200 g mol -1 and the nominal stoechiometry is PEO 19 -PPO 43 -PEO 19 . With 2 M added NaCl, microcalorimetric measurements [21] show that, for a 0.04 weight fraction, the cmt and the cloud point are at about 2 • C and 43 • C, respectively. The onset of micellar growth is about 30 • C [21,23]. The transition to the semidilute regime of wormlike micelles is estimated to occur at about 35 • C, when the zero-shear viscosity has increased by a factor 10. Stock solutions at 0.04 g cm -3 were prepared and stored in the dark at 4 • C for further use. The acoustic contrast for USV measurements was provided by monodisperse micrometric polystyrene latex particles (Microparticles GmbH, diameter 9 µm) added to the copolymer solution at a weight fraction of 0.01 [71]. To avoid settling of the particles, the solution was first seeded at room temperature. After the dispersion of the latices by vigorous stirring, the solution was brought to about 36 • C to increase its zero-shear viscosity. In these conditions, the latex particles remained suspended in the solution for the whole duration of the experiments, typically a few hours.

B. Rheology

Rheological measurements were performed with two home-built setups based on commercial rheometers. They have been described elsewhere [24,71]. In both cases the rheometers are stress-controlled but have a feedback loop that allows strain-controlled measurements. In the temperature range 30 • C-40 • C, the temperature regulation of the solution is straightforward and the evaporation is minimized thanks to the use of solvent traps.

In the case of the rheometer coupled with small-angle light scattering [24], the Couette device is operated by a RS1 rheometer (Thermo Haake). The inner rotor has an outer radius R 1 = 26 mm and the outer stator an The setup coupled with USV has very similar characteristics [71]. The inner rotor is operated by an AR1000 rheometer (TA Instruments). The rotor (R 1 = 24 mm) and the stator (R 2 = 25 mm) are made out of Plexiglas. The stator is surrounded by water whose temperature is kept constant to within ± 0.1 • C. The temperature is measured in the circulating bath and the offset with the actual temperature in the Couette cell is estimated to be about 0.5-1 • C in the range of temperatures probed in these experiments (36 < T < 41 • C).

Thus the experimental conditions of the shear flow can be well enough reproduced in the two setups to allow for a comparison of the results. However these conditions are not strictly identical. In particular, the different materials of the Couette devices might influence wall slip and small differences in the actual temperature of the samples will play an important role since temperature controls crucially the micellar growth.

We checked for the influence of added latex particles on the rheology with the setup of Ref. [24]. Figure 1 compares the flow curve at a given temperature T = 40.2 • C and the evolution of the zero-shear viscosity with temperature measured in the copolymer solution without latices to the same data obtained in the solution seeded with latices. Although the global temperature evolution is well reproduced in the two samples, the solution with added particles has a much higher zero-shear viscosity for a given temperature in the growth regime (see inset of Fig. 1). At T = 40.2 • C, close to the maximum of viscosity, the difference in the zero-shear viscosity is less marked. The difference in viscosity even vanishes progressively as the shear rate is increased so that the value and the boundaries of the stress plateau are almost unaffected by the presence of latices. Thus, although the behavior in the non-Newtonian regime, which is of interest in this paper, appears to remain unchanged by the addition of latex particles, the zero-shear viscosity cannot be used as an internal temperature calibration to correct for small differences in actual temperatures. Nevertheless, a qualitative comparison can still be achieved between the previous [24] and the present results.

In the following all the results were obtained with the setup of Ref. [71]. The combined rheological and velocimetry experiments were performed by imposing a constant shear rate to the solutions for increasing shear rate steps. The values reported for the stress correspond to its average value across the gap.

C. Ultrasonic speckle velocimetry

The USV setup has been described at length in Ref. [71]. This technique is based on backscattering from acoustic impedance inhomogeneities suspended in a fluid medium. When the liquid contains a lot of scatterers per unit volume, the backscattered signal results in an ultrasonic speckle built from the interferences of all the backscattered waves. To discriminate between different scatterers in space and to measure velocity profiles in a flow, a solution is to use short acoustic pulses that generate a series of backscattered echoes. The arrival times of the echoes are then directly linked to the positions of the scatterers along the acoustic beam. Cross-correlation of speckle signals corresponding to successive pulses yields the displacement of the scatterers projected along the acoustic axis. By using pulses with central frequencies larger than 20 MHz, velocity profiles can be measured in complex fluids sheared between two plates separated by a gap e = 1 mm with a spatial resolution of about 40 µm. The pulse repetition frequency is tuned according to the rotation speed of the rotor so that the maximum displacement between two successive pulses remains smaller than half the acoustic wavelength.

Velocity profiles were recorded after an equilibration time of 30 minutes. For γ < 15 s -1 , an "individual" velocity profile corresponds to an average over 1000 successive pulses. For a given number of pulses, the total averaging time is inversely proportional to the applied shear rate [71]. Typically, in the present experiments, for a small shear rate of γ ≈ 0.1 s -1 , an "individual" velocity profile was measured in 250 s. For γ ≥ 15 s -1 , we used averages over 20 bursts of 20 pulses. The period of the burst sequence was chosen so that the total averaging time for an individual velocity profile was equal to 1 s whatever γ ≥ 15 s -1 . The "time-averaged" velocity profiles shown below in Figs. 3 and6 profiles for the lowest shear rates to several hundreds as soon as γ ≥ 15 s -1 . For the purpose of the discussion below, it is important to keep in mind that USV measures the projection of the velocity vector along the acoustic axis. The reader is referred to Ref. [71] for full technical details.

III. RESULTS

A. T = 37.0 • C : homogeneous flow with wall slip

Figure 2 shows the flow curve measured by the rheometer at T = 37.0 • C (hereafter referred to as the "engineering" flow curve). As reported earlier [24], it shows an inflexion point but no stress plateau. At low shear rates, a zero-shear viscosity of 0.25 Pa s is measured. Figure 3 shows the corresponding time-averaged velocity profiles v(x) measured for different applied shear rates, where x is the distance from the rotor. The stator is located at x = e = R 2 -R 1 = 1 mm. For the whole range of shear rates investigated here, the time-averaged velocity profiles correspond to those expected for a Newtonian fluid and the time-averaged velocity gradient is constant across the whole gap.

However, for applied shear rates higher than about 3 s -1 , there is a noticeable difference between the velocity v 0 = γe imposed by the rotor and the true velocity of the fluid at the rotor (x = 0), i.e., wall slip occurs at the rotor. A quantitative estimate of the slip velocity is obtained through a linear fit ṽ(x) of the experimental data close to the inner wall over x = 0.08-0.20 mm. The slip velocity v s at the rotor is then simply given by v s = v 0 -ṽ(0). As seen in Fig. 4, wall slip does not set in abruptly for a well defined shear rate but the slip velocity rather appears to increase smoothly over the whole range of applied shear rates. Note the error bars on the first data points which indicate an experimental uncertainty on the determination of v s of about 0.01 mm s -1 . As long as γ < 3 s -1 , wall slip thus remains hardly detectable and is buried in experimental noise. The relative slip v s /v 0 increases until it reaches about 30% for γ > 100 s -1 (see inset of Fig. 4). For the purpose of further discussion we notice that no wall slip is observed at the stator (x = 1).

Finally, as long as the imposed shear rate remains smaller than 300 s -1 , the flow is stationary as shown by the very small standard deviation of the velocity measurements (of the order of the marker size in Fig. 3). For shear rates higher than 300 s -1 , the flow reaches a new, unstable regime, whose signature is the considerably larger statistical fluctuations of the instantaneous velocity profiles around their average value (velocity profiles not shown, see Sect. III B 5 for more details on this unstable flow regime). The flow profile becomes non-stationary although its mean shape for long time averages remains close to Newtonian in the bulk with about 30% wall slip at the rotor.

In Fig. 2 we marked by different background grey values the regions corresponding to Newtonian behavior (N), noticeable wall slip (S), and unstable flow (U). Figure 2 also shows the "true" flow curve, i.e., the shear stress σ plotted against the shear rate corrected for wall slip γtrue = γv s /e hereafter called the "true" shear rate (open symbols). Here and in the following the limits of the different flow regimes are defined according to the true shear rates.

B. T = 39.4 • C : complex banded flow
We now turn to measurements performed at a higher temperature where the flow curve was previously shown to present a stress plateau [24]. We first give a general overview of the rheological and flow behaviors and then describe more precisely the various flow regimes observed at T = 39.4 • C.

Rheological and flow behavior overview

As shown in Fig. 5, the flow curve at T = 39.4 • C exhibits an almost flat stress plateau at σ ≈ 1.2 Pa over more than two decades separating a Newtonian branch (N) for γ 0.03 s -1 (η 0 ≈ 39 Pa s) and a second increasing branch for γ 5 s -1 corresponding to unstable flow (U). From velocity profile measurements, the stress plateau can be divided into two parts: a first regime (S) for 0.03 < γ < 0.3 s -1 where only wall slip is observed, and a new regime (B) for 0.3 < γ < 5 s -1 where time-averaged velocity profiles reveal a three-band structure together with wall slip (see Fig. 6). Velocity profiles in regimes (N), (S), and (U) (not shown) are widely similar to those measured in the corresponding regimes at T = 37.0 • C. In particular, the high shear unstable branch (U) is again characterized by very large temporal fluctuations (δv/v 50%) around an average homogeneous velocity profile with significant wall slip at the rotor. The rheological and flow behaviors were found to be qualitatively the same for all investigated temperatures 5. γ = 0.5 (•), 1 (•), 1.5 ( ), and 2 s -1 ( ). The error bars correspond to the standard deviation of the measurements. Note that the corresponding slip velocities can be easily estimated from the applied shear rate and from the fluid velocity measured at the rotor using vs = γev(x = 0).

T ≥ 38 • C.

Rheological signals

Figure 7 shows that in regimes (N), (S), and (B), steady state is reached within about 10 minutes. A stress overshoot in the temporal evolution of the shear stress σ(t) marks the onset of nonlinear behavior for regimes (S), (B), and (U). In regime (U), the instability of the flow is revealed by the significant increase of the temporal fluctuations of σ(t): typically δσ/σ ≈ 20% whereas . The insets show that the feedback loop of the rheometer keeps the shear rate γ(t) constant to within 2% for γ = 0.02 s -1 , 0.6% for γ = 0.2 s -1 , and 0.3% for γ = 0.5 and 7 s -1 .

δ γ/ γ ≤ 0.3%. The Fourier spectrum of σ(t) exhibits a single characteristic frequency corresponding to that of the rotor rotation in an otherwise 1/f behavior. The evolution of the relative fluctuations of the applied shear rate depicted in the insets of Fig. 7 reflects the increasing difficulty to control the shear rate as it becomes very low. We could not detect any significant temporal correlation between the fluctuations of the applied shear rate and those of the measured stress.

Slip velocity

Once again no slip is detected at the stator for T = 39.4 • C. The evolution of the slip velocity at the rotor with the imposed shear rate is shown in Fig. 8. It strikingly differs from that observed at T = 37.0 • C (Fig. 4). After a sharp increase in regime (S), v s almost saturates in the shear banding regime (B). This behavior corresponds to a decrease of the relative slip v s /v 0 as γ is increased (see inset of Fig. 8) which starts at the onset of shear banding when v s /v 0 ≈ 0.8. The same criterion was fulfilled at all temperatures where shear banding could be observed (T ≥ 38 • C). Note that for T = 37.0 • C the ratio v s /v 0 remains always smaller than 0.3 although the absolute value of the slip velocity can be orders of magnitude larger at the lower temperature (see Fig. 4).

Shear banding regime

The most striking feature of Fig. 6 is the presence of three shear bands in the gap. Figure 9 shows the location of the two interfaces across the gap and the local shear rate values within the bands as a function of the true shear rate. Such data are easily extracted from piecewise linear fits of the time-averaged velocity profiles in regime (B). The width of the high shear band close to the rotor is seen to increase with the shear rate whereas no systematic trend can be given for the band at the rotor. Plotting the same data as a function of the local shear stresses at the interfaces does not provide evidence for any "stress selection" rule either. Moreover the central band remains very weakly sheared for most of the imposed shear rate values. The shear rate inside the band at the rotor is always at least twice as large as that in the band at the stator. These results clearly point to a complex shear banding scenario which will be discussed below in Sect. IV B. Note that temporal fluctuations of the velocity field remain small in regime (B) but not quite negliglible: standard deviations in Fig. 6 are δv/v ≈ 5-10% typically. A detailed study of these fluctuations is left for future work.

Unstable flow regime

Examples of instantaneous velocity profiles measured within about 1 s in regime (U) are plotted in Fig. 10. They clearly indicate that the flow is strongly nonstationary. The spatio-temporal representations of the velocity profiles v(x, t) shown in Fig. 11 illustrate the drastic difference between the stationary flow observed in regime (B) and the unstable flow regime (U) where huge temporal fluctuations of v(x) are recorded. Finally we shall argue in Sect. IV C that the flow becomes cannot remain purely tangential in regime (U).

IV. DISCUSSION

A. Wall slip

In complex fluids like colloidal suspensions, the possible break-down of the no-slip boundary condition has been shown for a long time [72,73]. Depending on the net interaction between the walls and the suspended colloids, different mechanisms are involved. For neutral or repulsive walls, wall depletion occurs and creates a lubricating layer of the suspending fluid close to the wall. The equilibrium concentration profile at rest can be further modified by the flow, in particular when the particles are soft and deformable [74]. This also occurs in polymer solutions [75]. For attractive walls, partly covered with adsorbed colloids, wall slip could be expected to be rather uncommon. One notable exception is the case of entangled polymer solutions and melts when entanglements between an adsorbed layer and the bulk system are present. Such phenomena have been investigated both theoretically and experimentally [76][77][78][79][80][81]. Near field laser velocimetry experiments on poly(dimethylsiloxane) (PDMS) melts sheared between sliding plates grafted with PDMS chains have shown three different slip regimes [78][79][80][81]. At low and high plate velocities, the slip velocity v s is proportional to the wall velocity v 0 , with a quasi plug flow, v s ≈ v 0 , at high velocities. In between these two limits, a nonlinear friction regime is observed, where v s increases more rapidly than linearly with v 0 . In this intermediate regime, the slip length b = ev s /(v 0 -v s ), i.e., the distance at which the velocity profile extrapolates to the wall velocity, increases almost linearly with the slip velocity (b ∝ v α s with α = 0.8-1.2). These results have been explained in terms of a dynamic decoupling between the surface and the bulk polymer chains [76,77]. More precisely, in the case of grafted chains, such a decoupling arises from the competition between the strongly entanglement-dependent frictional force that stretches the chains and the entropic elastic force that recoils them. The resulting steady-state extension of the surface and bulk chains increases with the shear rate until the stretched chains are close to disentangle. This is the onset of the intermediate slip regime which lasts until the shear rate becomes larger than the characteristic frequency at which the chains can reform new entanglements. Then the surface and the bulk chains become fully disentangled, the friction decreases by two orders of magnitude, and the slip velocity is high so that v s ≈ v 0 .

At the lowest temperature investigated in the present work, T = 37.0 • C, the behavior of the slip velocity as a function of shear rate is strongly reminiscent of the one reported for PDMS melts in contact with PDMS grafted surfaces [80]. Indeed, as shown by the lines in Fig. 4, a weak slip regime, with a behavior obscured by experimental noise, and a strong slip regime, where v s ∝ γ, are sep-arated by a region where the slip velocity increases faster than linearly with the imposed shear rate (v s ∝ γ1.7 ). In the strong slip regime, however, the slip velocity never reaches the limiting value v s = v 0 = γe (indicated by a dashed line in Fig. 4) but rather saturates at v s ≈ 0.3v 0 . The corresponding slip length is shown in Fig. 12. Due to the large uncertainty at small shear rates, we cannot give a reliable value for the slip length b in the weak slip regime, which cannot be distinguished from perfect Newtonian behavior using the present measurements. Still, although the power law behavior in our data is significantly different (b ∼ v α s with α ≃ 0.5), the slip lengths of several hundred of microns observed in the intermediate regime are close to those reported on PDMS (compare, e.g., Fig. 7 of Ref. [80] to Fig. 4 of the present paper).

The above similarity between our measurements at T = 37.0 • C and previous results on PDMS melts sheared between PDMS grafted plates prompts us to interpret wall slip in our experiments as a result of the dynamic decoupling scenario described above. Of course, in our case, micelles are not grafted at the cell walls. Nevertheless we may assume that PEO-PPO-PEO wormlike micelles adsorb on the surfaces. Indeed at least one experimental study has clearly demonstrated the adsorbtion of wormlike micelles on a substrate for poly(styrene)poly(isoprene) diblock copolymers in heptane, a selective solvent for poly(isoprene) [82]. No similar results have been reported for wormlike micelles made of PEO-PPO block copolymers. However it has been shown that spherical micelles formed from diblock PEO-PPO copolymers close to the bulk cmc adsorb on hydrophilic or hydrophobic silica substrates while retaining their shape [83,84]. Thus in principle we can expect similar effects in our system, namely adsorbtion of wormlike micelles on the walls.

For adsorbed polymers, the possibility of tearing off of the surface chains due to the shear stress should also play a role and could explain the difference in the scaling exponents and in the saturation value of v s . To the best of our knowledge, no theoretical prediction is available in the case of adsorbed chains. Moreover, in the case of adsorbed surface wormlike micelles, the scission of the micelles should also provide a mechanism for disentanglement. If the breaking time τ b is much smaller than the reptation time τ R , one expects to recover in a first approximation τ ∼ (τ b τ R ) 1/2 for the characteristic relaxation time [85]. Thus the transition to the strong slip regime is expected to occur for γτ 1. In our case, however, the bulk relaxation time τ at T = 37.0 • C is about 1 s [24] so that the onset of strong slip corresponds to γτ ≈ 100. Such a discrepancy is easily explained by the fact that adsorbed micelles should have a much smaller contour length left available to entangle with the bulk micelles. A more thorough investigation of the proposed scenario for wall slip at T = 37.0 • C is beyond the scope of this paper.

On the other hand, the v s vs γ data for the higher temperature T = 39.4 • C display a plateau-like behav-ior in the intermediate regime between weak and strong slip regimes (see Fig. 8). The difference with the lower temperature is even more striking when the slip length is considered (see Fig. 12). At T = 39.4 • C, after a large initial increase, b suddenly drops as the system enters the shear banding regime. Such a decrease was also observed in a lyotropic lamellar phase in the shear banding regime [86]. This behavior was interpreted in terms of a decrease of the thickness h of the lubricating layers that were assumed to be composed of perfectly aligned membranes. Indeed if one notes η s the viscosity of the fluid inside the slip layer and if one assumes that h ≪ e then stress continuity imposes η s v s /h = η(v 0 -v s )/e, where η is the bulk viscosity. In other words, one has h = bη s /η. Assuming η s to remain constant and equal to the solvent (brine) viscosity, a decrease of h in the shear banding regime was attributed to solvent migration from the walls to the shear-induced low viscosity fluid close to the rotor [86]. In the present case of PEO-PPO-PEO wormlike micelles, a constant η s seems like a rather strong assumption since one expects progressive disentanglement in the slip layer unless the strong slip regime is reached. An alternative explanation could be that the rheology of the slip layer itself presents a non-monotonic behavior leading to instability. Such a behavior was recently observed and linked to a stick-slip instability in nanotribology experiments on confined surfactant layers adsorbed on the mica surfaces of a surface force apparatus [87,88]. Finally we shall see below that the slip regime (S) in Figs. 5 and8 is also compatible with a very thin high shear band unresolved by our velocimetry setup, so that a definite interpretation of wall slip at T = 39.4 • C is still an open question.

To summarize our discussion on wall slip, the measurements performed at T = 37.0 • C and presented in Sect. III A are broadly consistent with the behavior expected for the dynamic decoupling at the interface between bulk and adsorbed surface chains. This supports the hypothesis that adsorbed wormlike micelles could be involved in the observed slip behavior. At T = 39.4 • C, the slip phenomena are markedly different, in correlation with the observation of shear banding in the bulk at this temperature, although the interplay between slip and shear banding remains unclear.

B. Shear banding

Shear-induced separation of complex fluids flow into bands of different viscosities has been predicted and interpreted for almost two decades in terms of an underlying non-monotonic flow curve which leads to the presence of a stress plateau at σ * between two critical shear rates γ1 and γ2 in the rheological experiments [25,26,49]. In the most simple scenario for shear banding, two or more shear bands of viscosities η 1 = σ * / γ1 and η 2 = σ * / γ2 coexist along the stress plateau and the proportion α of the highly sheared fluid is given by the so-called "lever rule" α = ( γ -γ1 )/( γ2 -γ1 ): as the imposed shear rate γ is increased from γ1 , the high shear bands progressively invade the whole sample until γ = γ2 . The plateau value was predicted to be σ * = 0.67G 0 (with G 0 the plateau modulus) [26], which was indeed observed in some experiments on wormlike micelles [49][50][51]. In the concentric cylinder geometry, the (small) stress inhomogeneity results in the presence of only two bands with the high shear band close to the inner cylinder. Moreover the lever rule was shown to hold in CPCl-NaSal semidilute wormlike micellar solutions [60,63,68]. More refined models involved constitutive equations based on the frame-invariant Gordon-Schowalter derivative [START_REF] Larson | Constitutive Equations for Polymer Melts and Solutions[END_REF]. In particular, the inclusion of diffusive terms in such models were shown to be a necessary ingredient to obtain stress selection in planar geometries [30,[33][34][35]. The possibility of non-uniform concentration in the gap was introduced by the means of a two-fluid description and the coupling between flow and concentration [37,[39][40][41]. This last model should be well adapted to describe our experimental system since it incorporates the flowconcentration coupling that can explain butterfly isointensity patterns parallel to the flow direction, measured earlier at T 37 • C [24], and allows for a phase separation in the system.

The flow curve of Fig. 5 looks typical of a simple shear banding phenomenon with σ * ≈ 1.2 Pa, γ1 ≈ 0.03 s -1 , and γ2 ≈ 5 s -1 . With a plateau modulus G 0 ≈ 1.8 Pa [24], the observed stress plateau is even nicely predicted by the most simple toy model, σ * = 0.67G 0 [26]. Note also that apparent wall slip in regime (S) may also be interpreted in the framework of a simple shear banding scenario. Indeed, if the lever rule is valid, the width of the high shear band remains smaller than the USV spatial resolution δ 0 ≈ 40 µm as long as γ < γ1 +δ 0 ( γ2 -γ1 )/e ≈ 0.23 s -1 . Thus due to the huge difference between η 1 ≈ 40 Pa s and η 2 ≈ 0.2 Pa s, the high shear band may be too thin to be detected over a significant fraction of the stress plateau, which would be consistent with the slip regime observed for γ = 0.03-0.3 s -1 .

However the velocity profiles measured along the stress plateau at T = 39.4 • C and shown in Fig. 6 clearly point to a much more complex shear banding scenario than predicted by even the most sophisticated models [37,[39][40][41] for the following reasons. (i) Stationary states with three shear bands are recorded over the whole stress plateau. (ii) Strong wall slip is detected at the rotor. (iii) The various shear rates inside the bands do not remain constant as the applied shear rate is increased. (iv) Even if the true shear rate is considered, the lever rule does not hold. To the best of our knowledge, even if transient three-banded velocity profiles were already reported in wormlike micelles [65], this is the first evidence for such profiles in the steady state. This observation should undoubtedly be confronted to the very recent model of Ref. [48]. Based on a bead-spring mechanism with non-affine motion [43], this model incorporates the same basic ingredients as the two-fluid diffusive Johnson-Segalman model [37,[39][40][41] and was shown to predict various flow structures depend- ing on the boundary conditions. If the conformation flux across the walls is set to zero (Neumann condition), twobanded steady states are found whereas when the micelles are aligned at the walls (Dirichlet condition) the model predicts a stationary three-band structures very similar to that observed in our experiments. Moreover, in the latter case, strong depletion is observed at both walls. Such a model may provide some keys to the unusual shear banding reported in the present study and more specifically to points (i) and (ii) above. Note that a similar approach was proposed in the framework of yield stress fluids and that similar results with multi-banded states depending on boundary conditions were obtained in the fluidity model of Ref. [START_REF] Picard | [END_REF]. Finally, concerning issues (iii) and (iv), let us note that a complex shear banding scenario with multiple successive transitions where the local shear rates do not remain constant and the lever rule does not hold was also reported very recently for solutions of self-assembled polymers [70]. Contrary to what we observed in regime U (see below), we did not notice any evidence for two-or three-dimensional flows in regime B.

C. Unstable flows

In all cases investigated in this work, the flow strongly fluctuates in time as soon as the high shear branch of the flow curve is reached. The individual velocity profiles of Fig. 10 even show portions with positive slope. More precisely, in spite of scatter due to differentiation, the local shear rate shown in Fig. 13 andcomputed

from γ(x) = v r - ∂v ∂r , (1)
where r = R 1 + x, becomes clearly negative for x ≈ 0.1-0.2 mm in one case and x ≈ 0.4-0.5 mm in the other. Since a negative local shear rate is not compatible with material stability, we conclude that the flow cannot remain purely tangential.

Indeed, as recalled in Sect. II C, USV only provides a measurement of the velocity vector projected along the acoustic axis. So far our interpretation of the velocity profiles has relied on the assumption that the flow is purely tangential. Here the only way to account for a negative apparent shear rate is to include a non-zero radial component v r in the velocity field so that the experimentally measured velocity actually reads

v(x) = v θ (x) + v r (x) tan θ , (2)
where v θ is the tangential component of the velocity and θ the angle between the acoustic axis and the radial direction. Since θ ≈ 15 • [71], the effect of radial velocity is amplified by a factor of about 4 and, depending on v r (x), Eq. ( 2) may lead to "negative local shear rates" when used in Eq. ( 1). Moreover the detection of velocities larger than the wall velocity v 0 close to the rotor is another strong indication for a non-zero radial component of the velocity (see Fig. 10). Note that, for γ = 7 and 15 s -1 , the rotation periods of the rotor are about 21 s and 10 s respectively, so that individual velocity profiles (recorded in about 1 s) only correspond to a fraction of the rotor revolution. Thus the present experiments do not allow us to discriminate between a two-dimensional non-axisymmetric flow, i.e. where v θ and v r are functions of both r and θ but the vertical velocity v z remains zero, and a fully three-dimensional flow with all three velocity components being non-zero. As a possible cause for instability, we may first rule out any inertial effect such as the Taylor-Couette instability. Indeed, even on the high shear branch of the flow curve at T = 39.4 • C where η ≈ 0.2 Pa s and γ ≈ 10 s -1 , the Reynolds number is Re = ρ γe 2 /η ≈ 0.05 and the Taylor number is T a = (e/R 1 )Re 2 ≈ 10 -4 so that the flow can be considered as inertialess. In the Newtonianlike case observed at T = 37.0 • C, one has η ≈ 10 mPa s for the highest effective shear rate γ ≈ 10 3 s -1 . Although these are only rough estimates, one gets T a ≈ 400 which remains smaller than the critical Taylor number 1712 for a Newtonian fluid.

We rather propose to interpret the observed unstable flows as a result of an elastic instability [91][92][93]. Such an instability is driven by negative normal stress differences that may cause a streamline to become unstable with respect to radial perturbations. As shown by theoretical analyses for viscoelastic fluids, a purely elastic instability may develop when (e/R 2 ) 1/2 W i > M c , where W i = γτ is the Weissenberg number and M c some critical number of order 1 whose exact value depends on the constitutive model assumed for the fluid. For instance, one finds M c ≈ 5.9 for both the Upper-Convected Maxwell model [92] and the Oldroyd-B model when the viscosity of the solvent is half that of the polymeric part [94].

Previous rheological measurements on the PEO-PPO-PEO Pluronic P84 in brine have shown that the relaxation time increases from τ ≈ 1 s at T = 37.0 • C to τ ≈ 25 s at T = 40.0 • C [24]. Assuming M c ≈ 5.9, these measurements lead to critical shear rates γc ≈ 30 s -1 at T = 37.0 • C and γc ≈ 1.5 s -1 at T = 39.4 • C for the onset of the elastic instability. The velocimetry experiments reveal non-stationary unstable flows above 300 s -1 and 5 s -1 for T = 37.0 • C and T = 39.4 • C respectively, well above the predicted thresholds for the elastic instability. Our observations are thus consistent with an elastic instability, although the early stages of such an instability, i.e., an initially toroidal stationary flow with Taylor-like vortices that undergo further bifurcation towards periodic states as the rotation speed is increased [91,93], should in principle be observed for γ = 30-300 s -1 and γ = 1.5-5 s -1 respectively. At T = 37.0 • C Newtonian-like velocity profiles are recorded all along the flow curve and no stationary toroidal flow was observed. At T = 39.4 • C, due to the complexity of the shear-banded velocity profiles, we cannot rule out the presence of a small timeindependent toroidal flow along the stress plateau. Moreover the exact viscoelastic properties of the fluid at high shear rates are unknown. In particular, the relaxation time of the sheared fluid is probably much smaller than that measured in the linear regime so that the instability threshold may be pushed towards higher shear rates. Finally the exact value of M c may differ significantly from 5.9 so that the present discussion can only remain rather qualitative.

V. CONCLUSIONS

We have investigated the flow of a PEO-PPO-PEO Pluronic P84 solution in brine using simultaneous velocity profile and rheological measurements. This study completes the small-angle light scattering measurements performed earlier on the same system [24]. It confirms the presence of shear-banding and instability in the flow of this triblock copolymer micellar solution. When the rheological flow curve is an increasing function of the shear rate, i.e., for T < T c ≃ 37.7 • C, the velocity profiles remain Newtonian but a transition from weak to strong wall slip is detected at the rotor. The evolution of the slip velocity was compared to previous results on PDMS and favors an interpretation in terms of micelles adsorbed at the wall that progressively untangle from the bulk fluid. For T > T c the flow curve displays a stress plateau that was correlated to the presence of shear banding. However our data clearly reveal a complex shear banding scenario with wall slip, stationary three-band structure and varying local shear rates in the bands. The presence of three shear bands in the gap was discussed in light of recent theoretical predictions accounting for the influence of boundary conditions on bulk behavior. In connection with wall slip, the possibility that micelles get strongly aligned at the walls could be at the origin of a three-band structure. Finally, in all cases, the flow was shown to become strongly non-stationary and either non-axisymmetric or even possibly three-dimensional at high shear rates. Such unstable flows are consistent with the occurence of an elastic instability on the high shear branch of the flow curve although information on the structure and rheology of the fluid in this regime is missing.

FIG. 2 :

 2 FIG. 2: Flow curve at T = 37.0 • C. The full symbols (•) correspond to the engineering data σ( γ) while the open symbols ( ) represent the "true" flow curve σ( γtrue) obtained once wall slip is taken into account (see text). The solid lines are σ = 0.25 γ in regime (N) and σ = 0.012 γ in regime (U).

FIG. 3 :

 3 FIG.3: Time-averaged velocity profiles at T = 37.0 • C. The standard deviation is of the order of the marker size. The dotted lines are the velocity profiles expected in a Newtonian fluid without wall slip. Inset: dimensionless data vs/v0 vs x/e. Top : γ = 0.5 (•), 1 (•), and 2 s -1 ( ). Bottom : γ = 20 (•), 50 (•), and 100 s -1 ( ).

FIG. 4 :

 4 FIG.4: Time-averaged slip velocity vs at the rotor at T = 37.0 • C vs applied shear rate γ. The error bars correspond to the standard deviation of the measurements. The solid lines have slopes 1.7 and 1 (see discussion in Sect. IV A). The dashed line is vs = v0. For γ < 1 s -1 , slip velocities are too small to show in logarithmic scales. Inset: relative slip vs/v0 at the rotor vs applied shear rate γ.

FIG. 5 :FIG. 6 :

 56 FIG. 5: Flow curve at T = 39.4 • C. The full symbols (•) correspond to the engineering data σ( γ) while the open symbols ( ) represent the "true" flow curve σ( γtrue) obtained once wall slip is taken into account. The solid lines are σ = 39 γ in regime N and σ = 0.65 γ0.55 in regime U.

FIG. 7 :

 7 FIG. 7: Rheological time series σ(t) recorded at T = 39.4 • C for four different imposed shear rates. (a) Newtonian regime (N): γ = 0.02 s -1 . (b) Wall slip regime (S): γ = 0.2 s -1 . (c) Shear banding regime (B): γ = 0.5 s -1 . (d) Unstable regime (U): γ = 7 s -1. The insets show that the feedback loop of the rheometer keeps the shear rate γ(t) constant to within 2% for γ = 0.02 s -1 , 0.6% for γ = 0.2 s -1 , and 0.3% for γ = 0.5 and 7 s -1 .

FIG. 8 :

 8 FIG.8: Time-averaged slip velocity vs at the rotor at T = 39.4 • C vs applied shear rate γ. The error bars correspond to the standard deviation of the measurements. The dashed line is vs = v0. Inset: relative slip vs/v0 at the rotor vs applied shear rate γ.

FIG. 9 :FIG. 10 :

 910 FIG. 9: Analysis of the three-band velocity profiles recorded in regime (B) at T = 39.4 • C. Positions of the interfaces inside the gap as a function (a) of the applied shear rate and (b) of the true shear rate. Local shear rates in the highly sheared bands at the rotor (•) and at the stator (•) and in the central weakly sheared band ( ) as a function (c) of the applied shear rate and (d) of the true shear rate.

FIG. 11 :

 11 FIG. 11: Spatio-temporal representation of the velocity profiles v(x, t) at T = 39.4 • C for (a) γ = 2 s -1 and (b) γ = 15 s -1 (color online).

FIG. 12 :

 12 FIG. 12: Slip length b as a function of slip velocity vs at the rotor for T = 37.0 • C (•) and T = 39.4 • C (•). The solid line is a power law with exponent 0.47.

FIG. 13 :

 13 FIG. 13: Local shear rate γ(x) computed from the individual velocity profiles recorded at T = 39.4 • C for γ = 15 s -1 and shown in Fig. 10.

Acknowledgments

The authors wish to thank L. Bécu for technical help on the experiments as well as the referees for fruitful comments on the manuscript.