Julie Diani
email: julie.diani@ensam.eu

Pierre Gilormini

Gerry Agbobada

Experimental study and numerical simulation of the vertical bounce of a polymer ball over a wide temperature range

Keywords: Polymers, Rebound, Viscoelasticity, Thermomechanics

The dependence to temperature of the rebound of a solid polymer ball on a rigid slab is investigated. An acrylate polymer ball is brought to a wide range of temperatures, covering its glass to rubbery transition, and let fall on a granite slab while the coefficient of restitution, duration of contact, and force history are measured experimentally. The ball fabrication is controlled in the lab, allowing the mechanical characterization of the material by classic dynamic mechanical analysis (DMA). Finite element simulations of the rebound at various temperatures are run, considering the material as viscoelatic and as satisfying a WLF equation for its time-temperature superposition property. A comparison between the experiments and the simulations shows the strong link between viscoelasticity and time-temperature superposition properties of the material and the bounce characteristics of the ball.

Introduction

The vertical bounce of a ball dropped on a flat rigid slab is a classical problem in Solid Mechanics, with obvious applications in sports, for instance ( [START_REF] Cross | The bounce of a ball[END_REF], [START_REF] Lewis | The dynamic behavior of squash balls[END_REF], [START_REF] Collins | Viscoelastic impact characterisation of solid sports balls used in the Irish sport of hurling[END_REF], among others). For symmetry reasons, this problem is equivalent to the frontal collision of two identical balls with exactly opposite velocities, unless such phenomena as acoustic emission are considered (as in [START_REF] Akay | Acoustic radiation from the elastic impact of a sphere with a slab[END_REF], for example), and thus is connected to a wide range of applications (from the collision of fruits [START_REF] Dintwa | Finite element analysis of the dynamic collision of apple fruit[END_REF] up to the stability of planetary rings [START_REF] Hertzsch | On lowvelocity collisions of viscoelastic particles[END_REF], for instance). A long list of scientific papers has followed the celebrated work of Hertz [START_REF] Hertz | Ueber di Berührung fester elastischer Körper[END_REF] on elastic balls, which took into account more complex mechanical behaviors like viscoelasticity ( [START_REF] Kuwabara | Restitution coefficient in a collision between two spheres[END_REF] and many others) or viscoplasticity [START_REF] Ismail | Impact of viscoplastic bodies: dissipation and restitution[END_REF]. A plastic behavior was considered essentially in the "dual" problem of the vertical bounce of a rigid ball on a deformable material, especially for the interpretation of the Shore hardness test on metals [10]. Polymers were also considered early in this dual problem ( [START_REF] Jenckel | Die Bestimmung von Relaxationszeiten aus der Rückprallelastizität[END_REF], [START_REF] Tillett | A study of the impact of spheres on plates[END_REF], [START_REF] Hunter | The Hertz problem for a rigid spherical indenter and a viscoelastic half-space[END_REF]), with attempts to correlate the coefficient of restitution to the damping factor of the viscoelastic material ( [START_REF] Lifshitz | Some experiments on anelastic rebound[END_REF], [START_REF] Calvit | Experiments on rebound of steel balls from blocks of polymers[END_REF], [START_REF] Raphael | Correlation of rebound tester and torsion pendulum data on polymer samples[END_REF]).

The present paper deals with the bounce of a viscoelastic solid homogeneous ball, which is simpler in a sense than the hollow (tennis, squash, for instance) or solid but heterogeneous (golf, baseball, etc.) balls used in sports. By contrast, a special effort is put here on a complete control of the ball, which is manufactured in the lab, and on a systematic identification of the mechanical behavior of its constitutive material. Moreover, the simple question "Does a ball rebound higher or lower when temperature is increased?" is addressed, and the influence of temperature from the glassy to the rubbery state is investigated. To the authors knowledge, this is the first study of the coefficient of restitution of a polymer ball over such a wide temperature range since the early experimental study of Briggs [START_REF] Briggs | Methods for Measuring the Coefficient of Restitution and the Spin of a Ball, Research Paper RP1624[END_REF] on golf balls. Jenckel and Klein [START_REF] Jenckel | Die Bestimmung von Relaxationszeiten aus der Rückprallelastizität[END_REF], Calvit [START_REF] Calvit | Experiments on rebound of steel balls from blocks of polymers[END_REF], Raphael and Armeniades [START_REF] Raphael | Correlation of rebound tester and torsion pendulum data on polymer samples[END_REF], Robbins and Weitzel [18], among others, covered very wide temperature ranges for the bounce of a rigid ball on polymer blocks instead. By contrast, recent measurements on baseballs ( [START_REF] Drane | Characterization of the effect of temperature on baseball COR performance[END_REF], [START_REF] Nathan | Corked bats, juiced balls, and humidors: the physics and cheating in baseball[END_REF]) or squash balls [START_REF] Lewis | The dynamic behavior of squash balls[END_REF], for instance, explored narrow temperature do-mains only. In addition, finite element simulations are also performed in this study to correlate the experimental results to the simulations using the identified mechanical behavior. This numerical technique is preferred here, for better reliability, to more or less simplified models ( [START_REF] Hunter | The Hertz problem for a rigid spherical indenter and a viscoelastic half-space[END_REF], [START_REF] Kuwabara | Restitution coefficient in a collision between two spheres[END_REF], [START_REF] Hertzsch | On lowvelocity collisions of viscoelastic particles[END_REF], [START_REF] Nagurka | A mass-spring-damper model of a bouncing ball[END_REF]) of the system.

The remainder of the paper is organized as follows. Section 2 presents how the material was synthesized, its mechanical characterization, how the balls were produced, and the experimental setup used to measure the coefficient of restitution, contact duration, and impact force. Section 3 explains how the finite element simulations were run. Finally, Section 4 details the experimental results and discusses the comparison with numerical simulations.

Material and methods

Synthesis of the material and production of the balls

The material is an acrylate network obtained by photopolymerization, with a composition given elsewhere [START_REF] Safranski | Effect of chemical structure and crosslinking density on the thermo-mechanical properties and toughness of (meth)acrylate shape memory polymer networks[END_REF]. The mix consists of 90% in mol of benzyl methacrylate (BMA) with 10% in mol of 550 g/mol molar weight poly(ethylene glycol) dimethacrylate (PEGDMA) used as crosslinker and 0.5% of 2-dimethoxy-2-phenylacetophenone (DMPA) as photoinitiator. Once mixed for 10 minutes at room temperature, the polymer solution was either injected between two glass slides in order to obtain rectangular plates of 1.3 mm thickness or injected into hollow spherical glass beads to obtain balls of 25 mm diameter. For polymerization, plates and beads were exposed to 365 nm ultra-violet (UV) light in UV chamber UVP CL-1000 during 50 minutes and 2 hours respectively. Notes that due to the transparent color of the glass and the mix, UV radiation penetrates the balls completely. The DMPA is a poor photoinitiator, which helps preventing early polymerization when preparing the mix, but which also increases the required duration of UV exposure for complete polymerization. In order to assess the complete polymerization of the balls, one ball has been cut into two identical hemispheres revealing a complete solid state.

Mechanical characterization of the material

A complete characterization of the material linear viscoelasticity and of its time-temperature superposition property were carried out in order to possibly relate these thermomechanical properties of the material with its bounce ability. After identifying the material glass transition temperature to 46 • C by differential scanning calorimetry when applying a heating ramp of 10 • C/min, plane samples were submitted to dynamic mechanical analysis in order to characterize the material viscoelasticity. For this purpose, rectangular samples of 40 × 12 mm 2 were cut from the 1.3 mm-thick plates. Theses samples were submitted to dynamic frequency sweep torsion tests on a MCR 502 rheometer from Anton Paar. The applied strain was set to 0.1% for frequencies ranging from 0.01 to 10 Hz. Frequency sweeps were performed at different temperatures, from 37 to 70 • C with 3 • C increments, and three frequency sweep tests were run at each temperature. Figure 1 shows the material storage modulus and damping factor dependences to temperature that were obtained.

In order to reach the material viscoelasticity over a wider range of frequencies, the time-temperature superposition principle was applied, which provides the material viscoelasticity master curves by mere horizon- tal shifts of the measured data in Fig. 1. Figure 2 shows the material master curves for the chosen temperature 2 shows also that this viscoelastic behavior is reproduced well by a generalized Maxwell model. In order to reach an accurate representation of the material viscoelasticity, 40 relaxation times τ i have been considered, with a fixed ratio of 1.87 between consecutive values. The fitting procedure of Weese [START_REF] Weese | A regularization method for nonlinear ill-posed problems[END_REF] was used to obtain the associated set of shear moduli G i , which applies a Tikhonov nonlinear regularization that is much more relevant than a mere least squares method. The stiffness of the elastic branch corresponding to elasticity at low frequencies was measured to G ∞ = 2 MPa, and the relaxation times and relaxation moduli are listed in table 1. As a consequence, the relaxation function in shear writes as

T ref = 64 • C. Figure
G(t) = G ∞ + 40 ∑ i=1 G i exp(-t/τ i ) .
(1) Finally, to characterize the material time-temperature superposition, the horizontal shift factors a T ref applied while building the master curves were found to satisfy the Williams-Landel-Ferry (WLF) equation [START_REF] Williams | The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids[END_REF]:

log 10 (a T ref (T )) = -C 1 (T -T ref ) C 2 + (T -T ref ) . ( 2
)
Figure 3 shows the shift factor values and their fit by the WLF equation ( 2) for

T ref = 64 • C, C 1 = 17 and C 2 = 124 • C.

Experimental setup

The ball was held by suction with a vacuum pump and was allowed to fall straight from a fixed height of h 0 = 81.5 cm onto a polished 14 × 14 × 10 cm 3 granite slab by turning the pump off. A simple calculation shows that air friction can be neglected during this free fall under gravity, and the velocity of the ball at impact is therefore

v 0 = √ 2gh 0 ≃ 4 m/s.
Two types of measures were carried out. First, several successive rebounds of the ball on the slab were allowed, and the noise was recorded with a microphone connected to a computer. This gave access to the time between the first two bounces (∆t) by a subsequent simple Matlab [25] treatment of the sound signal, and the upward ball velocity v after the first contact was deduced as v = g∆t/2 by neglecting air friction. This acoustic technique (see for instance Aguiar and Laudares [START_REF] Aguiar | Listening to the coefficient of restitution and the gravitational acceleration of a bouncing ball[END_REF], among many others), leads to the coefficient of restitution (COR) e = v/v 0 very simply.

During impact, the ball remains in contact with the slab during a time t c that varies according to the ball material behavior especially. In order to measure the contact duration of the ball during the first bounce, a thin piezoresistive sensor was stuck on the impacted surface of the slab in a second series of tests. Acoustic measurements confirmed that the COR was not significantly modified by the sensor laying between the ball and the slab during impact, and therefore a possible perturbation induced by the sensor could be neglected. A Flexiforce A401 sensor from Tekscan was integrated in a signal conditioning circuit using an Arduino Uno microcontroller board. In order to increase the recording frequency, the output analog signal was saved on a Dso5014A oscilloscope from Agilent Technologies. The Flexiforce sensor provides an easy technical solution to measure the contact duration. It can also give access to the contact force, but a low saturation threshold prevents the absolute measure of large forces, and only the shape of the contact force profile can be studied in such conditions.

The COR and the contact duration were estimated at various temperatures from room temperature to 180 • C. For each temperature, the polymer ball was placed in a thermal chamber for 15 minutes prior to testing. The temperature of the thermal chamber was controlled by a thermocouple run with myPCLab, and at least six rebound tests were performed at each temperature. An additional temperature of 0 • C was also considered for the COR, by cooling the ball in ice before testing.

3 Simulation technique

Finite element procedure

The Abaqus/Explicit finite element code [27] has been used in this study to perform the simulations of the experimental tests. This software applies an explicit integration scheme to solve highly nonlinear systems with complex contacts under transient loads. It offers a large choice of elements and wide material modeling capabilities. Concerning viscoelasticity, both the generalized Maxwell model and the WLF time-temperature superposition relation are available. Moreover, the finite strain extension of the generalized Maxwell model given by Simo [START_REF] Simo | On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects[END_REF] is also included, which uses the same relaxation times and shear moduli pairs as those defining the shear modulus master curve for small strains. Therefore, the 40 pairs listed in Table 1 were used. In addition, a reference elastic behavior is required, which was taken as neo-Hookean hyperelastic at high temperatures, with a shear modulus of G ∞ = 2 MPa and a bulk modulus of B = 3.11 GPa. The bulk modulus is assumed independent of temperature and, combined with the shear modulus, the above value leads to a Poisson's ratio close to 0.5 in the rubbery state and equal to 0.41 in the glassy state, consequently. The finite element simulations include finite strain.

The ball was meshed as shown in Fig. 4. Axial symmetry allows one half of the ball section to be modeled, and 2278 axisymmetric four-node elements with reduced integration were generated, with smaller element sizes in the vicinity of the contact area. Since inertia effects are essential in these simulations, a mass density of 1.21 g/cm 3 was used, as deduced from the weight and volume measured on the balls. The slab was modeled as a fixed rigid surface with a no-penetration condition imposed. This was allowed by preliminary simulations where the slab was meshed as a 10 cmhigh cylinder with a radius of 8 cm (in order to keep axisymmetry and have the same slab volume as in the experiments), using standard elastic constants for granite. No difference with the case of a mere rigid surface was found, which is due to a very large contrast of elastic properties between the polymer ball and the granite slab (the Young's modulus is multiplied by about 50 between granite and the glassy polymer, for example), but this equivalence would not apply for a steel ball, for instance. Moreover, no perceptible difference was obtained when the ball-slab friction conditions were varied, at either low or high temperature, from no friction at all to no slip, which is consistent with contact forces being essentially normal to the interface, and no friction was applied consequently.

At the beginning of the calculations, the ball is located slightly above the rigid surface, it has a uniform downward velocity of 4 m/s and its temperature also is uniform. Because of the initial position of the ball and because of its low weight (less than 10 grams), the simulations did not show any difference when gravity was taken into account or neglected. Therefore, no gravity loading was applied. The contact duration t c was defined by a non-zero reaction force applied to the fixed rigid surface. The COR e was computed as the ratio between the (upward) velocity of the ball center of mass at the contact end (i.e., when the reaction force returned to zero) and its initial velocity.

Preliminary simulations

The first preliminary simulation corresponds to the bounce of the polymer ball when its elastic glassy behavior is used. In these conditions, the COR should be unity since there is no dissipation in the system considered, except for a low artificial bulk viscosity required for the numerical stability of the integration scheme. As expected, the computed COR is found equal to 1 indeed. This problem also corresponds to a special case of Hertz's theory of impact [START_REF] Hertz | Ueber di Berührung fester elastischer Körper[END_REF], with well-known results ( [START_REF] Johnson | Contact Mechanics[END_REF], for instance): during contact time, the force applied on the rigid surface is given by

F = 4 3 E √ R 1 -ν 2 δ 3 (3)
with

δ = [ 15 m 16 1 -ν 2 E √ R (v 2 0 -δ2 ) ] 2/5 (4)
for a ball of radius R, mass m, Young's modulus E, Poisson's ratio ν, with an impact velocity v 0 . In these equations, δ denotes the difference between the ball radius and the height of the ball center of mass above the rigid surface. Numerical integration of the differential equation ( 4) yields a curve shown in Fig. 5, where the result of the finite element simulation also is presented.

Although the two curves are close, Hertz's theory can be observed to give a slightly longer contact duration and a smaller peak force. This theory is recalled to be based on an assumption, which replaces the dynamic problem by a series of static states of equilibrium where the ball is submitted to a different uniform body force at any moment of its contact with the slab. By contrast with this quasi-static approach, the finite element simulation does take the dynamic aspects of the problem into account, a non uniform acceleration is obtained, which gives a non uniform body force, with different results consequently.

In order to confirm the above explanation, a static analysis has also been performed, using the Abaqus/ Standard variant of the code, where a uniform vertical force density was applied instead of an initial velocity. When the pressure profile in the contact area is considered, Fig. 6 shows that a very good agreement is obtained between the finite element simulation and Hertz's theory. The latter involves a well-known elliptic distribution:

p(r) = 3F 2π r 2 c √ 1 - ( r r c ) 2 (5)
where denotes the radius of the contact area and F the total applied force. A value of F = 900 N was used to obtain the results of Fig. 6. The discernible difference near the end of the contact area in Fig. 6 can be attributed to the size of the elements used. As another preliminary test, Abaqus/Explicit has been used to simulate the bounce of the polymer ball in its elastic rubbery state. The ball is now much softer than in the first case considered above (and its behavior is nonlinear elastic). Consequently, it has to deform more in order to change its initial kinetic energy into elastic strain energy when the force reaches its maximum (and the first principal logarithmic strain reaches the value of 0.11). As a result, the ball crushes more significantly on the rigid surface, the maximum radius of the contact area is multiplied by about 3 (close to half the ball radius), and the underlying small deformation assumption in Hertz's theory is no more valid. This ex- plains a larger discrepancy with the (finite strain) simulation results in Fig. 7. It may be noted that the areas below the F (t) curves in this figure as well as in Fig. 5 are all equal to 2mv 0 (twice the initial momentum), as they should in a non-dissipative impact on a fixed rigid surface. As a result, a factor of about 10 on contact duration, which is observed when passing from the glassy (about 0.1 ms) to the rubbery state (about 1 ms), induces a factor of about 1/10 on the peak force. As mentioned previously, the rubbery behavior is very close to incompressible, but Fig. 7 also shows that using a value of ν = 0.475 (keeping the Young's modulus equal to 6 MPa) does not modify the results significantly. Thus, no bias is induced by the four-node reduced integration elements when quasi-incompressibility is considered in the bounce problem. This can be explained by the ball being quite free to deform, with very few constraints applied on its outer boundary. Unlike Hertz's theory, which predicts a perfect symmetry of the force history with respect to the force peak, some dissymmetry is evidenced by the finite element results in Fig. 7 (compare the curvatures at the beginning and end of contact, for instance). This was expected indeed, because the ball keeps vibrating after the contact is left, which differs from its rigid body motion before impact, and therefore the history of the velocity field (and all subsequent quantities) must be dissymmetric. This effect is enhanced when the elastic shear modulus of the ball is smaller (compare with Fig. 5), but it does not affect the COR, which remains very close to 1 (0.992, actually). By contrast, a sharp dissymmetry of the force history will be observed when the viscous part of the behavior will be taken into account in the next Section, but this will be due to dissipative effects.

r c = ( 3 4 1 -ν 2 E RF ) 1/3 (6)

Results and discussion

Coefficient of restitution

The experimental measurements of the COR with respect to temperature were very reproducible and they are presented in Figure 8. At both low and high temperatures, the COR is observed to plateau at its highest value of 0.94. First, it is interesting to notice that the polymer ball bounces as efficiently well at low temperature as at high temperature, since this result may seem somewhat counter intuitive according to everyday experience with "superballs". The latter (like in [START_REF] Cross | The bounce of a ball[END_REF], for instance) are in their rubbery state at room temperature, and one may have the feeling that a glassy ball would bounce less. This is not the case indeed, because the ball is close to elastic at low and at high temperatures as well, and the COR is able to reach equally high values. From 40 • C to 160 • C, the COR is strongly dependent on temperature and, as could be expected, it varies as the viscous component of the behavior increases, which expresses the delayed mechanical response of the material. More precisely, the COR starts decreasing when temperature increases because the relaxation times of the material increase. At higher temperature, the COR increases because the contact duration is increasingly exceeded by relaxation times, and viscous flow has less time to develop. With the vertical axis reversed, the shape of the experimental curve in Fig. 8 may recall the shape of the damping factor plotted as a function of temperature for a fixed frequency, which is somewhat similar to Fig. 2b. This is consistent with the latter curve characterizing energy loss in the material, whereas the COR reflects a fraction of non-dissipated energy, but no direct connection will be attempted here. The experimental results in Fig. 8 follow the same trends as the data of Briggs [START_REF] Briggs | Methods for Measuring the Coefficient of Restitution and the Spin of a Ball, Research Paper RP1624[END_REF] on golf balls, where a minimal COR was obtained around -40 • C. The right part of the experimental curve in Fig. 8 is also reminiscent of what is observed on squash balls [START_REF] Lewis | The dynamic behavior of squash balls[END_REF], and warming-up the ball before a game is a common practice that improves the bounce.

The experimental maximum e value is less than 1, which indicates some energy is lost during impacts at low and high temperatures. In the literature, there are many reports of COR values below 1 even though the behavior of the ball could reasonably well be assumed to be elastic, and the contributions of acoustic emission, vibrations, internal friction, slab plasticity or damage, surface imperfections, adhesion, etc. have been considered. For instance, COR values of 0.95 have been reported in [START_REF] Lifshitz | Some experiments on anelastic rebound[END_REF] for steel balls impacting steel slabs at moderate velocities. In the present case, it seems reasonable that some viscosity is still present at the extreme temperatures considered, inducing dissipation and a COR less than 1, which is similar to the contribution of internal friction invoked in the bounce of metal spheres. Nevertheless, the finite element simulations shown in Fig. 8 lead to COR values that are too high at the extreme temperatures considered in the experiments, which implies that the dissipation has been underestimated and that the model behavior is very close to perfect elasticity in these conditions. Additional simulations allowed to reject an explanation based on some uncertainty on the impact velocity, since substantial variations of the latter were not found to modify the COR significantly. One reason for overestimated COR values can rather be an imperfect rendering of the viscous dissipation in extreme cases by the Maxwell model that has been fitted over a limited frequency range and which may ignore a specific effect of high transient strain rates. Another reason can be the limited temperature range used in the identification of the behavior (from 30 to 70 • C), and the necessary extrapolation of the WLF equation to temperatures outside this range (from 0 to 180 • C). Finally, it should be mentioned that several (T ref , C 1 , C 2 ) triplets may fit the data of Fig. 3 equally well, but with a significant impact on the COR predictions.

The shape of the experimental curve in Fig. 8 is nevertheless reproduced well by the finite element simulations, and the minimum value of the COR (0.19 at 84 • C) is estimated quite well (0.22 at 87.5 • C), but some discrepancies remain. In addition to the reasons given above, a possible explanation could be some experimental errors in the temperatures. The temperature within the thermal chamber used to heat the ball was recorded with an accuracy of ±1 • C, but the ball sur-face may have cooled in the air at ambient temperature during the test duration. Actually, the latter is only a few seconds, which would lead to a temperature gradient located essentially near the free surface of the ball according to the analytical expressions given in [START_REF] Carslaw | Conduction of Heat in Solids[END_REF]. Such a temperature gradient has been found to have a negligible influence on the COR in additional finite element simulations that we performed. More importantly, the temperatures measured in the thermal chamber and in the rheometer have been found to disagree by about 1.8 • C when measured with the same thermocouple. This lead to a correction that has already been applied in Fig. 8 and cannot explain the differences, consequently. Therefore, the discrepancy between the experimental COR values and the simulations is believed to reflect the limits of the accuracy of the behavior provided by standard DMA tests, by the fit with a Maxwell model, and by the extrapolation of the timetemperature superposition.

Contact duration

The contact duration also is strongly dependent of temperature, as shown in Figure 9 where the experimental data obtained with the piezoresistive sensor are presented. Values between 0.13 and 1.2 ms are obtained over the temperature range considered, which confirm that high strain rates develop in the ball during impact. The contact duration appears to depend on the inverse of the material stiffness at impact, and it remains approximately constant below 40 • C and above 120 • C.

Figure 9 presents also the results of the finite element simulations, which overestimate by about 10% the contact durations at low and high temperatures. This encouraging result is consistent with a possible force threshold below which the sensor does not record contact, which induces underestimated experimental values. When temperature increases, the rapid transition between short and long contact durations is also reproduced quite well.

Force history during impact

The history of the voltage signal given by the piezoresistive sensor during impact has been recorded on the oscilloscope. As long as the force is moderate, this signal is proportional to the applied force, which allows to measure the actual value of the force with respect to time after calibration has been performed. For instance, Fig. 10 shows that the force history recorded at 165 • C is reproduced very well by the finite element simulation. Actually, the latter gives the same results as the simulation presented in 3.2 where an elastic ball in its rubbery state was considered (Fig. 7), which confirms that the viscous effects are absent in the model at moderately high temperatures.

Unfortunately, the force becomes so large below 120 • C that a saturation threshold of the sensor is reached. Actually, the points on the sensor surface where the applied pressure is too large saturate, and the totalized signal of the sensor is affected. As a consequence, the sensor returns the contact area (multiplied by the saturation value) rather than the force in such cases. This happens for instance at room temperature, where the simulations of 3.2 indicated an average pressure of 110 MPa for a glassy ball, i.e. about 100 times larger than in the rubbery case because of a force multiplied by about 10 and a contact area reduced by a factor of about 10. For this reason, the experimental results in Fig. 11 have been scaled vertically to reach the same peak value as the finite element simulation. In these conditions, the agreement is acceptable for the whole profile. It may be mentioned that the contact area resulting from ( 6) is proportional to F 2/3 , and thus would have a convex profile (without the concavities of the theoretical F (t) at the beginning and end of contact) even more similar in shape to the experimental sensor measure, but with a contact time still overestimated. Note finally that the symmetry of the experimental curve in Fig. 11 suggests that the sensor has no hysteresis, even though it saturates.

The normalization procedure of the experimental results has also been applied in Fig. 12, where two examples of force histories in the intermediate range of temperature is presented, i.e. when viscous dissipation is significant according to Fig. 8. Accordingly, the force vs. time profiles are now clearly unsymmetric. The trend in common with Fig. 10 and Fig. 11 is a decreasing peak force when temperature increases, i.e. the trend opposite to contact duration. Although the force peak is comparable with Fig. 10, some sensor saturation was present at 104 • C (Fig. 12a) because of a smaller contact area, but the agreement with the finite element results is still good when the force normalization is applied. By contrast, the simulated force history at 84 • C (Fig. 12b) differs more significantly from the sensor signal, even though the latter has been adjusted vertically to fit the force peak. A first reason to explain the difference is a strong sensor saturation due to a large pressure, which distorts the experimental force profile. It should also be noted that the temperature of 84 • C corresponds to the smallest experimental coefficient of restitution, and therefore to a maximum viscous dissipation. The discrepancy may thus also be due to the approximations and limitations involved in the model used to describe the viscoelastic behavior of the material and in its identification.

Conclusion

The effect of temperature on the vertical rebound of a polymer ball falling on a rigid slab has been shown experimentally to be substantial when a wide enough temperature range is considered. This allows to answer the question asked in the Introduction: the coefficient of restitution decreases when temperature increases if the ball is at low temperature, but the opposite trend applies if the ball is above a transition temperature. The contact duration and force history also have been measured, the former increases with temperature, whereas the contrary applies to the peak force.

The fabrication of the acrylate ball in the lab allowed a thorough mechanical characterization of the material by dynamic mechanical analysis which, combining a generalized Maxwell model and the WLF equation for time-temperature superposition, allowed in turn the finite element simulation of the rebounds without ad hoc adjustment. The simulations agreed globally well with the experiments. The trends were predicted correctly, which has shown the strong link between the viscoelastic and time-temperature superposition prop-erties of the material and the main characteristics of the bounce of a polymer ball.

Nevertheless, some discrepancies remain. Besides experimental difficulties like temperature control or piezoresistive sensor saturation, this study has shown that rebound testing enhances the effect of some uncertainties and extrapolations in the description of a temperaturedependent viscoelastic behavior. Moreover, it suggests that some features of the viscoelastic deformation of polymers at high transient strain rates may still be improperly implemented in simple models of the mechanical behavior.

Fig. 1

 1 Fig. 1 Temperature dependences of the material storage modulus (a) and of the damping factor (b) measured during dynamic torsion frequency sweeps

Fig. 2

 2 Fig. 2 The acrylate storage modulus (a) and damping factor (b) master curves at 64 • C, and their fits by a generalized Maxwell model.

Fig. 3

 3 Fig. 3 Time-temperature superposition shift factors and their fit by the WLF equation.

Fig. 4

 4 Fig. 4 Mesh used in the finite element simulations.

Fig. 5

 5 Fig.5History of the reaction force on the rigid surface during the bounce of the ball in its elastic glassy state. Finite element simulation (solid line) and Hertz's theory (broken line).

Fig. 6

 6 Fig.6Pressure profile in the contact area for a static force of 900 N applied to the ball in its elastic glassy state. Finite element simulation (symbols) and Hertz's theory (solid line).

Fig. 7

 7 Fig.7History of the reaction force on the rigid surface during the bounce of the ball in its elastic rubbery state. Finite element simulation (solid line) and Hertz's theory (broken line). The simulation using a Poisson's ratio of 0.475 is also shown (dotted line).

Fig. 8

 8 Fig. 8 Comparison between experimental measures and simulation estimates of the coefficient of restitution.

Fig. 9

 9 Fig. 9 Comparison between experimental measures of the contact duration and simulation estimates.

Fig. 10

 10 Fig. 10 Force history during impact at 165 • C.

Fig. 11

 11 Fig. 11 Force history during impact at 25 • C. The experimental results have been scaled vertically.

Fig. 12

 12 Fig. 12 Force history during impact at 104 • C (a) and 84 • C (b). The experimental results have been scaled vertically.

Table 1

 1 Relaxation times and associated shear moduli used in the generalized Maxwell model.

	τ i (s)	G i (Pa)
	1.259E-08	2.639E+07
	2.354E-08	2.256E+07
	4.401E-08	1.939E+07
	8.230E-08	1.690E+07
	1.539E-07	1.505E+07
	2.877E-07	1.380E+07
	5.380E-07	1.314E+07
	1.006E-06	1.310E+07
	1.881E-06	1.374E+07
	3.517E-06	1.514E+07
	6.576E-06	1.739E+07
	1.230E-05	2.046E+07
	2.299E-05	2.422E+07
	4.299E-05	2.822E+07
	8.038E-05	3.191E+07
	1.503E-04	3.475E+07
	2.810E-04	3.648E+07
	5.254E-04	3.718E+07
	9.824E-04	3.712E+07
	1.837E-03	3.632E+07
	3.435E-03	3.421E+07
	6.422E-03	2.963E+07
	1.201E-02	2.226E+07
	2.245E-02	1.421E+07
	4.198E-02	8.086E+06
	7.850E-02	4.416E+06
	1.468E-01	2.466E+06
	2.744E-01	1.480E+06
	5.132E-01	9.806E+05
	9.595E-01	6.962E+05
	1.794E+00 4.798E+05
	3.355E+00 2.875E+05
	6.272E+00 1.490E+05
	1.173E+01 7.616E+04
	2.193E+01 4.540E+04
	4.100E+01 3.494E+04
	7.667E+01 3.524E+04
	1.434E+02 4.427E+04
	2.680E+02 6.384E+04
	5.012E+02 9.740E+04

Acknowledgements The authors are grateful to several colleagues from PIMM laboratory: J. Lédion for lending his thermal chamber, M. Schneider for making his digital oscilloscope available, and F. Coste for his advice on signal acquisition. Moreover, M. Brieu from LML is acknowledged for giving access to his mechanical testing machine for sensor calibration.


J.-P Korb

L Patural

A Govin
email: govin@emse.fr

Ph Grosseau
email: grosseau@emse.fr.

NMR investigations of water retention mechanism by cellulose ethers in cement-based materials

We show how nuclear magnetic spin-lattice relaxation dispersion of proton-water (NMRD) can be used to elucidate the effect of cellulose ethers on water retention and hydration delay of freshly-mixed white cement pastes. NMRD is useful to determine the surface diffusion coefficient of water, the specific surface area and the hydration kinetics of the cement-based material. In spite of modifications of the solution's viscosity, we show that the cellulosic derivatives do not modify the surface diffusion coefficient of water. Thus, the mobility of water present inside the medium is not affected by the presence of polymer. However, these admixtures modify significantly the surface fraction of mobile water molecules transiently present at solid surfaces. This quantity measured, for the first time, for all admixed cement pastes is thus relevant to explain the water retention mechanism.

INTRODUCTION

It is known that the presence of cellulose ethers (CE) in mortars enhances drastically the water-retention (Mansoutre). Cellulose ethers (CE) are water soluble semi-synthetic polymers derived from cellulose, the most abundant polysaccharide in nature. This specific property of CE is crucial to achieve sufficient mortar-substrate adhesion when the mortar is applied in thin layers on highly absorbent substrates. Other properties are also affected by CE, such as the delay of hydration and setting (Peschard). A significant influence of the polymer molecular parameters like molecular weight and substitution degrees was evidenced [START_REF] Pourchez | HPMC and HEMC influence on cement hydration[END_REF], but no clear water retention mechanism could be clearly identified. During evaporation, no admixture loss is occurring, due to the very low vapor pressure of this admixture and a decrease in the water chemical potential occurs [START_REF] Pourchez | Some aspects of cellulose ethers influence on water transport and porous structure of cement-based materials[END_REF]. A decrease of water mobility due to the jamming of the diffusion space by the polymer molecules is even expected (Patural, 2011). However, a recent pulsed gradient field NMR study proved that the macroscopic bulk water self diffusion coefficient is not modified in CE solution or in admixed cement pastes [START_REF] Patural | A pulsed field gradient and NMR imaging invsertigations of the water retention mechanism by cellulose ethers in mortars[END_REF]. Moreover, the interdiffusion imaging experiments demonstrated also that the water diffusion at the paste interface is not affected by the presence of cellulosic admixture [START_REF] Patural | A pulsed field gradient and NMR imaging invsertigations of the water retention mechanism by cellulose ethers in mortars[END_REF]. So, the possibility of a potential microscopic diffusion barrier specifically at the cement hydrates interfaces is still controversial and the origin of the mechanism of water retention due to CE admixture is still an open question.

Our aim here is precisely to answer the latter question and to propose a realistic mechanism for the water retention in presence of CE in mortars and cements. For that purpose, we show how nuclear magnetic spin-lattice relaxation dispersion of protonswater (NMRD) can be used to elucidate the effect of cellulose ethers on water retention and hydration delay of freshly-mixed white cement pastes [START_REF] Patural | Nuclear magnetic relaxation dispersion investigations of water retention mechanisms by cellulose ethers in mortars[END_REF].

EXPERIMENTS

Mineral and organic products. Cement pastes were prepared with white cement in order to facilitate NMR relaxation experiments. However, we showed before by electronic spin resonance (ESR) that paramagnetic Fe 3+ ions were clearly present in white cement (see Figure 6 in [START_REF] Korb | Microstructure and texture of hydrated cement-based materials: A proton field cycling relaxometry approach[END_REF]). The admixtures used are cellulose ethers (CE) of chemical structures given in Figure 1. All the characteristics of the main physical-chemistry properties and labels used for the cellulose ethers studied are described in (Patural, Thesis 2011). Admixture amount (0.27%) was in addition to the total dry mixture (i.e. cement, sand and filler). The cellulose ether weight-average molecular mass ( ) was determined using size exclusion chromatography. We have prepared different samples with increased molecular mass labeled J1, J2 and J3, respectively. Others starch ethers were also investigated, those are polymers providing very different water retentions Water retention. For water retention measurements, mortars were prepared according to the CEReM mixture proportions consisting in 65% sand, 30% ordinary Portland cement (CEM I 52.5 R), 5% calcareous filler with a water to cement ratio w/c=1. The experimental device and way of probing water retention WR(%) have been described in (Patural, Thesis 2011). The cellulose ethers improved water retention up to WR=98.9% for HPMC J3. With respect to the water retention capacity of the nonadmixed mortar (64.5%), this represents a very large increase. However the starch ethers procure very different water retention capacities ranging from 66.2% to 92.6%.

Experiments of NMRD in admixed cement pastes. We performed proton nuclear magnetic relaxation dispersion (NMRD) on a fast-field cycling spectrometer from Stelar s.r.l., Mede, Italy, where the polarization magnetic field is 0.5 T while the evolution magnetic field ranges from 0.25 mT to 0.5 T. All samples were introduced into a 7 mm-tube which was inserted into a 10 mm-diameter standard NMR tube. Cement pastes were prepared with a water/cement ratio w/c=0.4 in order to have an appropriate consistency and for facilitating the NMR measurements at low fields. The NMRD sequence is very well documented (Kimmich). The experiments were repeated over a large range of proton Larmor frequencies ω Ι /2π (10kHz-20MHz) in order to obtain the complete dispersion profile of the longitudinal spin-lattice relaxation rate 1/T 1 (ω Ι ). The temperature was fixed at 298K.

Basically, proton NMR relaxation is a stimulated (non spontaneous) phenomenon driven by the coupling of the proton spins to the magnetic noise induced by molecular motions (translation, rotation, exchange, etc). Varying the magnetic field changes the proton Larmor frequency ω Ι /2π, and thus allows exploring the time scales of the magnetic fluctuations (noise) to which the nuclear spin relaxation is sensitive. For diffusive liquids, NMRD gives also a rich source of dynamical information over a large range of length scales, from localized and fast motions at large frequency to a delocalized and slow motions at low frequency.

We propose two kinds of NMRD experiments for white admixed cement pastes. The first experiment consists of measuring the magnetic field dependence of 1/T 1 (ω Ι ) of various admixtures of different CE (Figure 2).

Figure 2. Measured water 1 H magnetic relaxation dispersion profiles 1/T 1 (ω Ι ) for hydrated white cement paste (w/c=0.4) as a function of the proton Larmor frequency. The experiments have been realized in the early hydration period (0-45 min). Effect of three cellulose ethers (HPMC J1, J2 and J3) compared to the neat white cement paste. The continuous lines are the best fits obtained with our theory of surface relaxation (Barberon and Korb 2009).

The second experiment consists in probing continuously 1/T 1 (ω Ι =2π×10kHz, t hydr ) at a fixed and low frequency (10kHz) during the inducing period t hydr of cement paste with various admixtures of different CE (Figures 3a,4a). During this period, we observe a monoexponential longitudinal magnetization decay that rules out any distribution of T 1 , the non exponential relaxation only appearing after ten hours of hydration [see Figure 7 of [START_REF] Korb | Microstructure and texture of hydrated cement-based materials: A proton field cycling relaxometry approach[END_REF]]. The continuous measurement of 1/T 1 (ω Ι =2π×10kHz, t hydr ) in presence of various admixtures of cellulose ethers allows us to monitor the evolution of the specific surface area S(t hydr )/V, of the material during the hydration.

RESULTS AND DISCUSSION

We proposed previously a theoretical model necessary for interpreting all the NMRD relaxation features reported in Figure 2, 3a and 4a (Barberon and Korb 2009).

Here, we just outline the basic hypothesis of the model needed for probing the specific surface area and the water surface diffusion coefficients at the growing interface of the admixed and neat cement pastes. Basically, the molecular exchange between the solid and liquid phases is sufficiently fast compared to their respective individual proton relaxation times that a single 1/T 1,w (ω I ) exists given by a linear combination of a bulk and a surface contributions. The latter is weighted by the surface to volume ratio S/V present at a given time of hydration at the solid/liquid interface. 1/T 1,w (ω I ) for the confined proton-liquid has a bilogarithmic frequency dependence (Barberon and Korb 2009) that comes unambiguously from the two dimensional diffusion of the water molecules along the pore surface modulating the dipole-dipole interaction between the proton species and the paramagnetic Fe 3+ ions fixed at the surface of the pores. On the contrary, in the frequency range studied, the spin-lattice relaxation rate for the solid protons does not depend on the frequency and the bulk relaxation rate 1/T 1,bulk is frequency independent. The conservation of these frequency dependencies during the progressive hydration and the evolution of with the hydration time allowed us to renormalize the NMRD data to a single master curve (Barberon and Korb 2009), where S p,NMR is a NMR-based specific surface area of the hydrated cement that appears to be directly proportional to the degree of advancement of chemical reaction.

Surface diffusion coefficient.

The surface diffusion coefficient is obtained from a fitting procedure of the NMRD profiles of Figure 2 which shows the NMRD profiles for different admixed cement pastes with three HPMC (J1, J2 and J3) and neat (without CE) white cement pastes. In order to limit the kinetics effect during the measurements, we have decreased the number of frequency points on the profiles. This procedure allows a quite reasonable fit of the plateau dependence at low frequency and the bilogarithmic dependence at higher frequency (Figure 2). The correlation time for translational diffusion of water-proton is determined using the model described in (Barberon and Korb 2009). For these four different cement pastes, we find similar results for the translational surface diffusion τ m ≈1ns and for the time of residence τ s ≈10 µs. This suggests a pore scale invariance of water dynamics at the pore surface in CSH whatever the paste composition. This value of τ m is in agreement with previous studies performed on cement pastes or on mortars (Barberon and Korb 2009). The translational diffusion coefficient at the pore surface can be deduced from the Stokes-Einstein relationship:

where is the water-molecule size. For all cement pastes (admixed or not), the surface diffusion coefficient is about i.e. about 1/100 of that of bulk water at 23°C. In consequence, the presence of cellulose ethers in cement paste does not modify the surface diffusion coefficient at the pore surface. This result is similar to those obtained in the macroscopic scale by pulsed field gradient NMR showing that the CE does not modify the diffusion coefficient of water in the bulk [START_REF] Patural | A pulsed field gradient and NMR imaging invsertigations of the water retention mechanism by cellulose ethers in mortars[END_REF]. We found similar results for starch ethers that present very different water retentions (92.6% and 66.2% respectively). This demonstrates that the translational diffusion coefficient at the surface of the solid interfaces is not affected by starch ether's presence.

These experiments thus demonstrate that cellulose and starch derivatives do not change the surface diffusion coefficient of water. The water mobility at pore surface is thus not modified in presence of such polymers.

Specific surface area and relative quantity of water transiently present at pore surface. On Figure 2, one observes a constant value for 1/T 1 (ω I ) when the proton frequency becomes ω I< ω d =2π×30 kHz. Such a cross-over frequency ω d is of the order of the dipolar interaction in the solid state between two protons separated by 0.155 nm. On this low frequency range, the confined proton liquid explores relaxation processes induced by characterized long correlation times encountered in the solid state:

1/ . We have shown in (Barberon and Korb 2009) that owing to the cross relaxation process, the intermolecular dipole-dipole interaction then couples the water spin relaxation to that of the solid and the magnetic field dependence of the immobilized spin system. The observed data presented in Figure 2 confirms the plateau observed at low frequency (at 10 kHz) for 1/T 1 during the first hundred minutes. Then, relaxation rate increases due to cement hydration and development of the specific surface area. Moreover, a zoom on the Figure 3a shows that small but significative differences exist between each admixed cement pastes.

The observed plateau for frequency below ω d is thus characteristic of the rigid-lattice limit of the solid-proton hydrates. The absolute value of such a plateau, measured for instance at 10 kHz, is thus indicative of the specific surface area precisely at the solidliquid interface. The continuous measurement of 1/T 1 (ω Ι =2π×10kHz, t hydr ) in presence of various admixtures of cellulose ethers thus allows to monitor the evolution of the specific surface area S P,NMR (t hydr ), of the material during the hydration. How NMR relaxation can elucidate water retention mechanisms? In the very early aged of the hydration, the amount of CSH is not very large and the admixed polymer is still in the interstitial solution within the pores of the paste, even if the cellulose ethers are rather large. The relaxation of the mobile proton-water thus will be influenced by the presence of the polymer. We show that measuring the time evolution of the spin-lattice relaxation rate 1/T 1 (10 kHz, t hydr ) at 10 kHz, especially in the first minutes of hydration allows probing quantitatively the relative population of water transiently present at the solid hydrate surfaces N S /N(t hydr ).

On Figure 3a, the continuous lines exhibit the time evolution of 1/T 1 (10 kHz, t hydr ) for the non admixed (neat) white cement and with HPMC J1, J2, J3 admixtures. The periods during which the plateau remains constant are evidenced ( is related to neat cement) and ordered as followed:

. This period d CE , measured at low frequency, is a clear indicator of the delay of hydration induced by the CE polymer introduced into the material. The fact that d 0 stays much smaller than the other d CE is a clear indication of the delayed hydration and setting induced by the HPMC J1, J2, J3 admixtures. We show on Figure 3b, that one can correlate these delays of hydration measured by NMRD to the ones measured from isotherm calorimetry measurements (determined using the method described in Patural Thesis 2011) on similar materials. The quasi linear dependence (r 2 =0.99) shows the very good accuracy between the microscopic (NMRD) and macroscopic (calorimetry) measurements.

Another important parameter can be extracted from the behavior of the time evolution of 1/T 1 (10 kHz, t hydr ) during the first 100 minutes of hydration. This is the value of the plateau itself, noted, ( )

hydr observed CE t kHz T T , 10 / 1 / 1 , 1 , 1 =
which depends critically on the nature of the admixtures. The results found for the neat white cement paste and HPMC-admixed cement pastes J1, J2, J3 are ordered as followed:

1 3 , 1 1 2 , 1 1 1 , 1 1 0 , 1 6 . 37 / 1 3 . 36 / 1 9 . 35 / 1 30 / 1 - - - - = < = < = < = s T s T s T s T J J J
, where 1/T 1,0 is the value of the low-frequency plateau for neat cement. 1/T 1,CE is proportional to the correlation time and directly related to the relative amount of water transiently present during the first 100 minutes at proximity of solid interfaces that is proportional to the specific surface area. Owing to the observation that τ m ≈1ns and τ s ≈10µs are universal for all the cement-based materials, one obtains a proportion N S (t hydr =2 min)/N=1.06% for the neat white cement paste and N S (t hydr =2 min)/N=1.36% for the admixed cement pastes with HPMC J3. One obtains intermediate values for the other admixtures HPMC J1 and HPMC J2.

The fact to probe, in the first minutes of cement hydration, the fraction of mobile water molecules transiently present at solid interfaces in presence of different nature and quantity of cellulose ethers is a key result of this study that can be directly linked to the relative amount of water retained on the pores surface.

We have applied the same NMRD method for HPMC J1 with two different concentrations namely 0.1 and 0.4% giving 82.3 and 98.0% macroscopic water retention, respectively (Figure 4a). As expected, the value of the plateau 1/T 1,CE increases significantly with the concentration of admixture. Finally, on Figure 4b the linear correlation (r 2 = 0.93) of 1/T 1,CE and water retention of mortar admixed with three different concentrations 0.1, 0.27 and 0.4% of HPMC J1 proves that our microscopic NMRD measurements on cement pastes could be correlated with the macroscopic water retention on mortars.

CONCLUSION

An original method based on two different applications of proton-water magnetic relaxation dispersion (NMRD) has been proposed to elucidate the effect of cellulose ethers (CE) on water retention of freshly-mixed white cement paste.

The first application of NMRD probes directly the proton-water surface dynamics from the magnetic field dependence of the nuclear spin-lattice relaxation rate. In spite of modifications of the solution's viscosity, we find that the cellulosic derivatives do not modify the surface diffusion coefficient of water that is about two orders of magnitude smaller than that of bulk water.

The second application of NMRD concerns the continuous measurements of the spin-lattice relaxation rates at a fixed and very low Larmor frequency. At the very early ages, this application probes continuously and for the first time, the relative population of water transiently present at the surface of the time-growing solid hydrate interfaces. Compared to neat white cement, the CE modifies significantly the surface fraction of mobile water molecules transiently present initially at solid surfaces. Our results show that the larger the fraction, the better water-retention capacity of the hydrated cement or mortar samples at proximity of a solid support. This quantity measured for all admixed cement pastes is thus relevant to explain the water retention mechanism. Furthermore, a linear correlation is found between the NMR surface fraction and the standard ASTM method used by the mortar industry to estimate the water retention capacity of admixed mortars. However for very strong water retention, it seems that the ASTM is not able to distinguish polymer's effect while NMRD enables to make the difference between two admixtures.

Figure 1 .

 1 Figure 1. Structures of cellulose ethers (a) HPMC, (b): HEMC, (c): HEC

Figure 3 .

 3 Figure 3. (a) Water 1 H 1/T 1 (ω Ι ) of hydrated cement pastes (w/c=0.4) as a function of hydrating time measured at a Larmor frequency of 10kHz. Effect of three cellulose ethers (HPMC J1, J2 and J3). The inset represents a zoom on the frequency range of the plateau. (b) Correlation between the delays obtained by the NMRD and isotherm calorimetry.

Figure 4 .

 4 Figure 4. (a) Comparison of the water 1 H 1/T 1 (ω Ι ) of hydrated cement pastes (w/c=0.4) as a function of hydrating time measured at a Larmor frequency of 10kHz. Effect of the CE concentration of HPMC J1. (b) Correlation between the water retention evidenced from the NMRD technique and the macroscopic measurement described in the text (ASTM C1506-09 standard). The different concentrations of HPMC J1 are indicated.


Deep Kamal

Singh

Kandaraj Piamrat
email: kandaraj.piamrat@irisa.fr

Hyunhee Park
email: hyunhee.park@irisa.fr

César Viho
email: cesar.viho@irisa.fr

Jean-Marie Bonnin
email: jm.bonnin@telecom-bretagne.eu

Optimising QoE for Scalable video multicast over WLAN

Keywords: Quality of Experience (QoE), Scalable Video Coding (SVC), Wireless Multicast, Linear Programming

Quality of Experience (QoE) is the key to success for multimedia applications and perceptual video quality is one of the important component of QoE. A recent video encoding scheme called Scalable Video Coding (SVC) provides the flexibility and the capability to adapt the video quality to varying network conditions and heterogeneous users. In this paper, we focus on SVC multicast over IEEE 802.11 networks. Traditionally, multicast uses the lowest modulation resulting in a video with only base quality even for users with good channel conditions. To optimize QoE, we propose to use multiple multicast sessions with different transmission rates for different SVC layers. The goal is to provide at least the multicast session with acceptable quality to users with bad channel conditions and to provide additional multicast sessions having SVC enhancement layers to users with better channel conditions. The selection of modulation rate for each SVC layer and for each multicast session is achieved with binary integer linear programming depending on network conditions with a goal to maximize global QoE. Results show that our algorithm maximizes global QoE by providing highest quality videos to users with good channel conditions and by guaranteeing at least acceptable QoE for all users.

I. INTRODUCTION

One important share of multimedia market is video service, which is now available anywhere at any time. To support such service, a video service provider has to manage, store, and distribute content towards multiple types and scales of terminals, and over different and transient access technologies to reach the end user. A good user experience is the key to success for any video service. With multimedia applications, managing network resources with the goal of just optimizing QoS parameters can miss the essential point which is user satisfaction or user experience. User experience or currently called Quality of Experience (QoE) [START_REF]Definition of Quality of Experience (QoE)[END_REF] is defined as the overall acceptability of an application or service as perceived subjectively by the end-user. QoE is different from QoS network indicators in terms of bandwidth, loss rate, jitter, which are not sufficient to get a precise idea about the visual quality of a received video sequence. QoE instead focuses on the overall experience of the end user. It depends on the global system behavior, going from the source of a given service upto the final user, including the content itself and the network performance. It can be evaluated in terms of Mean Opinion Score (MOS) as shown in Table I. Therefore, QoE is more appropriate when dealing with multimedia service.

Video scalability seems to be one of the most interesting solutions to some of the multimedia needs as it provides the capability to adapt to varying network conditions and heterogeneous users. One of the most well-known scalable standards is the Scalable Video Coding (SVC) extension of H.264/MPEG-4 AVC video compression [START_REF] Schwarz | Overview of the Scalable Video Coding Extension of the H.264/AVC Standard[END_REF]. It enables the transmission and decoding of partial bitstreams to provide video services with lower temporal or spatial resolutions or reduced fidelity while retaining a reconstruction quality that is highly relative to the rate of the partial bit streams. Therefore, SVC provides functionalities such as graceful degradation in lossy transmission environments as well as bit rate, format, and power adaptation.

We can also observe that today a significant amount of multimedia traffic is transmitted over IEEE 802.11 based wireless networks as numerous access networks are available free of charge or at very affordable price. In wireless environment, different factors such as path loss, fading, or interference in the channel have direct impact on the variation of the received signal to noise ratio (SNR), which results in variation in Bit Error Rate (BER). The lower the SNR the more difficult to decode the received signal, resulting in higher BER. To overcome this problem, the standard has provided multirate transmission capability. For example: 1, 2, 5.5, 11 Mbps data rates are available in IEEE 802.11b; or [START_REF] Zhai | Three dimensional scalable video adaptation via user-end perceptual quality assessment[END_REF][START_REF] Piamrat | Q-DRAM: QoE-Based Dynamic Rate Adaptation Mechanism for Multicast in Wireless Networks[END_REF][START_REF] Kamal | Quality of experience estimation for adaptive HTTP/TCP video streaming using H.264/AVC[END_REF][START_REF] Xiao | Throughput and delay limits of IEEE 802.11[END_REF]24,36,48,54 Mbps are also available in IEEE 802.11a. These different data rates result from different modulation techniques and channel encoding schemes. With this adaptation capability, wireless resources can be managed more intelligently.

In this paper, we will focus particularly on layered multicast over wireless with multiple transmission rates. We propose a mechanism that uses both layering transmission in SVC and rate adaptation capability in IEEE 802.11 to optimize QoE in the network. The idea is use multiple multicast session by selecting an optimized modulation for transmission of each layer adapted to the channel conditions experienced by different users in the network environment. For that, the problem is formulated and solved with binary integer linear programming. The rest of this paper is structured as follows.

Section II provides background concepts covered by this paper and Section III describes related works. Section IV presents our proposed mechanism for QoE optimization. In section V, results obtained by the proposed mechanism are analyzed. Finally, section VI provides conclusions and perspectives.

II. OVERVIEW OF WIRELESS MULTICAST AND SCALABLE VIDEO CODING A. Wireless Multicast

In this paper, we are interested in multirate IEEE 802.11 based wireless network, more particularly for multicast transmission. It can be noticed that multicast over wireless networks is a fundamental communication function because wireless network is inherently broadcast by nature. A packet has to be sent only once to reach all receivers. Therefore, multicast is an efficient method to transmit the same data to a group as it allows transmission of data to multiple destinations using fewer network resources.

However, multicast applications have some constraints. Multicast traffic is typically set to the lowest transmission rate (basic rate) in order to reach all stations especially the further ones that are subject to important signal fading and interference. The lower rates are disadvantageous to transmission in terms of channel occupancy since they take longer time to send the same amount of data. Therefore, any slow host may considerably limit the throughput of other hosts roughly to its level of low rate [START_REF] Heusse | Performance anomaly of 802.11b[END_REF]. Another constraint in multicast transmission is the lack of acknowledgment and retransmission due to huge overhead of these packets. This is severe with multicast as their numbers are multiplied by the number of recipients in the group. This behavior could lead to feedback implosion.

In this paper, we will not focus on the unreliable problem of multicast since we assume that for real-time traffics like UDP-based streaming, it is preferable to lose a few packets than waiting for retransmission, which delays packets delivery. Hence, we focus here on the performance of the network with respect to user satisfaction and network utilization.

B. Scalable Video Coding (SVC)

For wireless multicast of video, we consider SVC as explained before. SVC has three fundamental types of scalabilities as depicted in Figure .1: spatial resolution (picture resolution), temporal resolution (frame rate), and quality (encoding quality). Its scalability is achieved via a layered approach. A typical SVC stream includes one base layer and one or several enhancement layers. The base layer in SVC is generally H.264/AVC compliant for backward compatibility, which delivers the minimum quality requirements of the decoded video stream. Enhancement layers can be added via progressive refinement slices that are generated using a finer quantization step-size than the base layer. Spatial resolution enhancement is achieved with region-of-interest (ROI) by adding enhancement layers to the base-layer resolution at the bit stream level. Finally, temporal enhancement can be achieved with increased frame rate though at the expense of coding delay. SVC delivers graceful degradation in decoded video quality that might arise due to channel fluctuations and losses. In such conditions, the removal of an enhancement layer still leads to reasonable quality of the decoded video at reduced temporal, spatial or/and SNR level.

III. RELATED WORKS Even many studies exist, to the best of our knowledge, none of them gather all the aspects (QoE, multirate, and wireless multicast) that we handle in this paper. We describe below some representative propositions with close objectives to ours.

A cross-layer approach introduced by Khalek et al. [START_REF] Khalek | A Cross-Layer Design for Perceptual Optimization Of H.264/SVC with Unequal Error Protection[END_REF] is an APP/MAC/PHY architecture that enables optimizing perceptual quality for delay-constrained scalable video transmission. The authors propose an on-line QoS-to-QoE mapping technique to quantify the loss visibility of packets from each video layer using the ACK history and perceptual metrics. At the PHY layer, they develop a link adaptation technique that uses the QoS-to-QoE mapping to provide perceptuallyoptimized unequal error protection per layer according to packet loss visibility. At the APP layer, the source rate is adapted by selecting the set of temporal and quality layers to be transmitted based on the channel statistics, source rates, and playback buffer state.

Another scheme proposed by Zhai et al. [START_REF] Zhai | Three dimensional scalable video adaptation via user-end perceptual quality assessment[END_REF] is QoEoriented, the authors studied the problem of how to optimally exploit the trade off among the three dimensional scalabilities in order to maximize user experience, given the available resources. For that, they propose a low-complexity algorithm that executes at resource-limited user end to quantitatively and perceptually assess video quality under different spatial, temporal and SNR combinations. Based on the video quality measures, the authors further propose an efficient adaptation algorithm, which dynamically adapts scalable video to a suitable three dimension combination.

Finally, Zhu et al. [START_REF] Xiaoqing Zhu | Distributed media-aware rate allocation for video multicast over wireless networks[END_REF] propose an optimization framework for rate allocation among multiple video multicast sessions sharing a wireless network. Their optimization objective is to minimize the total video distortion of all peers without incurring excessive network utilization. The system model explicitly accounts for heterogeneity in wireless link capacities, traffic contention among neighboring links, as well as different video rate-distortion characteristics.

IV. OPTIMISING QOE Quality of Experience (QoE) can be used as relevant indicator for managing network resource. Several studies have demonstrated good performances using QoE as metric, for example, packet scheduling or rate adaptation proposed by Piamrat et al. [START_REF] Piamrat | QoE-Aware Scheduling for Video-Streaming in High Speed Downlink Packet Access[END_REF], [START_REF] Piamrat | Q-DRAM: QoE-Based Dynamic Rate Adaptation Mechanism for Multicast in Wireless Networks[END_REF]. In this paper, optimising QoE is considered as an objective for managing SVC video transmission using multicast. SVC streams can be adapted to user environment when several users have diverse network conditions. We study the problem of choosing optimal rate modulations for different SVC layers in order to maximise global QoE considering different wireless conditions.

A. QoE estimation tool

A general technology called Pseudo-Subjective Quality Assessment (PSQA) has been proposed in [START_REF] Mohamed | A study of real-time packet video quality using random neural networks[END_REF]. It is based on a specific type of queuing network, called Random Neural Network (RNN) [START_REF] Gelenbe | Random neural networks with negative and positive signals and product form solution[END_REF], used as a learning tool. For every different context, such as when the video codec or the parameters affecting the QoE change, a new PSQA based module needs to be designed. This consists in the following steps: After that, the trained RNN will be used in real time to estimate the video quality. Thus, in order to use PSQA for SVC multicast, first the relevant parameters need to be identified and their effect on the perceived quality needs to be studied. Singh et al. [START_REF] Kamal | Quality of experience estimation for adaptive HTTP/TCP video streaming using H.264/AVC[END_REF] used PSQA to estimate QoE for dynamic HTTP streaming (DASH) having multiple levels of quality, but without considering multiple scalabilities of SVC. QoE estimation of SVC was provided in [START_REF] Kamal | Quality of experience measurement tool for SVC video coding[END_REF] with considered parameters such as frequency of Instantaneous Decoder Refresh (IDR) and Network Abstraction Layer Unit (NALU) losses per SVC layer. However, quality loss due to video compression was not considered. With SVC, the quality loss due to video compression changes progressively with different SVC layers. In the following text, we describe the parameters that are considered for QoE estimation in the context of SVC multicast and quality loss due to video compression is captured by QP (Quantization Parameter) in our parametric QoE model. In this work, we do not consider spatial scalability and thus the resolution of the video is fixed to 720p. We consider temporal as well as coarse grain quality layer scalability (CGS).

We consider the following parameters for QoE estimation: Quantisation Parameter (QP) and frames per second (fps). QP values ranging from 1 to 51 are used to quantize the transform coefficients obtained while encoding the video. The trade-off is that a higher value means more loss of information and lower quality, but a lower bitrate; and vice versa. For QoE estimation, we consider the average of QP values, over all macroblocks in all video frames present over the measurement time window. Frames per second affects the video quality such that lower frame rate results in low bitrate at the cost of perceived quality because users prefer higher frame rates. Note that we do not consider other signal and content based parameters such as amount of motion and spatial complexity in the video because we target a real time estimation without high complexity. The videos used to train the neural network are shown in Figure 2. JSVM (Joint Scalable Video Model) [START_REF]SVC reference software(JSVM software)[END_REF] encoder was used with interlayer prediction and IDR period as 32. CGS layers had same resolution 1280x720 with QP values from 16 to 48. Temporal layers had frame per second values of 7.5, 15, 30. A total of 40 videos were generated having different QP and fps values. A total of 15 users were used for video evaluations and neural network training. The QoE function obtained using PSQA methodology is shown in Figure 3 and RMSE (Root Mean Square Error) of 0.36 was obtained for real MOS scores vs the estimated PSQA scores, on the MOS scale of 1-5. Regarding only 15 users and subjective test conditions, we followed ITU R BT 500-11 [START_REF]Methodology for the subjective assessment of the quality of television pictures[END_REF] and ITU R P.910 [START_REF]Subjective video quality assessment methods for multimedia applications[END_REF]. P.910 recommends minimum 15 users and [START_REF] Winkler | On the properties of subjective ratings in video quality experiments[END_REF] explains that this number is reasonable and even 10 users can be enough if experiments are well designed.

B. QoE-driven resource management

Lets consider the case where each SVC layer can be transmitted using a different rate modulation thus effectively resulting in multiple multicast sessions for different clusters of users. As discussed in section II-A the trade-offs are described in the following lines. A layer transmitted with the lowest modulation requires highest channel occupancy for a given bitrate, but can be successfully decoded by a user far away from the Access Point (AP). Whereas, higher modulations with decreasing channel occupancy times may not be decodable by all users. Moreover, the SVC base layer has lowest bitrate requiring least resources and adding more enhancement layers plus base layer provide better quality at the price of higher overall bitrate.

Let us denote a SVC layer by l (1 ≤ l ≤ L), with total L layers, and let there be a set of K modulations m k with (1 ≤ k ≤ K). The AP can transmit multiple multicast sessions and the users with m k ≥ m l can correctly receive a SVC layer l transmitted using the rate modulation m l .

The problem is to find the optimal modulation set {m 1 , m 2 , ..., m l , ..., m L } corresponding to L SVC layers that will maximise Q, sum of QoE scores for all users, for a set of users {n 1 , n 2 , ...n k , ..., n K } such that n k corresponds to the number of users that can support a maximum modulation of m k . Note that some SVC layers can be transmitted using same modulation and also all layers need not be transmitted (m l = 0). Also consider the values {q 1 , q 2 , ..., q l , ..., q L } as the quality scores, corresponding to different layers, that are calculated by AP in real time by QoE estimation module described in next section. The value q 1 is the QoE score for the base layer, q 2 is the delta of QoE score obtained with the addition of layer 2 and so on.

C. Binary Integer Linear Programming

We solve the above problem using binary integer linear programming (BIP). Let us define an allocation matrix

Λ =        λ 1,1 • • • . . . . . . λ k,l . . . . . . • • • λ K,L       
such that λ k,l = 1, if layer l is transmitted with modulation m k and λ k,l = 0 otherwise.

The objective is to maximise Q, sum of QoE for all users, given by:

Q = K k=1 L l=1 q l n k k h=1 λ l,h (1)
Also, there are some constraints to consider. The most important constraint, in order to leave some resources for other traffic, is that we consider a channel occupancy limit ρ, equivalent to amount of total channel resources available for video multicast, such that the channel occupancy for all SVC transmissions should be less than the limit ρ with (0 ≤ ρ ≤ 1). Also, a layer l can only be transmitted using a modulation m k if and only if layer l -1 is transmitted using m h with h ≤ k. This ensures that the layer l can be successfully decoded as it depends on layer l -1. Finally we consider that a layer should not take more than a single multicast session as transmitting a layer already with a given modulation is accessible to all users supporting that modulation and transmitting it with a higher modulation also will waste resources.

Thus, the optimisation problem, to find Λ, can be stated as follows:

Objective Maximize K k=1 L l=1 q l n k k h=1 λ l,h (2)
Constraints K k=1 L l=1 r l t k λ k,l ≤ ρ 0 ≤ ρ ≤ 1 (3) λ k,l - k h=1 λ h,l-1 ≤ 0 ∀k ∈ K, 2 ≤ l ≤ L (4) K k=1 λ k,l ≤ 1 ∀l ∈ L ( 5
)
where ρ is the channel occupancy limit , r l is the rate of layer l in packets per second and t k is the time taken to transmit a packet with modulation m k . The constraints are as described before: the constraint (3) describes the channel occupancy limit, the constraint (4) ensures that a layer l is sent only if layer l -1 is at least sent once using an equal or lower modulation and the constraint (5) ensures that any given layer is sent only once. Also, in order to guarantee at least a basic minimum quality to all users, one may encode the first layer with the minimum required MOS score, lets say 3.0 or bunch the lower layers totaling up to this score. Then in order to force that all the users receive at least minimum QoE another constraint can be added: λ 1,1 ≥ 1 if there are non zero users supporting only m 1 modulation else constraint can be put on λ 2,1 and so on to force one of them to 1.

Note that the complexity of the BIP is reduced to a size L × K, independent of number of users. The above BIP is solved using GLPK (GNU Linear Programming Kit). The time required to solve it is small (≈10ms) for 8 SVC layers and 8 modulation rates on a 2 GHz processor with 1 GB RAM.

D. QoE estimation and optimisation

The AP implements a media aware network element (MANE) functionality as well as the function to estimate the maximum modulation rate supported by different users based on their SNR values. First the MANE estimates the MOS scores, q l ∀l ∈ L , for different SVC layers in real time using the PSQA tool trained for SVC. The layers not sent are dropped. The AP also stores the number of users that can support a given modulation based on the SNR values to determine the values n k ∀k ∈ K. We do not target highly mobile scenario and thus assume that the channel conditions of different users do not change fast. Thus, the AP can refresh the SNR statistics after a time period T which is of the order of some seconds. Such that the AP runs the binary integer program every T seconds to determine the optimal allocation matrix to transmit SVC video using multiple multicast sessions. We assume T to be a variable that can be changed according to the desire of the operator in order to adapt to changing user conditions. The parameter t k used in (3) is provided by Xiao et al. [START_REF] Xiao | Throughput and delay limits of IEEE 802.11[END_REF] for 802.11a:

t k = 2τ + t dif s + t sif s + cw min t slot 2 + t p + t phy + 8(L d + L h ) (10 6 * m k )
where different parameters are as follows: slot time t slot = 9µs, SIFS and DIFS time t dif s = 34µs, t sif s = 16µs, minimum back-off window size cw min = 15 , propagation delay τ = 1µs, transmission time of physical preamble t p = 16µs , transmission time of PHY header t phy = 4µs, average payload size L d = 1024 bytes and overhead size L h = 28 bytes. V. RESULTS In this section, we present some numerical results. We assume 802.11a and thus the available bitrates with different modulation are: 6, 9, 12, 18, 24, 36, 48, 54 Mbits/s. Multiple simulation runs are done to obtain the results. The distance of users from the AP is randomly picked to be D max d √ u meters where u is a uniformly distributed random variable between 0 and 1.0, D max is the maximum distance of a user from the AP, d (with 0 ≤ d ≤ 1) is the distance factor used to simulate the cases of users nearer or further from the AP such that a lower value of d means higher number of users are near to AP and higher value of d means users are uniformly distributed. Parameter d is a simulation parameter. It is used to generate cases varying from "small d: many users near to AP, hence good reception conditions" to "d = 1: all users spread uniformly with some users near the coverage edge". We simulate uniformly distributed users over a circle of radius D max * d meters. The default value of D max = 37m and an indoor scenario is considered in our simulations.

We generated cases corresponding to varied videos and different videos, for simulation and obtaining results, from a rate model provided by Ma et al. [START_REF] Ma | Modeling of rate and perceptual quality of compressed video as functions of frame rate and quantization stepsize and its applications[END_REF]: r = r max (q/q min ) -α (f ps/f ps max ) β (6

)
where q is 2 ((QP -4)/6) , q min corresponds to the value of q corresponding to the highest enhancement layer of SVC, r max corresponds to all the SVC layers for a given video and thus corresponds to q min , α is the rate decay factor that simulates the bitrates of lower layers and β impacts the bitrate depending on motion intensity in the video. In order to consider videos with different encoding complexities, we varied the values of r max : 44 Mbps and 32 Mbps and the rate decay factor α that is randomly assigned a value between 0.9 and 1.3 for a given simulation run. These values and ranges are obtained empirically after encoding the videos used in subjective testing.

We compare the proposed optimisation approach with 2 other approaches. First one transmits only the base layer of SVC with the lowest modulation (6 Mbits/s). In second scheme that we call "adaptive layer only" MANE uses the lowest modulation and adaptively selects some SVC layers such that the resulting bitrate satisfies the constraint in [START_REF] Heusse | Performance anomaly of 802.11b[END_REF]. Figure 4 shows the comparison and it can be seen that both the optimal and the second approach are significantly better than the first approach of transmitting only the base layer.

In order to compare the optimisation approach with the "adaptive layer only" approach, we plot CCDF (complementary cumulative distribution function) of the MOS scores obtained by different users in Figure 5 when the total number of users are 40. The approach "adaptive layer only" is denoted as "ref" and all other data points correspond to the optimisation approach with varying available channel resources ρ and distance factor d. It can be seen that the optimisation approach always provides better QoE to the users for example for ρ = 0.6 almost 95% users get a MOS score ≥ 4.0, whereas only 45% users are able to get MOS ≥ 4.0 with "adaptive layer only" when d = 0.8. This is because "adaptive layer only" approach doesn't adapt the modulation with respect to the channel conditions whereas optimisation approach adapts the number of SVC layers as well as modulation of different multicast sessions. It can be seen that in all the cases optimisation approach is better, however when lot of channel resources are available such that with ρ = 1.0 and when many users can only support the lowest modulation as they are far away, d = 1.0, then our approach provides only a slight gain. Moreover note that the data corresponding to "ref" does not change with d as in that scheme the modulation is not adapted to the channel condition of the users. Figure 6 compares the two approaches and shows the value of gain for the optimisation approach. Whereas, the gain(%) is defined as 100 • Qopt Q ref -100, where Q opt refers to sum QoE over all users for our approach and "ref" corresponds to the approach "adaptive layer only". It can be seen in the Figure that optimisation approach provides better gain with compared to the "ref" for different cases of ρ and d. It can be seen that gain increases with decreasing values of distance factor d and channel resources ρ and vice versa. This is because when users are far away that higher SVC enhancement layers cannot be transmitted as the modulation is low. Moreover, when high channel resources are available then almost all the SVC layers can be transmitted using the lowest modulation rate. The number of layers that can be transmitted using low modulation depends on the bitrate values and quality enhancements of different SVC layers. Thus we vary r max and rate decay α and the results are shown in Figure 7. It can be seen that when rate decay factor is high or r max is low then gain decreases. This is because lower bitrate values of different SVC layers permit the scheme "adapt layer only" to transmit many SVC layers using lowest modulation itself. However, the rate characteristics of the videos and modulation of the users change over time and hence our proposed scheme that is better in all cases with varying gain can optimise global QoE by adapting SVC layers as well as modulation of multicast sessions. Moreover as the time taken to solve the linear optimisation problem is of the order of 10ms, the algorithm can be run as soon as video bitrate characteristics or when the user modulation change.

VI. CONCLUSIONS AND FUTURE WORKS

We have proposed a mechanism that optimises QoE for transmission of SVC in wireless multicast environment. The results demonstrate performance improvements upto 55% in terms of quantified overall QoE as compared to the default basic-rate and the adaptive-layer approach. Moreover, the QoE optimising mechanism, which is based on binary linear programming, also includes the possibility of adapting the QoE objective according to the available resources.

For future work, we will continue working on QoE-driven resource management approach, more particularly, with more detailed simulations, more comparisons and a detailed loss model. For example, the trade-off of transmitting streams to a downstream network offering lower bandwidth or suffering from congestion or increased packet losses; this happens often in wireless access networks. One possible solution can use a Media Aware Network Element (MANE) to adjust the number of layers sent to an end user. Based on the QoE feedback, the MANE can take an action to drop some SVC layers. This will release more bandwidth for other traffic in the network. Furthermore, sometimes it is better not to decode the topmost SVC layer with high losses since QoE may be worst as compared to only decoding inferior layers.

Fig. 1 .

 1 Fig. 1. SVC scalabilities [4].

1 )

 1 Analysing the associated parameters and their impact on QoE. 2) Generating different distorted videos with different combination of the identified parameters. 3) Conducting new subjective tests to evaluate the distorted videos by a panel of human observers, 4) Training an RNN in order to capture the relation between the parameters and the perceived quality.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Videos used for simulation.

Fig. 4 .

 4 Fig. 4. Sum QoE comparison.

Fig. 5 .

 5 Fig. 5. CCDF of MOS scores with different values of ρ (available channel resources), and d (distance factor).

4 Fig. 6 .

 46 Fig. 6. Gain with different values of ρ (available channel resources), and d (distance factor).

Fig. 7 .

 7 Fig. 7. Gain with different values of Max SVC rate and Rate decay. The value of ρ = 0.6.

TABLE I

 I

		.	MEAN OPINION SCORE (MOS)
	MOS	Quality	Level of Impairment
	5	Excellent	Imperceptible
	4	Good	Perceptible but not annoying
	3	Fair		Slightly annoying
	2	Poor	Annoying
	1	Bad		Very annoying

This work was partly supported by the project CORRIDOR (ANR-11-VPTT-004) of the French national research organisation (ANR) and by the European project PROBE-IT.