Wael Eric Vezzoli

Ben Messaoud

Michel Amberg

Member, IEEE Betty Lemaire-Semail

Member, IEEE Frédéric Giraud

Marie-Ange Bueno

Physical and perceptual independence of ultrasonic vibration and electrovibration for friction modulation

Keywords: Tactile devices and display, Tactile stimulator, Electrovibration, Squeeze film effect, Ultrasonic devices, Electrostatic force, Friction modulation, Dynamic system response

Two different principles are available to modulate the user perceived roughness of a surface: electrovibration and ultrasonic vibration of a plate. The former enhances the perceived friction coefficient and the latter reduces it. This paper will highlight the independence of the two effects on the physical and perceptual point of view to confirm the increased range of sensation that can be supplied by the two coupled techniques. Firstly, a tribometric analysis of the induced lateral force on the finger by the two coupled effects will be presented, then a study on the dynamic of the two effects will be reported. In the end, a psychophysical experiment on the perception of the two coupled techniques will be shown.

INTRODUCTION

N the last years, a big interest has raised around the touch sense in the human-machine interaction, due to the relevance of this sense in the human perception through the identification and manipulation of objects. In the consumer world, the lack of the feedback of a real keyboard was one of the main issues during the spreading of the tactile screen technology in the beginning of the smartphone era. Nowadays, the tactile feedback problem is faced with the vibration of the device, which provides the user with some information, but is ineffective to simulate the tactile perception of the visualized surface on the screen. Nevertheless, a better immersion into virtual reality and a more efficient remote control will arise with the development of optimized tactile stimulation techniques. There are different approaches for the tactile simulation technique mediated by a flat surface: it is possible to simulate textures with controlled vibrations of a plate [START_REF] Wiertlewski | The Spatial Spectrum of Tangential Skin Displacement Can Encode Tactual Texture[END_REF] which recreate on the finger the pattern of vibration induced by the real texture. Then it is possible to control the friction coefficient between an active surface and a finger. Two techniques can be coupled with position sensing technology: the ultrasonic vibration of a plate and the electrovibration. The first is able to reduce the friction coefficient perceived by the user due to the creation of a film of pressurized air between the finger and the plate [START_REF] Biet | Squeeze Film Effect for the Design of an Ultrasonic Tactile Plate[END_REF][3] [START_REF] Ben Messaoud | Closed-Loop control for Squeeze Film Effect in Tactile Stimulator[END_REF], while the second increases it by the induced electrostatic force between the finger and the polarizing plate [START_REF] Ben Messaoud | Closed-Loop control for Squeeze Film Effect in Tactile Stimulator[END_REF][5] [START_REF] Mallinckrodt | Perception by the Skin of Electrically Induced Vibrations[END_REF] [START_REF] Bau | TeslaTouch: electrovibration for touch surfaces[END_REF]. Both of these technologies involve a global stimulation of the whole fingertip area touching the surface. The coupling of these technologies with finger position sensing technique allows establishing a spatio-temporal relation leading to the possibility of fine texture simulation [START_REF] Biet | Squeeze Film Effect for the Design of an Ultrasonic Tactile Plate[END_REF][3] [START_REF] Amberg | STIMTAC, a Tactile Input Device with Programmable Friction[END_REF]. The earliest attempt to couple ultrasonic vibrations and electrovibration was performed by Giraud et al. [START_REF] Giraud | Mergin two tactile stimulation principles: Electrovibration and Squeeze film effect[END_REF], confirming the technical possibility to use them together. The range of sensation can be enlarged due to their opposite influence on the perceived friction allowing a better rendering of simulated texture. The aim of this study is to show that the two principles can be added linearly in the static, dynamic and perceptual points of view. In this paper, both technologies will be explained with references to current models and the development of the haptic interfaces will be analyzed; then, the independence of the two tactile stimulation techniques will be shown in the static regime and an analysis of their dynamic behavior will be performed. In the end, a psychophysical experiment showing the linearity of the two effects in the perceptual point of view will be presented.

I ---------------- • E.V.

HAPTIC STIMULATOR

In this section, the two different tactile stimulators used in the experiments will be presented. The standalone stimulator will be used for static measure and psychophysical evaluation, because it performs simultaneously the two effects; whereas the fast response simulator will be employed in the dynamic behavior study.

Standalone stimulator

The haptic stimulator used for the static and the psychophysical experience is based on a modified USBstandalone version of the STIMTAC [START_REF] Amberg | STIMTAC, a Tactile Input Device with Programmable Friction[END_REF]. STIMTAC works according to ultrasonic vibrations of a plate, which can reduce friction. The tactile plate used in this experiment has a resonant frequency of 38220 Hz at ambient temperature. In the modified version, the stimulator has the inverter electrically uncoupled from the vibrating plate. In this way, it was feasible to introduce an amplifier to apply a high voltage to the tactile plate inducing the electrovibration effect. The elecrovibration force can be expressed as:

𝐹 ! = 𝜀 ! 𝐴 𝑣 ! ! 2 ! ! ! ! + ! ! ! ! 𝑇 ! + 𝑇 ! (1)
where 𝜀 ! is the vacuum permittivity, 𝐴 is the finger area in contact with the polarizing plate, 𝑇 ! and 𝑇 ! are the thicknesses of the insulator and the stratum corneum, 𝜀 ! and 𝜀 ! are the relative permittivities of the insulator and stratum corneum and 𝑣 ! is the effective voltage acting on the finger filtered by the human skin [START_REF] Vezzoli | Electrovibration modelling analysis[END_REF]. The utilized amplifier (UVA2kV, Ultravolt, USA) coupled with the signal generator inside is designed to provide a programmable voltage modulated at 440 Hz. In particular, it is possible to express the lateral force, 𝐹 ! , felt by the finger in an hybrid stimulator as:

𝐹 ! = 𝜇 -𝛥𝜇 𝐹 ! + 𝐹 ! ( 2
)
where 𝜇 is the friction coefficient between the finger and the plate, 𝛥𝜇 is the induced variation of the friction coefficient performed by ultrasonic vibrations [START_REF] Biet | Squeeze Film Effect for the Design of an Ultrasonic Tactile Plate[END_REF], 𝐹 ! is the normal force applied by the finger and 𝐹 ! is the induced normal force between the finger and the plate by electrostatic force from (1). To derive this formula, it is assumed that there is no mutual influence between the friction modulation induced by electrovibration and ultrasonic vibrations.

For static measure, in order to monitor the forces and the friction coefficient, a 6 axis force sensor (nano 43, ATI, USA) was used to ensure a resolution of 7.8 mN with a sampling period of 400 𝜇 s. To measure the vibration amplitude, an interferometer vibrometer was focused on the plate. To perform static measure, the tactile plate has been unmounted from the tactile stimulator and placed on the force sensor, figure 1a. At last, data are acquired with an oscilloscope (3014B, Tektronik, USA) then handled.

Fast response stimulator

To perform the dynamic behavior measurement, a specific, tactile stimulator has been designed to ensure a fast mechanical time response of the vibrating plate. It is needed to monitor the vibration amplitude of the plate itself because it is the effective parameter responsible for the generation of friction reduction [2] [11]. This plate has been equipped with five piezoelectric ceramics, four of them are used like actuators and one of them is used as a sensor. To calibrate the relationship between the signal provided by the ceramics and the effective vibration of the plate, a laser vibrometer has been used (OVF-5000, Polytech, Germany). To guarantee consistency with the previous setup, the spatial length of the vibration mode and the plastic film used to insulate for electrovibration are the same as the plate used for static measure, figure 1b. It may be noted that this plate was not designed to stimulate the finger with the two different technologies simultaneously.

PHYSICAL COUPLING

Static Independence

In this section, it will be demonstrated that there is no influence between the two tactile stimulation techniques while coupled. Electrovibration and friction reduction in ultrasonic devices act in opposition: electrovibration increases the normal force between the pulp of the finger and the plate whereas in the ultrasonic devices a reduction of the friction is experienced, see equation [START_REF] Biet | Squeeze Film Effect for the Design of an Ultrasonic Tactile Plate[END_REF].

The friction force and the normal force of a finger sliding on a tactile simulator have been recorded while the effects have been coupled with different phase shifts. The participant gave his informed consent to perform the experiment. This one has been carried out in free exploration by the user but data have been recorded after a period of training to assume the normal force constant as well as the speed. Before the experience, the subject was asked to wash his hands and completely dry them with talcum powder. To avoid the deposition of moisture on the plate, the exploration was allowed just 30 seconds before having a pause of 30 seconds to let the finger dry.

Visual feedback of the normal force applied on the plate was provided by the oscilloscope. The experiment was performed with 2 different phase shifts between the two effects (0 and 90 degrees) to study different possible interactions between the 2 principles, figure 2. It may be noted that before this measurement, the effect was a b singularly tuned on the participant finger to induce a similar reduction/increasing of the lateral force to highlight the possible mutual influence.

Figure 2a represents the case where the electrovibration effect and the ultrasonic vibrations are applied in phase. For a constant normal force applied by the finger while sliding, it has been recorded a constant lateral friction force.

On a different phase (90 degrees) between the two friction modulation techniques, a stair modulation appears on the measured lateral force, figure 2b. From all the performed measurements (about 50 tries), the friction coefficient has been computed for each considered case: electrovibration applied, ultrasonic vibration applied, no friction modulation and both effects applied. The average on the performed measurements is reported in figure 3. These measurements substantially confirm that the two techniques are fully mixable and that there is no mutual influence between the two in the static regime on the tribological point of view. Moreover, that is possible to find a level of the two effects to fully compensate each other and the resulting induced modulation of the lateral force.

Dynamic behavior

In this section, the possibility to couple the two effects also on a dynamic point of view will be discussed. For this experiment, the fast response simulator presented in section 2.2 was used. To compare the two effects, their rise time has been measured. This one is defined by the delay measured between the 10% and the 90% of the equilibrium value of the tangential force measure, after the application of the excitation. To perform this experiment, the tribometer detailed in [START_REF] Vezzoli | Electrovibration modelling analysis[END_REF] has been used due the larger bandwidth (10 kHz). A similar experience was performed by Meyer et al. [START_REF] Meyer | Dynamics of Ultrasonic and Electrostatic Friction modulation for Rendering Texture on Haptic Surfaces[END_REF]; they measured the response of the tangential force to a reference step either due to ultrasonic vibrations or electrovibration, for different finger speeds. They found a higher value of the time rise of the first compared to the latter worthy to be deeply investigated. To perform this measure, it was necessary to record the parameter generating tactile stimulation: the vibration amplitude for the ultrasonic devices and the voltage applied for the electrovibration. Moreover a unique surface to exploit both experiments is required to compare the two effects. For a similar plate design than the one used in [START_REF] Meyer | Dynamics of Ultrasonic and Electrostatic Friction modulation for Rendering Texture on Haptic Surfaces[END_REF] we have found a mechanical time response of the vibration mode of the plate of about 15 ms. This could be the cause of this discrepancy between the two stimulation principles. For this reason, the fast response stimulator described before has been used to conduct these experiments. This plate has a measured mechanical time response around 1.5 ms.

In figure 5 the measurements recorded on the same subject of the aforementioned analysis are presented. Each line represents the average of about 50 measurements. The tangential force response of both effects was analyzed for an increase and decrease of friction modulation. In the ultrasonic vibration related measure, the green line represents the vibration envelope of the plate, which has been measured through sensor ceramic. The voltage provided to the plate has been tuned to induce a vibration amplitude of 2 µμm peak to peak with the presence of the finger. To ensure a uniform mechanical behavior of the finger, the normal forces for the reported measurements are equal to 0.3 N with a standard deviation of 18%. At this point, it was possible to proceed with the identification of the model to link the parameter generating the friction force modulation. To do so, the MATLAB toolbox for Black-Box modeling has In the first graph is reported the presence of the effect, electrovibration (blue) and ultrasonic vibrations (red). In the second graph is reported the recorded lateral force. In the third graph is reported the measured normal force.

been utilized considering a second order model. The input parameter has been the vibration amplitude envelope for the ultrasonic device model identification and the applied voltage envelope for electrovibration. For each situation, the calculated model has been evaluated with a step function to investigate the actual behavior of the effect itself. In accordance with the approach reported in [START_REF] Meyer | Dynamics of Ultrasonic and Electrostatic Friction modulation for Rendering Texture on Haptic Surfaces[END_REF], the parameter evaluated to compare the effect has been the rise time of the response of the identified model to the step. The results are reported in figure 4.

PERCEPTUAL INDEPENDENCE

In this section, a psychophysical experiment will be presented, showing the possibility to find a level of compensation between the two effects, with which the user cannot distinguish between a surface with compensated friction modulation and a surface without any friction modulation. To perform so, the standalone stimulator was programmed to induce a friction reduction on the plate following a sinusoidal profile, with a spatial wavelength of 1 cm. Then, it has been implemented the possibility to have the same spatial profile function with the electrovibration modulation. It was possible to adjust the electrovibration maximal voltage applied to the plate on eight different levels, from 0 to 2000 V. The levels were equally spaced with a singular amplitude level of 287 V. The aim of the experiment is to know if for all the users, it was possible to a level of electrovibration where the friction modulation induced by the ultrasonic vibration was totally compensated and then the plate surface was perceived as uniform. It was asked to six subjects to freely explore the tactile plate while maintaining the normal force applied around 0.4 N. It was necessary to introduce this limitation to ensure that the changing force did not influence the friction reduction induced by the ultrasonic device [START_REF] Biet | Squeeze Film Effect for the Design of an Ultrasonic Tactile Plate[END_REF] and guarantee that the vibration damping of the plate induced by the presence of the finger was stable.

To avoid any noticeable difference into the dynamic of the two effects and the time response of the system that could induce distortion into the perception of the surface, it was asked to the participants to maintain a constant exploration speed (smaller than 40 mm/s). All of the participants gave their informed consent for the execution of the experiment. It was asked to the subjects to wear an anti-noise headset to prevent any influence of the sounds produced by the plate on their answer. The value of the normal force was provided via a graphical interface to the subjects. Before each experiment, the tactile plate was cleaned and the participants were invited to wash their hands and dry them with talcum. Before starting the experiment, the participants had the time to get accustomed to the friction modulation induced by the ultrasonic vibration with the compensation effect induced by electrovibration set to 0 (no effect of increased normal force). When the participant communicated that he was ready and exhibited a stable normal force while exploring the plate, the experiment started. The level of electrovibration was increased by one step and the subject was asked how the surface was and how it was in comparison with the set-up situation. The participant could describe the surface as non uniform or uniform, then the level of electrovibration compensation was increased by another step. To avoid moisture deposition, the exploration of the plate was allowed for 20 seconds with 20 seconds of pause to let the finger dry. After each participant, the tactile plate was cleaned to remove any remaining deposit. The starting level of compensation was always available to the users to compare the friction modulation. Every participant could find a level of electrovibration which fully compensates the friction modulation induced by the ultrasonic vibration. Whereas in other cases, the surface was perceived as non-uniform. After finding the level where the modulation was not perceived anymore, a different level of compensation was selected and it was asked to the participants if it was uniform or not. In case of non uniformity, the compensation level was reduced or increased by one step into the direction of the previous indicated compensation level. All the participants indicated the same level as before as the one exhibiting the uniform surface (see figure 6). The compensation level necessary to cancel the influence on the lateral force induced by the ultrasonic vibration exhibits a huge variability between the subjects. The variability of the compensation force can be explained by the variability of the thickness of the stratum corneum on the human fingertip [START_REF] Abel | Thickness of the Stratum Corneum of the Volar Fingertips[END_REF] or its water content, which strongly influences the induced electrostatic force on the finger [START_REF] Vezzoli | Electrovibration modelling analysis[END_REF], as well as the possible variability of the friction modulation induced by ultrasonic vibration which has not been analyzed yet.

DISCUSSION

The static dependence measure confirmed that it is possible to constructively couple the two effects to enhance the range of lateral force modulation experienced by the finger.

Moreover, it has been demonstrated that if the influence on the lateral force by the two effects is set as similar, it is possible to obtain a friction modulation equal to zero while coupling them in phase. These measurements confirm the physical independence of the two effects and confirm the validity of the equation [START_REF] Biet | Squeeze Film Effect for the Design of an Ultrasonic Tactile Plate[END_REF]. In this range of induced modulation, it is possible to conclude that the two effects are purely additive, and there is not any mutual influence.

The measured dynamic of the two effects is an indicator of their behavior in the varying regime considering a real finger. The experiment substantially confirms that in most of the cases considered, the rising times of the two effects are comparable. Thus, they can be successfully coupled additively also in dynamic. An exception is the case of rising friction for ultrasonic vibrations with low exploration speed (20 mm/s), this is probably due to some stick-slip effect that happened after the increasing of the friction. This phenomenon can be seen in figure 5b for the lower speed of exploration. For this reason, the rising time is higher than in the other measured cases. For a hybrid tactile stimulator, it would be possible to introduce a pulse of electrovibration to increase the rising time in this low speed situation to realign the time response to the other cases and obtain a linear stimulator in dynamics. To obtain a linear coupled tactile stimulator it is necessary to design the mechanical plate and chose the amplifier for the electrovibration to match the dynamic of the two effects. To obtain the effective rise time of the friction for the developing of a linearized hybrid tactile stimulator, it has to be considered the mechanical rise time of the plate for ultrasonic vibrations and the rise time of the amplifier for the electrovibration. The former is dependent on the Q factor of the plate and can be reduced to 1-2 ms where the latter is just dependent on the amplifier and, with the right amplifier choice, could be neglected.

These results are obtained considering all the device parameters which affect the stimulation principle itself and are consistent between the two different techniques. As suggested by [START_REF] Meyer | Dynamics of Ultrasonic and Electrostatic Friction modulation for Rendering Texture on Haptic Surfaces[END_REF], fingertip mechanical behavior plays an important role in the determination of these forces and the analysis of these rising times is consistent with the analysis proposed by Wiertlewski et al. [START_REF] Wiertlewski | Mechanical Behavior of the Fingertip In The Range of Frequencies and Displacements Relevant to Touch[END_REF] for the mechanical response of the fingertip. Their analysis predicts a fingertip response in the order of milliseconds, which is in good agreement for all the considered friction modulation changes.

The psychophysical experiment on the two coupled effects confirms that it is always possible to find a level where the two effects fully compensate each other. The experiment also shows that the two friction modulations are perceived in the same way by the users. The compensation level reported in figure 6 shows that there is no difference, from the point of view of the user perception, between a real surface and a surface where the friction modulation is compensated. This does mean that the users cannot perceive the presence of the effects on the surface, but perceive just the impact of those effects on the lateral force.

With these techniques, it is possible to change the friction coefficient of a surface, without influencing all the other parameter concurring to generate the user perception of the surface itself.

CONCLUSION

This paper analyzed the possibility to successfully couple two friction modulation techniques, electrovibration and ultrasonic vibrations, to expand the range of sensation provided to the users. An analysis of the time rise of these two effects decoupled has been proposed to analyze the possibility to couple them also in the dynamic domain. The psychophysical evaluation confirms that the effects are fully mixable and the user cannot distinguish the presence of the two compensated effects on the surface to the nonmodulated one because he/she perceives this surface as uniform.

ACKNOWLEDGMENT

The author would like to thank Clement Nadal for the useful discussions, FP7 Marie Curie Initial Training Network PROTOTOUCH, grant agreement No. 317100 for founding and IRCICA laboratory for hosting.

Fig. 1 .

 1 Fig. 1. a. Static measurement system. The red spot on the plate is the focus of the laser of the vibrometer. b. cartography of the fast response stimulator.

Fig. 3

 3 Fig.3Calculated 𝜇 for the different modulations in the considered cases. When the effects are regulated to induce the same friction modulation, their coupling results in a compensation of the induced friction modulation.

Fig. 4 .

 4 Fig. 4. Rise time of the step response for the identified models (ultrasonic vibration UV et electrovibration EV) based on the previous measures for the two different friction cases considered: mounting friction (left) and descending friction (right).

Fig. 2 .

 2 Fig. 2. Example of measure of coupling of electrovibration and ultrasonic vibrations in phase a, and in quadrature b.In the first graph is reported the presence of the effect, electrovibration (blue) and ultrasonic vibrations (red). In the second graph is reported the recorded lateral force. In the third graph is reported the measured normal force.

Fig. 5 .

 5 Fig. 5. Friction modulation for different finger speeds related to the measured plate vibration or voltage envelope reported in the figure. The shadowed area represents the measure standard deviation. a, descending friction ultrasonic devices, b increasing friction ultrasonic devices, c decreasing friction electrovibration and d increasing friction electrovibration.

Fig. 6 .

 6 Fig.6. Compensation level of electrovibration necessary to cancel the friction modulation induced by the ultrasonic devices and have a uniform perception of the surface.


Eric Vezzoli
email: eric.vezzoli@ed.univ-lille1.fr

Ddzidek Brigyda

Thomas Sednaoui
email: thomas.sednaoui@st.com

Member, IEEE Frédéric Giraud
email: frederic.giraud@polytech-lille.fr

Adams Michael

Member, IEEE Betty Lemaire-Semail

Brygida Dzidek
email: b.m.dzidek@bham.ac.uk

Michael Adams
email: m.j.adams@bham.ac.uk

Role of Fingerprint Mechanics and non-Coulombic Friction in Ultrasonic Devices

  L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Current touch based user interfaces lack natural or at least credible haptic feedback, for example, when interacting with a virtual keyboard, which was one of the main issues during the initial commercialization of smartphones. Haptic interaction is still limited to the vibration of the whole device, but is ineffective in recreating the sensation expected from an image on the screen. Few approaches have shown a realistic potential for coupling with the flat capacitive position sensing displays that are employed in consumer devices. One involves the use of a lateral vibrating surface to simulate the vibrotaction signal of the fingertip induced by sliding over a real texture [START_REF] Wiertlewski | The Spatial Spectrum of Tangential Skin Displacement Can Encode Tactual Texture[END_REF]. Another is based on the modulation of the friction between a sliding finger and the active surface. The possibility of tracking the position of a finger led to the generation of a spatio-temporal relationship that is useful for simulating texture [START_REF] Biet | Discrimination of Virtual Square Gratings by Dynamic Touch on Friction Based Tactile Displays[END_REF]. Stimulating the whole fingertip area in the same manner led to the classification of this approach as global stimulation. Electrovibration and ultrasonic vibration are the two leading technologies for coupling with capacitive touch screens. The former is based on the polarization of the finger pad with a high voltage supplied plate covered with an insulator [START_REF] Vezzoli | Electrovibration Modeling Analysis[END_REF]- [START_REF] Kaczmarek | Polarity Effect in Electrovibration for Tactile Display[END_REF], the resulting attractive force results in an increased friction while interacting with the display. The latter involves out-of-plane ultrasonic vibration to attenuate the friction between the contacting finger pad and the display. The generation of a squeeze film effect between a vibrating plate and the finger pad was the first mechanistic interpretation [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF], [START_REF] Watanabe | A method for controlling tactile sensation of surface roughness using ultrasonic vibration[END_REF]. It relied on a simplified model of the interaction between two smooth rigid surfaces because of the enormous complexity of a real finger pad-display interaction e.g. the large range of Knudsen numbers involved and the viscoelastic deformation of a finger pad. There is evidence that squeeze flow is an important contributory factor [START_REF] Vezzoli | Pressure dependence of friction modulation in ultrasonic devices[END_REF], modeled by Biet et al. [START_REF] Biet | Implementation of tactile feedback by modifying the perceived friction[END_REF], but an analysis of the mechanical and tribological interactions between a finger pad and an ultrasonic vibrating plate has never been performed. This is essential because the coefficients of friction associated with such devices can be > 0.1, which is greater than that which would be expected by an effective acoustic levitation mechanism (< 0.01); intermittent finger pad contacts have been confirmed recently [START_REF] Dai | LateralPaD: A surface-haptic device that produces lateral forces on a bare finger[END_REF]. The aim of the current work is to introduce a first order mechanical model of the finger pad that is applicable to the ultrasonic domain. It is clear that the model is not complete, especially it neglects inertia and viscosity of the fingerprints. Nevertheless it can help to establish key principles to be further extended. Firstly, the mechanical interplay between the proposed model with a lateral vibrating ultrasonic plate is evaluated and a psychophysical assessment is proposed to assess the relevance of the approach. Secondly, the interaction with a normal ultrasonic vibrating plate is analyzed and the reduction of the friction with a smooth surface is estimated.

II. FINGER PAD TRIBOLOGY

Provided that a finger pad is sufficiently moist, particularly due to occlusion of moisture secreted from the sweat pores in the finger print ridges when in contact with a smooth impermeable surface such as glass, the friction may be described by stochastic molecular theories that have been developed for elastomers [START_REF] Gent | Adhesion of elastomers: Dwell time effects[END_REF]. Friction arises from the intermittent extension and rupture of molecular junctions at the sliding interface, which involves dangling macromolecular chains; such bonds are continually pinning and depinning due to thermal activation. A characteristic of elastomers is that they exhibit aging such that the adhesion to a rigid substrate increases with the dwell time. This arises because molecular rearrangements occur at the interface and the associated relaxation times are coupled to those for pinning. Thus disruption of the pinning process will result in a reduction of the friction. Frictional aging for finger pads is an extreme example that occurs over tens of seconds due to the plasticization of the asperities on the finger pad ridges by the secreted moisture, which results in an increase in the contact area [START_REF] Dzidek | Role of Occlusion in Non-Coulombic Slip of the Finger Pad[END_REF]. It is reasonable to suppose that this is accompanied by aging associated with the formation of molecular bonds.

III. LATERAL ULTRASONIC VIBRATIONS

A. Introduction

In this section, the influence of lateral vibration on a finger pad under a small tangential load, which is less than that required for slip, is considered. It is proposed that the friction is reduced by micro-slip due to the disruption of the molecular bond network. Such slip requires the elastic retractive force induced in the finger pad by the periodic displacement to be greater than the frictional force. It is probable that, at low vibrational frequencies, this displacement will be completely accommodated by the gross deformation of the finger pad. However, the finger pad is viscoelastic so that the stiffness increases with frequency [START_REF] Wiertlewski | Mechanical behavior of the fingertip in the range of frequencies and displacements relevant to touch[END_REF]; at sufficiently high frequencies it will be assumed that (a) the response is entirely elastic and (b) the finger pad is sufficiently rigid that the deformation is localized to the finger print ridges.

B. Bed of springs approximation

The elastic response of a finger pad will be modelled as a bed of springs as shown schematically in Fig. 1. Different models have been proposed based on a combination of springs or springs and damper [START_REF] Ho | A novel model for assessing sliding mechanics and tactile sensation of human-like fingertips during slip action[END_REF], but no one focused on the behavior of the fingertip in the ultrasonic domain. The springs may be treated as finger print ridges for current purposes. The normal stiffness, 𝑘 " , is defined by:

𝑘 " = 𝐸 % 𝐴 ℎ % (1)
where 𝐸 % is the Young modulus of the stratum corneum, 𝐴 is the contact area between the plate and the ridges and ℎ % is the undeformed height of the ridges. The tangential stiffness, 𝑘 ( , is defined by:

𝑘 ( = 𝐺 % 𝐴 ℎ % and 𝐺 % = 𝐸 % 2(1 + ν) (2)
where 𝐺 % is the shear modulus and ν is the Poisson modulus of the ridges. The dependence of 𝐴 on the normal force, 𝐹 " , is assumed to be:

𝐴 = 𝑘(𝐹 " ) 2 3
(3

)
where 𝑘 is equal to 138.1 mm 8 /N 2 3 (this value is the result of a best fit to the experimental data reported in [START_REF] Adams | Finger pad friction and its role in grip and touch[END_REF]). The other parameters considered in the analysis are given in table 1. Thus, the value of 𝑘 ( could be calculated from (2) using that of 𝐴 obtained from (3).

C. Modelling slip

The criteria for stick and slip are given by the following expressions:

𝑘 ( Δx >?@ ≤ 𝜇 C 𝐹 "
No slip ( 4)

𝑘 ( Δx >?@ > 𝜇 E 𝐹 " Slip
where Δx >?@ is the maximum displacement of the springs induced by the lateral vibrating plate, 𝜇 C is the static coefficient of friction and 𝐹 " is the applied normal force across the bed of springs. If the slip condition is satisfied, there is relative displacement when 𝑘 ( Δ𝑥 > 𝜇 C 𝐹 " until the relative velocity between the finger pad and the plate is zero, 𝑉 = 0, Fig. 2. In the represented case, 𝜇 C is 1, the dynamic coefficient of friction 𝜇 E is 0.7, and the applied normal force is 0.5 N. The calculation ignores the offset tangential preload force, which could not be measured in the current work. If it is greater than the force induced by the periodic displacement, then no slip will occur in the direction of the offset force and slip will be enhanced in the other direction, so that the model will correspond to a lower bound solution. In Fig 2, the case for an ultrasonic harmonic vibration of 20 kHz of 5 µm for an applied force of 0.5 N is represented.

Figure 2: Steady state lateral sliding where x is the normalized position of the plate, V is the normalized speed of the plate and 𝐹 ( is the lateral force experienced by the finger. In the gray shaded area, the relative motion between the ridges and the plate is zero. In the green crossed area the finger is sliding on the plate.

A glide threshold is defined as the minimum amplitude of vibration necessary to induce a transition from stick to slip as represented in Fig. 2:

Δx J = 𝜇 C 𝐹 " 𝑘 ( (5)
where Δx J is the critical threshold displacement. The contact area between the finger pad and the plate is dependent on the applied normal force (3) and this dependence influences the value of 𝑘 ( (2). It is possible to compute the dependence of the threshold displacement on the applied normal force by solving (4) for small time steps to calculate the associated lateral displacements, 𝛿𝑥, with the limit defined by (5) as shown in Fig. 3a.

(a) (b)

D. Psychophysical validation 1) Experimental setup

In order to validate the model, it is necessary to evaluate the induced tactile response of a lateral vibrating plate. A Langevin actuator (FBL28302SSF-FC, Fuji Ceramics Corporation, Japan) was employed with a closed-loop vibration amplitude stabilization system to ensure the stability of the vibration amplitude during the experiment. The employed actuator has a resonance frequency of 27.83 kHz at ambient temperature. The control of the piezoelectric actuator was implemented with piezoceramic sensor glued on the body. A DSP (EZDSPF2812, Texas Instrument, USA) was programmed do obtain the desired behavior of the device. The actuator has two parallel surfaces at the top of the device (2x1.4 cm 8 ) on both sides for making contact with the finger pad, Fig 3b.

2) Methododolgy and results Four participants (male, age between 26 and 41) were asked to describe their tactile perception while holding the actuator as in Fig. 3b for vibrational amplitudes of 0, 2 and 4 µm after imposing a small tangential force on the vibrator that was insufficient to induce gross slip. All participants gave their informed consent to perform the experiment. They were equipped with a headset to cancel the influence of environmental noise and they were asked to close their eyes during the experiments. The frequency of the stimulation was the resonance of the actuator at 27.83 kHz. The amplitudes of vibration were changed randomly and the participants reported their corresponding tactile evaluation. None of the participants could identify a difference between the 2 µm amplitude and the static condition. However, all reported a sensation of slip for an amplitude of 4 µm with comments such as "it felt more slippy." This amplitude corresponds to the glide threshold for an applied normal force of about 0.4 N, which is typical of the values used in touch [START_REF] Adams | Finger pad friction and its role in grip and touch[END_REF].

IV. NORMAL VIBRATIONS

A. Introduction

In this section, the interaction of the proposed model with a normal vibrating plate is proposed, and the friction reduction arising from the interaction with a smooth surface is estimated. It has been shown that an ultrasonic vibrating plate in the normal direction can induce intermittent contact with a finger pad for vibrational amplitudes of a few micrometres [START_REF] Dai | LateralPaD: A surface-haptic device that produces lateral forces on a bare finger[END_REF]. This attach/detach phenomenon could lead to a friction reduction during the sliding motion of the finger due to the influence on the interfacial molecular bonding interactions in addition to the development of a gaseous squeeze film. Here, the loading/unloading cycle of a finger pad again will be represented by a bed of equivalent elastic springs to investigate the mechanism in more detail. A key factor is that the detachment will lead to rupture of the interfacial bonds and thus finite time is required for repinning. It is assumed that the viscoelastic retardation time of finger pad is infinitely long during the unloading cycle of an out-of-plane imposed vibration compared to the reciprocal frequency of the vibration. The finger print ridges are much stiffer than the finger pad and also it is assumed that they will behave elastically during unloading.

B. Bed of springs model

Following a compression cycle of a continuous out-of-plane vibration, the springs will unload and detachment will require the vibration amplitude to be greater than the unloaded length of the spring. The critical amplitude to induce detachment, 𝑊 N , is given by the following expression for the current model:

𝑊 N = Δz " = 𝐹 " 𝑘 " (6)
where Δz " is the compression induced by the normal load and 𝑘 " was calculated from [START_REF] Wiertlewski | The Spatial Spectrum of Tangential Skin Displacement Can Encode Tactual Texture[END_REF]. The dependence of the necessary vibrational amplitude to induce detachment of the fingertip as a function of the applied force in the range relevant to touch is reported in Fig 4 . The analysis is valid for frequencies greater than 20 kHz.

C. Non-Coulombic friction

In the previous section, it was shown that detachment was possible for a range of normal vibrational amplitudes that are compatible with those commonly employed in ultrasonic devices. Assuming that there is not frictional aging, then it is possible to calculate the minimum effect on the periodic frictional force by taking account of the fact that the coefficient of friction decreases with increasing normal force for smooth surfaces [START_REF] Dzidek | Role of Occlusion in Non-Coulombic Slip of the Finger Pad[END_REF]; this analysis is relevant for the possible application of the principle to the screen of smartphone and tablets. Finally, a comparison with experimental data will be made to examine the significance of the phenomenon. At equilibrium, the integral of the imposed force from the the bed of springs must be equal to the total reaction force of the plate over a vibration cycle: 𝐹 P%?N 𝑑𝑡 = 𝐹 " 𝑇 NTN(% [START_REF] Watanabe | A method for controlling tactile sensation of surface roughness using ultrasonic vibration[END_REF] where 𝐹 P%?N is the normal reaction force of the plate in contact with the springs and 𝑇 is the vibrational period of the plate. The vibration of the plate is described by 𝜙 sin (2𝜋𝑓𝑡), where 𝜙 is the vibrational amplitude and 𝑓 is the frequency. In this case, the frequency is 20 kHz leading to a period of 50 µs. With this relationship it is possible to compute the instantaneous compressive force, 𝐹 P%?N , of the springs induced by the vibrating plate; the case shown in Fig. 5 is for an applied normal force of 0.5 N. The corresponding frictional force, 𝜇 E 𝐹 P%?N , is plotted in Fig. 6. It may be seen that the contact is frictionless for an increasing proportion of the unloading cycle as the amplitude of vibration increases. The radial contact pressure profile for a finger pad in contact with a smooth flat plate is not uniform as assumed above but can be approximated by the Hertzian function:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 -5 -0.2 0 0.2 0.4 0
𝑃(𝑟) = 3 2 𝐹 " 𝜋𝑏 8 1 - 𝑟 8 𝑏 8 (8)
where 𝑏 is the radius of contact and 𝑟 is the radial coordinate with an origin at the center of contact. From a knowledge of 𝑃 𝑟 , it is possible to integrate the reaction force across the contact over the cycle period and hence compute the mean normal force as a function of time. The frictional force can then be calculated from the following integral:

𝐹 ( = 𝜇 E (𝐹 P%?N )𝑃(𝑟) b c 𝜋 8 𝑑𝑟 (9)
where 𝜇 E is a function of the normal force as given in Fig. 7. Fig. 8 shows that the calculated value of the relative coefficient of friction, 𝐹 ( (𝜙) 𝐹 ( (0), decreases with increasing vibration amplitude.

D. Experimental comparison

It was possible to compare the simulated friction reduction recovered by this method with data taken from [START_REF] Sednaoui | Experimental evaluation of friction reduction in ultrasonic devices[END_REF] to highlight the compatibility of the effect with the general results observed for ultrasonic devices. In that work, the friction reduction induced by the normal ultrasonic vibrating device has been measured with a tribometer for different index fingers and for different loads applied. Fig. 9 shows a comparison of the reduction in the coefficient of friction measured for three fingers with an applied normal force of 0.35 N for different vibrational amplitudes. The model predicts the trend of decreasing friction with increasing amplitude as observed experimentally, even though the starting slope is mismatched. This phenomenon is due to the full elastic implementation of the simulation, the lack of a viscoelastic term leads to a poor simulation for small amplitude of vibration. It may be noticed that this approach, at the contrary of the squeeze film hypothesis, leads to a finite value of friction modulation without predicting full finger levitation.

Figure 9: Experimental comparison between the modeled phenomena and experimental results taken from [START_REF] Sednaoui | Experimental evaluation of friction reduction in ultrasonic devices[END_REF] for normal vibration; the considered frequency was 25 kHz.

V. DISCUSSION

The proposed elastic model is a considerable simplification of the real system, but nevertheless can explain part of the behavior of a finger pad sliding over an ultrasonic vibrating plate. The reduction in friction has been used for the creation of different tactile stimulators [START_REF] Amberg | STIMTAC: A Tactile Input Device with Programmable Friction[END_REF], [START_REF] Winfield | T-PaD: Tactile Pattern Display through Variable Friction Reduction[END_REF], but the underlying mechanisms are still unresolved. The squeeze film effect has been shown to be a contributory factor [START_REF] Dai | LateralPaD: A surface-haptic device that produces lateral forces on a bare finger[END_REF] but the behavior of the friction modulation is not in complete agreement with the analysis proposed in [START_REF] Biet | Squeeze film effect for the design of an ultrasonic tactile plate[END_REF], which suggests that there are other concomitant contributions. The currently proposed elastic model of the fingerprint ridges is consistent with a psychophysical assessment of the influence of lateral ultrasonic vibrations. In particular, it is clear that ultrasonic perturbation frequencies are essential to induce a sufficiently stiff response of the finger pad and also cause micro-slip in a temporal domain that is at least comparable to the times required to thermally reactivate a molecular junction. Published values of such times are sparse but it is interesting that, for thin molecular layers in aqueous media, repinning times are ~ 200 µs [START_REF] Drummond | Friction between two weakly adhering boundary lubricated surfaces in water[END_REF], which is comparable to the period (50 µs) of the vibrational frequency (20 kHz).

In the case of normal vibrations, the results of the model are consistent with the trends observed experimentally [START_REF] Dai | LateralPaD: A surface-haptic device that produces lateral forces on a bare finger[END_REF]. Taking account of the non-linearity of the coefficient of friction was an important factor in the model. Again, high frequencies are a prerequisite for ensuring that the gross retardation time of the finger pad is much longer than the cycle time of the vibration. While the reduction in friction calculated with the current model was based on the separation time between a finger pad and a virtual vibrating plate, ultrasonic frequencies will induce a synergistic effect of ensuring that the disrupted molecular junction network remains at least partially disrupted even during contact. This would result in a further reduction in the coefficient as well as a possible contribution from a squeeze flow effect. It is also possible that an additional effect could be attributed to a reduction in the occlusion arising from the secreted moisture from the sweat pores (deocclusion). Essentially, the intermittent loss of contact could allow some moisture to diffuse from the contact region and hence result in asperity hardening compared with a continuous intimate contact Simple models of the type described in the current work are useful in order to establish key principles for formulating models that are more complete. The non-linear viscoelastic behavior of the finger pad is particularly relevant since it controls the gross temporal response of the finger pad. In addition, it is important to consider the details of the contact mechanics because detachment involves radial crack propagation in which an annulus of failure grows from the periphery of a contact [START_REF] Wiertlewski | Mechanical behavior of the fingertip in the range of frequencies and displacements relevant to touch[END_REF]. Consequently, different mechanisms may dominate across the contact region e.g. the squeeze flow mechanism may be localized at the periphery of the contact.

VI. CONCLUSIONS

A simple model of the interaction of a finger pad with a plate vibrating at ultrasonic frequencies, which is based on representing the finger print ridges as elastic springs, captures some of the critical factors that govern the performance of vibrational haptic displays. In particular, it is concluded that ultrasonic frequencies are essential for ensuring that the finger pad is sufficiently stiff to cause periodic contact separation for out-of-plane vibrations. More work is required to establish the relative contributions of this mechanism, squeeze flow and deocclusion. For both normal and lateral vibrations, it is also necessary to consider the disruption of the network of molecular junctions that are responsible for friction.

Figure 1 :

 1 Figure 1: Bed of springs model.

Figure 3 :

 3 Figure 3: Glide threshold as a function of the applied force (a) and experimental setup for psychophysical evaluation (b).

Figure 4 :

 4 Figure 4: Detach condition of the fingerprint in function of the applied force: as an example, a 0.15 N applied load with a vibrational amplitude of 1.2 µm is required to induce detachment.

Figure 5 :

 5 Figure 5: Plate reaction force as a function of the time over a vibration cycle for different vibrational amplitudes. The amplitude of vibration varies from 0.1 µm (blue) to 8 µm (red). During the detachment of the springs from the plate, the plate reaction force is zero.

Figure 6 :

 6 Figure 6: Instantaneous frictional force as a function of the time over a vibration cycle for different vibrational amplitudes. The amplitude of vibration varies from 0.1 µm (blue) to 8 µm (red). During the detachment of the springs from the plate, the frictional force is zero.

Figure 7 :

 7 Figure7: Coefficient of friction between a finger pad and a smooth surface as a function of the applied normal force[START_REF] Gent | Adhesion of elastomers: Dwell time effects[END_REF].

Figure 8 :

 8 Figure 8: Relative coefficient of friction as a function of the vibrational amplitude induced by the non-linearity of the coefficient of friction.

VII. ACKNOWLEDGEMENTS

The authors would like to thank Sofiane Ghenna for the useful assistance with the piezoelectric actuator. The authors would like to acknowledge FP7 Marie Curie Initial Training Network PROTOTOUCH, grant agreement No. 317100 as well as Nord Pas-de-Calais region, and FEDER within framework of Touch-it project for founding.

* This work was funded by the European Union under the FP7 programs FP7-PEOPLE-317100 PROTOTOUCH.


F Gruy
email: gruy@emse.fr

M Cournil

Light Scattering Cross Section of Hydrophilic and Hydrophobic Silica Aggregates

Keywords: Light scattering, aggregates, hydrophilic, hydrophobic, silica

The aggregation dynamics of solid particles in liquid media is currently followed by optical based particle sizing methods. Because it can be used in situ and applied to a wide particle size range, turbidimetry is acknowledged as one of the best methods for this characterization.

Although much work has been done about aggregation, some aspects are less known and require additional experimental and theoretical research. This is particularly the case of aggregation of hydrophobic particles. Corresponding aggregates are 3-phase objects (solidliquid-gas) the morphology and optical properties of which are not known. Present work rests on the turbidimetric study of hydrophilic and hydrophobic silica samples in stirred aqueous solutions. Modelling involves different aspects: aggregate morphology, aggregate optical properties, and aggregation dynamics. This paper particularly emphasizes the second one.

Fractal-like models are proved to be representative of the aggregate morphology even at small size. Light scattering cross-section of the aggregates is calculated from their averaged projected area; effective refractive index is proved to be a good parameter for modelling their optical properties both for hydrophilic and hydrophobic aggregates. Classical models of porous aggregate formation (Kusters theory) are used for describing the aggregation dynamics.

Introduction

Aggregation of hydrophilic particles in stirred liquid media can be considered as a relatively well understood process in spite of the variety and complexity of its aspects: physicochemical, hydrodynamic, morphological [START_REF] Elimelech | Particle Deposition & Aggregation Oxford[END_REF]. On the contrary, aggregation of hydrophobic particles in aqueous media or more generally solid particles in non wetting media is less known, at least on certain aspects, particularly the aggregate structure and properties.

Experimental works have confirmed the existence of strong long range (20-200 nm) attractive forces between hydrophobic surfaces in water [START_REF] Yushchenko | Interaction between particles in a nonwetting liquid[END_REF][START_REF] Parker | Bubbles, cavities, and the long-ranged attraction between hydrophobic surfaces[END_REF]. This hydrophobic interaction was quantified from direct force measurements and proved to be, in most cases, much larger than the van der Waals forces. Previous experimental results have given rise to intensive theoretical work about the origins of the hydrophobic interaction. Several interpretations have been explored in some details [START_REF] Attard | Nanobubbles and the hydrophobic interaction[END_REF]. It seems that the most likely explanation focuses upon the bridging of nanobubbles which are present on the hydrophobic surface. The existence of these bubbles was first deduced from force measurements and then confirmed by direct observations [START_REF] Parker | Bubbles, cavities, and the long-ranged attraction between hydrophobic surfaces[END_REF]Yabukov et al., 2000;[START_REF] Attard | Nanobubbles and the hydrophobic interaction[END_REF]. Although experimental evidence and thermodynamic interpretation conclusively prove that gas bridging from pre-existing nanobubbles is of major importance in hydrophobic aggregation, some theoretical aspects are less understood. In particular, volume fraction and spatial repartition of gas in hydrophobic aggregates are not known.

Here a few results are presented from an experimental study of aggregation of two silica samples coming from the same lot: the original sample is naturally hydrophilic; the other has been made hydrophobic by surface chemical processing. Aggregation process is followed by in situ turbidimetry, the interpretation of which requires knowledge of the light scattering cross section of aggregates.

Experimental Methodology

Materials and methods

Original solid samples consist of monodisperse silica spheres (0.5 µm in diameter, Geltech Inc product code: S0501). They are highly insoluble in water and can be made hydrophobic by superficial grafting of carbon chains. The grafting procedure consists first of an hydroxylation process which produces highly reactive silanol (-SiOH) groups, then of a wetting step with a N,N dimethyloctadecylaminosilane solution which produces the surface silanisation reaction, i.e. the grafting of a carbon chain via a strong Si-O-Si bond. The contact angle on this processed surface is 125° in pure water. Both silica particles, processed or unprocessed, have a zero point charge pH of 3.2 [START_REF] Cugniet | Etude de l'agrégation de particules solides en milieu non mouillant Ecole des Mines de Saint-Etienne[END_REF][START_REF] Gruy | Influence of non-wetting on the aggregation dynamics of micronic solid particles in turbulent medium[END_REF].

The reactor is a double jacketed 4-baffled 1.45 L vessel. It is provided with a watertight cover so that it can be completely filled with liquid, avoiding in this the presence of a free interface which could entrap hydrophobic particles. Turbulent agitation is ensured by a four bladed 45° teflon impeller. Temperature is kept constant at 25.0°C. Each aggregation experiment consists of the following steps: preliminary dispersion by ultrasonics of 0.246 g of solid sample in an ethanol-rich small volume; filling-up of the reactor with water; injection of the silica suspension and destabilization by nitric acid addition (pH 2 to 4). Aggregation immediately starts and can be followed by turbidimetry.

Over the last ten years, we have developed several optical devices to determine in situ the particle size distribution (PSD) or the moments of particulate systems. For diluted suspensions (solid volume fraction φ < 10 -4 ), an optical sensor based on spectral turbidimetry principles can be used to determine the PSD of suspensions composed of submicronic and micronic particles (Crawley et al., 1997). The turbidity probe is vertically located at the twothirds of the vessel radius halfway between two baffles, and mounted at third liquid height above the bottom of the tank. Light emitted by a Xenon UV light source is led by fibre optics to the probe window. The emerging light beam is collimated by a lens. When crossing the 1 cm optical path, light is scattered by the different particles and aggregates. Emerging light is collected by the same system (similar lens and fibre optics) and led to a commercial spectrophotometer which finally delivers the turbidity spectrum in the wavelength range [350 nm-800 nm]. Turbidity τ which expresses the extinction due to scattering by particles is defined by the relation:

( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 0 0 0 0 ln 1 λ λ λ τ I I L opt (1)
in which λ 0 is the wavelength, I 0 (λ 0 ) the intensity of the incident beam and I(λ 0 ) the intensity of the transmitted beam after an optical path of length L opt .

Principle of exploitation of the experimental data

In situ turbidimetry has been successfully applied by the authors to several systems in order to determine their aggregation dynamics. Details can be found in [START_REF] Cugniet | Etude de l'agrégation de particules solides en milieu non mouillant Ecole des Mines de Saint-Etienne[END_REF] and [START_REF] Saint-Raymond | Turbulent aggregation of alumina in water and n-heptane[END_REF].

For a polydisperse population of particles, turbidity contains the contribution of each class of aggregate:

( ) i N i i ext N C ∑ = = 1 , 0 λ τ (2)
where N i is the number concentration of aggregates consisting of i primary particles. Particle extinction section C ext,i is derived from the Mie theory (van de Hulst, 1957); C ext,i depends on the particle diameter, on the refractive index ratio and on the wavelength. Its calculation is relatively easy for spherical, compact, or large particles. However, it is much more intricate for small, non compact aggregates. From equation (2) it can be seen that variations in the different aggregate concentrations N i result in turbidity variation, thus the interest of turbidimetry to follow the aggregation process. When aggregation takes place in a liquid suspension, turbidity either increases or decreases versus time [START_REF] Saint-Raymond | Turbulent aggregation of alumina in water and n-heptane[END_REF][START_REF] Gruy | Influence of non-wetting on the aggregation dynamics of micronic solid particles in turbulent medium[END_REF]. From a general point of view, decreasing or increasing trend is determined both by the primary particle size and the refractive index ratio between the liquid phase and the aggregates. In both cases, amplitude of turbidity variation indicates the global extent of the whole process. Initial slope is proportional to the aggregation rate whereas non-zero height of the final plateau reveals the existence of aggregate maximum size possibly due to fragmentation.

Experimental results

Figures 1 and 2 represent respectively turbidity change with time for hydrophilic and hydrophobic silica (λ = 550 nm) in ethanol (3.45 %)-water (96.55 %) mixture. This composition corresponds to the smallest ethanol quantity leading to a high dispersion of hydrophobic silica suspension during sonication. Effect of the stirring rate on turbidity variation with time has been examined in both cases. Relatively simple calculation using Mie theory (van de [START_REF] Van De Hulst | Light Scattering by Small Particles New York[END_REF] shows that, for the present initial radius and refractive index values, turbidity should increase at least initially when aggregation proceeds. This is observed for the unprocessed particles. The opposite trend which is noted for hydrophobic silica can be explained by a decrease in the aggregate refractive index and a change in the index contrast between the liquid phase and the aggregate. Precisely, presence of gas (air or vapour) in an aggregate may considerably change this contrast and be responsible for the behaviour modification we noted. Referring to the usual models of hydrophobic interaction [START_REF] Attard | Nanobubbles and the hydrophobic interaction[END_REF], this is a strong indication for the presence of gas bridges between the particles.

Another confirmation can be found in the final value of the turbidity (Figures 1 and2) which is very low in the case of hydrophobic particles and consistent with the formation of few large aggregates. In the case of unprocessed particles, the relatively high level of final turbidity indicates the presence of numerous small aggregates. The hydrophobic aggregates are apparently much less or even not at all sensitive to fragmentation in the same hydrodynamic conditions. This behaviour is as well consistent with the presence of gas bridges between the particles which strongly strengthen the aggregates.

Interpretation and Models

Aggregate morphology

Most recent experiments of aggregation in turbulent medium tend to prove that the aggregates have a fractal structure [START_REF] Kusters | Aggregation kinetics of small particles in agitated vessels[END_REF][START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF][START_REF] Cugniet | Etude de l'agrégation de particules solides en milieu non mouillant Ecole des Mines de Saint-Etienne[END_REF]. An aggregate containing i identical primary particles of radius a 1 is characterised by its fractal dimension D f and outer radius a i which are linked by the relation:

1 1 Df i i a a Fs ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ (3)
where Fs is a structure factor, which obeys the relation [START_REF] Gmachowski | Hydrodynamics of aggregated media[END_REF]:

0.42 0.22 f Fs D = - (4)
According to the different authors, the fractal dimension corresponding to turbulent (local shear flow) aggregation is equal to 2.35 ± 0.15.

The fractal-like models are normally applicable only to large aggregates; [START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF], however, has introduced the notion of weak fractal dimension (based on projected area of an aggregate on any plane) which can be extended to small aggregates and may make equations (3,4) suitable for them.

Aggregate optical properties

In a previous paper [START_REF] Cournil | Experimental study and modelling of inclusions aggregation in turbulent flows to improve steel cleanliness[END_REF], it was shown that hydrophobic particles are probably covered by a gas layer which invades their superficial porosity and that this gas layer may play an essential part both in the aggregate formation and fragmentation. In particular, aggregate fragmentation is considerably reduced due to the presence of gas bridges between particles whereas effect of hydrophobicity is less noticeable on aggregation kinetics, i.e. the change of aggregate concentrations ( ) i N t . As decrease in turbidity occurs as early as the start of aggregation, it is believed that new optical properties of the aggregates should be invoked at the first instance to explain the trend inversion in the turbidity variation.

Sedimentation of large aggregates also occurs later on in the process.

Previous work on microbubbles-particles suspension

Suspensions with small particles and air bubbles (or microvoids) in a liquid or resin are met in the paints or coatings industry [START_REF] Ross | Theoretical computation of light scattering power ; Comparison between TiO 2 and air bubbles[END_REF][START_REF] Kerker | Pigmented microvoid coatings; Theooretical study of three models[END_REF]. Generally a white paint film is composed of titania (TiO 2 , rutile) particles uniformly covered by a shell of silica, alumina or other materials. Opacity or hiding power is one of the main properties of a paint. It is due to multiple scattering of light by small particles. The higher the concentration of particles and the higher the optical contrast, the better the opacity. However, if the TiO 2 particle concentration is too large, particles can form aggregates, that are less efficient for scattering the light [START_REF] Auger | Scattering efficiency of clusters composed by aggregated spheres[END_REF]. In order to minimize the cost of white paint, one replaces rutile particles by small air bubbles, which are, however, less efficient scatterers than TiO 2 particles [START_REF] Ross | Theoretical computation of light scattering power ; Comparison between TiO 2 and air bubbles[END_REF]. So, a paint film can contain single rutile particles, microvoids or air microbubbles and aggregates of rutile particles. It has been suggested that the scattering efficiency could be improved by introducing rutile particles inside the microbubbles, due to the increased optical contrast (TiO 2 n = 2.8; air n = 1; resin n = 1.5). This situation is very similar to the system under study (SiO 2 n = 1.44; air n = 1; water n = 1.34).

Turbidimetry for diluted suspension rests on the "single scattering event" theory. For a monodisperse suspension, turbidity is expressed by:

(5)

1 ext sca l NC NC τ - = = =
where l is the mean free path of photon. For a concentrated dispersion such as a paint, optical properties will be calculated in the framework of the radiative transfer theory. Simpler modelling of scattering in paints is also realized by using Kubelka-Munk theory [START_REF] Ross | Theoretical computation of light scattering power ; Comparison between TiO 2 and air bubbles[END_REF].

The main physical parameter for the two theories is the transport mean free path defined as:

* l ( ) ( ) S N l sca φ μ = - = - 1 1 * (6) with ( 1 1 sca p S C V ) μ = - (7)
where , , p V μ φ

are respectively the particle anisotropy factor, the volume fraction of particles and the volume of one particle. and S characterize, respectively, the suspension and the particle. and S are also called the scattering efficiency of the film and of the particle. φ and S are dependent on ν . This result was confirmed by Auger [START_REF] Auger | Optical properties of an eccentrically located pigment within an air bubble[END_REF]. A similar conclusion, by using / sca p C V as scattering efficiency, was attained by Auger (Figs 4, 5, 6. in Auger et al., 2001) for 10 ν < . When 2 ν >> , Kerker et al. (1975) observed a synergistic effect: the replacement of a part of the particle by a microvoid concentric spherical shell results in an enhancement of the scattering efficiency S (but, not for / sca p C V ). This effect, though weaker, was confirmed by [START_REF] Auger | Optical properties of an eccentrically located pigment within an air bubble[END_REF] for 5 ν > . The transition value for ν between the two behaviours is dependent on the system: TiO 2 which is characterized by an high refractive index presents a maximum of scattering efficiency for a particle radius equal to about 0.2 µm (1 2 / a 2 ν π λ < = < ). Then, minimum for scattering efficiency may occur for a spherical particle -air shell set, the size of which is larger than the one of a single particle having a maximal scattering efficiency. [START_REF] Kerker | Pigmented microvoid coatings; Theooretical study of three models[END_REF] suggested that the minimum of scattering efficiency can be explained in the framework of Rayleigh theory, i.e. by introducing the polarisability of a coated sphere [START_REF] Kokhanovsky | Optics of Light Scattering Media 2 nd Ed[END_REF]. Then, this is equivalent to the definition of an effective refractive index for small coated sphere:

cs m ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ( ' 1) 2 ' ( ' ) 2 ' 1 ( 1) ( 2) ( ' 2) 2 ' ( ' ) 2 ' 2 p cs cs p m m m m m m m m m m m m m m φ φ - + + - + -= + + + + - - (8)
m and m' are respectively the relative (compared to the continuous medium) refractive index of particle and gas. Minimum of scattering efficiency corresponds to this value of .

cs m

The displacement of the TiO 2 particle from the centre to the edge of the air bubble can lead to either an increase or a decrease of scattering cross section by a factor two [START_REF] Auger | Scattering properties of rutile pigments located eccentrically within microvoids[END_REF]. However, the effect of location of particle inside the air bubble on scattering properties is less pronounced than the effect of the particle location that can be inside or outside the air bubble (Fig 4. in Auger et al., 2001). This brief survey shows that the presence of microbubbles in suspension may lead to a decrease of its scattering properties.

To characterize the optical properties of silica suspensions, it is proposed to calculate now the extinction cross section of the hydrophilic, then hydrophobic aggregates. Exact methods could be used for calculating scattering properties of aggregates (see, for instance, [START_REF] Xu | Electromagnetic Scattering by an Aggregate of Spheres[END_REF][START_REF] Xu | Efficient Evaluation of Vector Translation Coefficients in multiparticle light scattering theories[END_REF][START_REF] Auger | Scattering properties of rutile pigments located eccentrically within microvoids[END_REF][START_REF] Auger | Scattering efficiency of clusters composed by aggregated spheres[END_REF]. Due to time-consuming computations, these calculations are restricted to very small aggregates and only used in the case of simple aggregate morphology. As the aggregation process leads to a wide population of aggregates with various morphologies, one needs approximate methods for extinction cross section estimates.

Hydrophilic silica aggregates

Each aggregate is characterised by its radius a i and its mean inner volume fraction a φ (deduced from (3)). Two kinds of modelling are successfully used to calculate optical properties of small silica aggregates [START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF].

-Effective Refractive Index method

The easiest way to determine the optical properties of an aggregate is to calculate its effective refractive index m a (see, for instance, Born and Wolf (1980); [START_REF] Vargas | Light scattering in pigmented coatings: experiments and theory[END_REF]. The equation derived by Maxwell-Garnett has been proved to be suitable [START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF] for this system:

) 2 ( ) 1 ( ) 2 ( ) 1 ( 2 2 2 2 + - = + - m m m m a a a φ (9)
m and m a are the relative refractive indices respectively for primary particles and aggregates.

Given the radius and the effective refractive index of an aggregate, the Mie theory (van de Hulst, 1957) can allow us to calculate the scattering cross section C sca for a given wavelength λ.

-Anomalous Diffraction method

Another simple way to calculate the optical properties of small silica aggregates considers interferences between electromagnetic waves scattered by each primary particle of the aggregate. In present case of 0.5 μm silica particles in water-ethanol solution, optical parameters have the following properties:

1 1〈〈 - m and ( ) 1 1 4 1 ≈ - m a λ π . When ( ) 1 4 1 - m a λ π
is not too small, light scattering by particles follows the anomalous diffraction model (van de Hulst, 1957), i.e., globally, straight transmission and subsequent diffraction. In this case, scattered intensity is concentrated near the original direction of propagation and the extinction cross section is given by relation:

∫∫ - - = p S p ext dS m C )) 1 ( 2 cos 1 ( 2 δ λ π (10)
Integration is performed over the object projected area S p on a plane perpendicular to propagation direction. δ is the optical path length through the object. This calculated path is a function of the projection coordinates. As aggregate can randomly orientate, the optical properties are obtained after calculating an average over all orientations.

Hydrophobic silica aggregates

Refractive indices of silica, water and gases are respectively 1.44, 1.34 and 1.0. Scattering properties of particles in a medium mainly depend on the index contrast between the media and on the particle size. According to the works of [START_REF] Kerker | Pigmented microvoid coatings; Theooretical study of three models[END_REF] and [START_REF] Auger | Scattering properties of rutile pigments located eccentrically within microvoids[END_REF][START_REF] Auger | Optical properties of an eccentrically located pigment within an air bubble[END_REF], presence of gas (air or vapour) in an aggregate may considerably change this contrast and be responsible for the behaviour modification that was noted in Figures 1 and2.

Calculation of the scattering properties of a silica-water-air aggregate requires assumptions on the three-phase (silica, water, gas) location. The two cases which have been envisaged -surrounding thin layer (Figure 3a) or homogeneous density in the porous particle (Figure 3b) [START_REF] Cugniet | Etude de l'agrégation de particules solides en milieu non mouillant Ecole des Mines de Saint-Etienne[END_REF] -give similar results.

In this paper, considered hydrophobic aggregates are a set of primary particles, each covered by a thin layer of gas. Then, one can apply the anomalous diffraction theory in order to calculate the extinction cross section:

( ∫∫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + - - = p S p ext dS m m C ) 1 ' ( ' ) 1 ( 2 cos 1 2 δ δ λ π ) (11)
δ and δ' are, respectively, the light path (function of the projection coordinates) through silica and gas. m and m' are, respectively, the relative refractive index of silica and gas. Figure 4 represents the ratio of scattering cross sections without and with gas versus gas content P (ratio of gas volume over silica volume) for different silica small aggregates (D f = 2.7). One observes the decrease of scattering cross section with P whatever the aggregate size. This decrease is due to ' 1 m < in equation ( 11). ' i a ( )

1 1 ' 3 1 1 f D i i a a P Fs ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ + ) (12a) 2 i > ( 1 ' 3 1 1 1 a a P = + (12b)
( ) Another benefit of this procedure is that the effect of the turbidimeter acceptance angle can be easily taken into account in the framework of Mie theory for sphere [START_REF] Gruy | Formation of small silica aggregates by turbulent aggregation[END_REF]. This procedure was used recently in [START_REF] Cugniet | Etude de l'agrégation de particules solides en milieu non mouillant Ecole des Mines de Saint-Etienne[END_REF] to interpret the main aspects of hydrophilic and hydrophobic silica aggregation. We give here two typical examples of simulations. Figure 5 shows the experimental and simulated curves obtained for hydrophilic silica aggregation. Figure 6 relates to hydrophobic silica aggregation. Relevant simulation parameters are aggregate fractal dimension, respectively, 2.4 and 2.67 for hydrophilic and hydrophobic aggregates, and maximum particle number per aggregate, respectively, 68 and 2048 in this case. For hydrophobic aggregate, a volume fraction of 0.3 in gas corresponds to the best agreement between experimental and calculated curves.

1 ' 3 2 1 1.364 1 a a = +P (12c) ( ) 2 2 2 2 2 2 ( ' 1) ( 1) ( ' 1) ( ' 2) ( 2) ( ' 2) a a a m m m A P m m m φ - - - = + + + + a B φ (13a) with 0.3 i a A φ - = (
From a strict optical point of view, this example confirms as well as many others [START_REF] Cugniet | Etude de l'agrégation de particules solides en milieu non mouillant Ecole des Mines de Saint-Etienne[END_REF] that the inversion in turbidity variation which was observed experimentally is due to the presence of gas in the hydrophobic silica aggregates.

Conclusion

The influence of stirring rate on aggregation of hydrophilic and hydrophobic silica in water was studied experimentally by turbidimetry. The strong differences in behaviour which are clearly observed have an optical origin which expresses itself through modifications in the structure of the aggregates. In addition to its interest relatively to original conditions of aggregation, this paper presents new calculation procedures of light scattering cross-section of small aggregates of micron sized silica particles in a liquid phase and of three-phase (gasliquid-solid) aggregates. The different models are validated using the available aggregation experimental data.

Figure 1 .

 1 Figure 1. Influence of stirring rate on hydrophilic 0.5 μm silica aggregation. 1: 200 rpm; 2:

  [START_REF] Kerker | Pigmented microvoid coatings; Theooretical study of three models[END_REF] studied a single titania particle coated by an air layer embedded in resin.They calculated its scattering efficiency as a function of the size parameter 2

Figure 3 .

 3 Figure 3. Different gas-liquid-solid configurations in the hydrophobic aggregates: a) thin gas

  into account the size change with the gas content. Equation 13 is a corrected Maxwell-Garnett expression: it contains two empirical parameters A and B. A is only effective for very small aggregates ( 4 i ≤ ). In B expression, it appears two parameters and which are simple functions of D f[START_REF] Cugniet | Etude de l'agrégation de particules solides en milieu non mouillant Ecole des Mines de Saint-Etienne[END_REF].

Figure 4 .

 4 Figure 4. Ratio of scattering cross sections without and with gas versus gas content P for

Figure 5 .

 5 Figure 5. Comparison between experimental and simulated (D f = 2.4; L = 68) turbidity plots for 0.5 μm hydrophilic silica particles aggregation at stirring rate 200 rpm.

Figure 6 .

 6 Figure 6. Aggregation of 0.5 μm hydrophobic silica at 200 rpm: experimental and simulated