T Michálek
email: tomas.michalek@fel.cvut.cz.

Z Hurák

A Bolopion
email: aude.bolopion@femto-st.fr.

M Gauthier

Electrorotation of arbitrarily shaped micro-objects: modeling and experiments

Tomáš Michálek, Aude Bolopion, Zdeněk Hurák, and Michaël Gauthier Abstract-In this paper, we study the electrorotational behavior of non-spherical micro-objects. We extend a controloriented model of general dielectrophoresis to incorporate also the hydrodynamics so that we can predict the motion of nonspherical micro-objects in fluidic environments. Such mathematical (computational) model enables model-based feedback control of a position and orientation of particles by real-time (online) computation of voltages applied to the electrodes. We use the measured data from experiments with electrorotation of an artificial micro-object having a Tetris-like shape, to evaluate the performance of the proposed model. We also demonstrate the qualitative difference in behavior from the commonly performed electrorotation of a sphere advocating the necessity for modelbased control. Further analysis of the simulation results for other than the experimentally explored scenarios provides additional useful insight into the electrorotational behavior of non-spherical objects.

Index Terms-electrorotation, non-spherical objects, controloriented model

I. INTRODUCTION

E LECTROROTATION (ROT), as one of the AC elec- trokinetic effects, provides a way to induce a rotational motion of micro-objects by an external electric field in a non-contact fashion. The background mechanism of ROT is closely related to the mechanisms of electro-orientation (EO) or dielectrophoresis (DEP) forming a family of phenomena also called as generalized dielectrophoresis (gDEP). First description of these phenomena were provided by Pohl in [START_REF] Pohl | Dielectrophoresis: The Behavior of Neutral Matter in Nonuniform Electric Fields[END_REF] and a subsequent detailed theoretical background of all the mentioned gDEP effects can be found for example in [START_REF] Morgan | AC Electrokinetics: Colloids and Nanoparticles, ser. Microtechnologies and microsystems series[END_REF]- [START_REF] Pethig | [END_REF], short summary in [START_REF] Jones | Basic theory of dielectrophoresis and electrorotation[END_REF] and a review of gDEP applications in [START_REF] Pethig | Review article-dielectrophoresis: Status of the theory, technology, and applications[END_REF].

Mostly, ROT was used in an open-loop regime for characterization of micro-objects (often biological cells) through measurements of their so-called electrorotational spectra [START_REF] Han | An electrorotation technique for measuring the dielectric properties of cells with simultaneous use of negative quadrupolar dielectrophoresis and electrorotation[END_REF]- [START_REF] Gasperis | Automated electrorotation: dielectric characterization of living cells by real-time motion estimation[END_REF]. In [START_REF] Benhal | System identification and stochastic estimation of dielectric properties of a spherical particle using ac-induced electro-rotation[END_REF], a feedback system was proposed for a stochastic estimation of dielectric properties of spherical particles.

ROT can be, however, used also for a closed-loop feedback orientation control of micro-objects, similarly as the DEP was already used for their position control [START_REF] Zemánek | Phase-shift feedback control for dielectrophoretic micromanipulation[END_REF]- [START_REF] Kharboutly | 2d open loop trajectory control of a micro-object in a dielectrophoresisbased device[END_REF]. Non-contact manipulation, including both positioning and orientation, is a sought skill for example in biology or micro-assembly Fig. 1. Foreseen application of gDEP-micro-electrodes at the junction of two microfluidic channels generate controlled electric field for non-contact assembly of micro-objects into the desired products. application domains. In [START_REF] Jiang | Planar cell orientation control system using a rotating electric field[END_REF], Jiang et al. demonstrated the feedback orientation control on a case of spherical yeast cells. Most of the objects of interest in the mentioned application domains are, however, often non-spherical (e.g., red blood cells or virus bodies in biology or completely artificial shapes in micro-assembly). An example task solved in the field of micro-assembly is shown in Fig. 1. Consider a device formed by two microfluidic channels merging in a junction. Through each of these two channels, objects of two distinct shapes are conveyed to the junction where a set of microelectrodes can dielectrophoretically actuate them. Here, ROT could be used to orient suitably both of the objects, one w.r.t. to the other, and DEP would simultaneously bring them near to each other so that they assemble. The finished product would then continue along the flow through the output channel.

Although the use of spheres, as the manipulated objects, is quite common in DEP and ROT experiments, when it comes to non-spherical objects the results are mostly limited to less or more prolate ellipsoids. Miller and Jones [START_REF] Miller | Electro-orientation of ellipsoidal erythrocytes. theory and experiment[END_REF] used EO to characterize the electrical properties (internal conductance and permittivity) of human and llama erythrocytes. In [START_REF] Minoura | Dielectric measurement of individual microtubules using the electroorientation method[END_REF], EO was used for characterization of microtubules. Both ROT and EO were used in [START_REF] Morganti | Characterization of non-spherical polymer particles by combined electrorotation and electroorientation[END_REF] for characterization of surface conductance and permittivity of artificial SU-8 micro-rods with variously functionalized surfaces. In all of these works, the objects were approximated for analysis purposes as (possibly layered) ellipsoids. In [START_REF] Egger | Electrorotation of dumb-bell shaped particles[END_REF], Egger, et al. derived a model for ROT of dumb-bell shaped objects (two connected spheres) and applied it in their experiments with single erythrocytes and pollen. Arcenegui et al. presented in [20] a theory for EO and ROT of slender cylindrical objects and validated it in experiments with metal nanowires. Both positive and negative 0000-0000/00$00.00 c 2019 IEEE DEP manipulation of rod-shaped viruses were demonstrated in [START_REF] Morgan | Dielectrophoretic manipulation of rod-shaped viral particles[END_REF]. A quantitative agreement with an expression for DEP force acting on a homogeneous dielectric cylinder was obtained. Apart from the experimental results, there exist several purely numerical simulation studies [START_REF] Liu | Immersed electrokinetic finite element method[END_REF], [START_REF] Di Biasio | Numerical simulation of dielectric spectra of aqueous suspensions of nonspheroidal differently shaped biological cells[END_REF].

Except for the missing theory and experiments involving objects of shapes other than spheres, ellipsoids or cylinders (often modeled as ellipsoids), it is also the hydrodynamics governing the interaction of these objects with the surrounding fluidic medium, which is usually left without discussion in gDEP experiments. Mathai et al., in their work [START_REF] Mathai | Simultaneous positioning and orientation of a single nano-object by flow control: theory and simulations[END_REF] on position and orientation control using electroosmotic flows, dealt with modeling of hydrodynamic forces and torques influencing the motion of ellipsoidal objects. A complete and exhaustive treatment of hydrodynamics at low Reynolds number are given by Happel and Brenner in [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF].

Contribution

In this paper, we couple our recently published controloriented model of DEP for arbitrarily-shaped objects [START_REF] Michálek | Control-oriented model of dielectrophoresis and electrorotation for arbitrarily shaped objects[END_REF] with control-oriented model of hydrodynamics working again with arbitrarily-shaped objects based on [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF]. The resulting model then allows us to predict both the translational and rotational velocities of the micro-object of interest and thus simulate its motion in a fluidic environment. Such model, evaluable in real-time (in fractions of a second), is motivated by and can be used for design and implementation of model-based control algorithms (e.g., for micro-assembly applications).

We demonstrate the performance of the model by comparing its outcomes against experimental observations. We perform an electrorotation with Tetris-shaped micro-object as a task in which both gDEP and hydrodynamics play a crucial role.

II. MATHEMATICAL MODEL

The purpose of the mathematical model is to predict the motion of an arbitrarily shaped micro-object in a fluidic medium under the influence of an external electric field generated by a set of microelectrodes. Our ultimate goal is to make the model a part of a control algorithm that sets the voltages applied to the electrodes so that an object follows a planned trajectory. To this purpose, we require that the model be evaluable in real-time (in fractions of a second) similarly as in [START_REF] Michálek | Control-oriented model of dielectrophoresis and electrorotation for arbitrarily shaped objects[END_REF]. Despite its simplicity, which is necessary for its fast evaluation, such a model should still take into account all the relevant physical phenomena for accurate enough results. In our case, the electrokinetics and hydrodynamics represent the dominant effects, and we will further deal with them in the following subsections.

We will neglect the other, unobserved or not so influential, effects to keep the model fast to evaluate. A feedback controller will then compensate for the discrepancies between the model and the reality. These neglected effects include, for example, the thermally induced fluid flows or friction forces between the object and other solid parts of the experimental chamber.

A. AC electrokinetics

For actuation, we drive the micro-electrodes with harmonic signals having a frequency of 25 kHz, which is high enough to allow us to consider just the gDEP forces and torques and to neglect all the other effects like electrolysis, electroosmosis, etc [START_REF] Wong | Electrokinetics in micro devices for biotechnology applications[END_REF]. We further decided to ignore any electrical interaction of the object with the walls or the bottom floor of the experimental chamber, since a sufficiently large distance separates them during the experiments.

The critical enabling principle used in gDEP is the electric polarization of the object of interest, which takes place when it is exposed to an external electric field. From the various polarization mechanisms (electronic, atomic, orientational, etc.), it is the interfacial polarization caused by a longrange charge transport in dielectrics, whose effect prevails in the AC electrokinetics applications (micro-objects located in fluidic mediums). The polarization is not instantaneous, but the charge rather gradually builds up at the outer as well as the inner interfaces of the object with speed characterized by a relaxation time τ (a period, in which the maximum polarization takes place).

Thus the polarization depends on the frequency of the external AC electric field. With frequencies sufficiently below f co = 1/ (2πτ ), the charges manage to follow the changing field whereas with frequencies sufficiently above this threshold, they will not be able to 'catch up' the field at all and no polarization will take place. The frequency response is strongly influenced by the material properties of the object and the medium. At lower frequencies, these are the conductivities, which have the major influence over the polarizability. At higher frequencies, the impact of permittivity takes over. All of the above characteristics are captured in the so-called Clausius-Mossotti factor [START_REF] Jones | Basic theory of dielectrophoresis and electrorotation[END_REF].

The resulting charge structure emerging at the interface of the object then interacts through Coulomb forces with the original electric field exerting forces and torques on the object itself. Based on whether the object is more polarizable than the medium (at the given frequency) or vice versa, it will be attracted to (so-called positive DEP) or repelled from (socalled negative DEP), respectively, the places of high electric field intensity. A non-spherical object will tend to orient along the electric field lines. In case of a rotating electric field that continuously changes its orientation, the object will keep rotating as the charge distribution on its interface attempts to 'catch up' with the field (so-called cofield rotation) or keeps being pushed ahead of the field (so-called antifield rotation) as repelled by the charges on the electrodes. These phenomena are, in the given order, the already mentioned DEP, EO, and ROT.

To compute the gDEP forces and torques, we use our recently published model described in [START_REF] Michálek | Control-oriented model of dielectrophoresis and electrorotation for arbitrarily shaped objects[END_REF], which, unlike others, can also deal with non-spherical objects. Precomputing a "basis table" for the given specific shape of the object, it then provides a way for calculation of its multipolar representation in real time (in fractions of a second). As a consequence, also the gDEP force and torque can be computed this fast. An implementation of the described model is freely available at https://goo.gl/eGZY7P.

B. Hydrodynamics

Except for the electrokinetic forces, the other significant forces and torques that act on the object are the ones caused by hydrodynamics. As the objects moves through and revolves inside the fluidic medium, it will experience hydrodynamic drag force and torque. They not only oppose the object's motion slowing down its translation or rotation but in case of a non-orthotropic object (it does not have three perpendicular planes of symmetry), they also couple the translational and rotational motion. From a merely translational movement at the beginning, a hydrodynamic drag torque can emerge forcing the object to rotate or vice versa-a rotational motion can propel the object throughout the fluid. This effect is the basic locomotion principle of the robotic microswimmers [START_REF] Abbott | How should microrobots swim?[END_REF], [START_REF] Klumpp | Swimming with magnets: From biological organisms to synthetic devices[END_REF].

The nonlinear partial differential Navier-Stokes equations governing the fluid flow are generally hard to solve. Timeand memory-demanding finite element method (FEM) has to be usually deployed. Since we aim at constructing a control-oriented model evaluable in real time, we pose a few simplifying assumptions valid for microfluidic systems. For low Reynolds numbers (small dimensions and low velocities typical for microfluidic systems) and incompressible fluids of constant viscosity μ, the governing Navier-Stokes equations simplify to the so-called Stokes flow (or creeping flow) equations [25]

∇ 2 v = 1 μ ∇p, ∇ • v = 0. (1)
Here, v is the vector field representing fluid velocity, p is the scalar pressure field, and ∇ is the vector differential operator. Furthermore, to simplify the explanation, we first consider the fluid to be unbounded. This is reasonable if the object does not take place in close vicinity of the bottom of the chamber or its walls. We will revisit this assumption at the end of this section.

Although the Stokes flow equations are generally solvable just by using FEM, they are at least linear. Omitting the details discussed in [START_REF] Happel | Low Reynolds number hydrodynamics: with special applications to particulate media[END_REF], this allows us to formulate the model for hydrodynamic drag force, F, and torque T O around a point O inside the object as

F drag = -μKv -μC T O ω, T drag = -μC O v -μΩ O ω. ( 2
)
Here, v is the translational velocity of the object, ω is its angular speed and K, C O , and Ω O are translational, coupling and rotational tensors, respectively. The choice of O is arbitrary, but if it is the center of the hydrodynamic reaction, (2) poses the most symmetric form (C O is then symmetric). We chose O to be at the geometric center of the object.

These tensors contain all the shape-specific information and can be computed beforehand using FEM just once for each specific shape of the object. We do so by solving (1) for six different sets of boundary conditions. The first three of them are

v = U i on S p , i = 1, 2, 3, (3)
where v → 0 as r O → ∞, S p denotes the surface of the object and r O is a position vector having an origin at O. We set

U 1 = [1, 0, 0] T , U 2 = [0, 1, 0] T and U 3 = [0, 0, 1]
T so that they correspond to translational movements of the object in a quiescent fluid along the three coordinate axes. The remaining three boundary conditions are then

v = ω i × r O on S p , i = 1, 2, 3, (4)
with v → 0 as r O → ∞, and vector of angular velocities set as

ω 1 = [1, 0, 0] T , ω 2 = [0, 1, 0] T and ω 3 = [0, 0, 1]
T corresponding to rotation of the object in a quiescent fluid about the three coordinate axes. The total computation time was under 30 min using a regular PC. We denote the individual solutions in the order in which the distinct boundary conditions were given as velocity and pressure fields v i and p i , i = 1, 2, . . . , 6, respectively. Based on these, we then compute the position dependent quantities

P i 1 = ⎡ ⎢ ⎢ ⎣ -p i μ + 2 ∂v i x ∂x ∂v i y ∂x + ∂v i x ∂y ∂v i z ∂x + ∂v i x ∂z ⎤ ⎥ ⎥ ⎦ , P i 2 = ⎡ ⎢ ⎢ ⎣ ∂v i x ∂y + ∂v i y ∂x -p i μ + 2 ∂v i y ∂y ∂v i z ∂y + ∂v i y ∂z ⎤ ⎥ ⎥ ⎦ , P i 3 = ⎡ ⎢ ⎢ ⎣ ∂v i x ∂z + ∂v i z ∂x ∂v i y ∂z + ∂v i z ∂y -p i μ + 2 ∂v i z ∂z ⎤ ⎥ ⎥ ⎦ .
(

The desired tensors K, C O , and Ω O are then obtained as

K = [K 1 , K 2 , K 3 ] ,
with K i = -

Sp P i 1 , P i 2 , P i 3 dS, C O = [C 1 , C 2 , C 3 ] ,
with

C i = - Sp r O × P i 1 , r O × P i 2 , r O × P i 3 dS, Ω O = [Ω 4 , Ω 5 , Ω 6 ] ,
with

Ω i = - Sp r O × P i 1 , r O × P i 2 , r O × P i 3 dS.
Please notice that although the expressions for C i and Ω i are the same, the tensors C O and Ω O will differ, because different values of indices i (and thus data from differing simulations) are used for their computation.

For cases when the object moves in the vicinity of solid obstacles (e.g., walls or bottom of the experimental chamber), we can not consider the fluid to be unbounded. In such situations, tensors K, C O , and Ω O need to be computed repeatedly for various positions of the object to the walls and stored in a lookup table from where they can be accessed and interpolated in real time.

Our numerical simulations for the Tetris shape revealed that below ∼130 μm the error in drag torque if we use the unbounded fluid assumption, is larger than 1 %. Therefore we utilized the described approach to take into account the proximity of the bottom floor of the chamber when comparing the simulations to experimental behavior in section V.

C. Resulting model

Now, that we know both the propulsion and frictional forces and torques caused by the electrokinetics and hydrodynamics, respectively, we can state the general model describing the motion of the object

ẋ = v o vo = 1 m F DEP + 1 m F sed - 1 m F drag θ = ω ω = I -1 T DEP -I -1 T drag , ( 6
)
where x = [x, y, z] T is a vector representing the position of the center of the object,

v o = [v x , v y , v z ] T is its translational velocity, F DEP = [F DEP,x , F DEP,y , F DEP,z ] T is the DEP force, F sed = [0, 0, (ρ medium -ρ object ) V g] T is
the sedimentation force (buyonacy and gravity effect) and

F drag = [F drag,x , F drag,y , F drag,z ] T is the hydrodynamic drag force. Similarly, θ = [ψ, θ, φ]
T is a vector comprised of Euler angles representing the orientation of the object in space (successive rotations about the x-,y-and z-axes of the coordinate frame), ω = [ω x , ω y , ω z ] T is then the rotational velocity and T DEP and T drag are the DEP and hydrodynamic drag torques, respectively. The mass of the object is m = V ρ object , with V being the object's volume and ρ object its density. Finally, ρ medium is the density of the fluidic medium, g is the acceleration due to the gravity and I is the object's moment of inertia.

In practice, however, the inertia of the object is negligible due to its low weight, and the object achieves its final translational and rotational velocity immediately without any noticeable transients. Analogously, the object immediately stops moving or rotating due to the DEP when the voltage applied to the electrodes is suddenly switched off. The general model represented by [START_REF] Jones | Basic theory of dielectrophoresis and electrorotation[END_REF] can therefore be simplified using

vo = 0 ω = 0, (7)
from which the steady state values of velocities follow and the new motion equations are

ẋ = v o,final θ = ω final , ( 8
)
where v final and ω final are the mentioned final translational and rotational velocities occurring when all the forces and torques are balanced. This amounts to solving the following linear system of equations based on (2)

-μ K C T O C O Ω O v o,final ω final = F drag T drag = - F DEP + F sed T DEP ,
which leads us to

ẋ θ = 1 μ K C T O C O Ω O -1 F DEP + F sed T DEP . ( 9
)

III. DESCRIPTION OF THE USED EXPERIMENTAL SETUP

A. Micromanipulation workplace

To induce DEP motion, we used a quadrupolar electrode array shown in Fig. 2A, which is commonly used for electrorotation. It consists of four identical electrodes that are mutually rotated by 90 • . They are made of gold (500 nm gold coating deposited on top of a 20 nm underlying chromium layer) on a glass substrate using a photolithography technology. The object of interest is placed in the area between the four electrodes. Around this workplace, there is a plastic basin of a square shape attached by epoxy glue. Figure 3 depicts the whole chip including the connection pads.

The basin is filled with ethanol and enclosed from above by a glass coverslip to prevent its rapid evaporation and onset of the related fluid flows. This fluidic environment has a density just slightly lower than the density of the microobject. This enables us to keep the micro-object levitating during the experiments in an equilibrium formed by the balance of the downwards directed sedimentation force and the upwards oriented DEP force. Therefore we do not have any friction between the micro-object and the floor of the basin, which would otherwise complicate the modeling task. Furthermore, the low conductivity of ethanol prevents the onset of electrolysis-usually unwanted phenomena occurring during DEP micromanipulation.

As the manipulated micro-object we used a Tetris-shaped piece also shown in Fig. 2A cleanroom applications to manually arrange the object in place before the experiment starts. Figure 4 depicts the electrode array including its dimensions and defines the coordinate system used for the description of the micro-objects position and orientation in subsequent sections.

We summarized all the relevant physical properties of the used materials in Table I.

B. Instrumentation

The glass chip with the electrode array is plugged into a spring connector, and it is placed under an optical tube with an objective (10×) attached to its bottom and a camera to its top. The captured video is sent to a regular Linux PC and processed in real-time (we use a custom C++ application taking advantage of the real-time application interface (RTAI) and OpenCV libraries). Although the real-time requirement on image processing is not necessary for analytical purposes, it will come handy when implementing any control algorithm. Since, in the described experimental settings, the object located in the space between the electrodes rotates approximately just around the z-axis, we aim at measuring the x and y position of its center and the orientation angle φ with respect to time. To this purpose, we threshold the captured image, crop it and compute its centroid and the second order central moments. The centroid gives us the planar position and eigenvector associated with the largest eigenvalue of the covariance matrix constructed from these image moments gives us the direction of greatest intensity. For a known shape this can be mapped to its orientation. For symmetrical objects, like the Tetris shape we use, special care has to be taken to solve unambiguity issues. We do so by imposing a constraint that the orientation of the object can not change abruptly.

We generate the harmonic voltage signals using NI 6733 analog output card, then amplify them by a couple of Krohn-Hite 7600M high-voltage amplifiers and finally apply them to the electrodes. In the described experiments we used signals with a frequency f = 25 kHz. The whole setup is schematically shown in Fig. 5 including an indication of the voltage signals applied to the electrodes in typical ROT experiments (90 • mutually shifted harmonic signals). To validate our theoretical model against the experimental observations, we will also use other values of phase-shifts, and we will also vary the amplitudes between the neighboring electrodes. Note, however, that the signals on opposite electrodes are necessarily always mutually inverted (we use normal and inverted outputs of the amplifiers) and therefore we can set neither the amplitude nor the phase-shift between individual signals independently on all the four electrodes.

IV. EXPERIMENTAL OBSERVATIONS

In the very first experiment, we placed the Tetris-shaped micro-object close to the center of the electrode array (i.e., the origin of the coordinate system) and applied four harmonic signals shifted in phase by exactly 90 • to the four electrodes it consists from. As a consequence, the Tetris-shaped microobject positions itself in the center of the electrodes and simultaneously moves upwards to a stable equilibrium where it levitates and starts rotating.

The upwards and centering motion is caused by the negative DEP force, which repels the object from high-intensity electric field regions along the edges of the electrodes. With the increasing distance from the electrodes, this force becomes weaker and at some point it is balanced by the sedimentation force. We denote this equilibrium as its levitation height. The described variation of phase on the electrodes creates a rotating electric field inducing the ROT motion of the micro-object.

The experiment revealed that the object rotates around its vertical axis with a nonconstant rotational speed. Figure 6 shows the measurement of the angle φ and the corresponding angular velocity ω = dφ/dt obtained by numerical differentiation. Note that we utilized the apparent periodicity of the motion and plotted ω with respect to φ rather than time, which will be more suitable for subsequent comparisons against the simulation outcomes. Measurements of the x and y positions of the center of the object presented in Fig. 7 show that apart from rotation, the object is also subjected to a slight, almost periodic, oscillatory motion in the xy-plane. Note also that the rotation did not take place precisely at the center of the electrode array (at the origin of the coordinate system). The reason for this might be a non-symmetry in the design of the experimental chamber or of the paths contacting the individual electrodes. Nonetheless, even if the object's center point laid on the z-axis, its rotational speed would not be constant as in the case of a sphere. Although the electric field is ideally rotating along the z-axis, anywhere outside this centerline the rotating character of the field deteriorates. This is apparent from Fig. 8, where an eccentricity of the polarizing electric field is plotted at a typical levitation height. Since the field is not precisely rotating in any neighborhood of z-axis, let alone in an area that is spanned by the micro-object of interest, the polarization of the object depends on its orientation φ. Therefore also the ROT torque and subsequently the angular velocity ω varies with φ. This is in sharp contrast with the ROT behavior of a sphere, which achieves due to its geometric symmetry always constant angular velocity. This demonstrates that even in such simple setting the resulting behavior of the micro-object might be non-trivial and every attempt for a precise and accurate control of its motion will necessarily require a suitable controloriented model.

V. COMPARISON OF MODEL AND EXPERIMENTS

In this section, we compare the ROT experimental measurements with the proposed control-oriented model of gDEP and hydrodynamics.

To evaluate the model and get the rotational speed of the micro-object, we need to know its position and orientation. As it was already mentioned in section III-B and presented in Figs. 6 and7, all we can measure is just its x and y coordinates and its orientation φ around the z-axis. From the captured video, we also see that the orientations of the micro-object around the remaining two axes (i.e., angles ψ and θ) are zero. The z-coordinate of the micro-object (its levitation height) can not be, however, directly measured and has to be at least experimentally estimated. We did so by bringing the object to focus for two situations: while rotating and levitating, and while lying in rest at the bottom floor of the experimental chamber. Subtracting the corresponding readings on the focusing micrometer-screw and adding half of the object's thickness (since it is the position of the center of the object which we are interested in), the levitation height was determined to be z = 136 μm. Since the measured x and y planar coordinates also evince a periodic behavior as it was demonstrated in the previous section, we used their averaged value for each specific orientation of the micro-object.

The comparison is made using a set of experiments. In each of them we slightly adjust the voltage signals applied to the electrodes, which results in a different rotational behavior. The voltage waveforms were, starting from the top electrode in Fig. 2 and continuing in a clockwise direction, given by (A -Δa 2 ) cos(2πf t), (A + Δa 2 ) cos(2πf t -π 2 + Δb π 180 ), (A-Δa 2 ) cos(2πf t-π) and (A+ Δa 2 ) cos(2πf t-3 2 π+Δb π 180 ), where A = 35V is the base amplitude, t represents the time and Δa and Δb are the parameters differing for each individual experiment. Δa basically introduces a difference in the amplitudes of the waveforms between the neighboring electrodes and similarly Δb changes their mutual phase-shift from the default 90 • . This particular choice of the set of experiments was motivated by the need to stay close enough to the standard ROT settings and prevent the object from escaping from the center of the manipulation area. The resulting rotational behavior is shown in Fig. 9 always with the corresponding Δa and Δb. The individual plots are arranged such that Δa increases from the bottom to the top rows and Δb increases from the left to the right columns.

Note that for Δa = Δb = 0 (all the voltage signals have equal amplitude and successive 90 • phase shift) the rotational velocity should be, because of the symmetry of the field and micro-object, exactly π 2 periodic. This holds for simulation, but not for the experiment. The model outcomes for such situation better fit the measurements made for Δa = 0 and Δb = 2 • and generally that also the other model outcomes better suit the measurements made for Δb increased by around 2 • . We denote such better fitting model as "corrected model". This indicates that the system is quite sensitive even to small changes of the input parameters. As a consequence, another source of the modeling discrepancies might arise also from any asymmetries in the arrangement of the electrodes (due to asymmetric design of the connection paths or fabrication imprecisions).

VI. SIMULATION ANALYSIS OF OTHER SCENARIOS

Now when we saw that the proposed model can explain the particular observed experimental behavior, we will use a simulation outcomes to analyze different possible scenarios.

Specifically, we explore how the rotational behavior of the micro-object would change with its varying levitation height. As mentioned above, the levitation height is quite sensitive to even slight variations in the properties of the used materials, mainly their densities. In Fig. 10 we demonstrate this for three different settings of Δb and Δa. Note that with the increasing levitation height the speed of rotation decreases since also the electric field intensity gets weaker. Depending on the particular settings, there is an interesting transitional zone at lower levitation heights (below 50 μm). If the object is levitating near the electrodes, it might not revolve, but it will rather just orient itself with respect to the electrodes by means of EO. The reason for this behavior is that here the electric field is not so nicely rotating as it is in higher levitation heights where it is weaker but smoothed out.

Experimenting with other various shapes of objects and of the electrodes, we further observed, that the spatial (w.r.t. orientation φ) period of angular velocity ω observed in Figs. 6 and 9 depends on both the spatial periodicity of the source electric field and on the rotational periodicity of the object's shape. Namely, it is equal to their greatest common divisor.

VII. CONCLUSION

In this paper, we extended the previously developed controloriented model of gDEP [START_REF] Michálek | Control-oriented model of dielectrophoresis and electrorotation for arbitrarily shaped objects[END_REF] for arbitrarily-shaped objects to include also the hydrodynamics. The resulting model can compute the translational and angular velocities of microobjects under gDEP in fractions of a second.

The comparison of the simulation predictions with experimental measurements showed that the model reflects the complex ROT behavior of the Tetris-shaped micro-object.

Since the presented experimental system is very sensitive to even slight changes in the input voltage signals, any control task might be challenging. As a first step towards the simultaneous position and orientation control of micro-objects motivated by the micro-assembly and biology applications, we would like to use the presented model in a controller achieving a steady angular velocity of the arbitrarily-shaped objects.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Electrode array and the micro-object having a shape of one of the Tetris pieces.

Fig. 4 .

 4 Fig. 4. Dimensions of the electrode array and Tetris-shaped micro-object used in the validation simulations.

Fig. 5 .

 5 Fig. 5. Scheme of the whole experimental setup.

Fig. 6 .

 6 Fig.6. Experimentally measured rotational behavior of a Tetris-shaped microobject using a standard electrorotational laboratory settings (four electrodes supplied with 90 • -shifted harmonic voltages).

Fig. 7 .

 7 Fig. 7. Experimentally measured x and y positions of the center of a Tetrisshaped micro-object.

Fig. 8 .

 8 Fig. 8. Eccentricity of the polarizing electric field at z = 136 μm. Ideally rotating electric field has eccentricity value of zero.

Fig. 9 .

 9 Fig.9. Dependence of an angular velocity of the micro-object on its orientation for a set of 13 experimental measurements and their comparison with simulation results.

Fig. 10 .

 10 Fig. 10. ROT torque acting on the Tetris-shaped object located on the z-axis for varying values of the z-coordinate and angle φ. Each row of graphs was generated using different voltages on the electrodes encoded by Δa and Δb in the respective titles. Two different color scales (unsaturated and saturated) are used for each data set to capture the behavior for lower as well as higher values of z.

ACKNOWLEDGMENT This work has been supported by the EIPHI Graduate School (contract "ANR-17-EURE-0002"), by the French RE-NATECH network and its FEMTO-ST technological facility, by the ANR and FNS project CoDiCell (grant No ANR-17-CE33-0009), by the MiMedi project funded by BPI France (grant No. DOS0060162/00) and the European Union through the European Regional Development Fund of the Region Bourgogne-Franche-Comte (grant No. FC0013440) and by the Czech Science Foundation within the project P206/12/G014 (Center for advanced bioanalytical technology, http://www. biocentex.cz).


Pierre-Henri Cornuault

Luc Carpentier

Marie-Ange Bueno
email: marie-ange.bueno@uha.fr

Jean-Marc Cote

Guy Monteil

Influence of the physicochemical, mechanical and morphological fingerpad's properties on the frictional distinction of sticky/slippery surfaces

Keywords: Tactile, biotribology, friction, skin, hydrolipid film, finger

This study investigates how the fingerpad hydrolipid film, shape, roughness and rigidity influence the friction when it rubs surfaces situated in the slippery psychophysical dimension.

The studied counterparts comprised two "real" (physical) surfaces, and two "virtual" surfaces.

The latter were simulated with a tactile stimulator named STIMTAC. Thirteen women and thirteen men rubbed their right forefingers against the different surfaces as their arms were displaced by a DC motor providing constant velocity and sliding distance. Tangential and normal forces were measured with a specific tribometer. The fingerpad hydrolipid film was characterised by Fourier transform infrared spectroscopy. The shape and roughness of fingers were extrapolated from replicas. Indentation measurements were carried out to determine fingerpad's effective elastic modulus. A clear difference was observed between women and men in terms of friction behaviour. The concept of tactile frictional contrast (TFC) which was introduced quantifies an individual's propensity to distinguish two surfaces frictionally. The lipids/water ratio and water amount on the finger skin significantly influenced the TFC. A correlation was observed between the TFC and fingerpad's roughness, i.e. the height of the fingerpad ridges. This is essentially due to gender differences. A significant difference between men and women's finger topography was also noted since our results suggested that men have rougher fingers than women. The friction measurements did not correlate with the fingerpad curvature nor with the epidermal ridges' spatial period.

INTRODUCTION

Human tactile perception has been the subject of many studies over the last years and has aroused interest from various fields of application such as remote surgery, virtual reality, product designing, tactile deficiencies detection and rehabilitation, e-shopping, smart surfaces…etc. Simultaneously, the recent and dazzling development of touch interfaces for interactive and mobile devices has led to the exciting challenge of developing tactile stimulators. These devices aim at giving the illusion of rubbing various textures when scanning an active and programmable single surface with one's finger [START_REF] Biet | Implementation of tactile feedback by modifying the perceived friction[END_REF]. Although tactile stimulators are still in their infancy, some prototypes have been developed in recent years both as useful tools for comprehension improvement of tactile perception as well as for surface and texture simulation. Depending on the considered apparatus, the stimulation can be applied either by means of mechanical stretching of the skin [START_REF] Sylvester | Effects of longitudinal skin stretch on the perception of friction[END_REF], setting up of an air-bearing between the finger and a plate (squeeze effect) [START_REF] Biet | Implementation of tactile feedback by modifying the perceived friction[END_REF], or generating of electrostatic forces [START_REF] Meyer | Fingertip friction modulation due to electrostatic attraction[END_REF].

While different principles and technologies are used, all stimulators operate with the same main objective: modulating friction forces between the fingerpad and a counterpart. Therefore, one can define these stimulators as friction force modifiers which consequently induce tactile perception modulation. Indeed, tactile perception originates in the mechanical activation of sensorial receptors located beneath the skin surface, in the dermis [START_REF] Vallbo | Properties of mechanoreceptors in the human hand related to touch sensation[END_REF][START_REF] Cohen | Relationships between touch sensations and estimated population responses of peripheral afferent mechanoreceptors[END_REF][START_REF] Johansson | First spikes in ensembles of human tactile afferents code complex spatial fingertip events[END_REF]. When exploring surfaces with the finger, the induced dynamical friction forces are thus transmitted toward these receptors, producing signals transferred to the central nervous system by means of a mechano-electrical transduction process. Consequently, the finger/surface friction behaviour has a critical influence on tactile perception. This statement, which is assessed by psycho-perceptual studies [START_REF] Bergmann Tiest | Tactual perception of materials properties[END_REF][START_REF] Ekman | Roughness, smoothness, and preference : a study of quantitative relations in individual subjects[END_REF][START_REF] Smith | Role of friction and tangential force variation in the subjective scaling of tactile roughness[END_REF][START_REF] Skedung | Tactile perception: Finger friction, surface roughness and perceived coarseness[END_REF], justifies the operating concept of tactile stimulators.

Moreover, it underlines the necessity of a better understanding of the fingerpad's friction behaviour on commonly touched "real" surfaces in order to improve stimulators' efficiency.

A usual criterion for quantifying friction during sliding is the coefficient of friction (COF), i.e. the ratio of the tangential force versus the normal load. From a tribological perspective, the factors influencing the friction between a surface and the finger could be classified according to three categories: counterpart features (material properties, roughness and texture…), sliding conditions (speed, motion, normal load, environment…), and finger properties (mechanical behaviour, skin surface chemistry, roughness of fingerprint ridges…). These factors and their influences on the COF between a surface and the finger (or another body region) are discussed below.

The influence of the rubbing surface features on the COF has been extensively studied [START_REF] Derler | Tribology of skin: Review and analysis of experimental results for the friction coefficient of human skin[END_REF].

First, materials in contact have a strong influence. The COF between the skin and the smooth surface of a homogeneous material can be classified in the following decreasing order: soft polymers such as rubber, hard polymers (except PTFE: polytetrafluoroethylene) [START_REF] Gee | A new friction measurement system for the frictional component of touch[END_REF], metals, and PTFE [START_REF] Hendriks | Influence of surface roughness, material and climate conditions on the friction of human skin[END_REF]. The rubbing against glass induces a lower or a higher friction coefficient than hard polymers [START_REF] Adams | Finger pad friction and its role in grip and touch[END_REF] and metals, depending on the moisture amount in the skin/glass interface.

Second, the extensively studied effect of the counterbody's roughness on friction has shown different behaviours. The friction regime can be dominated by adhesion on smooth surfaces or by deformation of the fingerpad on rougher surfaces. Consequently, the COF decreases with increasing roughness since Ra (the arithmetic average of asperities height's absolute values) is smaller than 11.5 µm [START_REF] Hendriks | Influence of surface roughness, material and climate conditions on the friction of human skin[END_REF][START_REF] Skedung | Finger friction measurements on coated and uncoated printing papers[END_REF]. This effect has been explained by the decrease of the tangential force owed to adhesion [START_REF] Derler | Friction of human skin against smooth and rough glass as a function of the contact pressure[END_REF]. But for higher Ra, the COF rises when roughness increases due to the increase of skin deformation [START_REF] Tomlinson | Human finger friction in contacts with ridged surfaces[END_REF]. Moreover, for well-defined regular surfaces produced by laser texturing with a similar Ra, the COF slightly depends on texture parameters such as asperities radius and spacing [START_REF] Van Kuilenburg | An experimental study on the relation between surface texture and tactile friction[END_REF].

As regards sliding conditions, although rotary set-ups are sometimes used [START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF], linearalternate motion of the finger is carried out in most experiments. In this case, the COF is dependent on the finger angle relative to the scanning surface [START_REF] Warman | Fingerprints are unlikely to increase the friction of primate fingerpads[END_REF] due to the variations of the contact area. Moreover, the evolution of the COF between finger skin and various materials decreases with an increase of the normal load. The COF commonly follows a linear relationship with 1 -n W , where W is the normal load and n a coefficient ranging from 0.66 to 1 [START_REF] Bowden | Friction of diamond, graphite, and carbon and the influence of surface films[END_REF]. Nevertheless the relationship between W and the COF is considered linear in certain cases [START_REF] Adams | Finger pad friction and its role in grip and touch[END_REF]. When rubbing on smooth counterfaces, some studies point out an increase of the COF for low sliding speed (until 10 mm/s) and a decrease for higher velocities [START_REF] Derler | Stick-slip phenomena in the friction of human skin[END_REF]. These changes have been linked to the dynamic of occlusion mechanism (moisture accumulation) under the fingerpad [START_REF] Pasumarty | Friction of the human finger pad: Influence of moisture, occlusion and velocity[END_REF]. Finally, as some studies have highlighted, the climate (dry, damp…) has a strong effect on COF values [START_REF] Pasumarty | Friction of the human finger pad: Influence of moisture, occlusion and velocity[END_REF][START_REF] Cua | Frictional properties of human skin: relation to age, sex and anatomical region, stratum corneum hydration and trans-epidermal water loss[END_REF].

Fingerpad properties are related to their complex structure and composition. The outer layer of the epidermis, namely the stratum corneum (SC), is composed of numerous stacked corneocytes (flattened protein-enriched cells) which are embedded in a lipid matrix [START_REF] Merle | Physical and chemical perturbations of the supramolecular organization of the stratum corneum lipids : in vitro to ex vivo study[END_REF].

Although most of the body regions show SC thickness ranging from 10 to 30 µm [START_REF] Merle | Physical and chemical perturbations of the supramolecular organization of the stratum corneum lipids : in vitro to ex vivo study[END_REF][START_REF] Menon | The structure and function of the stratum corneum[END_REF], the fingerpad's SC thickness reaches several hundred micrometers by considering epidermal ridges [START_REF] Pasumarty | Friction of the human finger pad: Influence of moisture, occlusion and velocity[END_REF]. The SC's elastic modulus has been extensively measured on various regions of the body [START_REF] Van Kuilenburg | Contact modelling of human skin: What value to use for the modulus of elasticity[END_REF][START_REF] Pailler-Mattei | In vivo measurements of the elastic mechanical properties of human skin by indentation tests[END_REF][START_REF] Geerligs | In vitro indentation to determine the mechanical properties of epidermis[END_REF]. Nevertheless, in vivo measurement of the fingerpad elastic modulus were performed, to the knowledge of the authors, in only one study [START_REF] Wang | In vivo biomechanics of the fingerpad skin under local tangential traction[END_REF], which reported values ranging from 0.9 to 4 MPa. Another of the finger's crucial surface property is its physicochemistry. A brief overview of the chemistry of the skin surface could be depicted as a mixture of lipids and water covering the SC [START_REF] Stefaniak | Review: Dissolution of materials in artificial skin surface film liquids[END_REF]. But the origins of these two main components are even more complex: sebum, free fatty acids and water due to the desquamation process, environmental moisture, adsorbed lipids, water diffusion through the SC…etc [START_REF] Merle | Physical and chemical perturbations of the supramolecular organization of the stratum corneum lipids : in vitro to ex vivo study[END_REF][START_REF] Cua | Skin surface lipid and skin friction: Relation to age, sex and anatomical region[END_REF]. A more accurate denomination of skin physicochemistry than sweat should involve the notion of cutaneous hydrolipid film (CHF). The water amount in the CHF has a strong effect on the skin's mechanical behaviour [START_REF] Van Kuilenburg | Contact modelling of human skin: What value to use for the modulus of elasticity[END_REF][START_REF] André | Fingertip moisture is optimally modulated during object manipulation[END_REF] since the plasticizing action of water decreases the SC elastic modulus [START_REF] Adams | Friction and lubrication of human skin[END_REF]. The relationships between fingerpad physicochemistry and the COF have rarely been studied [START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF]. Nevertheless numerous studies have related the friction of other body regions on various counterbodies to the CHF. Skin hydration or moisturization has been extensively highlighted to increase the COF [START_REF] Derler | Tribology of skin: Review and analysis of experimental results for the friction coefficient of human skin[END_REF][START_REF] Hendriks | Influence of surface roughness, material and climate conditions on the friction of human skin[END_REF][START_REF] Adams | Finger pad friction and its role in grip and touch[END_REF][START_REF] Derler | Friction of human skin against smooth and rough glass as a function of the contact pressure[END_REF][START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF][START_REF] Cua | Skin surface lipid and skin friction: Relation to age, sex and anatomical region[END_REF][START_REF] André | Fingertip moisture is optimally modulated during object manipulation[END_REF]. This effect has been explained by capillary adhesion forces and an increase of the contact area leading to more adhesion [START_REF] Adams | Friction and lubrication of human skin[END_REF][START_REF] Pailler-Mattei | A new approach to describe the skin surface physical properties in vivo[END_REF]. From a tribological perspective, CHF lipids play an unclear role. Since adhesion decreased after that skin was delipidized with ether [START_REF] Pailler-Mattei | A new approach to describe the skin surface physical properties in vivo[END_REF], it could be assumed that lipids tend to increase the COF. Nevertheless, no overall correlation has been found between the lipids of the skin surface and the COF for most of the body regions except for the forehead and postauricular region which are rich sebum areas [START_REF] Cua | Frictional properties of human skin: relation to age, sex and anatomical region, stratum corneum hydration and trans-epidermal water loss[END_REF][START_REF] Cua | Skin surface lipid and skin friction: Relation to age, sex and anatomical region[END_REF]. Differences of lipids composition and quantity depending on the body region could explain the contradictory correlations observed between lipids and the COF [START_REF] Mavon | Sebum and stratum corneum lipids increase human skin surface free energy as determined from contact angle measurements: A study on two anatomical sites[END_REF][START_REF] Sivamani | Review: Coefficient of friction: tribological studies in man -an overview[END_REF].

Because of the natural biological diversity of humans, each fingerpad is unique. This elementary statement assumes that friction forces' differences are due to the diversity of finger properties. Moreover, it underlines the necessity of performing in vivo friction tests with a rather large number of individuals [START_REF] Veijgen | Variables influencing the frictional behaviour of in vivo human skin[END_REF]. The diversity of fingerpad's properties is a major concern in the field of tactile stimulators development. Indeed, such a device must be adapted and equally efficient from one individual to another. Another crucial issue is the stimulator's capacity to generate similar friction forces to those measured when the finger rubs "real" surfaces. This last point underlines the kind of surfaces which need to be considered. Compared with precedent studies, the present one follows the original approach of picking out counterparts which are defined from a perceptual perspective rather than from an engineering one. In the framework of psycho-perceptual researches, it has been shown that textures could be classified according to four dimensions depending on the perception of individuals when touching a surface. These dimensions of tactile perception have been defined as: "softness", "warmth", "roughness" (eventually divided in the two sub-dimensions: fine and macro roughness) and "slipperiness" [START_REF] Hollins | Perceptual dimensions of tactile surface texture: A multidimensional scaling analysis[END_REF]. Among them, only the latter listed dimension has been entirely related to friction forces occurring at the skin/surface interface.

As friction modifiers, tactile stimulators are consequently convenient to simulate surfaces situated in a sticky/slippery scale (slipperiness dimension).

This study aims at characterizing the fingerpad properties of a rather large number of individuals and provides COF measurements of these fingers when they scan surfaces situated in the slipperiness dimension. Some counterfaces are "real" (physical) surfaces, and others are "simulated" ones by the use of a tactile stimulator named STIMTAC [START_REF] Biet | inventors Interface tactile vibrante[END_REF]. The goal of this study is to contribute giving answers to the following questions: What are the fingerpad properties' differences between individuals? When rubbing the same surface, what are the COF differences between individuals? What fingerpad's properties govern the COF when they rub against physical surfaces? Do the COFs measured on a stimulator have the same range than the COFs encountered on physical surfaces?

EXPERIMENTS AND METHODS

Experiments overview

Participants

Twenty-six French volunteers (13 women and 13 men) participated in the experiments which were carried out. All procedures performed in studies involving human participants were in accordance with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. The subjects gave informed consent and provided information about their date of birth. None of them had been subjected to a similar experiment before and they were neither physiologically nor cognitively deficient. Individuals' age ranged from 34 to 56 years with an average of 42, and the mean age for men was 41.2 while the mean age for women was 42.8.

Global experiments procedure

Every participant was requested to follow the same global experiments procedure during approximately 45 min. This procedure was chronologically organised as follows. First, the subject had to stay calm for 5 min in a room where the temperature was 191°C before washing his/her right forefinger with a common hydro-alcoholic gel (Aniosgel 85 NPC) for 30 seconds in order to prevent pollution due to product handling or sebum excess resulting from eventual physical activities just before the experiments. After the cleaning and drying of the subject's hands, SC's physicochemistry analysis was performed (see §2.2.3). Then, friction experiments were conducted in a second room where temperature was kept at 200.5°C (see §2.3). As the chemical composition of SC changes during the day [START_REF] Puttnam | Attenuated total reflectance studies of the skin[END_REF], an important task was to perform friction experiments just after the characterization of the fingerpad's physicochemistry. Indentation tests were performed afterwards (see §2.2.2).

Finally, the right forefinger of the participant was moulded to obtain its morphology and its roughness (see §2.2.1). The finger replica was made later, after the subject was released.

Fingerpad properties characterization

Shape and roughness

After hands cleaning and drying, the right forefinger's replica of each individual was realized in three steps. First, the subject pressed his/her finger on a high viscosity silicone paste in order to create a roughly-defined mould. Second, a low viscosity silicone was trapped between the participant's finger and the mould. During this step, the mould was put in to contact with the hinge of the participant's phalange but not against the fingerpad, in order to obtain a precise mould and prevent dermatoglyphs' deformation. Then, a polyurethane replica was made. The replica's topography was made by using an optical apparatus based on variations of focal length (InfiniteFocus G4, Alicona, Austria). The data were processed with the open source software Gwyddion 2.30. Two kinds of measurements were made: (a) the fingerpad's curvature radii Rmain and Rperpendicular (respectively along and across the finger's principal axis) led to calculate the geometric mean of the curvature radius:

lar perpendicu main gm R R = R  (1)
(b) after extraction of 5 profiles, the mean spatial period SP and the mean epidermal ridges Rt, i.e. the sum of the largest profile peak height and the largest profile valley depth (see Figure 1) were calculated. Note that Rt's mean value has been preferred to Ra in order to be more accurately related to the dermatoglyphs' height.

To validate the replica's process, comparison of a grooved aluminium plate (with dimensions similar to fingerpad in terms of spatial period and depth of valleys) and its replica was carried out. There was a 4% error for Rt and 1% for SP between the original and the replica.

Effective elastic modulus

A specific indentation device was implemented to evaluate the fingerpad's rigidity. This apparatus was mainly composed of a ring, an inductive displacement transducer rigidly connected to the ring, and a load gauge sensor (see Figure 2). During one of the tests, the subject placed her/his finger on the ring which was vertically guided with very low-stiffness elastic blades (7.8×10 -2 N/mm), and pressed her/his finger on the load sensor. As the ring had a 22 mm inner diameter, it stayed in contact with the phalange's extremities (see drawings included in Figure 2), thus only the fingerpad was pressing the load sensor.

This was an active test: the participant pressed her/his fingerpad by her/himself on the load sensor's surface (loading and unloading). The only direction given to the subject was to reach a maximum load which had to be as close as possible to 0.5 N (similar to the loading encountered during friction tests, see §2. Normally, the well-defined geometry of the indenter is used for determining the projected contact area and then the reduced elastic modulus of the tested material. In the present method, as the indenter is a flat surface, the current method cannot be used. Considering the beginning of the loading curve to be a purely elastic deformation, the loading measurement points until 0.4 N were fitted with Hertz's theory (see Figure 2) to calculate the reduced elastic modulus E * using:

2 3 2 1 3 4 h R E P= *    (2)
with:

P the normal load (N), E * the reduced elastic modulus (Pa),

h the displacement (m),
R the radius of the equivalent sphere (m).

P and h were obtained from the indentation measurements. The radius R was extracted from the analysis of the finger's morphology (see §2.2.1): R was chosen to be equal to Rgm, which gave an identical contact area for an equivalent penetration depth. Then, the elastic modulus of the finger Efinger was determined according to the following equation: As pointed out by van Kuilenburg et al [START_REF] Van Kuilenburg | Contact modelling of human skin: What value to use for the modulus of elasticity[END_REF], this elastic modulus has to account for the multilayered and non-homogeneous structure of the skin. Therefore, Efinger was more accurately defined as the fingerpad's effective elastic modulus Eeff.

steel steel finger finger * E ν E ν E 2 2 1 1 1     (3)

Cutaneous hydrolipid film (CHF) composition

The physicochemistry of the fingerpad's SC was characterized by Fourier transform infrared spectroscopy with attenuated total reflection (ATR-FTIR). Briefly, in ATR-FTIR spectroscopy analysis, the sample is placed on an infrared transparent crystal of high refractive index (the internal reflectance element or IRE) whose geometry allows total internal reflection. The internal reflectance results in an evanescent wave which is attenuated in regions where the sample absorbs the IR radiation.

In this study, the fingerpad's SC absorbance was measured in the mid-IR region (1000 to 4000 cm -1 ) with a 2 cm -1 resolution using 8700-Nicolet spectrometer equipped with a ZnSe IRE displaying a trapezoidal cut at 45°. Between each analysis, the 0.8 cm² surface of the IRE was gently cleaned with high purity ethanol. In ATR, the IR radiation penetration depth depends on the contact pressure between the sample and the IRE [START_REF] Puttnam | Attenuated total reflectance studies of the skin[END_REF][START_REF] Brancaleon | Attenuated total reflection-Fourier transform infrared spoectrocsopy as a possible method to investigate biophysical parameters of stratum corneum in vivo[END_REF]. To minimise penetration depth variations from one subject to another, the experimenter applied with his own forefinger a nearly constant loading of 5 N on the subject's fingernail. The reason for holding the load in this way (lack of mechanical loading system) was to avoid causing the participant pain or discomfort. Pre-tests have shown that loading exceeding 3 N did not change spectrum peak intensity and position.

The spectra obtained in this study (see Figure 3) were very similar to those presented in the literature for human fingers or other parts of the human body in vivo [START_REF] Brancaleon | Attenuated total reflection-Fourier transform infrared spoectrocsopy as a possible method to investigate biophysical parameters of stratum corneum in vivo[END_REF] and for SC biopsies in vitro [START_REF] Merle | Physical and chemical perturbations of the supramolecular organization of the stratum corneum lipids : in vitro to ex vivo study[END_REF].

ATR-FTIR spectra's interpretation is much discussed in the literature in order to characterise the structure and composition of the SC. As regards the characterisation of water quantity in the SC, the most appropriate technique is to calculate the as long as the water content was lower than 0.2 g/g [START_REF] Potts | Stratum corneum hydration : experimental techniques and interpretations of results[END_REF]. As regards the characterisation of lipids, it must be noticed that lipids' IR absorbance signal is representative of a mixture of both intercellular lipids and skin surface lipids [START_REF] Merle | Physical and chemical perturbations of the supramolecular organization of the stratum corneum lipids : in vitro to ex vivo study[END_REF]. To compare the skin of different subjects, it is convenient to calculate the ratio of the integrated intensities     The IR radiation depth of penetration dp has been defined elsewhere [START_REF] Brancaleon | Attenuated total reflection-Fourier transform infrared spoectrocsopy as a possible method to investigate biophysical parameters of stratum corneum in vivo[END_REF] as:

    2 1 2 1 2 2 1 sin 2 1 n n θ πνn d p    ( 4
)
where ν is the frequency of the incident radiation, θ is the angle incidence (45°), n1 = 2.43 and n2 = 1.55 are the refractive indexes of the IRE crystal and the skin, respectively [START_REF] Potts | Stratum corneum hydration : experimental techniques and interpretations of results[END_REF].

Considering the wavenumbers used, the penetration depth of the evanescent field in the fingerpad ranged from 0.6 to 1.1 µm. ATR-FTIR analysis was also located in the outer layer of the SC.

Friction tests

Scanned surfaces

Two sets of surfaces were considered in this study.

The first set of surfaces consisted of two 40×40 mm² flat slabs of 4 mm thickness made of polymer: PTFE and acrylonitrile butadiene styrene (ABS) with a sticky coating. The roughness Ra of these slabs was 0.70 µm for the PTFE slab and 1.56 µm for the ABS slab.

Prior to the friction experiments, all individuals have judged the ABS surface as being much more sticky than the PTFE one (thus perceived more slippery than the coated ABS slab). In the following, the coated ABS and PTFE slabs will be thus respectively defined as "sticky surface" and "slippery surface". These two surfaces were adjoined on both sides of a larger plate in order to be scanned by the finger in a continuous way.

The second set of surfaces was based on the use of a tactile stimulator named STIMTAC [START_REF] Biet | inventors Interface tactile vibrante[END_REF].

The plate of STIMTAC (in beryllium copper coated with a polymer film with Ra = 1.23 µm) looks like a laptop computer touchpad (see Figure 4), but it vibrates with the help of piezoelectric actuators. The key concept of this stimulator's operating is based on the squeeze film air-bearing theory which is detailed in [START_REF] Wiesendanger | Squeeze Film Air Bearings using Piezoelectric Bending Elements[END_REF]. When one's finger rubs the vibrating plate, a thin layer of air is trapped between the plate and the fingerpad and leads to the lowering of friction force. The greater the vibration amplitude is (up to 2.3 µm), the thicker the layer of air between the plate and the fingerpad is, and thus the lower the friction force is. It must be mentioned that the lowering of the friction force has not been attributed to roughness modification due to vibration but to an air-lubrication mechanism [START_REF] Biet | Implementation of tactile feedback by modifying the perceived friction[END_REF]. The friction force is modulated with time: at a given moment, vibration amplitude is the same all over the plate.

However, as STIMTAC is equipped with sensors measuring the finger's position, the lowering of friction forces (i.e. plate's vibration) can be changed depending on the fingerpad's position on the plate. Consequently, one can program STIMTAC to lower the friction force differently from one position to another. In this study, the device was programmed to vibrate at the maximal amplitude when the finger was on one side of the plate while no vibration was applied when the finger scanned the other side. Simply put, these two sides of the vibrating plate will be respectively defined as "slippery surface" (lowest friction force) and "sticky surface" (highest friction force), even if STIMTAC does not produce different surfaces but different friction conditions.

To easily distinguish them, the first set of surfaces (polymer slabs) and the second set (STIMTAC use) were named "real surfaces" and "simulated surfaces", respectively.

COF measurements

A specific tribometer was implemented to measure the friction coefficient between the subject's finger pad and a counterpart. This apparatus was constituted of a homemade sensors' box allowing for simultaneous measurements of normal and tangential forces at a rate of 1000 samples/s. The normal load resolution was about 10 mN and the tangential sensor had a 5 mN resolution and high stiffness. STIMTAC was rigidly fixed on the sensors box in order to perform friction tests with the simulated surfaces (see Figure 4). When performing friction tests with the real surfaces, the sticky and slippery adjoined slabs were placed on STIMTAC's vibrating plate which was switched off in this case (no vibration). With a constant speed of 4 mm/s, a linear DC motor moved the subject's arm which was held in a trough. The finger slid in the medial radial direction, from right to left and left to right (see Figure 4), and one test consisted of four cycles of 40 mm. Due to the linear motor operating limits, the finger scanning speed was lower than the more spontaneous velocities encountered during natural active tactile exploration which are close to 40 mm/s [START_REF] Tanaka | Contact force and scanning velocity during active roughness perception[END_REF]. Nevertheless, the scanning speed used in this study was similar to those classically performed during passive tactile tests [START_REF] Van Kuilenburg | An experimental study on the relation between surface texture and tactile friction[END_REF][START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF].

As the participant's finger was driven by the DC motor, the test was performed in passive touch conditions in terms of speed and sliding distance. Nevertheless, the subject exerted by her/himself a force normal to the test surface, pressing her/his finger with a load of about 0.5N. A gauge on computer screen allowed the visualization of the actual loading in real time to help the subject staying in control. The angle between the finger and the scanned surface was about 25°. The friction tests were performed between the participant's right forefinger against real and then simulated surfaces. Before each test, the rubbed surfaces were cleaned with distilled water. for the same subject and the same set of surfaces) the difference in the two paired data is considered and the t-test is applied to the mean difference using the null hypothesis. In order to test the linear correlation between two populations, the Pearson's coefficient of correlation R is calculated and its square R², the coefficient of determination, demonstrates the correlation's strength in terms of percentage.

RESULTS AND DISCUSSION

Gender's influence on the fingerpad's properties

Substantial differences of fingerpad properties between women and men were encountered in this study. They are summarized in the following paragraphs which successively deal with the fingerpad's roughness, effective elastic modulus and then hydrolipid film composition.

Concerning the fingerpad's shape and roughness, the mean values of Rgm, SP and Rt calculated from the replicas' analysis of the 26 individuals are respectively 13±3 mm, 458±71 µm and 102±26 µm. These values are consistent with these encountered in the literature [START_REF] Maneo | Relationship between the structure of human finger tissue and the location of tactile receptors[END_REF].

Whereas other studies have shown women have less spaced epidermal ridges than men [START_REF] Acree | Is there a gender difference in fingerprint ridge density?[END_REF], there is no difference of SP between them in the present study. However, our results suggest that men have respectively higher epidermal ridges than women, with Rt values of 118±25 µm and 86±16 µm. This difference between women and men can be considered very significant (p-value<0.001). To the best of our knowledge, this gender difference in fingerprint depth has never been published.

The fingerpad's effective elastic modulus Eeff of all the participants reaches 70±20 kPa with a very significant difference between men and women (F-test, p<0.01). Women's fingerpad are stiffer than those of men: 82±20 kPa compared with 59±13 kPa. These values are in agreement with previous studies [START_REF] Derler | Tribology of skin: Review and analysis of experimental results for the friction coefficient of human skin[END_REF][START_REF] Pailler-Mattei | In vivo measurements of the elastic mechanical properties of human skin by indentation tests[END_REF], but again, to the best of our knowledge, this difference of the fingerpads' elastic modulus between men and women has never been reported. The range of elastic modulus values suggests that the maximum shear stress is located between 1.5 and 2.3 mm beneath the finger's surface. This observation justifies the denomination of effective elastic modulus used, which takes into account the stiffness of the SC, the epidermis, the dermis, the hypodermis and the underlying tissues also.

Figure 6 highlights the calculated DW and DL/W values. For all the participants, the results confirm the decrease of water quantity when the ratio of lipids to water rises. Clearly, women generally have a lower water content (and also greater lipids proportion) in the outer layer of the SC than men. The influence of gender on the amount of skin lipids and on skin hydration has not [START_REF] Cua | Frictional properties of human skin: relation to age, sex and anatomical region, stratum corneum hydration and trans-epidermal water loss[END_REF][START_REF] Cua | Skin surface lipid and skin friction: Relation to age, sex and anatomical region[END_REF] or has rarely been observed in the literature [START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF]. The water descriptor index DW is valued at 0.061±0.055 (a.u.) with a significant difference between women and men (Ftest, p<0.01). Men have higher water content with 0.094±0.062 (a.u.) than women with 0.029±0.014 (a.u.) which agrees with Veijgen et al [START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF]. The lipids/water index DL/W reaches 0.122±0.075 with a significant difference between men and women (F-test, p<0.05). The hydrolipid film in women has a higher lipid to water ratio (0.15±0.07) than that of men (0.09±0.07). Moreover, men can have much larger values of DW (up to 0.2) than women (no higher than 0.052), but DL/W ranges are similar for men (from 0.03 to 0.25) and women (from 0.07 to 0.27). Some studies point out the softening of skin when hydration increases [START_REF] Derler | Tribology of skin: Review and analysis of experimental results for the friction coefficient of human skin[END_REF][START_REF] Van Kuilenburg | Contact modelling of human skin: What value to use for the modulus of elasticity[END_REF][START_REF] Geerligs | In vitro indentation to determine the mechanical properties of epidermis[END_REF]: this result is consistent with the measurements obtained in the present study which show lower effective elastic moduli and higher water contents for men compared to women.

COF's variability

The first line of the Table 1 highlights the mean and standard deviation values of the COFs related to the sticky and slippery surfaces (both real and simulated). As expected, µstick is higher than µslip. The difference between µstick and µslip is very high (t-test. paired data. p<0.001) for both sets of surfaces.

As depicted in Figure 7, µstick and µslip can reach very different values from one subject to another. This dispersion supports the assumption that the diversity of the fingerpad's properties induces different friction behaviours between individuals even though the same surface is scanned. Most participants have µstick and µslip ranging respectively from 0.5 to 3 and from 0.1 to 2.4, whereas these COFs rise up to 5.5 and 3.7 for a few subjects. Although being extremely high, these values have nevertheless been reported elsewhere [START_REF] Van Kuilenburg | An experimental study on the relation between surface texture and tactile friction[END_REF].

Women show lower COFs than those of men for each kind of surfaces, both for real and simulated surfaces (see Table 1 and Figure 7). This difference is very significant (t-test, p<0.01) for the four studied surfaces. This gender's influence on the COF has been not [START_REF] Cua | Frictional properties of human skin: relation to age, sex and anatomical region, stratum corneum hydration and trans-epidermal water loss[END_REF][START_REF] Cua | Skin surface lipid and skin friction: Relation to age, sex and anatomical region[END_REF][START_REF] Acree | Is there a gender difference in fingerprint ridge density?[END_REF] or rarely indicated in the literature as far as the finger is concerned [START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF].

Concept of Tactile Frictional Contrast (TFC)

An overview of the 52 friction cycles collected (26 participants and the two surfaces' sets) leads to the observation of large variations of their global shapes. Without considering the algebraic values of µstick and µslip, some individuals show very large differences between these two COFs whereas others show very close levels of µstick and µslip. While the counterparts and sliding conditions are the same, this means that subjects' fingers are characterized by their propensity to frictionally distinguish two surfaces situated in the slipperiness dimension. To quantify this observation, the authors introduce the concept of Tactile Frictional Contrast. By analogy with the optical contrast which quantifies the brightness difference between the darkest and the brightest levels in a picture, the Tactile Frictional Contrast (TFC) aims at quantifying the COF's difference when one's finger rubs two different surfaces. For a welldefined couple of surfaces, the TFC is thus an individual's feature. As this study focalizes on surfaces' slipperiness, the TFC can be here defined as:

stick slip TFC     1 (7)
The TFC has the convenience of being a dimensionless index ranging from 0 (no difference between µstick and µslip) to 1. The more µstick and µslip are different, the higher the TFC is. As the participant's TFC ranges from 0.03 to 0.89, it can be concluded that the differentiation of the friction force in the slipperiness dimension can be extremely different between individuals. It can be seen from Table 1 that women's TFC is significantly higher than that of men both for real and simulated surfaces (t-test, p<0.01). Moreover, results suggest that the highest COF's values induce the lowest TFC's values.

Friction related to fingerpads' properties

For both real and simulated surfaces, µstick and µslip are significantly negatively correlated to

Eeff. The coefficients of determination R² range from 0.14 to 0.22 (t-test, p<0.05), with a slightly higher correlation for the simulated surfaces. This correlation has been pointed out by Liu (49) with the help of spherical-shaped artificial fingers. His study shows the stiffer the material is, the lower the COF. Besides, the TFC is significantly correlated to Eeff only for the simulated surfaces with R² = 18% (t-test, p<0.05). For the real surfaces the correlation is not significant (p<0.1).

The height of epidermal ridges Rt is significantly correlated with µstick and µslip for the two sets of surfaces. The coefficients of determination R² range from 0.32 to 0.42 (t-test, p<0.01). This shows that at least 32% of the COF's variance can be linearly related to the height of epidermal ridges when one's finger scans a more or less sticky surface. However, no correlation has been highlighted between Rgm or SP with both the COFs and the TFC for the four studied surfaces (t-test, p>0.05).

While some papers examine separately the effect of skin hydration [START_REF] Hendriks | Influence of surface roughness, material and climate conditions on the friction of human skin[END_REF][START_REF] Derler | Friction of human skin against smooth and rough glass as a function of the contact pressure[END_REF][START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF][START_REF] André | Fingertip moisture is optimally modulated during object manipulation[END_REF][START_REF] Sivamani | Review: Coefficient of friction: tribological studies in man -an overview[END_REF] and of the amount of lipids [START_REF] Cua | Frictional properties of human skin: relation to age, sex and anatomical region, stratum corneum hydration and trans-epidermal water loss[END_REF][START_REF] Cua | Skin surface lipid and skin friction: Relation to age, sex and anatomical region[END_REF] on the COF, the present study investigates these two parameters together. As regards lipids, Cua et al studies [START_REF] Cua | Frictional properties of human skin: relation to age, sex and anatomical region, stratum corneum hydration and trans-epidermal water loss[END_REF][START_REF] Cua | Skin surface lipid and skin friction: Relation to age, sex and anatomical region[END_REF] reveal a linear relationship between the COF and the amount of lipids in the skin. Nevertheless, our results do not show any correlation between the COFs and DL/W. As regards the fingerpad's hydration, the literature points out an increase of the COF with skin's moisturization [START_REF] Veijgen | Relating Friction on the human skin to the hydration and temperature of the skin[END_REF][START_REF] André | Fingertip moisture is optimally modulated during object manipulation[END_REF]. In the present study, DW is correlated with µslip but not with µstick. Moreover, very significant correlations (R 2 =0.7, ttest, p<0.001) have been found between the two physico-chemical descriptors and the TFC, both for simulated and real surfaces. As shown in Figure 8, a continuous decrease of the TFC is observed when DW increases. Moreover, results point out an almost linear increase of the TFC with the lipids/water ratio until DL/W is approximately 0.12 before reaching a plateau.

Therefore, for surfaces situated in the slipperiness dimension, the present study concludes to the predominance of fingerpads' hydrolipid film composition on the individuals' frictional differentiation of surfaces.

The absence of significant correlation between the COFs (µstick as µslip) and Eeff, SP or DL/W could be explained by the studied tribocontact's complexity. As tests have been performed

with human subjects, no fingerpad properties remained constant from one friction test to another. Thus, one can assume that the COFs and the fingerpad's properties are linked by a very complex transfer function including various couplings. Indeed, these properties are dependent on one another: DW and Eeff, for instance [START_REF] Derler | Tribology of skin: Review and analysis of experimental results for the friction coefficient of human skin[END_REF][START_REF] Van Kuilenburg | Contact modelling of human skin: What value to use for the modulus of elasticity[END_REF][START_REF] Geerligs | In vitro indentation to determine the mechanical properties of epidermis[END_REF].

STIMTAC efficiency in the slipperiness dimension

Considering Table 1, it can be seen that real surfaces lead to slightly higher COFs than simulated ones, whatever the participants' subclass (all subjects, or women and men separately). However, Figure 7 highlights that the COFs' measurement ranges are practically the same between the real and the simulated surfaces, both for µstick and µslip. Besides, the above mentioned gender's influence (higher COF for men than for women) is similarly encountered on both sets of surfaces. These results conclude to STIMTAC's efficiency in the simulation of sticky/slippery surfaces: it can achieve stimuli which induce similar friction forces to these encountered on this kind of surfaces.

The two sets of surfaces show similar TFC's values (see Table 1). Whatever the considered subclass of participants, the TFC is non-significantly different for the real and the simulated surfaces (t-test, p>0.1). The TFCs of the two sets of surfaces are plotted in Figure 9 where each point along the graph corresponds to a given participant. The two TFCs are linked with a linear relationship going through the origin with a slope close to 1 and R² = 0.73. Assuming that the finger/surface friction behaviour has a critical influence on tactile perception [START_REF] Bergmann Tiest | Tactual perception of materials properties[END_REF], STIMTAC could be exploited as a useful device, adapted to most individuals, to give the illusion of rubbing two distinct surfaces.

Moreover, Figure 8 shows a very similar influence of the CHF's composition on the TFC for the two sets of surfaces. Reminding that the TFC is mainly dependent on the CHF's composition (see §3.4), the latter result indicates that STIMTAC's accurately reproduces the impact of fingerpad's properties on the individuals' frictional differentiation of sticky/slippery surfaces.

CONCLUSION

This study has provided in vivo friction measurements of twenty-six fingerpads with rubbing conditions which were rather similar to those encountered when spontaneous natural touching occurs. The scanned surfaces were situated at both ends of the slipperiness psycho-perceptual dimension (highly sticky and highly slippery surfaces). Physical surfaces and simulated ones with STIMTAC have been rubbed. In parallel, mechanical, physicochemical and morphological features characterization of the twenty-six studied fingerpads has been carried out. To the best knowledge of the authors, noticeable differences of the fingerpads' properties between men and women have been highlighted for the first time. Women present lower epidermal ridge's height and water content than men but a higher forefinger's elastic modulus and lipids/water ratio.

With the same tribological modalities, the measured friction coefficients have been shown

very different from one individual to another due to large differences in their fingerpad's properties. Nevertheless, difficulties have been encountered to correlate the coefficients of friction (COFs) with the fingerpad's features. Furthermore, the newly concept of Tactile Frictional Contrast (TFC) has been defined. With fixed tribological conditions and the same surfaces' couple, this index accounts for one individual's feature and has revealed large differences between the studied subjects. For sticky/slippery surfaces, the TFC has been correlated with the hydrolipid film composition of the fingerpad's skin. As a differential measurement, the TFC could be an interesting index in the field of tactile stimulators' efficiency quantification as well as tactile perception.

For most participants, STIMTAC has allowed to reach similar TFCs than those obtained with the physical surfaces, thus validating its use for research on tactile tribology and friction perception. and TFC mean values and standard deviations of all subjects, women and men. Data are given for simulated and real surfaces separately.

FIGURE CAPTIONS

  3.2). During the test, a gauge on a computer screen indicated continuously the normal load applied. The angle between the finger and the surface was about 25° and loading speed (imposed by the subject) ranged from 0.9 to 3.5 mm/s. Prior to the real test, a 5 min training period took place in order to habituate the subject to use the indentation device.During the indentation, two simultaneous measurements were carried out: (a) the normal force P applied by the fingerpad on the polished stainless steel flat surface of the load sensor; (b) the displacement h of the ring relative to the load sensor surface. These measurements led to draw a loading-unloading indentation curve for each subject.

Figure 2

 2 Figure 2 highlights the indentation curve typically obtained. It shows a classically observed

  with: finger  the Poisson's ratio of the fingerpad fixed to 0.5, as for a perfectly incompressible material, steel  the Poisson's ratio of the stainless steel plate = 0.3, steel E the Young's modulus of the stainless steel = 210 GPa,

  and gravimetric measurements of SC in vitro has shown that the two measures were proportional

  lipids/water proportion in the SC[START_REF] Brancaleon | Attenuated total reflection-Fourier transform infrared spoectrocsopy as a possible method to investigate biophysical parameters of stratum corneum in vivo[END_REF].Consequently, integrated intensities (peak surface under the baseline) present on the fingerpad and a lipids/water descriptor index the lipids/water proportion were calculated.

Figure 5 2 . 4

 524 Figure5shows the typical curve of the COF versus the displacement obtained during one

Figure 1 :

 1 Figure 1: 3D overview of the fingerpad's surface. SP is the spatial period.

Figure 2 :

 2 Figure 2: Fingerpad indentation setup and indentation curve (grey points) typically obtained.

Figure 3 ::

 3 Figure 3: Typical ATR-FTIR spectrum obtained. Arrows indicates bonding's vibration bands

Figure 4 :

 4 Figure 4: Friction measurement device during a test with STIMTAC.

Figure 5 :

 5 Figure 5: Representation of a friction cycle (COF versus displacement of the finger) during a

Figure 6 :

 6 Figure 6: Water descriptor index DW and lipids/water ratio descriptor index DL/W

Figure 7 :

 7 Figure 7: COFs measured on sticky surfaces (µstick) relative to COFs measured on slippery

Figure 8 :

 8 Figure 8: Tactile Frictional Contrast (TFC) relative to DW (top) and DL/W (bottom) of women

Figure 9 :

 9 Figure 9: Tactile Frictional Contrasts (TFCs) on simulated surfaces against TFCs on real

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6

 123456 Figure 1

Figure 7 Figure 8

 78 Figure 7

Figure 9

 9 Figure 9

TABLE CAPTIONS Table 1 :

 CAPTIONS1 COFs (measured on sticky surfaces: µstick, and measured on slippery surfaces: µslip)

Table 1

 1

			Simulated surfaces			Real surfaces
		µstick	µslip	TFC	µstick	µslip	TFC
	All subjects	1.6±0.7	0.9±0.8	0.5±0.3	2.4±0.9	1.5±1.0	0.5±0.3
	Women	1.1±0.4	0.4±0.4	0.7±0.2	1.9±0.7	0.8±0.6	0.6±0.2
	Men	2.2±0.6	1.5±0.7	0.4±0.3	3.0±0.8	2.2±0.9	0.3±0.2

ACKNOWLEDGEMENTS

The authors thank the IRCICA/L2EP of the University of Lille 1 for providing the stimulator STIMTAC and for their help and advice. The authors are deeply grateful to all the volunteers who participated and gave their time freely, Jean du Verger for proofreading the English version, and Fred Bobrie for fingers drawing. This work was supported by the DEFISENS program of the National Organisation for Scientific Research (CNRS, France).