S A E Boyer

M Baba

J-M Nedelec

J-P E Grolier
email: j-pierre.grolier@univ-bpclermont.fr

Polymer Microstructures. Modification and Characterisation by Fluid Sorption 1

Keywords: scanning transitiometry, vibrating wire weight sensor, thermoporosimetry, temperature modulated DSC, polymer foam, mesh size distribution, pore size distribution

Polymer micro-organisation can be modified by combination of the three constraints, thermal, hydrostatic and fluid sorption. In selecting the fluid's nature, chemically active or inert, and its physical state, liquid or supercritical, new "materials" can be generated. In addition, the interplay of temperature and pressure allows tailoring the obtained material structure for specific applications. Several complementary techniques have been developed to modify, analyze and characterize the end products: scanning transitiometry, vibrating wire (VW)-PVT coupling, thermoporosimetry, temperature modulated DSC (TMDSC), sorptometry. The great variety of possible applications in materials science is illustrated with different polymers which can produce materials from soft gel to rigid foams when submitted to fluid sorption, typical fluids being methane, or a simple gas (CO 2 or N 2 ). Absorption of an appropriate fluid in a cross linked polymer leads to a swelling phenomenon. Thermoporosimetry is a calorimetric technique developed to measure the shift by confinement of thermal transition temperatures of the swelling fluids, which can be currently used solvents or mercury. Application of thermoporosimetry to a swollen cross linked polymer allows to calculate the mesh size distribution and to evaluate the degree of reticulation of the polymer. The same technique can be applied to characterise the pore size distribution in a foamed polymer.

INTRODUCTION

Selection of materials is usually made according to their thermophysical properties and their structural characteristics. In numerous industrial applications, in nano-science at large, making use of advanced nano-technologies, polymers are playing a major central role. Due to their intimate native structures to the many modified structures ??? polymers offer a large spectrum of useful all purpose materials. Among possible tailored polymer structures most of them are obtained by modifying the polymer micro-organizations through foaming processes.

The polymer foaming sector is certainly one of the most active branches of the industrial activity in materials science. Efficient foaming is developed by using selected blowing agents.

As a result of the negative impact on the environment of most of the blowing agents traditionally employed up to now, an international regulation has imposed a ban on all chlorofluorohydrocarbons types blowing agents. Replacement agents, including simple gases like CO 2 or N 2 , are currently proposed and experienced [1]. There from ongoing active researches are focused on gas/polymer interactions in terms of energy of interaction, solubility as well as polymer swelling under gas sorption. As a matter of fact, the micro-organization of polymers can be modified by combination of three constraints, thermal, hydrostatic and fluid sorption. In selecting the fluid's nature, chemically active or inert, and its physical state, liquid or supercritical, new "materials" can be generated. Furthermore, the interplay of temperature and pressure allows obtaining material structures to demand for specific applications. In this context, several complementary techniques have been developed to analyze, characterize and modify the polymers'organization: scanning transitiometry [2], vibrating wire (VW)-PVT coupling [3], thermoporosimetry [START_REF] Randzio | French Patent 91-09227[END_REF] and temperature modulated DSC (TMDSC) [START_REF] Randzio | [END_REF]. In what follows, the different techniques are shortly described and selected examples are used to illustrate their pertinent contribution.

SCANNNG TRANSITIOMETRY

The use of state variables (p, V and T) in scanning calorimetric measurements has lead [2,3] from P V T-calorimetry to the now well established scanning transitiometry technique [START_REF] Randzio | French Patent 91-09227[END_REF].

With this technique the simultaneous determination of thermal and mechanical responses of the investigated system, pertubated by a variation of an independent thermodynamic variable while the other independent variable is kept automatically constant, allows the determination of thermodynamic derivatives over extended ranges of pressure and temperature, impossible to obtain by other known techniques. Four thermodynamic situations are thus possible to realize in the instruments based on such technique, namely, pVT-controlled scanning calorimeters or simply scanning transitiometers, since they are particularly adapted to investigate transitions by scanning one the three thermodynamic variables. The four possible thermodynamic situations are obtained by simultaneous recording of both heat flow (thermal output) and the change of the dependable variable (mechanical output). Then, making use of the respective related Maxwell relations one readily obtains the main thermophysical properties as follows: a) scanning pressure under isothermal conditions yields the isobaric thermal expansivity α P and the isothermal compressibility κ T as functions of pressure at a given temperature; b) scanning volume under isothermal conditions yields the isochoric thermal pressure coefficient β V and the isothermal compressibility κ T as functions of volume at a given temperature; c) scanning temperature under isobaric conditions yields the isobaric heat capacity C P and the isobaric thermal expansivity α P ; d) scanning temperature under isochoric conditions yields the isochoric heat capacity C V and the isochoric thermal pressure coefficient β V .

A detailed description of a basic scanning transitiometer (from BGR TECH, Warsaw, Poland) used in the present applications to polymers is given elsewhere [START_REF] Randzio | [END_REF]. It consists of a calorimeter equipped with high-pressure vessels, a PVT system, and a LabVIEW based virtual instrument (VI) software. Two cylindrical calorimetric detectors (ext. diameter 17 mm, length 80 mm) made from 622 thermocouples chromel-alumel each are mounted differentially and connected to a nanovolt amplifier. The calorimetric detectors are placed in a calorimetric metallic block, the temperature of which is directly controlled with an entirely digital feedback loop of 22-bit resolution (~10 -4 K), being part of the transitiometer software. The calorimetric block is surrounded by a heating-cooling shield connected to an ultracryostat (Unistat 385 from Huber, Germany) and the temperature difference between the block and the heating-cooling shield is set constant (5, 10, 20 or 30 K) as controlled by an analogue additional controller. The whole assembly is placed in a thermal insulation enclosed in a stainless steel body and placed on a stand, which permits to move the calorimeter up and down over the calorimetric vessels. The actual operating ranges of scanning transitiometry are respectively 173 K < T < 673 K and 0.1 MPa < p < 200 MPa (or 400 MPa).

Figure 1 here

According to its basic principle a scanning transitiometer can then be operated in different scanning situations. In particular two main scanning situations, as described in Fig. 1, have been used: pressure-controlled scanning transitiometry (PCST) and temperature-controlled scanning transitiometry (TCST). In addition, the differential mounting of the calorimetric detector permits to measure the differential heat flux, between the reference and the measuring cells, resulting from the physicochemical effects occurring in the investigated systems/phenomena. The advantage of PCST is well illustrated by the study of gas/polymer systems, in particular during sorption (compression by gas-pressurizing fluid)/desorption (by decompression of gas-pressuring fluid) pressure scans. Taking advantage of the respective role of the measuring and reference cells, two modes have been preferably used, as shown Fig. 2 (a et b), the thermal differential comparative in which two different polymers can be directly compared and the thermal differential in which a polymer sample is characterized vs an inert reference (steel for example). For each thermal mode, the measured heat rate with polymer was corrected for the heat effects produced by pressurization/depressurization of the hydraulic fluid (expansibility) and the volume sample (asymmetry of the cells). The thermal differential comparative mode was chosen to directly compare the difference in behavior of two polymers submitted simultaneously to the sorption/desorption of the same gas under the same T and P conditions. For this, samples of identical size and shape of two polymers, medium density polyethylene (MDPE) and poly(vinylidene fluoride) (PVDF) were placed in the two cells both connected to the gas line. Isothermal pressure scans were performed using CO 2 as pressure transmitting fluid. Measurements were taken on polymers in the solid state, i.e. between glass and melting transitions temperatures, at different temperatures between 353 and 403 K and at each temperature pressure was scanned up to 100 MPa. The calorimetric signal, that is the comparative {(MDPE-PVDF)/CO 2 } differential heat flux, is then proportional to the thermal effect due to the difference of the {polymer/gas} interactions between the two polymers. Interestingly, below 40 MPa, the calorimetric signal dQ {(MDPE -PVDF)/CO 2 }/dP < 0} is endothermic thus PVDF exhibits larger interactions with CO 2 than MPDE. Above 40 MPa the reverse situation is observed, the calorimetric signal dQ {(MDPE -PVDF)/CO 2 }/dP > 0} is exothermic showing that MDPE exhibits larger interactions with CO 2 [6].

Figure 2 here

The simple thermal differential mode, as shown Fig. 2 (c), where the differential heat flux is measured between the measuring cell containing a polymer sample and the reference cell empty (and not connected to the pressure line) acting only as a thermal reference was used in the temperature-controlled scanning transitiometry (TCST) situation. Typically, TCST was chosen to investigate transitions and modifications of polymers submitted to isobaric temperature scans. For example, measurements were taken on polymer samples in cooling or heating between 223 and 473 K at different pressures up to 100 MPa. Under such possible P-T conditions melting/crystallization and glass transitions can be fully characterized. The hydraulic fluid can either be a neutral fluid like mercury or a simple gas like carbon dioxide, nitrogen or methane. In this context, several studies have been successfully conducted; namely: modification of polystyrene glass transition high pressure methane [7], isobaric thermal expansion of polymers (MDPE, PVDF) in absence or in presence of solubilized gases (CO 2, N 2 ) [1,8]. Another advantage of TCST is certainly the possibility, through the rigorous thermodynamic control of the process, to foam polymers. The foam structure (type, size, polydispersity of microcavities) is monitored by controlling both T and P and also the nature of the pressurizing fluid (actually the hydraulic fluid itself) [9]. Regarding nanoscale selfassembled molecular organization in liquid crystalline type amphiphilic diblock copolymers, the use of TCST is well described [6,10,11] by the systematic investigation of PEO m -b-PMA(Az) n type polymers (with m and n degrees of polymerization) containing hydrophilic polyethylene oxide (PEO) and hydrophobic azophenyl (PMA(Az)) moieties. Isobaric scans over the isotropic transition of PEO 114 -b-PMA(Az) 20 using either mercury or supercritical CO 2 as pressurizing fluids allowed to demonstrate that "inert" mercury and "chemically active" CO 2 have quite opposite effect on the Clapeyron slope of this transition. That is a lowering of the transition temperature T iso and a concomitant significant decrease of the transition entropy ∆S iso showing that CO 2 sorption into the polymer is facilitated; therefore the molecular self organization at the interface polymer/liquid crystal becomes much easier to yield nano-scale ordered structure, that is PEO cylinders organized in an hexagonal structure into the liquid crystal domain (see schematic illustration Fig. 3), as confirmed by atomic force microscopy analysis [10].

VIBRATING WIRE -PVT COUPLING

Processing of polymer foams is easily controlled when using blowing agents acting chemically to lower the glass transition (plasticization effect) and physically to promote the foaming process (nucleation effect). As a matter of fact, tailoring of foam structures, that is to say adjusting the type of cells, their sizes and shapes, is possible through a quantitative evaluation of the amount of gas, very often in supercritical state, absorbed in the polymer and also through a fine tuning of both temperature and pressure during processing (usually extrusion-expansion). When dealing with gas sorption in porous materials [12], that is investigating gas sorption/desorption phenomena depending on the pores distribution, again the quantitative evaluation of the amount of gas involved is before all necessary. A double technique has been developed to meet such requirements. This new concept [13] combines two techniques, a vibrating wire (VW)-sensor and a pressure decay PVT technique. The VWsensor acts as a force sensor to weigh the material sample during sorption while concomitantly the PVT technique allows to quantitatively estimating the corresponding number of moles of gas absorbed. In the case of a polymer, the buoyancy force exerted by the pressurized fluid on the polymer depends on the swollen volume, ∆V pol , of the polymer due to the gas sorption. The VW-sensor consists essentially of a high pressure cell in which the polymer sample is placed in a holder suspended by a thin tungsten wire (diameter 25 µm, length 30m) in such a way that the wire is positioned in the middle of a high magnetic field generated by a square magnet placed across the high pressure cell. Through appropriate electric circuitry and electronic control the tungsten wire is activated to vibrate. The period of vibration which can be accurately measured is directly related to the mass of the suspended sample. The pressure decay PVT-method allows calculating the number of moles, n sol , of gas [which is initially kept in a high-pressure calibrated cell] absorbed in the polymer, from the measurement of pressure at its initial value p i when the gas enters the measuring cell and its final value p f after the gas-polymer system has returned to thermodynamic equilibrium. The combined VW-PVT apparatus is designed to measure sorption of the gas and the concomitant volume change of the polymer at pressures up to 100 MPa from room temperature to 473 K.

The experimental method consists in a series of successive transfers of the gas by connecting the calibrated cell to the equilibrium cell which contains the polymer. The initial, p i , and final, p f , pressures are recorded between each transfer. The detailed methodology and the revisited iterative calculation to simultaneously calculate the solubility together with the volume change of the polymer due to sorption [14,15] are using two rigorous working equations.

The working equation ( 1) for the PVT-technique gives the amount of gas entering the polymer sample during the first transfer once equilibration is attained

( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + - + ∆ = pol f f f i i i g pol f f f g sol V V V T Z p V T Z p R M V Z p T R M m 3 2 3 (1)
The term m sol is the mass of gas dissolved in the polymer, M g is the molar mass of the dissolved gas, Z i with Z f are the compression factors of the gas entering the polymer respectively at the initial (index i ) and final (equilibrium sorption, index f ) conditions. Volume of the degassed polymer and the volume change due to sorption are represented by V pol and ∆V pol respectively. And the total amount of gas absorbed by the polymer after completion of the successive transfers is given by equation (2)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - ∆ - - + + ∆ = ∆ - - - - k f k f pol k f k f k f k pol p k f k i k i k i g k f k f k pol k f g k sol T Z V V V p T Z V V V p T Z V p R M T Z V p R M m 3 2 ) ( 1 1 1 2 ) 1 ( 3 ) ( ) ( ) ( ) ( ) ( (2)
where

) (k sol m ∆
is the increment in dissolved gas mass resulting from the transfer k and

( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + - + ∆ = pol C S B pol sol V V g R L V m ρ π ρ ω ω ρ 2 2 2 0 2 4 (3)
The terms ω 0 with ω B represent the natural (angular) frequencies of the wire in vacuum and under pressure respectively, V C the volume of the container. The symbols L, R and ρs are respectively the length, the radius and the density of the wire.

A common term appears in both equation (2) and equation ( 3), the density, ρ, of the gas

f f g gas Z p T R M = ρ (4)
Then equation ( 3) can be written

d V m pol gas k sol + ∆ = ∆ ρ ) ( (5)
The term d represents the apparent concentration of gas in the polymer, i.e. when the change in volume, ∆V pol , is zero. However, despite the different terms appearing in the two working equations ( 2) and ( 3), these two equations can be both expressed by the same reduced equation ( 5) having the slope given by equation ( 4). The vibrating-wire sensor VW is described with rigorous models yielding a working equation in which all parameters have a physical meaning: ω 0 and ω B the natural frequencies [Hz], V C [m 3 ], L [m], R [m] and ρs [kg m -3 ] of the wire. The pressure decay technique requires Z, M g [g mol -1 ] for the gas and the volumes of the cells [m 3 ]. The volume of the polymer sample, V pol , with its associated change, ∆V pol , and the total mass of dissolved gas, m sol , are the only unknown terms.

It appears that the vibrating-wire sensor technique is more precise than the PVT -technique since there are no cumulative errors like in the case of the PVT -method, when the successive transfers are performed during an isothermal sorption. The technique does not require extensive calibrations. Essentially, uncertainties come from the experimentally measured resonance frequencies. Errors are reduced with the new data acquisition permitting the simultaneous recording of the phase with the frequency: effectively, the phase angle is better suited than the amplitude to detect the natural resonant frequency (and also the half-width) [15]. The main source of uncertainty affecting the evaluation of the gas concentration data comes from the term of equation ( 5) which contains the density of the gas and the change in volume of the polymer. At this stage, it was then necessary to elaborate a new procedure to unambiguously obtain the apparent solubility of the gas in the polymer and the associated change in volume. For this, the Sanchez-Lacombe equation of state was selected to estimate the change in volume of the polymer at different pressures and temperatures [15].

THERMOPOROSIMETRY

Since the early work of Thomson [16] it is well established that a fluid confined in the pores of a material exhibits a noticeable shift ∆T of its transition temperature from its liquid to solid state. Furthermore it has been shown that this ∆T-shift is directly related to the sizes of the pores, as expressed by their radius R p , in which the fluid is trapped [17][18][19]. Then the knowledge of the R p (∆T) function allows the determination of the sizes of the pores of a given material simply by measuring ∆T for a selected fluid [20,21]. This method, called thermoporosimetry becomes essential for the determination of the pore size distribution in porous materials. The experimental technique which is relatively simple consists in measuring the crystallization temperatures of the fluid confined in the pores and of the ("free") fluid surrounding the material. Actually, the investigated system is made of the porous material over saturated by the fluid and the technique makes use of a sensitive differential scanning calorimeter (DSC). In this context, our original contribution was to use thermoporosimetry to investigate the microstructure of polymeric materials.

Nanoporous sol-gel derived silica monoliths were used to "calibrate" the solvents used for thermoporosimetry. Careful control of the ageing procedure allowed the production of silica gel matrices with tailored textural properties. A number of solvents [22] including linear alkanes, cyclohexane, substituted benzene, CCl 4 , acetone, dioxane were tested. Various polymers [23] like polystyrene (PS), ethylene-propylene rubber (EPR), ethylene-propylenediene-monomer (EPDM), polybutadiene, polyisoprene, poly(dimethyle syloxane) (PDMS), cross linked by UV or gamma irradiation, by heating or ageing degradation, were characterized and their mesh size distributions calculated.

Figure 4 here

To illustrate the use of thermoporosimetry technique we choose the cyclohexane/PDMS system. PDMS sample is a cross linked (by peroxide reaction) and heavily silica loaded one (30% w/w). It is used as a part of an electrical insulator device. Cyclohexane was supplied by Aldrich and used without further purification. Cyclohexane, as shown by DSC (see Fig. 4 ), exhibits two typical thermal transitions. The cooling from room temperature to -150°C at -0.7°C/min shows two mean exothermal transitions. The first one (at 3.47 °C and releasing 30.38 J/g as specific energy) is the liquid-to-solid normal transition marked by a slight supercooling effect of about 3 °C. The second transition takes place at -88.63 °C and releases 81 J/g namely 2.5 times more than the crystallisation. The latter corresponds to the transition from allotropic phase I to allotropic phase II. It is well established that cyclohexane undergoes conformational transitions between the chair and boat isomeric forms. As indicated by X-ray diffraction the boat isomer proportion is less than 1% at room temperature and smaller at lower temperature [22,24]. Then the phase I-to-phase II transition cannot be assigned to this conformational change. The signal of the I-to-II transition exhibits two peaks expressing the fact that the observed transition proceeds in two steps. When heating from -150°C to room temperature, the DSC plot shows inverse transitions of those recorded during the cooling. The first one represents the II-to-I transition and occurs at -86.34°C with same specific energy.

Unexpectedly, only a single peak is present. At 7.15°C the melting transition occurs releasing 29.6 J/g. When cyclohexane is confined inside a porous material or a swollen polymeric gel, its thermal transition temperatures are consequently shifted; thermoporosimetry uses this temperature shift to calculate the pore or mesh size distributions of these divided media. In this context our contribution, as illustrated in what follows, was to extend the thermoporosimetry in particular towards soft matter characterisation [23,25].

Figure 5 here

Effectively, based on its two thermal transitions cyclohexane can be used as a thermoporometry probe. Fig. 5 shows the DSC exotherms related to confined cyclohexane inside well known textured four different silica gel samples. The characteristics of four silica gels are given in Table 1. According to our calibration procedure [XXX], numerical relationships can be established between the radii of the pores where the cyclohexane undergoes the thermodynamical transitions and the related subsequent temperature depressions. The thermoporosimetry formalism can then be used to calculate the pore or the mesh size distributions (PSD or MSD respectively), namely:

for the liquid to solid transition 31.99 2.02 exp( )

p R T = -∆ (6)
and for the solid to solid transition 1 0.075 1.69 12.5exp( ) 0.0205

p T R ∆ + = + - (7)
where R p is the radius (in nm) of the pore in which the transition takes place and ∆T= T p -T 0 , the depression temperature. T p being the temperature in pore and T 0 the transition temperature of the bulk cyclohexane (7.15 and -88.63 °C for the solid to solid and for the solid to liquid transitions respectively).

Application to cross linked and filled polymers

PDMS samples of around 10 mg have been left to swell for 48 hours soaking in cyclohexane.

A swollen sample was sealed in a DSC aluminum pan and submitted to a cooling temperature program from 10 down to -115 °C with 0.7°C/min temperature rate. Fig. 6 shows the recorded DSC traces. Two groups of exotherms can be pointed out.

Figure 6 here

The first group located at high temperature (between 5 and -20°C) is attributed to the liquid to solid transition (crystallization). In this group the first peak is that of the bulk cyclohexane, the second one is attributed to the confined solvent inside the PDMS macromolecular network. The second group of two peaks appearing at lower temperature represents the solid I to solid II transition undergone by cyclohexane. As expected the first peak is related to the bulk solvent whereas the second one has to be attributed to the trapped cyclohexane. The solid to solid transition, as shown in Figs. 4 and6, is three times more energetic than the crystallization phenomenon. Peaks of cyclohexane confined in the swollen PDMS sample give a first idea of the mesh size distribution in the polymeric network however, it was demonstrated that the heat of thermal transition is strongly temperature dependent. Without taking in account this dependence, the smallest pore should be underestimated. Following the calculation procedure described elsewhere [23,25], the DSC thermograms can be converted to the mesh size distributions. Fig. 7 shows the mesh size distribution MSD of the cross liked silica filled PDMS sample under study. The calculated MSD obtained from the liquid to solid transition is less marked ??? than that derived from the solid to solid one but the satisfactory overlapping of the two distributions has to be noticed. Evidently, the absorbed solvent can be used as a textural probe. Its thermal transitions are highly sensitive to its surrounding constraints. The polymeric meshes of the swollen network limit considerably the degrees of freedom of the cyclohexane molecules macromolecular chains leading to a large depression affecting the phase transition. Thermoporosimetry appears as a unique tool to investigate the cross linked (insoluble) and filled (opaque) polymers.

TEMPERATURE MODULATED DIFFERENTIAL SCANNING CALORIMETRY (TMDSC)

The rapid evolution of both methodology and technology has led to recent applications of modulated temperature calorimetry, particularly the combination of the technique with differential scanning calorimetry [26]. Different terms such as « periodic », « oscillating », « dynamic » or « alternating » have thus been used to describe similar techniques based on different types of instruments. The use of temperature modulated DSC to characterize polymers deals mostly with detection of weak transitions, determination of heat capacities in quasi-isothermal mode and separation of superimposed phenomena. The mathematical development necessary to describe the technique is well known [26] and different approaches have been proposed [27]. The total differential heat flow obtained after deconvolution of the modulated heat flow represents the sum of two distinguishable contributions, because the response to the imposed temperature modulation is different depending on the phenomena submitted to the temperature changes. One component, called reversing heat flow, is linked to the heat capacity change; the modifications that depend on the temperature scanning rate can be cycled by alternating heating and cooling effects. The second component is linked to the kinetics and is called non-reversing heat flow, by opposition to the first one; modifications appearing in this signal depend only on the temperature.

The specific heat capacity c p measured with a conventional DSC, under the conditions of negligible temperature gradient within the sample, is approximately proportional to the temperature difference between the sample and the reference or to the heat flow difference.

Calculation of the sample heat capacity is possible through calibration data, at the working temperature, via the following relation (m and m Al 2 O 3 are respectively the mass of the sample and the mass of sapphire used as standard calibrating substance) :

m c m c a a a a p Al O p Al O s b c b = - - 2 3 2 3 ( ) (8)
where the quantities a represent the amplitudes of the heat flow (or temperature) differences signals for different situations corresponding to : sample, calibration and baseline runs designated respectively by 's', 'c' and 'b'. Usually, the baseline run (a b ) is obtained with two empty aluminum pans. In a measuring run (a s ), the sample is in one of the aluminum pans, whereas in the calibration run (a c ) the sample is replaced by the standard sapphire (Al 2 O 3 ), having a well known specific heat capacity. In the case of TMDSC the calibration equation takes the form ( )

C C A A K C s r Ts r - = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ∆ ω 2 2 (9)
where C s and C r represent the heat capacities of the sample and of the reference respectively (these quantities include the contribution of the sample and of the pans which can be regarded as identical on the two sides), A ∆ is the temperature difference amplitude between sample and reference, A Ts the sample temperature amplitude (in Kelvin) and ω designates the modulation frequency (=2π/p with p being the modulation period in s). K is the temperature-dependent Newton's law constant. When calibrating with sapphire, equation ( 9) leads to the following equation :

( ) C C K A A s r Cp HF Ts - = ω (10)
where A HF refers to the amplitude of the differential heat flow, and K Cp is the calibration constant for heat capacity measurements. The heat capacity of the sample obtained using equation (10) allows the determination of the contributions of the reversing and of the nonreversing heat flows respectively, noted HF rev and HF non-rev , to the total heat flow HF tot using the following relations :

HF rev signal = average temperature scanning rate × heat capacity signal (conventionally, on heating a negative sign is necessary because an endothermic effect in the sample, i.e. heat consumption, creates a negative ∆T between the sample and the reference) then, HF non-rev signal = HF tot signal -HF rev signal One of the major advantages of the technique described above is most probably the possibility to access directly in a single run to overlapping effects. In this respect, temperature modulated DSC appears to be a more powerful technique than standard DSC. The thermal analyzer used in the present work was a Mettler DSC 821 equipped with a refrigerating cooling system RP 100 (from LabPlant, UK) allowing to work from 200 to 700 K. Taking as an example a polystyrene sample modified by high pressure methane, Fig. 8 illustrates the high potentiality of the technique to detect unambiguously polymer glass transitions. On Fig. 8 the glass transition clearly appears in the HF rev signal, whereas the HF tot signal does not allow detecting the typical inflexion reflecting the glass transition which is hidden in the main fusion peak.

CONCLUSION

To modify and characterize polymer microstructures, several techniques relevant to rigorous According to the respective role of the measuring M and reference R cells, three thermal modes are described. In PCSC situation: a, the thermal differential comparative mode, where both measuring and reference cells contain a polymer and are connected to the gas line; b. thermal differential mode, where the measuring cell contains the polymer and the reference cell contains an inert sample of equal volume, both cells are connected to the gas line. In TCSC situation: c, the simple thermal differential mode, where the measuring cell contains the polymer and is connected to the gas line while the reference cell acts as a thermal reference. The endotherms are related to the bulk cyclohexane whereas the exotherm signals represent the thermal transitions (liquid to solid I at higher temperatures and solid I to solid II at lower ones) of the confined solvent. R1 to R4 are the silica gel samples calibration described in Table 1. The scanning rate is 0.7°C/min. TCSC run P = f(t)

Figure 3

 3 Figure 3 here

  change in volume after transfer k. The working equation (3) for the vibrating-wire sensor VW relates the mass, m sol , of gas absorbed in the polymer to the change in volume, ∆V pol , of the polymer. The natural angular frequency of the wire, through which the polymer sample holder is suspended, depends on the amount of gas absorbed. The physical characteristics of the wire are accounted for in equation (3) as:

  thermodynamics have been recently developed. Calorimetric techniques, particularly PVTcalorimetry, or scanning transitiometry, and temperature modulated differential scanning calorimetry allow through temperature (or pressure) scans to either modify the polymers micro-organization or to investigate the various transitions undergone by the materials. In addition to the role played by the two variables P and T, a fluid either gaseous (often in supercritical state) or liquid can be advantageously used to promote and/or characterize the polymer nano/microstructure. A complementary technique which combines a weighing technique and a pressure decay PVT technique allows evaluating the amount of gas penetrating a polymer sample or a porous material during sorption under a precise control of the different parameters.

Figure 1 .

 1 Figure1. Schematic view of PVT-controlled scanning calorimetry showing two main scanning situations, the Pressure-controlled scanning calorimetry (PCSC) situation and the Temperature-controlled scanning calorimetry (TCSC) situation. The PCSC situation, under isothermal P-scans, yields through the thermal output the isobaric thermal expansivity (here α pol-g-int is for a polymer sample interacting with a gas). The TCSC situation, under isobaric Tscans, yields through combination of both thermal and mechanical outputs the entropy change.

Figure 2 .

 2 Figure 2. Schematic representation of PVT-controlled scanning calorimetry showing the calorimetric detector housing the twin cells connected to the high-pressure PVT line.According to the respective role of the measuring M and reference R cells, three thermal modes are described. In PCSC situation: a, the thermal differential comparative mode, where both measuring and reference cells contain a polymer and are connected to the gas line; b. thermal differential mode, where the measuring cell contains the polymer and the reference cell contains an inert sample of equal volume, both cells are connected to the gas line. In TCSC situation: c, the simple thermal differential mode, where the measuring cell contains the polymer and is connected to the gas line while the reference cell acts as a thermal reference.

Figure 3 .

 3 Figure 3. Schematic picture showing the liquid crystal PMA(Az) n domain surrounding the PEO m nanoscaled cylinders in the smectic phase; the PEO m domain can expand under super critical CO 2 sorption.

Figure 4 .

 4 Figure 4. Thermal transitions of bulk cyclohexane observed under 0.7°C/min scanning rate.

Figure 5 .

 5 Figure 5. DSC curves of cyclohexane. The endotherms are related to the bulk cyclohexane whereas the exotherm signals represent the thermal transitions (liquid to solid I at higher temperatures and solid I to solid II at lower ones) of the confined solvent. R1 to R4 are the silica gel samples calibration described in Table1. The scanning rate is 0.7°C/min.

Figure 6 .

 6 Figure 6. DSC traces obtained in the case of cyclohexane swelling a PDMS sample. The scanning rate is of 0.7°C/min.

Figure 7 .

 7 Figure 7. Mesh size distribution (MSD) of a cross linked silica loaded PDMS sample swollen by cyclohexane, calculated either from the liquid to solid transition or from the solid to solid transition respectively.

Figure 8 .

 8 Figure 8. Thermograms obtained with temperature modulated differential calorimetry for a polystyrene sample modified by sorption of methane under high pressure (150 MPa). The reversing heat flux HF rev shows clearly the glass-transition inflexion curve which is otherwise hidden in the main fusion peak of the total heat flux HF tot .

  Figure 2

Figure 8

 8 Figure 6

Table 1 .

 1 Textural characteristics of samples used for calibration: SSA, the specific surface area; V p , the porous volume; σ, the uncertainty on V p ; R p , the pore radius; MaxPSD, the maximum of the pore size distribution.

	Sample	SSA (m 2 /g)	Vp (cm 3 /g)	Rp (Å)	Max PSD (Å)
	R4	532	0.696	26.2	24
	R3	473	0.922	39	34.2
	R2	166	0.991	119.2	87
	R1	183	1.327	144.9	142.5


Ali Mokh

Julien De Rosny

George C Alexandropoulos

Ramin Khayatzadeh

Mohamed Kamoun

Abdelwaheb Ourir

Arnaud Tourin

Mathias Fink

Time Reversal for Multiple Access and Mobility: Algorithmic Design and Experimental Results

Keywords: Iterative time reversal, low-complexity reception, mobility, multiple access, UWB, precoding

Time Reversal (TR) has been proposed as a competitive precoding strategy for low-complexity wireless devices relying on Ultra-WideBand (UWB) signal waveforms. However, when TR is applied for multiple access, the signals received by the multiple users suffer from significant levels of inter-symbol and inter-user interference, which requires additional processing for mitigation by each receiving user. In this paper, we present an iterative Time-Reversal Division Multiple Access (TRDMA) approach that aims to dim the latter interference levels. The performance of iterative TRDMA is evaluated experimentally in a reverberation chamber that mimics a rich scattering indoor wireless propagation environment. The improved efficiency, in terms of the number of algorithmic iterations, of the proposed approach compared to conventional TRDMA, is demonstrated. We also consider a mobile user configuration, where the position of the receiver changes between the channel estimation and data transmission steps. It is showcased, even for this experimental setup, that the proposed iterative TRDMA approach is more efficient than conventional precoding schemes.

I. INTRODUCTION

Ultra-WideBand (UWB) signal waveforms have ushered a new era in short-range data transmissions for wireless sensor networks, by enabling both robust communications and accurate ranging capabilities [START_REF] Zhang | UWB systems for wireless sensor networks[END_REF]. One central issue facing UWB waveforms for low complexity devices refers to effective energy collection that is usually dispersed in rich scattering wireless environments. The Time Reversal (TR) technique appears to be a paradigm shift in exploiting rich multipath signal propagation, which is capable of focusing UWB pulses both in time and space [START_REF] Lerosey | Time reversal of electromagnetic waves[END_REF], while shifting the signal processing complexity to the transmitter side.

With TR precoding, the receivers can benefit from high levels of the intended received signals enabling, in this way, simple, low cost, and low power reception modules [START_REF] Wang | Green wireless communications: A time-reversal paradigm[END_REF], [START_REF] Chen | Time-reversal wideband communications[END_REF]. Moreover, the offered temporal pulse compression and spatial focusing can partially mitigate the Inter-Symbol Interference and the Inter-User Interference (IUI), resulting from the multipath signal propagation [START_REF] Oestges | Time reversal techniques for broadband wireless communication systems[END_REF]. The above characteristics render TR precoding a promising scheme for wireless communications and localization applications. Very recently, [START_REF] Alexandropoulos | Indoor time reversal wireless communication: Experimental results for localization and signal coverage[END_REF] demonstrated indoor cm-level localization accuracy with TR at 3.5 GHz using up to 600MHz bandwidth channel sounding signals. In [START_REF] Mokh | Time reversal precoding at subTHz frequencies: Experimental results on spatiotemporal focusing[END_REF], considering a leaking reverberation cavity, the authors experimentally verified the ultra-accurate spatiotemporal focusing capability of TR at 273. [START_REF] Alexandropoulos | Indoor time reversal wireless communication: Experimental results for localization and signal coverage[END_REF] GHz using transmission bandwidths up to 4 GHz.

In [START_REF] Han | Timereversal division multiple access over multi-path channels[END_REF], a TR-based Division Multiple Access (TRDMA) scheme for multi-user Multi-Input Single-Output (MISO) systems was presented. It was shown that the proposed scheme provides a cost-effective waveform design enabling energy-efficient Internet-of-Things (IoT) nodes [START_REF] Wang | Green wireless communications: A time-reversal paradigm[END_REF]. Thanks to this simple precoding strategy, base stations can support a large number of simultaneous links with moderate complexity [START_REF] Han | A multiuser TRDMA uplink system with 2D parallel interference cancellation[END_REF]. A theoretical analysis for the average Signalto-Interference Ratio (SIR) of the intended and unintended receivers in a distributed TR-based transmission scheme was carried out in [START_REF] Wang | SNR analysis of time reversal signaling on target and unintended receivers in distributed transmission[END_REF], while an experimental evaluation has been recently presented in [START_REF] Mokh | Indoor experimental evaluation of ultrawideband MU-MISO TRDMA[END_REF]. However, it is known that when operating in low Signal-to-Noise Ratio (SNR) values, the interference level can affect the TR performance. Indeed, one major issue with TRDMA is that the SIR level cannot be tuned. For instance, it has been shown that in an ideal Rayleigh channel, the averaged SIR is equal to 0 dB.

To overcome the latter problem with TR while keeping the receiver as simple as possible, the iterative TRDMA (ITRDMA) technique was proposed in [START_REF] Montaldo | Telecommunication in a disordered environment with iterative time reversal[END_REF], aiming to reduce the ISI and IUI. This technique belongs to the family of prerake processing techniques, which were originally introduced in [START_REF] Imada | Pre-RAKE diversity combining for UWB systems in IEEE 802.15 UWB multipath channel[END_REF]. It is noted that iterative processing is a popular approach for equalization. It has been applied for single and multi-carrier code division multiple access (CDMA) systems [START_REF] Liu | Iterative detection of chip interleaved cdma systems in multipath channels[END_REF]. Very recently, a low complexity iterative rake detector for orthogonal time-frequency space modulation has been presented in [START_REF] Thaj | Low complexity iterative RAKE detector for orthogonal time frequency space modulation[END_REF], [START_REF]Low complexity iterative rake decision feedback equalizer for zero-padded OTFS systems[END_REF]. Although the ITRDMA scheme proposed in [START_REF] Montaldo | Telecommunication in a disordered environment with iterative time reversal[END_REF] deals with ISI and IUI, it is quite difficult to implement. It requires multiple feedback between the transmitters and the receivers, which makes it non-applicable for very low duty cycle nodes.

Motivated by the latter limitation, in this paper, we present an ITRDMA algorithm that is based on the iterative processing of the Channel Impulse Responses (CIRs) of the multiple users, which are acquired during the channel estimation phase. The proposed algorithm is implemented at the transmitter side as a precoder, instead of being implemented at the receiver as a post-processing step. Hence, the ITRDMA computational complexity is concentrated at the transmitter (e.g., the base station or access point) instead of the receiver, when downlink communication is considered. The perform-ance of the proposed algorithm, in terms of ISI and IUI mitigation, is experimentally evaluated in a multi-user scenario inside a reverberating room mimicking a strongly scattering indoor environment. We specifically validate the performance using channels that have been experimentally acquired at the carrier frequency 2 GHz using an 100 MHz communication bandwidth. It is showcased, even for an experimental setup with mobility, that the proposed ITRDMA approach is more efficient than conventional TRDMA schemes.

II. TIME REVERSAL FOR MULTI-USER PRECODING

In this section, we first introduce the TR precoding and then present the proposed ITR scheme for multi-user wireless systems.

A. System Model and Time Reversal Precoding

We consider a multi-user MISO wireless communication system comprising one base station that is equipped with M antenna elements, and which wishes to communicate in the downlink direction with N single-antenna users. We represent by h i,m [k] the baseband CIR at discrete time k between the m-th (m = 1, 2, . . . , M ) transmitting and i-th (i = 1, 2, . . . , N ) receiving antennas, i.e., at the base station and the i-th user, respectively. The TR precoding technique utilizes the time-reversed CIR, i.e., h * i,m [L -k], to focus the transmitted information-bearing electromagnetic field on the i-th user location. As a consequence, the signal sent by the m-th transmit antenna at discrete time k to focus the baseband message x i on each individual receiving antenna is given by:

s m [k] = N i=1 L l=1 x i [l]h * i,m [L + l -k] M m=1 L l=1 |h i,m [l]| 2 , ( 1
)
where it is assumed that the CIR is composed of L significant taps. The normalization in this expression ensures that the power emitted toward each user is the same. Using (1), the baseband received signal at each j-th (j = 1, 2, . . . , N ) user at time k can be expressed as follows:

y j [k] = L l=1 N i=1 x i [l]R j,i [L + l -k] + n j [k]. (2)
where the expression of the correlation function R j,i [k] between the CIRs of users j and i is defined as

R j,i [k] M m=1 +∞ k =-∞ h * i,m [k -k ]h j,m [k ] M m=1 L l=1 |h i,m [l]| 2 , (3)
and n j represents the zero-mean additive white Gaussian noise with standard deviation σ at user j.

In multipath channels, when applying TR precoding using the CIR of user i, the time-reversed field focuses in time and space in the location of this user [START_REF] Fink | Acoustic time-reversal mirrors[END_REF]. This means that a signal peak given by the CIR's auto-correlation function

R i,i [0] = M m =1 L l=1 |h i,m [l]| 2
appears at user i; at this time and location, M L signals add up coherently. However, this term can be corrupted by incoherent contributions of the auto-correlation function R i,j [k]. In fact, the contributions when k = 0 but i = j are due to imperfect time focusing on the targeted user/antenna. This implies that around the main focused peak, some secondary peaks are still present that generate ISI. Note that the elements of the auto-correlation function when i = j happen due to the sides lobes. Hence, TR precoding does not perfectly focus the wave on the intended user, inducing interference to the other users.

B. Proposed Time-Reversal-Based Multiple Access

To reduce the levels of ISI and IUI appearing in (3), we propose the ITRDMA algorithm. In contrary to the scheme presented in [START_REF] Montaldo | Telecommunication in a disordered environment with iterative time reversal[END_REF], our algorithm is based on the measurement of the CIRs of the multiple users. Its main principle is to iteratively remove the secondary peaks and lobes in the autocorrelation function. To this end, the time position and the antenna location of the targeted lobe are identified iteratively. In particular, a pre-processing timereversed waveform is used to focus on that position at that time index, but with a peak that is the opposite of the complex amplitude of the lobe. Hence, if this time-reversed waveform is added to the actual CIR, thanks to the linearity, the secondary lobe will be canceled. The same procedure can be used iteratively to remove all secondary peaks and lobes from the autocorrelation function. The algorithmic steps of the proposed ITRDMA scheme are summarized as follows:

1) The transmit precoder

s i0,m [k] at the discrete time k is initialized with h * i0,m [L-k]
to focus the received signal on the user/antenna i 0 at the time instant (i.e., channel tap) k = L. The notation h stands for the normalized CIR, which is computed as follows:

hi,m [k] h i,m [k]   M m=1 L k =1 |h i,m [k ]| 2   -1 . (4)
2) The differences ∆ i [k] ∀i = 1, 2, . . . , N between the resulting fields f i [k] on the N receiving users/antennas, which are given by the autocorrelations Ri0,i [L -k] and their targeted values δ[L -k]δ[i -i 0 ] (i.e., for the fields without lobes), are computed for each receiving antenna. The notation Rj,i [k] with i, j = 1, 2, . . . , N represents the autocorrelation function using the normalized CIRs, which is given by the expression:

Rj,i [k] = M m=1 +∞ k =-∞ h * i,m [k -k ] hj,m [k ]. (5)
3) The tap index k and the antenna index î where the maximum of |∆ i [k]| occurs are found. This corresponds to the highest (undesired) secondary peak in the autocorrelation function. 4) The ITRDMA precoder is then updated with the timereversed CIRs, which are shifted in time to focus on the î-th user/antenna at the k-th tap with a complex amplitude opposite in sign with ∆ î[ k], in order to cancel the secondary peaks on each receiving user. 5) The differences ∆ i [k] ∀i are also updated to take into account the new contributions to the precoder field. it will hold that ∆ î[ k] = 0. The algorithmic steps of the proposed ITRDMA scheme are summarized in Algortihm 1. It can be seen that the iterative Steps 6 to 8 remove the secondary taps via TR-based precoding. The iterations stop either when their maximum allowable number n max is reached or when 3:

|∆ i [k]| becomes
Set ∆ i [k] = Ri0,i [L -k] -δ[L -k]δ[i 0 -i].
4:

Set n = 0.

5:

while max|∆ i [k]| > and n < n max do 6: Find k, î = arg max k,i |∆ i [k]|. 7: Set s i0,m [k] ← s i0,m [k] -∆ î[ k] h * î,m [L -k + k]. 8: Set ∆ i [k] ← ∆ i [k] -∆ î[ k] Rî ,i [L -k + k].
9:

Set n = n + 1.

10:

end while

11: Set s i0,m [k] ← s i0,m [k] k ,m |s i0,m [k ]| 2 -1
.

12: end for smaller than a target value . It is noted that this value can be advantageously tuned to meet a target SNR level or to tune the complexity and the processing latency at the transmitter side. C. Signal-to-Interference-Plus-Noise (SINR) Performance

The complex-valued information symbols x i [l] intended for each i-th user are convoluted with the precoding sequences s i,m [k] computed from Algortihm 1. As a consequence, the baseband received signal at each j-th user can be expressed as follows:

y j [k] = L l=1 N i=1 x j [l]w i,j [k -l] + n j [k], (6)
where w i,j [k] represents the precoding signal given by

w i,j [k] L l =1 M m=1 s j,m [l ]h i,m [k -l ]. (7)
In the case of TR precoding, by inspection of expression (2), it can be concluded that

w i,j [k] = R i,j [L -k].
Therefore, the sequence w i,j [k] can be interpreted as a virtual channel that includes the TR-based precoding. This equivalent channel approach has been already use for TR in [START_REF] Phan-Huy | Make-It-Real" precoders for MIMO OFDM/OQAM without inter carrier interference[END_REF]. By assuming that the power of the information symbols is normalized, it is easy to show that SINR of the equivalent channel at each receiving user/antenna i is obtained as follows:

SINR i |w i,i [0]| 2 I i + σ 2 . ( 8
)
where

I i L l=1,l =0 |w i,i [l]| 2 + L l=1 N j=1,j =i |w j,i [l]| 2 represents the intereference term.

III. EXPERIMENTAL VALIDATION RESULTS

A. Experimental Setup

To evaluate the performance of the proposed ITRDMA precoding scheme, we have implemented an experimental setup to first sound the channel and then to TR-precode the transmitted signals, as shown in Fig. 1. The setup consists of a transmitter array with M = 8 antennas, all connected to a solid-state Radio Frequency (RF) switch. The input of this 1 × 8 switch is connected to a power amplifier that is fed by an Arbitrary Waveform Generator (AWG) from Tektronix (specifically, the AWG-7012). This AWG was used to generate transmitted signals with a carrier frequency at 2 GHz. The transmitter (Tx) was placed inside a reverberating cavity to create a rich multipath signal propagation channel. This cavity has an opened window facing the Receivers (i.e., Rx1 and Rx2), which are placed outside the cavity. The AWG and DSO were connected to an Ethernet network switch and controlled by a desktop computer. The AWG triggered the DSO to ensure synchronization. The sampling rate of the DSO and the AWG was 12.5 and 10 GHz, respectively.

Two different setup configurations were used for two different sets of measurements. The first setup intended to evaluate the ITRDMA precoding scheme in a two-user MISO scenario, where two dipole antennas were connected to two low noise amplifiers, and then, to the inputs of an RF Digital Storage Oscilloscope (DSO) (specifically, the Tektronix TDS6604B). The second setup was used to evaluate the TR precoding in a mobile setting, considering a singleuser MISO scenario. In particular, the receiving user Rx1 was mounted on a linear motorized bench in order to be able to move its receiving antenna without strongly modifying the channel property inside the reverberating cavity. The bench was placed parallel to the cavity's opening, resulting in a displacement range equal to 30cm.

B. Spatiotemporal Focusing Capability

In the first experiment, we ran the transmission chain with a bandwidth B = 100 MHz to focus the precoded signal on the user Rx1. We applied both the conventional TR and the proposed ITR precoding algorithms using 50 iterations. The radiated field was probed not only on the user Rx1, but also on the user Rx2. The amplitude of the baseband received fields versus the discrete time index is demonstrated in Fig. 2. We first clearly observe a strong time and spatial focusing with the both TR-based precoding algorithms. However, the side lobes on Rx1 and the secondary lobes on Rx2 are smaller when ITRDMA is applied. One can also see that the ITRDMA-precoded signals have slightly lower amplitude at time t = 0. To quantify the improvement of the ITRDMAbased focusing, we plot in Fig. 3 the SIR performance of the received signal, as computed via [START_REF] Han | Timereversal division multiple access over multi-path channels[END_REF] by setting σ = 0, as a function of the number of the algorithmic iterations for a B = 100 MHz bandwidth signal. Note that the SIR of conventional TR precoding is obtained when the number of iterations with ITR equals to 0. It can be seen that, thanks to the proposed iterative scheme, an improvement up to 6.5 dB is obtained with ITR for 400 iterations. Evidently, the improvement is not linear with the number of iterations. We actually observe that the SIR improves with the square root of this number, as soon as the iterations are more than 50. We believe that those improvement and behavior strongly depend on the channel characteristics and transmission bandwidth.

C. The Effect of Mobility

The TR-based precoding is known to be sub-optimal, especially in the case of low-to-moderate noise levels. However, it is also known to be less sensitive to channel fluctuations and receiver mobility. Hence, one may wonder to what extent the proposed ITRDMA processing is more sensitive than conventional TR, when considering a displacement of a receiving user between its position when channel estimation was performed and its new position during the data transmission phase. We experimentally investigate this question, as follows. We consider the user Rx1 lying on the motorized bench, and a first step, the 8 CIRs are estimated for an initial position of Rx1 at the point x = 8 cm. Using those estimates, the ITRDMA precoder is derived. In a second step, the Rx1 is moved between the positions x = 1 and 15 cm with a step size of 1 cm. For each of those positions, the same precoder was applied and the measured signal in baseband was recorded and stored. This signal's amplitude, considering 50 algorithmic iterations for ITRDMA, is illustrated as a function of time and RX1 position in Fig. 4. As expected the maximum of the spatiotemporal focusing is obtained for the Rx1 position at x = 8 cm, where the CIR was estimated. It is also shown that the maximum amplitude reaches a minimum when the receive antenna is moved by 5 cm from the CIR measurement position. For TR preprocessing, it is well known that the width of the focal spot is directly given by the spatial coherence length, which is equal to half a wavelength for a diffuse field. In our experiment at 2 GHz carrier frequency, this coherence length is 3.4 cm. The larger value seen in this figure happens because the receiving antenna is placed outside the reverberation cavity, leading to an increase of the spatial coherence length.

We next investigate the effect of the user receiver displacement between the channel sounding and the data transmission phases in the SINR performance, by evaluating the expression [START_REF] Han | Timereversal division multiple access over multi-path channels[END_REF]. The Doppler effect was not taken into account. However, it is expected to have a moderate impact on the data transmission since we perform wideband processing in the time domain. The SINR curves versus the Rx1 displacement in cm for the SNR value 30 dB are demonstrated in Fig. 5. We clearly observe that, for small distances, the larger the number of iterations is, the faster is the decay of the SINR with respect to the distance. Nevertheless, the ITRDMA algorithm seems to inherit the TR robustness. Indeed, once the SINR improvement due to the 50 algorithmic iterations is lost, its behavior becomes identical to the one obtained with only 20 iterations. Hence, by optimizing the number of iterations in our algorithm, one can get better performance with the proposed ITRDMA scheme than using conventional TR. We hereinafter set as τ the time duration between the channel estimation process and the actual data transmission, ans as v the speed of mobility of the user Rx1. Based on the findings in Fig. 4, one can assess the maximum speed that Rx1 can reach before its SINR performance reduces by a factor of 2 for different CIR estimation time intervals τ . Indicative results for this metric are provided in Table I. The shown time duration values are completely compatible with our indoor applications of the ITRDMA algorithm. For outdoor applications, the ITRDMA processing has the same limitations with any precoding algorithm, where the channel needs to be estimated by the transmitter. Note that the speed values listed in Table I are probably overestimated, because the outdoor coherence length can be larger. In this table, we have calculated the minimum speed of the user Rx1 in order to reach half of the signal strength for the corresponding time duration τ . As clearly shown in the figure, a lower value of τ will result in SINR being more robust to mobility.

Finally, in the SINR performance evaluation results in Fig. 6, we investigate the effect of the noise on the behavior of the different TR precoding schemes, when considering varying mobility speeds for the user Rx1. In this figure, the time duration τ between the CIR estimation and data transmission is set to 1 ms, and we plot the SINR values of the baseband received signal when applying conventional TR, as well as ITRDMA with 20 and 50 iterations. Performance curves for operating SNR values equal to 2 and 10 dB are depicted. It is shown that, the lower the SNR is, the closer are the performances of the different TRbased precoding schemes. This is due to the fact that for high noise levels, the interference becomes lower than the noise level, and then the interference cancellation process becomes useless. For low-to-moderate user speeds, which are reasonable for indoor applications, it is demonstrated in the figure that ITRDMA outperforms conventional TR. However, for very large speeds, conventional TR, which is obtained from ITRDMA for zero iterations, is sufficient.

IV. CONCLUSION

In this paper, we have presented indoor experimental results for an UWB multiple access scheme based on the TR technique. We have proposed a novel iterative algorithm to confront the effects of ISI and IUI in TR-precoded wireless communication systems. The efficiency of the presented ITRDMA scheme has been validated through real experiments in a highly scattering medium. The presented approach includes a multi-user MISO channel sounding process to estimate the CIRs, which was used to derive the precoding scheme. Then, the TR-precoded received intended signal as well as the interference were measured in baseband and used for the performance assessment. Our performance results showcased that the proposed ITRDMA algorithm is capable to reduce the interference in the received signal, and therefore increase the SINR level. We also investigated the effect of mobility of a receiving user using the same ITRDMA scheme. It was demonstrated that the proposed precoding scheme can still be more efficient than conventional TR at high SNRs, and up to moderate speed values that are reasonable for indoor wireless communications. In future works, we will investigate the performance of the ITRDMA algorithm in imperfect channel estimation settings, and will perform comparisons with other spatial precoding techniques (e.g., regularised zero-forcing). We also intend to experiment data communication considering reconfigurable intelligent surfaces [START_REF] Alexandropoulos | Reconfigurable intelligent surfaces for rich scattering wireless communications: Recent experiments, challenges, and opportunities[END_REF] for reprogrammable rich scattering conditions.

Algorithm 1

 1 The Proposed ITRDMA Precoding Scheme Initialize: Set the values for M , N , and L. 1: for i 0 = 1, 2, . . . , N do 2: Set s i0,m [k] = h * i0,m [L -k].

  Finally, at the Step 11 of the algorithm, the TR-based precoding signals s i,m [k] at each time discrete index k are normalized in energy.

Figure 1 .

 1 Figure 1. The designed and realized experimental setup for the proposed TR-based multi-user MISO scheme.

Figure 2 .

 2 Figure 2. Amplitude in Volts of the baseband received field probed by the antenna of Rx1 (top figure) and the antenna of Rx2 (bottom figure), when TR and ITRDMA precoding are considered by the 8-antenna transmitter array in order to spatiotemporally focus the transmitted signal at Rx1.

Figure 3 .

 3 Figure3. The SIR performance of the proposed ITRDMA precoding algorithm with respect to the number of iterations using B = 100 MHz. The inner plot is the same plot, but its horizontal axis is in log scale.

Figure 4 .

 4 Figure 4. Amplitude [V] of the baseband received signal with respect to time and the position of the antenna of Rx1. The ITRDMA-precoded signals have been obtained using 50 algorithmic iterations and the transmission bandwidth B = 100 MHz.

Figure 5 .

 5 Figure 5. The estimated SINR at Rx1 with respect to the distance traveled by this user between the CIR estimation process and the application of the ITRDMA precoder. The performance curves were obtained for 3 different precoders: i) conventional TR, as well as ITRDMA with ii) 20 and iii) 50 algorithmic iterations. The transmission bandwidth and operating SNR were set to 100 MHz and 30 dB, respectively.

Figure 6 .

 6 Figure 6. The SINR performance versus the speed of the mobile user Rx1 lying in the motorized bench for transmission bandwidth B = 100 MHz, time mismatch duration τ =1 ms, and operating SNR equal to 2 and 10 dB.

Table I THE

 I REQUIRED SPEED OF USER RX1 TO REACH HALF OF THE SIGNAL STRENGTH FOR DIFFERENT CHANNEL ESTIMATION DURATIONS.

	τ (msec)	50	10	1
	v (km/h)	2.16	10.8	108