• Aucun résultat trouvé

TEMPO Mixed SAMs: Electrocatalytic Efficiency versus Surface Coverage

N/A
N/A
Protected

Academic year: 2022

Partager "TEMPO Mixed SAMs: Electrocatalytic Efficiency versus Surface Coverage"

Copied!
5
0
0

Texte intégral

(1)

TEMPO Mixed SAMs: Electrocatalytic E ffi ciency versus Surface Coverage

Pierre-Yves Blanchard, Olivier Alévêque, Tony Breton,* and Eric Levillain*

LUNAM Université, Universitéd’Angers, CNRS UMR 6200, Laboratoire MOLTECH-Anjou, 2 Bd Lavoisier, 49045 Angers Cedex, France

ABSTRACT: Electrocatalytic behavior of TEMPO derivative SAMs on gold has been studied in the presence of benzyl alcohol. The results demonstrate that interfacial activity of the SAMs can be enhanced by diluting the TEMPO moiety with an alkyl passive matrix. The absolute catalytic activity exhibits a maximum for an intermediate value of the surface coverage of catalytic centers. The most significant feature is the monotonic increase of the turnover (relative activity) until a limit value reached for low TEMPO surface concentrations. The electrocatalytic performances seem to be governed by a combination of two factors: the physical accessibility (by

alcohol molecules in solution) and the regeneration (via the comproportionation of oxoammonium and hydroxylamine before electrochemical reoxidation) of the catalytic centers.

INTRODUCTION

Self-assembled monolayers (SAMs) of thiolate derivatives have been widely studied and are known to form well-organized structures on various substrates.1 The self-organization of monolayers leads to packed structures and allows high surface coverage.2 Employing redox SAM-modified electrodes for catalysis, recognition, and sensing has resulted in some elegant examples that incorporate sophisticated receptors on the electrode surfaces.3,4However, one of the limitations of such architecture is the low structural quality exhibited by the monolayers when the structure-building forces are disturbed by the use of bulky active elements. In such cases, the strong interactions between the head groups result in poorly organized SAMs and possible ungrafted zones.5 A second disadvantage can come from the strong mutual interactions of the functional groups in confined spaces. Indeed, the transposition of the properties of an active moiety from the solution to the surface can only be successful if the available space around the immobilized active center is sufficient.6 Few works have resorted to the elaboration of mixed SAMs in order to favor active molecule accessibility7 or enhance its stability,8 but, to the best of our knowledge, quantification of the interfacial activity of an immobilized center as a function of its surface concentration has not been examined in detail. The present study investigates the reactivity of TEMPO derivative SAMs and mixed SAMs toward the electrocatalytic oxidation of benzyl alcohol in solution. Our goal was to demonstrate that interfacial activity of the SAMs can be enhanced by diluting the TEMPO moiety with an alkyl passive matrix.

EXPERIMENTAL SECTION

Au substrates were prepared by deposition of ca. 5 nm of chromium followed by ca. 100 nm of gold onto a glass substrate through a

shadow mask (MECACHIMIQUE/France) using physical vapor deposition system (PVD ME300 PLASSYS/France) and were made immediately before use. This protocol provides reproducible Au(111) surfaces with high crystallographic quality, low roughness (Ra less than 2 nm). They do not undergo post-treatment after completion.

Mixed SAMs were elaborated by successive adsorptions of aminoxylalkanethiol C15T (Scheme 1) and dodecanethiol C12.

Synthesis and characterization of C15T were described in ref 9 and dodecanethiol was used as received from Sigma-Aldrich.

Cyclic voltammetry (CV) was performed in a three-electrode cell controlled at a temperature of 293K. Working electrodes were functionalized Au substrates. Counter electrodes were platinum plates.

Reference electrodes were Ag/AgNO3 (0.01 M CH3CN). Cyclic voltammograms were recorded in dry HPLC-grade dichloromethane (CH2Cl2) and the supporting electrolyte was tetrabutylammonium hexafluorophosphate (Bu4NPF6).

X-ray photoelectron spectroscopy (XPS) data have been collected using a Kratos Axis Ultra spectrometer. The X-ray source is Al K working at 1486.6 eV and using a spot size of 0.7 × 0.3 mm2. Semiquantitative XPS analysis has been performed using a pseudo- Voigt function constrained by full width at half-maximum (fwhm)

Received: April 5, 2012 Revised: July 19, 2012 Published: September 4, 2012

Scheme 1. Derivative of TEMPO Radical: C15T

pubs.acs.org/Langmuir

(2)

ranges typical of each element and all spectra are calibrated taking Au4f as a reference binding energy of 83.95 eV and 87.63 eV.

RESULTS AND DISCUSSION

C15T SAM was prepared by immersion of the gold substrate in a millimolar C15T solution in dichloromethane for 30 min.

Cyclic voltamograms of CH2Cl2/Bu4NPF6 (0.1 M) exhibit a fully reversible one-electron process, closed to 0.47 V vsAg/

AgNO3, corresponding to the oxidation of the radical structure into the oxoammonium ion (Figure 1). The shape of the

voltammetric waves and the linear dependency between peak intensities and scan rates are characteristic of surface-confined redox species (Figure 1, inset). The TEMPO surface coverage, estimated by integration of the voltammetric signal, was 4.38× 10−10 mol·cm−2, which is in good agreement with the one expected for such structure considering a monolayer.10

In order to investigate the electrocatalytic behavior of the TEMPO derivative SAMs as function of their surface concentration, the preparation of randomly distributed C15T in the C12 passive matrix appears more suitable than segregated SAMs. As recently reported by our group, information on the distribution of the redox centers can be extracted from the voltametric data, and especially from the full width at high maximum (fwhm) and peak potential (Ep).11,12 A linear dependency of fwhm andEpwith surface coverage indicates a random distribution of the redox centers whereas a non linear dependence characterizes segregated phases. This electro- chemical method has confirmed that the preparation protocol of the mixed SAMs plays a key role on their surface organization (coadsorption lead to segregated domains and successive adsorption preferentially lead to randomly dis- tributed redox entities).13

Thus, mixed SAMs were prepared by the successive adsorption way by immersing a C15T SAM in a millimolar solution of C12 in dichloromethane. The C15T surface coverage (determined by cycling voltametry) was tuned by varying the immersion time in the C12 solution (15 min to reach 2.86 × 10−10 mol·cm−2, 40 min to reach 1.91 ×10−10 mol·cm−2 and 3 h to reach 0.86 × 10−10 mol·cm−2). The TEMPO surface coverages of these SAMs correspond respectively to 65%, 45%, and 20% of a full TEMPO monolayer. The voltamograms of the mixed SAMs (Figure 2) were found characteristic of surface confined electroactive species. fwhm and Ep were extracted from voltametric curves

and plotted versus surface coverage. The linear dependency observed (Figure 2 insets) is consistent with a random distribution of the C15T molecules in the C12 matrix. The electron transfer rate constants were evaluated by alternative current voltametry using the method developed by Creager and Wooster.14This rate was equal to 50 s−1for the C15T SAM and was slightly lowered by the dilution of the TEMPO units (comprised between 20 and 40 s−1for the mixed SAMs). This result agrees with the decrease of the transfer rate reported by Chidsey et al. for ferrocene terminated mixed SAMs prepared by coadsorption.15

The composition of a C15T SAM and a 45% C15T/C12 mixed SAMs were also analyzed by X-ray photoelectron spectroscopy in order to obtain information on the C15T/

C12 exchange step. Au4f, S2p, and N1s signals were closely examined and the relative concentration of each element was determined by integration of all the components of the signals.

Table 1 summarizes the results obtained.

Considering the Au/S ratios, the data indicate a 20% increase of the sulfur concentration in the layer during the C15T−C12 exchange (Table 1). This result traduces the fact that the C15T molecule is replaced by more than one dodecanethiol molecule.

This effect was expected because of the steric hindrance difference between the two thiols. Interestingly, the Au/S ratio obtained for the mixed SAM is very close to the one of a pure dodecanethiol monolayer, indicating the relatively high compacity of the mixed SAM prepared via the successive adsorption protocol. Au/N ratio give access to a desorption yield of C15T during the exchange step calculated at 53%, value very close to the one obtained previously by voltammetry (55%). N/S ratio calculated for the C15T/C12 SAM (i.e., 0.89) give access to the proportions of C15T and C12 which are, respectively, equal to 47 and 53%. All these data are in good Figure 1. Cyclic voltamogram of a C15T SAM having a TEMPO

surface coverage of 4.38×10−10mol·cm−2recorded at 100 mV·s−1in 0.1 M nBu4NPF6/CH2Cl2. Inset: variation of oxidation peak current as function of scan rate.

Figure 2.Cyclic voltamograms of a C15T SAM (a), and C15T/C12 mixed SAMs having a TEMPO surface coverage of 2.83 × 10−10 mol·cm−2(65%) (b), 1.91×10−10mol·cm−2(45%) (c), and 0.86× 10−10 mol·cm−2 (20%) (d) recorded at 100 mV·s−1 in 0.1 M nBu4NPF6/CH2Cl2. Top inset: variation of oxidation peak potential as function of surface coverage. Bottom inset: variation of full width at high maximum as function of surface coverage.

Table 1. Calculated Ratios of the Elements of the SAM from Integration of the XPS Core Level Spectra Signals

Au/S ratio Au/N ratio N/S ratio

C15T SAM 25.1 11.3 2.22

C15T/C12 mixed SAM 21.4 24.0 0.89

C12 SAM 22.1

(3)

agreement with the TEMPO surface coverages calculated by voltammetry and confirm the preparation of densely packed mixed SAMs.

Figure 3 shows the S2pspectra recorded for a full TEMPO monolayer, a 45% C15T/C12 mixed monolayer and a pure

dodecanethiol monolayer. All spectra werefitted by combinai- sons of two pseudo-Voight peaks (S2p3/2/ S2p1/2) with afixed intensity ratio of 0.5, an energy splitting of 1.2 eV and a fwhm of 1.6 eV. No oxidized sulfur species (S2p > 166 eV) were detected on any of the samples. For all the SAMs, two main series were identified at 161.9/163.1 eV and 163.4/164.6 eV, corresponding to 2 sulfur species. Thefirst series is consistent with the sulfur atoms bound to the gold surface as thiolate species.16,17The second one is commonly assigned to unbound S state18even if some authors attributed this series to another bound thiolate species.19 The bound/unbond sulfur ratio was found equal to 1.3 for the C15T and C12 SAMs, whereas this ratio reached 2 for the C15T/C12 mixed SAM. All attempts to reduce the proportion of unbounded species by rinsing or ultrasonication were unsuccessful and no change in the voltammetric response was observed, which is consistent with a strong incorporation of the unbound species in the layer. This result is consistent with an increase of the surface organization after replacement of the C15T units by C12 ones and suggests that the unbounded C15T are preferentially desorbed during the exchange. An additional S species was identified at BE of 161.2/162.4 eV only for the C15T (6% of the sulfur signal) and C15T/C12 (10% of the sulfur signal) monolayers. This feature is assigned to the different state of S bond on the Au (111) surface20and could be caused by the organization constraints in the SAMs containing C15T as mentioned by Ito et al. in the case of sterically hindered structures.21

The interfacial reactivity of the four SAMs was then tested in the presence of benzyl alcohol. An excess of 2,6-lutidine was added to promote the regeneration of the catalytic centers (via the comproportionation of oxoammonium and hydroxylamine before electrochemical reoxidation; see Scheme 2).22As far as the electron transfer rate is high (vide infra), it was not considered as a limiting step during the electrocatalytic processes.

After addition of benzyl alcohol a well-defined electro- catalytic response was observed for the four SAMs (Figure 4), and was typical of a catalytic reaction occurring on an

electroactive SAM.23For a fast electron transfer, the catalytic current corresponds to the algebraic sum of the charge transfer (i.e., redox sites on SAM, Figure 1) and the catalyst reaction (i.e., interfacial reaction). In the case of the unmixed C15T monolayer, we can clearly see the monolayer component, especially the reduction peak in the reverse scan. When the C15T molecules are progressively mixed with dodecanethiol (Figure 2b and c), the component due to the monolayer becomes less apparent with disappearance of its reduction peak.

As the CV shape depends on numerous correlated parameters,23 we thus choose the catalytic charge (Qcatalytic) exchanged during thefirst vertex to study the relative and the absolute catalytic activity. Indeed, in addition to classical limiting stages as the chemical rate constant or the alcohol diffusion, the regeneration efficiency of the oxoammonium via comproportionation is assumed to be dependent on the surface coverage.Qcatalyticcharacterizes the absolute performance of the electrocatalytic oxidation of benzyl alcohol and is calculated by subtraction of QSAM of the total charged exchange in the presence of benzyl alcohol. The relative performance is obtained with the turnover, corresponding to the number benzyl alcohol molecules transformed per TEMPO and per second, TO = (1/tvertex)((Qcatalytic/2)/QSAM) (two TEMPO are necessary for the oxidation of one alcohol). Table 2 summarizes the results obtained.

Table 2 shows a monotonic increase of the turnover when the surface coverage of catalytic centers decreased, with a limit value reached for surface coverage down to 20%. ForQcatalytic, a nonmonotonic variation is observed, presenting a maximum value for intermediate TEMPO coverage.

The electrocatalytic results of the SAMs can be interpreted as the steric constraints occurring in confined phase as organized monolayers. For high TEMPO surface coverage the number of catalytic centers is maximum but the space available around each center is minimum. In such configuration, the reaction between oxoammonium and benzyl alcohol appears limited, as illustrated by the very low value of Qcatalytic. An important fraction of the TEMPO head remains chemically inactive and a turnover of only 1.34 molecules·s−1·TEMPO−1 is calculated during the positive sweep.

Figure 3.S2pcore level spectra for a C12 SAM on gold, a C15T/C12 mixed SAM having a TEMPO surface coverage of 45% of a full C15T SAM prepared by successive adsorption and a C15T SAM.

Scheme 2. Electrocatalytic Sequence of Oxidation/

Regeneration of the TEMPO Unit via Comproportionation in the Presence of 2,6-Lutidinea

aThe R substituent corresponds to the chain detailed in Scheme 1.

(4)

The results obtained for the 65% C15T/12 mixed SAM clearly highlights the effect of the dodecanethiol incorporation:

an increase (+15%) of the exchanged charged during the positive scan despite fewer TEMPO (−35%) and consequently an 80% increase of the turnover traducing a reduction of the number of chemically inactive TEMPO head.

This result unambiguously demonstrates the positive effect of the TEMPO random dilution in the passive C12 matrix on the interfacial reactivity, probably due to the decrease of the steric constraints between catalytic centers. This effect is much more pronounced when the C15T concentration is lowered to 45%

by a longer immersion in the C12 solution (Qcatalytic, +59% ; TO, 3.4 times higher).

For higher dilution ratio (20% C15T/C12 SAM), different trends are observed. First, the number of TEMPO immobilized becomes rate limiting and thereforeQcatalyticstarts to decrease.

Second, the turnover continues to increase (6.12 times higher) which is consistent with a high ratio of active catalytic centers.

When the C15T ratio is decreased to 15%, the voltametric shape does not undergo significant modifications (not shown).

The exchanged catalytic charge continues to decrease and the turnover tends to reach a limit value around 8 mole- cules·s−1·TEMPO−1. This stabilization probably corresponds to an equilibrium between the increase of the accessibility of the catalytic centers and the decrease of the comproportiona- tion rate (due to TEMPO dissemination).

The efficiency of electrocatalysis seems to be linked to a combination of the number of active nitroxyl radicals immobilized and their accessibility by target molecules in solution. The increase of the turnover when decreasing C15T

surface coverage suggests the limitations due to the inaccessibility of the catalytic centers.

CONCLUSION

We report here a study focused on the impact of the nitroxyl surface coverage of SAMs on their interfacial reactivity toward alcohol oxidation. All the results were obtained for TEMPO derivative monolayers or randomly distributed mixed SAMs with dodecanethiol. It is established that the maximum catalytic efficiency is not obtained for a full TEMPO monolayer but for a mixed SAM having a 45% TEMPO surface concentration. This behavior clearly demonstrates that the reactivity of a surface can not only be linked to the concentration of immobilized active moiety as generally admitted. Finally, our work evidenced a marked increase of the turnover when the TEMPO surface concentration is lowered until a limit for coverage inferior to 20%. This result outlines the effect of the steric environment of a device molecule on its interfacial reactivity in confined state as self-assembled monolayers.

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (T.B.); eric.levillain@univ- angers.fr (E.L.).

Notes

The authors declare no competingfinancial interest.

ACKNOWLEDGMENTS

This work was supported by the “Centre National de la Recherche Scientifique” (CNRS France), the “Agence Nationale de la Recherche” (ANR France), and the “Regioń des Pays de la Loire”(France).

DEDICATION

Dedicated to the memory of Dr. Nuria Gallego-Planas.

(1) Love, J. C.; Estroff, L. A.; Kriebel, J. K.; Nuzzo, R. G.; Whitesides,REFERENCES G. M. Self-Assembled Monolayers of Thiolates on Metals as a Form of Nanotechnology.Chem. Rev.2005,105, 1103−1170.

(2) Schreiber, F. Structure and growth of self-assembling monolayers.

Prog. Surf. Sci.2000,65, 151−257.

(3) Flink, S.; van Veggel, F.; Reinhoudt, D. N. Sensor functionalities in self-assembled monolayers.Adv. Mater.2000,12, 1315−1328.

(4) Zhang, S.; Cardona, C. M.; Echegoyen, L. Ion recognition properties of self-assembled monolayers (SAMs). Chem. Commun.

2006, 4461−4473.

Figure 4.Cyclic voltamograms of a C15T SAM (a), and of C15T/C12 mixed SAMs having a TEMPO surface coverage of 2.83×10−10mol·cm−2 (65%) (b), 1.91×10−10mol·cm−2(45%) (c), and 0.86×10−10mol·cm−2(20%) (d) recorded at 100 mV·s−1in 0.1 M nBu4NPF6/CH2Cl2without benzyl alcohol (- - -) and after addition of benzyl alcohol (10 mM) and 2,6-lutidine (80 mM) ().

Table 2. Catalytic Activities of Unmixed and Mixed TEMPO SAMs

QSAM (μC)a

Qcatalytic (μC)b

TO

(molecule·s−1·TEMPO−1)c

C15T SAM 8.4 90.5 1.34

C15T/C12 (65%) 5.4 104 2.41

C15T/C12 (45%) 3.9 144 4.61

C15T/C12 (20%) 1.6 105 8.20

C15T/C12 (15%) 1.2 77 8.28

aIntegrated charge exchanged during the positive sweep in 0.1 M nBu4NPF6/CH2Cl2.bCalculated by subtraction ofQSAMof the charged exchange in the presence of benzyl alcohol.cTurnover calculated by the formula: TO = ((Qcatalytic/QSAM)/2)/4. The numbers 2 and 4, respectively, correspond to the number of electrons exchanged when the benzyl alcohol is converted into benzaldehyde and to the time duration of the positive scan).

(5)

(5) Frey, S.; Shaporenko, A.; Zharnikov, M.; Harder, P.; Allara, D. L.

Self-assembled monolayers of nitrile-functionalized alkanethiols on gold and silver substrates.J. Phys. Chem. B2003,107, 7716−7725.

(6) Jensen, L. G.; Nielsen, K. A.; Breton, T.; Sessler, J. L.; Jeppesen, J.

O.; Levillain, E.; Sanguinet, L. Self-Assembled Monolayers of Mono- Tetrathiafulvalene Calix[4]pyrroles and Their Electrochemical Sensing of Chloride.Chem.Eur. J.2009,15, 8128−8133.

(7) Amatore, C.; Genovese, D.; Maisonhaute, E.; Raouafi, N.;

Schöllhorn, B. Electrochemically driven release of picomole amounts of calcium ions with temporal and spatial resolution.Angew. Chem., Int.

Ed.2008,47, 5211−5214.

(8) Fuchigami, T.; Shintani, T.; Konno, A.; Higashiya, S.; Nonaka, T.

Preparation of a stable self-assembled thiol-modified electrode having immobilized 2,2,6,6-tetramethylpiperidinyl-1-oxyl (TEMPO) and its application to electrocatalysis.Denki Kagaku1997,65, 506−507.

(9) Aleveque, O.; Seladji, F.; Gautier, C.; Dias, M.; Breton, T.;

Levillain, E. Nitroxyl Radical Self-Assembled Monolayers on Gold:

Versatile Electroactive Centers in Aqueous and Organic Media.

ChemPhysChem2009,10, 2401−2404.

(10) Aleveque, O.; Blanchard, P. Y.; Breton, T.; Dias, M.; Gautier, C.;

Levillain, E. Revisiting the determination of full steady-state coverage of redox centers on self-assembled monolayers.Electrochem. Commun.

2012,16, 6−9.

(11) Aleveque, O.; Blanchard, P. Y.; Breton, T.; Dias, M.; Gautier, C.;

Levillain, E.; Seladji, F. Nitroxyl radical self-assembled monolayers on gold: Experimental data vs. Laviron’s interaction model.Electrochem.

Commun.2009,11, 1776−1780.

(12) Aleveque, O.; Blanchard, P. Y.; Gautier, C.; Dias, M.; Breton, T.;

Levillain, E. Electroactive self-assembled monolayers: Laviron’s interaction model extended to non-random distribution of redox centers.Electrochem. Commun.2010,12, 1462−1466.

(13) Aleveque, O.; Breton, T.; Levillain, E. Electrocatalytic activity of nitroxyl mixed self-assembled monolayers: combined effects of the nanoscale organization and the composition. Soft Matter 2012, 8, 3875−3880.

(14) Creager, S. E.; Wooster, T. T. A new way of using ac voltammetry to study redox kinetics in electroactive monolayers.Anal.

Chem.1998,70, 4257−4263.

(15) Chidsey, C. E. D.; Bertozzi, C. R.; Putvinski, T. M.; Mujsce, A.

M. Coadsorption of ferrocene-terminated and unsubstituted alka- nethiols on gold: electroactive self-assembled monolayers. J. Am.

Chem. Soc.1990,112, 4301−4306.

(16) Heister, K.; Zharnikov, M.; Grunze, M.; Johansson, L. S. O.

Adsorption of alkanethiols and biphenylthiols on Au and Ag substrates: A high-resolution X-ray photoelectron spectroscopy study.J. Phys. Chem. B2001,105, 4058−4061.

(17) Shaporenko, A.; Brunnbauer, M.; Terfort, A.; Johansson, L. S.

O.; Grunze, M.; Zharnikov, M. Odd-even effects in photoemission from terphenyl-substituted alkanethiolate self-assembled monolayers.

Langmuir2005,21, 4370−4375.

(18) Castner, D. G.; Hinds, K.; Grainger, D. W. X-ray photoelectron spectroscopy sulfur 2p study of organic thiol and disulfide binding interactions with gold surfaces.Langmuir1996,12, 5083−5086.

(19) Zubragel, C.; Deuper, C.; Schneider, F.; Neumann, M.; Grunze, M.; Schertel, A.; Woll, C. The presence of 2 different sulfur species in self-assembled films of n-alkanethiols on Au and Ag surfaces.Chem.

Phys. Lett.1995,238, 308−312.

(20) Ishida, T.; Hara, M.; Kojima, I.; Tsuneda, S.; Nishida, N.;

Sasabe, H.; Knoll, W. High resolution x-ray photoelectron spectros- copy measurements of octadecanethiol self-assembled monolayers on Au(111).Langmuir1998,14, 2092−2096.

(21) Ito, E.; Noh, J.; Hara, M. Steric effects on adsorption and desorption behaviors of alkanethiol self-assembled monolayers on Au(111).Chem. Phys. Lett.2008,462, 209−212.

(22) Breton, T.; Liaigre, D.; Belgsir, E. Allylic oxidation: easy synthesis of alkenones from activated alkenes with TEMPO.

Tetrahedron Lett.2005,46, 2487−2490.

(23) Saveant, J. M. Molecular catalysis of electrochemical reactions.

Mechanistic aspects.Chem. Rev.2008,108, 2348−2378.

Références

Documents relatifs

Under low-frequency US irradiation (20 kHz), the benzyl alcohol conversion rate and benzaldehyde yield are higher than in silent conditions, especially at lower concentrations

Based on the density functional theory (DFT) calculation, Silva et al. [15] explained that when Pd loading was high, the activity was enhanced because the increased number of Pd

Les SAMs mixtes TEMPO/éther couronne présentent une plus faible modification des propriétés électroniques après injection de cations par rapport aux SAMs mixtes

The Laviron's interaction model, dedicated to randomly distributed electroactive adsorbed species, was extended to a non-random distribution in order to extract

At 1 M ethanol concentration (Figure 7.4 e and f) the measured currents are less stable than at the other concentrations. This is especially salient for the highest measured

On these observations there is general agree- ment. Controversy starts when this experience is being used in the decision making on how to treat a specific patient suffering from

This article, based on a result of BORCH and an extension of BUHLMANN, gives a complete characterization of Pareto optimal risk exchanges by a system of differential

An important assumption presented in this work is that “the relation between the predominant perceived tempi and the resonant tempo of the model could be used to predict the