crystals

Communication A Phononic Crystal-Based High Frequency Rheometer

Maxime Lanoy 1, Alice Bretagne 2, Valentin Leroy 3 and Arnaud Tourin 2,*

1 Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2, Canada; [email protected] 2 ESPCI Paris, PSL University, CNRS, Institut Langevin, 75005 Paris, France; [email protected] 3 CNRS, Laboratoire Matière et Systèmes Complexes, Université Paris Diderot, 75013 Paris, France; [email protected] * Correspondence: [email protected]; Tel.: +33-180-963-063

 Received: 16 March 2018; Accepted: 29 April 2018; Published: 1 May 2018 

Abstract: Dynamic Mechanical Analysis (DMA) allows for the measurement of the complex shear modulus of an elastomer. Measurements at frequencies above the frequency range of the device can be reached thanks to the Time–Temperature Equivalence principle. Yet, frequencies higher than a few kHz are not attainable. Here, we propose a method exploiting the physics of bubble phononic crystals to measure the complex shear modulus at frequencies of a few tens of kHz. The idea is to fabricate a phononic crystal by creating a period arrangement of bubbles in the elastomer of interest, here PolyDiMethylSiloxane (PDMS), and to measure its transmission against frequency. Fitting the results with an analytic model provides both the loss and storage moduli. Physically, the shear storage modulus drives the position of the dip observed in transmission while the loss modulus controls the damping, and thus the level of transmission. Using this method, we are able to compare the high-frequency rheological properties of two commercial PDMS and to monitor the ageing process.

Keywords: acoustics; phononic crystals; ; ; shear storage modulus; shear loss modulus

1. Introduction A series of recent works have shown that phononic crystals, or metamaterials, based on bubbles trapped in a soft elastic matrix can exhibit exotic properties such as negative refraction [1], superabsorption [2], or subwavelength focusing [3]. In all these works, the key is that a spherical air cavity in an elastic matrix is an oscillator, much like a bob on a spring, which resonates at a natural frequency given by q 0 ω0 = 1/a 3βg + 4G /ρ (1) with βg the bulk modulus of the inside the bubble, ρ and G’ the mass density and shear storage modulus of the host elastic matrix, respectively, and a the air cavity radius. This formula tells us that the restoring force is due to both the of the gas and the shear elasticity of the while the inertia is that of the solid. Provided that the matrix is soft enough, the resonance is associated to a wavelength much larger than the air cavity size as is the case for the so-called Minnaert frequency resonance of an air bubble in a (i.e., with G0 = 0). Based on periodic arrangements of air cavities trapped in PolyDiMethylSiloxane (PDMS), metascreens with adjustable transmission properties can be realized [4,5]. A simple analytic model proposed in [6] allows the transmission and reflection of such metascreens to be predicted. In this short communication, we show reciprocally that this model can be used to infer the high-frequency rheological properties of cross-linked soft polymers in which bubbles are trapped with a controlled

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Crystalsshort2018 communication,, 8, 195 we show reciprocally that this model can be used to infer the high-frequency2 of 7 rheological properties of cross-linked soft polymers in which bubbles are trapped with a controlled periodicity. We demonstrate that this kind of “phononic crystal-based rheometer” allows for high periodicity. We demonstrate that this kind of “phononic crystal-based rheometer” allows for high frequency measurements, which would be not attainable with classical rheometers. frequency measurements, which would be not attainable with classical rheometers. 2. Materials and Methods 2. Materials and Methods The behavior of viscoelastic materials can be described by their complex shear modulus, The behavior of viscoelastic materials can be described by their complex shear modulus, G(ω) = ”, defined as the ratio of the shear to the shear strain. The storage modulus, , G0(ω) + iG00(ω), defined as the ratio of the to the shear strain. The storage modulus, represents the in-phase (elastic) stress response, while the shear loss modulus, ”, is related to the G0, represents the in-phase (elastic) stress response, while the shear loss modulus, G00, is related to dynamic through ” . Dynamic mechanical analysis (DMA) is a standard the dynamic viscosity η through G00(ω) = ηω. Dynamic mechanical analysis (DMA) is a standard technique for measuring the frequency, temperature dependence, or both, of both moduli. For technique for measuring the frequency, temperature dependence, or both, of both moduli. For example, example, Figure 1 shows a typical variation of both the storage and the loss modulus of a PDMS Figure1 shows a typical variation of both the storage and the loss modulus of a PDMS sample sample measured at 20 °C. It was obtained with an imposed of 5% with frequencies measured at 20 ◦C. It was obtained with an imposed deformation of 5% with frequencies varying varying between 0.01 Hz and 20 Hz. between 0.01 Hz and 20 Hz.

FigureFigure 1. Typical 1. Typical variation variation of storage of storage (∆) ( and∆) and loss loss (o) (o) moduli moduli of PolyDiMethylSiloxaneof PolyDiMethylSiloxane (PDMS) (PDMS) at 20at ◦20C. °C.

By repeating the same kind of measurement at lower temperatures, the behavior of PDMS at By repeating the same kind of measurement at lower temperatures, the behavior of PDMS at higher frequencies can be inferred beyond the frequency range of the apparatus. The idea is to use higher frequencies can be inferred beyond the frequency range of the apparatus. The idea is to use the principle of Time–Temperature Equivalence (TTE) to generate a master curve of storage modulus the principle of Time–Temperature Equivalence (TTE) to generate a master curve of storage modulus vs. frequency at a particular temperature. The TTE principle states that the change in temperature vs. frequency at a particular temperature. The TTE principle states that the change in temperature from the measurement temperature, T, to the reference, T0, is equivalent to multiplying the frequency from the measurement temperature, T, to the reference, T0, is equivalent to multiplying the frequency scale by the so-called horizontal translation factor aT(T,T0). For example, Palade et al. [7] were able to scale by the so-called horizontal translation factor aT(T,T0). For example, Palade et al. [7] were able determine the viscoelastic properties of polybutadienes over 15 decades of frequencies window by to determine the viscoelastic properties of polybutadienes over 15 decades of frequencies window performing rheological tests over a temperature range of 200 °C. However, for some polymers the by performing rheological tests over a temperature range of 200 ◦C. However, for some polymers application of the TTE is limited to a temperature range fixed by the crystallization temperature (~−40 the application of the TTE is limited to a temperature range fixed by the crystallization temperature °C for PDMS). Moreover, for most polymers, the translation factor takes different values for different (~−40 ◦C for PDMS). Moreover, for most polymers, the translation factor takes different values for relaxation (terminal or segmental) modes [8,9], thus making the TTE difficult to apply. different relaxation (terminal or segmental) modes [8,9], thus making the TTE difficult to apply. Here, we propose an alternative method to perform high frequency measurements. The idea is Here, we propose an alternative method to perform high frequency measurements. The idea is to to form a one-layer crystal of bubbles by trapping air cavities in the elastomer of interest (here PDMS), form a one-layer crystal of bubbles by trapping air cavities in the elastomer of interest (here PDMS), to measure its acoustic transmission and to adjust the results by the simple analytic model presented to measure its acoustic transmission and to adjust the results by the simple analytic model presented in [6]. The one-layer crystal of bubbles is fabricated using soft lithography techniques (Figure 2). One in [6]. The one-layer crystal of bubbles is fabricated using soft lithography techniques (Figure2). first creates a mold consisting of a two-dimensional square arrangement of cubes. The cube side One first creates a mold consisting of a two-dimensional square arrangement of cubes. The cube side length is acube = 590 μm and the lattice constant is b = 2 mm. PDMS is soft enough such that the cube length is a = 590 µm and the lattice constant is b = 2 mm. PDMS is soft enough such that the cube resonancecube is associated to a wavelength much larger than the air cavity size and pretty much the resonance is associated to a wavelength much larger than the air cavity size and pretty much the same same as the one of a sphere with the same volume [4,10,11]. The radius of the equivalent sphere is as the one of a sphere with the same volume [4,10,11]. The radius of the equivalent sphere is given by q given3 3 by . A thin layer of PDMS is then deposited on the mold◦ and cured at 65 °C a = 4π acube. A thin layer of PDMS is then deposited on the mold and cured at 65 C during 45 min. We thusduring obtain 45 min. a patterned We thus PDMSobtain layera patterned that we PDMS close bylayer a pure thatPDMS we close layer. by a pure PDMS layer.

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FigureFigureFigure 2. 2. 2.PracticalPractical Practical realization realization realization of of of a a monolayera monolayer monolayer bubble bubble bubble crystal crystal crystal in in in PDMS: PDMS: PDMS: ( (aa )() aThe The) The monomer-crosslinking monomer-crosslinking monomer-crosslinking ◦ mixturemixturemixture is is ispoured poured poured onto onto onto a a mold;a mold; mold; ( (bb ())b After After) After sealing, sealing, sealing, the the the sample sample sample is is iscure cured cured ford for for 45 45 45 min min min at at at65 65 65 °C. C.°C. The The The operation operation operation isis isrepeated repeated repeated with with with a a second,a second, second, non-textured non-textured non-textured mold; mold; mold; ( (cc )() cAfter After) After curing, curing, curing, the the the polymer polymer polymer becomes becomes becomes solid solid solid and and and can can can be be be unmolded;unmolded;unmolded; ( (dd ())d The The) The two two two blocks blocks blocks of of of PDMS PDMS PDMS are are are joined joined joined so so so that that that the the the bubbles bubbles bubbles are are are isolated isolated isolated from from the the outside. outside.

Then,Then,Then, the the the transmission transmission transmission coefficient coefficient coefficient of of of such such such a one-layera one-layer crystal crystal is ispredicted predicted to to be be [6]: [6 [6]:]: 1 1 iKa = + √ (2)(2) t 1  √  (2) ω2 √ 0 − + a − i( + Ka) ω2 1 2 π b δ wherewhere is isthe the dissipative dissipative damping damping constant constant of of the the bubbles, bubbles, and and with with the the 2 wavelengthwavelength for for longitudinal longitudinal waves waves in in PDMS. PDMS.  λ  ka where δ is the dissipative damping constant of the bubbles, and Ka = b 2π with λ the wavelength ForFor a agas gas cavity cavity in in a asoft soft elastic elastic medium medium dissipative dissipative damping damping is isdominated dominated by by viscous viscous losses losses for longitudinal waves in PDMS. suchsuch that that 4/ 4/ where where and and are are the the viscosity viscosity and and density density of of PDMS, PDMS, respectively respectively [12] [12]. . For a gas cavity in a soft elastic medium dissipative damping is dominated by viscous losses is isassumed assumed to to be be independent independent of of frequency frequency in in the the frequency frequency range range of of interest. interest. Ka Ka is isa asuper- super- such that δ = 4η/ρa2ω where η and ρ are the viscosity and density of PDMS, respectively [12]. η is radiativeradiative term term telling telling us us that that the the bubble bubble layer layer radiates radiates /2 /2 times times more more than than one one single single bubble, bubble, assumed to be independent of frequency in the frequency range of interest. Ka is a super-radiative term 2 wherewhere represents represents the the number number of of bubbles bubbles that that oscillate oscillate in in phase. phase. We We would would like like to to stress stressλ  telling us that the bubble layer radiates N/2π times more than one single bubble, where N = that Equation (2) incorporates all multiple scattering events in the bubble layer and that it predictsb a representsthat Equation the number(2) incorporates of bubbles all multiple that oscillate scattering in phase. events We in would the bubble like to layer stress and that that Equation it predicts (2) a minimumincorporates in transmission all multiple at scattering /1 events 2√ in/ the bubble layer and that it predicts a minimum in minimum in transmissionq at√ /1 2√/ transmission at ω / (1 − 2 πa/b). TransmissionTransmission0 measurements measurements were were carried carried out out in in a alarge large water water tank tank (Figure (Figure 3). 3). A A piezoelectric piezoelectric Transmission measurements were carried out in a large water tank (Figure3). A piezoelectric transducertransducer generates generates a ashort short pulse pulse that that propagates propagates through through water, water, traverses traverses the the bubble bubble crystal, crystal, and and transducer generates a short pulse that propagates through water, traverses the bubble crystal, reachesreaches the the hydrophone. hydrophone. To To mimic mimic a aone-dimensional one-dimensional configuration, configuration, the the sample sample is isplaced placed in in the the far far and reaches the hydrophone. To mimic a one-dimensional configuration, the sample is placed in the fieldfield of of the the transducer. transducer. Edge Edge effects effects were were overcome overcome by by mounting mounting the the sample sample on on the the aperture aperture (5 (5 cm cm far field of the transducer. Edge effects were overcome by mounting the sample on the aperture (5 cm diameter)diameter) of of an an acoustically acoustically opaque opaque screen. screen. diameter) of an acoustically opaque screen.

FigureFigure 3. 3.Setup Setup for for acoustic acoustic transmission transmission measurements. measurements. A Ashort short ultrasonic ultrasonic pulse pulse is isemitted emitted from from the the Figure 3. Setup for acoustic transmission measurements. A short ultrasonic pulse is emitted from transducertransducer and and the the transmissi transmissionon is isprobed probed using using a ahydrophone. hydrophone. Sample Sample parameters parameters are are the transducer and the transmission is probed using a hydrophone. Sample parameters are a = 590 μm, 2 mm, 0.8 mm, 5 mm cube 590 μm, 2 mm, 0.8 mm, 5 mm. . 590 µm, b = 2mm, e1 = 0.8mm, e2 = 5mm. 3. 3.Results Results WeWe studied studied two two different different commercial commercial PDMS: PDMS: Sylgard Sylgard 184 184 and and RTV615. RTV615. The The composition composition of of these these twotwo products products is iscarefully carefully kept kept secret, secret, but but the the RTV615 RTV615 is isknown known to to be be softer softer than than Sylgard Sylgard 184 184 and and thus thus

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3. Results

We studiedCrystals 2018 two, 8, x different FOR PEER REVIEW commercial PDMS: Sylgard 184 and RTV615. The composition4 of of 7 these two products is carefully kept secret, but the RTV615 is known to be softer than Sylgard 184 and thus more convenientmore for convenient the fabrication for the offabrication microvalves of microvalves in microfluidic in microfluidic devices. devices. We haveWe have tested tested these these twotwo materials materials with our method. Following the process described in the previous section, we have with our method. Following the process described in the previous section, we have fabricated two bubble fabricated two bubble crystals with the same structural parameters and measured their transmission crystals withcoefficients the same (Figure structural 4). parameters and measured their transmission coefficients (Figure4).

Figure 4. TransmissionFigure 4. Transmission coefficients. coefficients. Measurements Measurements (full (full circles) circles)are are adjustedadjusted by by Equation Equation (2) (2)(full (full lines). lines).

We wouldWe likewould to like point to point out thatout that the the gap gap observedobserved in in Figure Figure 4 does4 does not originate not originate from Bragg from Bragg scattering but rather from hybridization between the free space propagation mode and the bubble scattering but rather from hybridization between the free space propagation mode and the bubble Minnaert resonance [13,14]. Such a gap is known to survive disorder [15,16]. The position of the Minnaerttransmission resonance dip [13 is,14 driven]. Such by the a storage gap is modulus known wh toile survive its height disorderis related to [ 15the, 16loss]. modulus. The position The of the transmissioninferred dip values is driven are listed by in the Table storage 1. The modulusresults confirm while that itsRTV615 height is softer is related and less to viscous the loss than modulus. The inferredSylgard, values which are explains listed why inTable it is preferably1. The resultsused for confirmthe fabrication that of RTV615 microvalves. is softer and less viscous than Sylgard, which explains why it is preferably used for the fabrication of microvalves. Table 1. Values of storage modulus and viscosity for two kinds of PDMS.

Table 1. ValuesPDMS of storage modulusG’ (MPa) and viscosityη for (Pa.s) two kinds of PDMS. RTV615 0.60 ± 0.05 1.5 ± 0.1 Sylgard 1.8 ± 0.1 2.5 ± 0.2 PDMS G’ (MPa) η (Pa.s) RTV615 0.60 ± 0.05 1.5 ± 0.1 Note that the uncertainty measurement originates from the margin of error in the cavity size (of Sylgard 1.8 ± 0.1 2.5 ± 0.2 the order of 10 μm). Given these promising results, we decided to use our method to monitor the ageing of PDMS, a phenomenon the users of PDMS are familiar with. Although appreciated for its Notegreat that versatility, the uncertainty this material measurement causes difficulties originates in terms from of repeatability. the margin The of chain error crosslinking in the cavity size (of the ordermechanism of 10 µism). a complex Given phenomenon these promising that depends results, on working we decided conditions to and use is ouroften methodincomplete. to monitor A very simplified image consists in picturing the polymer as a complex entanglement of chains with the ageinga ofdegree PDMS, of cross-linking a phenomenon which increases the users during of PDMS the polymerization. are familiar As with. a consequence, Although it quickly appreciated for its great versatility,becomes difficult this materialfor free chains causes to reach difficulties the reactive in terms sites. A of well-known repeatability. manifestation The chain of this crosslinking mechanismphenomenon is a complex is the phenomenon evolution of thatthe interfacia dependsl properties on working of PDMS. conditions While andbeing is oftennaturally incomplete. A very simplifiedhydrophobic, image PDMS consists is often in treated picturing with theoxygen polymer asto aobtain complex good entanglementconditions of wetting. of chains with However, this treatment does not resist in the long-term because of the migration of low molecular a degree ofweight cross-linking chains to the which surface. increases However, if during these ch theains polymerization.move under the effect As of a diffusion, consequence, they are it quickly becomesalso difficult required for to freecontinue chains crosslinking to reach according the reactive to very specific sites. time A well-knownscales. manifestation of this phenomenon isTo theillustrate evolution this phenomenon, of the interfacial we have properties performed of time-lapse PDMS. Whilemeasurements being naturallyof the acoustic hydrophobic, PDMS istransmission often treated through with a periodic oxygen layer plasma of bubbles to obtain embedded good in RTV615. conditions Figure of 5 shows wetting. the acoustic However, this transmission curves obtained respectively 1 h, 1 day, and 9 days after curing. Ageing manifests itself treatmentin does the shift not of resist the resonance in the long-term towards higher because frequency, of the which migration reflects of an low increase molecular in the storage weight chains to the surface.modulus. However, if these chains move under the effect of diffusion, they are also required to continue crosslinking according to very specific time scales.

To illustrate this phenomenon, we have performed time-lapse measurements of the acoustic transmission through a periodic layer of bubbles embedded in RTV615. Figure5 shows the acoustic transmission curves obtained respectively 1 h, 1 day, and 9 days after curing. Ageing manifests itself in the shift of the resonance towards higher frequency, which reflects an increase in the storage modulus. Crystals 2018, 8, 195 5 of 7 Crystals 2018, 8, x FOR PEER REVIEW 5 of 7

FigureFigure 5. Transmission 5. Transmission through through the bubblethe bubbl phononice phononic crystal crystal at differentat different times times reveals reveals ageing ageing of PDMS.of MeasurementsPDMS. Measurements (full circles) (full are circles) adjusted are byadjusted Equation by Equation (2) (full (2) lines). (full lines).

This result is consistent: the higher the degree of crosslinking, the less chain rearrangements are Thislikely result to occur. is consistent: The response the to higher a solicitation the degree then of becomes crosslinking, essentially the lesselastic chain and rearrangementsPDMS hardens. are likelyFrom to occur. a more The quantitative response point to aof solicitation view, we obtain thened becomes a good agreement essentially with elastic the model and for PDMS the values hardens. Fromlisted a more in Table quantitative 2. point of view, we obtained a good agreement with the model for the values listed in Table2. Table 2. Evolution in time of storage modulus and viscosity for PDMS RTV615. Table 2. Evolution in time of storage modulus and viscosity for PDMS RTV615. Time G’ (MPa) η (Pa.s) 1 h 0.4 ± 0.05 1.4 ± 0.01 Time G’ (MPa) η (Pa.s) 1 day 0.5 ± 0.05 1.4 ± 0.01 9 days 1 h 0.6 ± 0.4 0.05± 0.05 1.4 1.5± 0.01± 0.01 1 day 0.5 ± 0.05 1.4 ± 0.01 4. Discussion 9 days 0.6 ± 0.05 1.5 ± 0.01 For some polymers, the crystallization phenomenon limits the temperature range which can be 4. Discussionexplored such that the determination of the shear modulus remains limited to low frequencies. ForSeveral some strategies polymers, based the on the crystallization measurement phenomenon of both the velocity limits and the the temperature attenuation of range shear whichwaves can be exploredhave been such proposed that the in determinationthe past for reaching of the higher shear frequencies modulus remains up to the limited MHz range. to low Reflection frequencies. measurements are preferably used due to the strong attenuation of shear waves in viscoelastic Several strategies based on the measurement of both the velocity and the attenuation of shear materials at these frequencies. For example, in [17] the authors exploit the echoes reflected at the waves have been proposed in the past for reaching higher frequencies up to the MHz range. interface between the sample of interest and a piece of quartz, the latter being used as a delay line. ReflectionThe echoes measurements measured in absence are preferably of the sample used (i.e., due at the to quartz/air the strong interface) attenuation are taken of as shear a reference. waves in viscoelasticThe storage materials and loss at theseshear frequencies.modulus can Forthen example, be determined in [17 ]from the authorsthe ne echo exploit by taking the echoes the ratio reflected of at theits interface amplitude between to the reference the sample and the of interestdifference and between a piece its of arrival quartz, time the and latter the one being of the used reference as a delay line.(Figure The echoes 6). measured in absence of the sample (i.e., at the quartz/air interface) are taken as a reference.The The method storage however and loss requires shear modulusfrequencies can sufficiently then be determined high such that from the the successive ne echo echoes by taking do the rationot of itsoverlap. amplitude The length to the of referencethe delay line and can the be difference increased to between overcome its this arrival difficulty, time andthe price the oneto pay of the referencebeing (Figure diffraction6). effects that blur the results. Furthermore, a good coupling between the viscoelastic Thematerial method and the however delay line requires is crucial. frequencies sufficiently high such that the successive echoes do not Our alternative approach avoids these difficulties. We have shown that bubbles trapped in a overlap. The length of the delay line can be increased to overcome this difficulty, the price to pay being viscoelastic material to form a bubble phononic crystal can be used to probe its rheological properties diffraction effects that blur the results. Furthermore, a good coupling between the viscoelastic material in a frequency domain (a few tens of kHz) hard to reach with either DMA or acoustic reflection and themeasurements. delay line is Of crucial. course, some further tests will be required to fully validate this approach as a new Ourrheology alternative measurement approach technique. avoids Such these a method difficulties. should be We useful have to shownprobe the that rheological bubbles properties trapped in a viscoelasticof cross-linked material polymers to form in a an bubble intermediate phononic frequency crystal canrange be (i.e., used from to probea few tens its rheologicalof kHz up to properties a few in a frequency domain (a few tens of kHz) hard to reach with either DMA or acoustic reflection measurements. Of course, some further tests will be required to fully validate this approach as a new rheology measurement technique. Such a method should be useful to probe the rheological Crystals 2018, 8, 195 6 of 7

Crystals 2018, 8, x FOR PEER REVIEW 6 of 7 properties of cross-linked polymers in an intermediate frequency range (i.e., from a few tens of kHz MHz)up to not a few accessible MHz) not by accessiblecommon techniques. by common Interestingly, techniques. Interestingly,it appears as itan appears unexpected as an outcome unexpected of theoutcome work we of have the work done wefor the have last done ten years for the on last the tenexotic years transmission on the exotic properties transmission of bubble properties phononic of crystals.bubble phononic crystals.

FigureFigure 6. 6. AcousticAcoustic measurement ofofthe the complex complex shear shear modulus modulus of aof viscoelastic a viscoelastic material. material. A short A short pulse pulseis sent is fromsent from a shear a shear wave wave piezo-electric piezo-electric transducer transducer through through a delay a line.delay The line. storage The storage and loss and moduli loss modulican be can deduced be deduced from the from amplitude the amplitude and arrival and arrival time of time the nofe echothe n comparede echo compared to the reference.to the reference.

Author Contributions: M.L., V.L., and A.T. conceived and designed the experiments; A.B. and M.L. performed theAuthor experiments Contributions: and analyzedM.L., V.L.,the data; and A.T. conceivedwrote the paper. and designed the experiments; A.B. and M.L. performed the experiments and analyzed the data; A.T. wrote the paper. Acknowledgments: This work was supported by LABEX WIFI (Laboratory of Excellence ANR-10-LABX-24) Acknowledgments: This work was supported by LABEX WIFI (Laboratory of Excellence ANR-10-LABX-24) withinwithin the the French French Program Program “Investments “Investments for for the the Future” Future” under under Reference Reference No. No.ANR-10-IDEX-0001-02 ANR-10-IDEX-0001-02 PSL*. PSL*.We thankWe thank Helene Helene Montes Montes (SIMM, (SIMM, ESPCI ESPCI Paris) Paris) for DMA for DMA measurements. measurements. ConflictsConflicts of of Interest: Interest: TheThe authors authors declare declare no no conflict conflict of of interest. interest. The The founding founding sponsors sponsors had had no no role role in in the the design design ofof the the study; study; in in the the collection, collection, analys analyses,es, or or interpretation interpretation of of data; data; in in the the writing writing of of the the manuscript; manuscript; and and in in the the decision to publish the results. decision to publish the results.

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