XPCS Investigation of the Dynamics of Filler Particles in Stretched Filled Elastomers

XPCS Investigation of the Dynamics of Filler Particles in StretchedFilled ElastomersFrancoise Ehrburger-Dolle,*,† Isabelle Morfin,† Francoise Bley,‡ Frederic Livet,‡ Gert Heinrich,§

Sven Richter,§ Luc Piche,∥ and Mark Sutton∥

†Univ. Grenoble 1/CNRS, LIPhy UMR 5588, Grenoble F-38041, France‡SIMaP, UMR 5266 Grenoble INP/CNRS/UJF, 38402 Saint Martin d’Heres, France§Leibniz-Institut fur Polymerforschung Dresden, 010169 Dresden, Germany∥Physics Department, McGill University, Montreal, Quebec H3A 2T8, Canada

ABSTRACT: The complexity of the mechanical behavior offilled elastomers can be partly attributed to the fact that theduration of an applied strain plays a crucial role. In order tobring new insights into this still incompletely solved problem,we look for relationships between the macroscopic mechanicalrelaxation and the relaxation of the filler particles at the nano-to mesoscale. To this end, X-ray photon correlation spectros-copy (XPCS) in homodyne and heterodyne configurationscombined with tensile stress relaxation is employed. The paperis devoted to the study of the role of the filler−filler and the filler−matrix interactions in a cross-linked elastomer on the agingmechanisms under strain. The fillers investigated are carbon black, as an example of strong filler−matrix interactions, andhydroxylated silica for which the filler−filler interaction is strong (H-bonds). Homodyne XPCS correlation reveals features ofjammed systems (compressed exponential and ballistic motion) for both systems. The exponents characterizing the aging of thehomodyne relaxation times are not the same in the carbon black and in the silica filled samples. For both systems, the decrease ofthe particle velocity determined by heterodyne detection with aging time follows a power law. The silica sample is characterizedby a slow decrease of the velocity during aging. For the carbon black sample, the velocity remains small and decreases faster thanfor the silica sample. The reverse is observed for the behavior of the tensile force.

1. INTRODUCTION

Over the past several years, X-ray photon correlation spectro-scopy (XPCS), originally named intensity fluctuation spectro-scopy (IFS),1 has been shown to be an important technique forinvestigating slow dynamics in soft and hard condensed mattersystems.2−4 It provides a tool complementary to dynamic lightscattering for the study of small-scale high-q dynamics. Even forlength scales accessible to light scattering, it can be used foropaque samples. A few years ago, we developed a heterodynetechnique (HD-XPCS) for XPCS measurements5,6 which yieldsinformation about the phase shift in the scattering signal. Thisallows us to measure the velocity of filler particles during therelaxation after an initial tensile deformation of un-cross- andcross-linked carbon black filled elastomers. It was shown thatthe velocities were larger for the un-cross-linked sample thanfor the cross-linked one. In both cases, however, the velocitydecreased with the aging time as 1/t. This first series ofmeasurements proved that HD-XPCS was very well suited forinvestigating the mechanisms of aging of filled elastomers afterrelease of a strain. Therefore, we decided to perform extendedmeasurements in which tensile force and XPCS measurementson strained samples would be combined. Such measurementswere expected to be particularly interesting for at least tworeasons.

The first reason concerns the possibility to relate mechanicalrelaxation (i.e., a relaxation occurring at the macroscopic scale)and relaxation of the filler particles at nano- to mesoscales.Relating such macroscopic and microscopic properties is naturalin the physics of disordered out-of-equilibrium systems like gelsor glasses.7 Dynamic light scattering (DLS) combined withrheology was used to relate internal dynamics and elasticity infractal colloidal gels8 local dynamics and nonlinear rheology ofsoft colloidal paste9 or ultraslow dynamics in aging of a gelcomposed of multilamellar vesicles10 or in a suspension of claylaponites.11,12 This methodology allowed researchers to finduniversal features in the fluid-to-solid transitions, the jammingstate, or the sol-to-gel transition in colloidal suspensions.13,14 Forseveral years, XPCS has been used to complement DLS wheninvestigating the aging behavior of colloidal gels,15−18 silicaparticles dispersed in unvulcanized rubber,19,20 or the dynamicsof colloidal particles in ice21 or that of magnetic particlesdispersed in water and submitted to a magnetic field.22,23 Thecombination of XPCS and rheological properties is also reportedin the case of gel-forming silica suspension in decalin24,25 or in

Received: July 3, 2012Revised: September 5, 2012Published: October 16, 2012

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© 2012 American Chemical Society 8691 dx.doi.org/10.1021/ma3013674 | Macromolecules 2012, 45, 8691−8701

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polymer melts26,27 and in several other systems recentlyreviewed.3 The common feature of all the above quoted experi-ments uses signatures in the relaxation, such as ballisticmovements and particular exponents in the power law aging,to try and put a given complex out of equilibrium system intoa given class of materials (e.g., gels, glasses, jammed systems,etc.).7,28 As a consequence, it becomes possible to learn moreabout which are the controlling parameters (particle−particle orparticle matrix interactions, mechanical history, and so on) and,eventually, to extend the underlying theories.The second reason for carrying out a new series of HD-

XPCS measurements is the similarity of some of our results onfilled elastomers [refs 5, 6 and unpublished results] with thosereported on jammed systems. Robertson and Wang29,30 gaveexperimental evidence suggesting the existence of an analogybetween dynamic strain-induced nonlinearity in the modulus offilled rubbers, the physics of the glass transition of glass-formingmaterials, and the jamming transition of vibrated granularmaterials. Actually, the mechanical behavior of filled elastomersand its time involvement is an active research area since morethan 50 years. Bhattacharya et al. [ref 31 and references therein]reviewed several “classical” models attempting to take intoaccount the complexity of the problem. These authors alsoreported an extensive study of the nonlinear, time-dependentmechanical behavior of a cross-linked carbon black filledelastomer. They concluded that none of the existing modelswere able to capture the complexity of these systems, and theyunderlined the need for the introduction of nonlinear stressrelaxation in the models. Combination of transmission electronmicroscopy, mechanical measurements, and small-angle scatter-ing (neutrons and X-rays) recently highlighted the role of thepolymer−filler interfacial interaction32 and that of the fillerdispersion33 on the mechanical reinforcement in model nano-composites. Our goal was to investigate the role of theseparameters on the dynamics of filled elastomers.For this new series of HD-XPCS experiments, well-defined

samples have been prepared in order to investigate the role ofthe filler volume fraction (below and above the percolationthreshold) and the role of filler−filler and filler−matrixinteractions on the dynamics and tensile properties of a filledelastomer (un-cross-linked or cross-linked). HD-XPCS measure-ments were performed on strained samples while the stressrelaxation was simultaneously measured. This experimentalapproach is particularly suitable for systems for which theduration of the application of a strain (the “waiting time”) plays acrucial role. Preliminary homodyne and heterodyne XPCSresults34 have shown that the aging behavior under 20% tensilestrain of the cross-linked elastomer filled with carbon black wasdifferent from the one filled with hydroxylated silica. The presentpaper is aiming to describe the experimental results obtained forthe same cross-linked elastomer filled either with carbon black orwith hydroxylated silica stretched at 60% elongation. A particularattention will be paid to the methods used for the analysis of themechanical and the XPCS (homodyne and heterodyne) data.

2. MATERIALS AND EXPERIMENTAL METHODS2.1. Samples. The present study involves two model samples

consisting of an elastomer (ethylene propylene diene monomer,EPDM, rubber) filled with carbon black (N330) or hydoxylatedpyrogenic silica (Aerosil 200, Degussa). The morphological character-istics of both fillers have been previously described in detail.35−37

The filled elastomers (40 parts of rubbers i.e., 40 phr) were preparedby mixing 40 g of filler with 100 g of rubber and 3 g of dicumyl

peroxide in a two-roll mill for 5 min at 50 °C. Afterward, the mixtureswere cured in molds at 160 °C under pressure during 10 min. Thethickness of the samples was about 1 mm. The volume fraction offiller corresponding to 40 phr is close to 0.18 for the carbon blackfilled sample and to 0.16 for the silica one, i.e., in both cases, abovethe “percolation” threshold. In our measurements, the 1 mm thickplates are punched out to the classical dumbbell shape (width =4 mm).

2.2. Force Measurements. An apparatus that permits simulta-neous stress and scattering measurements has been developed.34 Thesetup is sized as to fit in a vacuum can with X-ray aperture windowsand mounted on a translational stage on the IMMY/XOR-CAT(8-ID) beamline at APS (Argonne, IL). The sample placed per-pendicular to the beam axis was subjected to a rapid vertical extensionstep which was then held constant. A strain gauge and a digitizing unitwere used to measure the resulting evolution of the stress on thesample every 2 s. By using a symmetric stretch, the instrument wasdesigned so that there is a nearly fixed point on the sample for theX-ray beam to pass through. This setup permitted measurements of thesame section of material during the entire range of strain conditionsapplied during the course of a particular experiment. A completedescription of the apparatus can be found in the master thesis of L. J. S.Halloran.38 The aim of these measurements was to determine the stressrelaxation which is the time-dependent reduction in stress, i.e., the forceneeded to keep the sample under constant deformation.

2.3. XPCS Measurements. XPCS measurements were performedon the IMMY/XOR-CAT (8-ID) beamline at APS in similar experi-mental conditions as for the previous series.5,6 The X-ray wavelength λwas equal to 1.663 Å (7.448 keV). The 20 μm × 20 μm beam size wasselected by means of carefully polished slits placed 0.64 m before thesample. Guard slits were added, 0.16 m before the sample, in orderto limit the scattering from slit diffraction in the SAXS region. Theincident beam intensity I0 (∼109 photons s−1) was measured by adiode located in the beamstop. A heterodyne signal was obtained froma powder of fume silica (Aerosil 200), compacted in one of the twoholes (diameter and depth 1 mm) drilled in a plate placed immediatelyupstream the filled elastomer sample. The total thickness of material inthe scattering volume was kept close to 2 mm, and interferences fromscattering in this volume could be observed. The requirement for thisreference sample was that it occupies the same coherence volume asthe specimen to be investigated. The second hole was empty in orderto permit homodyne measurements. The sample holder was mountedon an x−z stage, perpendicular to the beam. By translating the sampleholder, measurements could be made either directly on the fluctuatingsample alone or on the hybrid sample (sample + reference), thusallowing for an easy comparison of homodyne and heterodyne results.In the latter case the beam intensity was reduced by a factor of 13 owingto the absorption in the reference. The sample-to-detector distance was2.8 m. A direct-illumination deep-depletion CCD (PI 1152 × 1242,22 μm resolution) camera was used as an area detector. Figure 1 shows atypical speckle pattern. The beam center and beamstop were placed inthe upper right corner. Reading of each frame takes about 2 s. At eachelongation step, the duration of X-ray illumination was 100 ms per framein the homodyne and the heterodyne setup for the carbon black filledsample. The heterodyne (het) and homodyne (hom) sequences wereorganized as follows: 100 frames-het, 100 frames-hom, 500 frames-het,500 frames-hom. The total duration of the measurement, includingmeasuring transmission and dark, was 2805 s. For the silica filled sample,the exposure time was 500 ms in the heterodyne setup and 100 ms inthe homodyne one. The sequences (total duration equal to 4126 s) wereorganized as follows: 100 frames-het, 100 frames-hom, 200 frames-het,200 frames-hom, 500 frames-het, and 500 frames-hom.

The multispeckle analysis of the experimental data was providedby the software “coherent” (Matthew A. Borthwick and Peter Falus,22 Nov 2003 MIT, 1996)39 available at beamline 8-ID at APS. Thissoftware is designed to permit the determination of the static SAXScurve I(q) normalized by the transmission and the incident beamintensity, by means of the application Sqphi. The experimental XPCScorrelation functions G(τ) is obtained by means of a multiple taucorrelation calculation in the application g2qphi. In order to determine

Macromolecules Article

dx.doi.org/10.1021/ma3013674 | Macromolecules 2012, 45, 8691−87018692

the correlation functions in heterodyne and homodyne configurations,the delay per tau level (dpl) is fixed at 150. Because of the anisotropyof the correlation functions, images are partitioned as shown inFigure 1. Fitting of the experimental curves is achieved by nonlinearregression procedures by means of the Marquardt−Levenbergalgorithm (SigmaPlot 10.0).

3. ANALYSIS OF THE EXPERIMENTAL DATA3.1. Force Measurements. The elongation of the samples

was increased by steps of 20% from 0 to 60%, as shown inFigure 2. For the 60% step investigated here, the duration was

4126 s for the silica filled sample and 2805 s for the carbonblack one.A detailed analysis of the force (stress) relaxation at a given

constant deformation measured for all samples and all elonga-tions reached from below (“up”) or from above (“down”) willbe presented elsewhere [manuscript in preparation]. In brief,it appears that, in all cases for which noise was not a factor, theexperimental rate of change of the force |ΔF(t)|/Δt decreases

with time as follows:

|Δ |Δ

∝ α−F tt

t( )

(1)

The exponent α is larger than 1 for hydroxylated silica filler atany strain, smaller than 1 for carbon black filler at strains below60% elongation. It is equal to 1 for carbon black or nonhydroxylated silica fillers at larger strains. It follows that theexperimental curves F(t) obtained for 60% elongation shown inFigure 3 can be fitted to the following equations:• for the silica filled sample

= +α− +∞F t At F( ) ( 1)

(2a)

• for the carbon black filled sample

= +F t A t F( ) log( ) 1 (2b)

The result obtained for the silica sample (Figure 3) indicatesthat the force reaches an asymptotic limit F∞ = 2.413 N. On thecontrary to the strained silica sample, the force measured forthe carbon black sample does not tend to an asymptotic limit.It is slowly decreasing from F1 = 3.19 N (when t = 1 s). Thisconclusion, however, results from observations during less than3000 s.For filled elastomers, stress relaxation is generally associated

with reorientation of the polymer network under strain, withthe rearrangement of the chain entanglements and with thebreaking of bonds between chains, between filler particles orbetween chains and filler particles (debonding). Gent40 reportsthat the stress relaxation generally decreases with a logarithmicbehavior, as observed here in Figure 3, for the carbon blackreinforced samples. A logarithmic decrease of the stress hasbeen also reported in the case of tensile deformation of semi-crystalline polymers.41 Theoretical models and simulations haveshown that logarithmic relaxation at the macroscopic level is oftenassociated with strongly interacting elements at the microscopicscale.42,43 Very recently, Amir et al.44 have observed that, in manycases, the relaxation of very different systems in nature islogarithmic. These authors propose a generic model for slowrelaxations in relation with a broad distribution of relaxation rates.Power law stress relaxations similar to those observed for the

silica filled sample for which the exponent (α − 1) equals 0.17were reported for cross-linked polymer networks mechanicallydeformed (stretch, shear, or creep) by Curro and Pincus45 and

Figure 1. Typical speckle pattern obtained by the direct illuminationCCD camera. Each image is partitioned into 30 angular sectors of5.65° (angles φ vary between 140° and 306° counterclockwise) and 30q-domains ranging from 2.7 × 10−3 to 3.2 × 10−2 Å−1. The direction ofthe strain is vertical and corresponds to φ = 270°.

Figure 2. Mechanical history of the hydroxylated silica and the carbonblack filled elastomers. The curves in red correspond to the stressrelaxation at 60% elongation investigated in this article. The arrowsindicate the origin of the time scale for aging (ta = 0).

Figure 3. Time decrease of the force requested to maintain anelongation of 60% for the carbon black and the silica filled samples.Red lines are the results of the fit by eq 2.



later by Heinrich and Vilgis.46 This apparent power law behavioris better represented by the empirical Chasset and Thirionequation47,48 into which eq 2a can easily be transformed. Thevalue of the exponents ranging between 0.05 and 0.20 wererelated to the degree of cross-linking. Ng and McKinley49 haveshown that the shear modulus of gluten gels at finite straindecreases as t−n with n = 0.175 ± 0.005. Thin films of anaqueous suspension of laponite previously subjected to a creepflow field undergo a time-dependent enhancement of the elasticmodulus G′ ∝ t0.22.50 An exponent equal to 0.4 was reportedby Guo et al.24,25 for the growth of the storage modulus of gel-forming colloidal suspensions at large waiting times. Theseexamples suggest that the stress relaxation of the strained silicafilled sample is typical of a gel-like viscoelastic material. Theforce tends to an asymptotic limit which may correspond to anew stable arrangement of the H-bonded silica network in thestretched sample. On the contrary, the logarithmic relaxation ofthe strained carbon black sample does not reveal a limiting valuefor the force with the same time domain. This system couldrelax under strain as an elastoplastic solid and/or according toviscoelastic damage models.31

3.2. Static Scattering Measurements. Figure 4 showsthe SAXS curves deduced from the speckle patterns by means

of the application Sqphi in the software “coherent”. The firstgraph (Figure 4a) displays SAXS curves obtained at differentazimuthal angles φ. It appears that the scattering by bothsamples is nearly isotropic down to the lowest q-value measured(2.7 × 10−3 Å−1). We have previously shown37 that a cross-linked sample of ethylene−propylene−rubber (EPR) filled withthe same volume fraction of the same Aerosil 200 silica displays

a butterfly pattern when stretched at 52%. This anisotropyappeared between 7 × 10−4 and about 3 × 10−3 Å−1, i.e., in aq-domain that was not reached in the present experiment. Inthe intermediate q-domain, the shape of the SAXS curves is thesame as previously described. For the carbon black filledsample, the curve I(q) scales as q−p with p = 3.76. This featurecorresponds to the scattering of the surface of the primaryparticles characterized by a surface fractal dimension DS = 6 −p = 2.24.35 For the silica filled sample, the curve I(q) scales asq−m with m = 2.26. This power law, characterized by anexponent which absolute value is smaller than 3, describes thescattering of fractal silica aggregates with Dm = 2.26.35,37

Figure 4b indicates that aging of the samples under straindoes not involve any structural change at mesoscopic lengthscales (approximately 230 to 21 nm).The SAXS intensity curves obtained for the two samples in

homodyne and heterodyne configuration are compared inFigure 5. These curves yield an experimental estimation of the

mixing coefficient x(q) = Is(q)/[Is(q) + Ir(q)] where Is(q) is theintensity scattered by the sample (in the homodyne config-uration) and Ir(q) the intensity scattered by the compactedAerosil 200 used as reference in the heterodyne configuration(Figure 6). It may be noticed that, above about 8 × 10−3 Å−1, forthe silica (Aerosil 200) filled sample x(q) remains constant. Atsmaller q values, the scattering of the silica dispersed in theelastomer increases more than in the compacted state becausefractal aggregates are less interpenetrated at a large scale than at ashorter one.35,37

3.3. XPCS in Heterodyne Regime. Theoretical back-grounds and experimental conditions of X-ray heterodyningwere discussed in details previously.5,6,51 Owing to an expectedrelative velocity v between the filler particles in the strainedsample and the immobile silica particles in the reference, thecorrelation function acquires a phase factor of exp(iq·vτ). Wecan define

ω φ φ= ·⎯→ = −q v qv cos( )0 (3)

where φ is the angle given by q and φ0 is the angle for thedirection v.In this case the heterodyne correlation function can be

written as5

Figure 4. Comparison between the SAXS curves obtained for the silica(open triangles) and the carbon black (closed circles) filled samples at60% elongation: (a) for different angles φ and (b) for different agingtimes.

Figure 5. Comparison between SAXS curves measured for the carbonblack and the Aerosil 200 silica filled sample in homodyne and inheterodyne geometries. The green triangles correspond to the curvemeasured for the compacted Aerosil 200 sample used as reference(heterodyne geometry).



τ τ τ

ωτ τ τ

= + +

+

g A B g

C g

( ) 1 [ ( / )]

cos( ) Re[ ( / )]het(2) (1)

02

(1)0 (4)

with A = β(1 − x)2, B = βx2, and C = 2βx(1 − x).Typically we use g(1)(τ/τ0) = exp[−(τ/τ0)μ]. The term β is

from partial coherence and β = 0.30 as determined by ourexperimental setup. The exponent μ introduced in the dampingterm results from the assumption of a possible compressedexponential behavior observed in jammed systems.14,15 For thesuspension of colloidal latex particles in glycerol investigatedearlier,5 μ was equal to 1 and ω equal to zero as expected for adilute system at equilibrium. The heterodyne correlationfunction (eq 4) implies that the reference sample does notdisplay any dynamic component. Thus, it was necessary todetermine its homodyne correlation over a large period of time,equal at least to the delay time involved in the heterodynemeasurements. The experimental homodyne correlations G(τ)obtained for the reference sample for two q-values plotted inFigure 7a indicate that our setup is stable over at least 1000 s.Figure 7b shows the correlation curves obtained for the silicafilled sample at φ = 208° for different q-values. These curves arededuced from 50 frames, i.e., 122 s, among 500 frames obtainedin the third series of heterodyne measurements. The value ofaging indicated in these figures corresponds to the mean valueof the aging time for the first and the last frame considered forthe correlations. Each block of φ, q was fit to eq 4. Figures 8and 9 show that eq 3 is valid, and a single |v| and φ0 explainsall blocks.The inset in Figure 8 shows that, in the range of q

investigated, the value of the exponent μ fluctuates between0.5 and 1.5 with quite a large statistical error primarily becausethe delay window is not broad enough to precisely determinethe damping of the oscillation. The two-times correlationmatrices in ref 51 show that the velocity varies with aging understrain. Thus, the delay window must be narrow in order toavoid spurious effects in the correlations. On the other hand, itis possible to determine quite precisely the velocity within adelay window that needs not to be larger than half a period.This possibility will be used to determine the variation of thevelocity as a function of the aging time under strain. In suchconditions, information about the damping of the oscillations(characterized by μ and τ0) is lost. Nevertheless, Figure 7cclearly shows that, even in a narrow time window, the amplitudeof the oscillations decreases much faster in carbon black filledsample than in the silica one. This feature appears as well for

similar values or different values of ω (in the case of the silicafilled sample). At large q values, the damping of the oscillationamplitude becomes strong enough to eventually make difficult afit with eq 4 over the whole delay window, as already reported.34

Figure 6. Variation with q of the mixing coefficient x in the case of thecarbon black and the silica filled sample.

Figure 7. (a) Homodyne correlations obtained for the referencesample. (b) Examples of heterodyne correlations obtained for the silicafilled sample at different q values at an angle φ = 208° and (c) for thecarbon black and the silica filled samples for q = 4.7 × 10−3 Å−1.Continuous and dashed lines are fits of the experimental data with eq 4in which μ is a free parameter or fixed to 1.5, respectively.

Figure 8. Fluctuations with q of ω/q deduced from the fit of thecorrelation curves partly shown in Figure 7b. The inset shows thefluctuations of the exponent μ with q.



The above comments based on the aspect of the curves plottedin Figure 7b are confirmed by the values of the parameters τ0 andμ deduced from the fit with eq 4 given in Table 1. The strong

damping of the oscillations in the strained carbon black filledsample is characterized by a shorter relaxation time than for thehydroxylated silica one, and the exponent μ remains close to 2 forthe former. Now, the influence of the value of μ on the qualityof the fit and the value of the other parameters must be checked.To this end, μ is no longer considered as a free parameter in thefitting equation but fixed to 1.5 in both cases (dashed lines inFigure 7b). As shown in Table 1, the only significant effect is theexpected lowering of τ0 for the silica filled sample. The value ofthe particle velocity vfl remains the same in all cases. It followsthat the obvious difference in the damping of oscillations havingnearly the same frequency results from the difference in τ0 whichis much smaller for the strained carbon black sample than for thesilica one.3.4. XPCS in Homodyne Regime. The homodyne correla-

tion function is written as follows:

τ β ττ

= + −μ⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥g a( ) exp 2hom

(2)

0 (5)

where a, supposed to be equal to 1, will be considered as a freeparameter in the nonlinear regression. It appears that, in mostcases, the value a differs from 1 by less than ±0.005.It has been verified experimentally5 that, in the case of the

dispersion of latex particles in glycerol in an equilibrium state,the values of β, τ0, and μ (μ = 1) are the same as in the hetero-dyne correlation function. It was also shown that τ0 wasisotropic and scaling as q−2 as expected for diffusion. In thepresent case, the homodyne correlations change with theangle φ (Figure 10) and the characteristic time τ0 scales as q

−1

(Figure 11).

Anisotropic dynamics has been reported, for example, inferroglasses under an applied magnetic field.23 In this case, themagnetic field induces a structural anisotropy evidenced bySAXS in the same q-domain. In the q-domain investigated here,the SAXS curves show no structural anisotropy induced byelongation of the sample. It follows that, from this point ofview, our homodyne correlations are supposed to be isotropic.Cipelletti et al.14 have observed in concentrated emulsionsrelaxation times that are larger for q perpendicular tocentrifugation acceleration than for q parallel. This anisotropywas attributed to the internal stress as the sample is loaded andcentrifuged. Anisotropic correlations have also been reported inXPCS experiments for probing the diffusive dynamics ofPMMA colloidal particles in a shear flow.52−54 These authorshave observed that the correlation curves indicate a fasterdecorrelation (i.e., a smaller apparent τ0) in the direction of theshear flow (q∥) than in the direction perpendicular (q⊥). In ourcase (Figure 10) it appears that decorrelation is faster for φ =180°, i.e., in the direction perpendicular to the stretch, than inthe direction nearly parallel to the stretch (φ = 266°). Thecomparison between the two homodyne situations suggests theexistence of a shear flow in the stretched samples perpendicularto the stretching direction. In the case of the microfluidicexperiments performed on solutions of PMMA colloidal

Figure 9. Constancy of the modulus of the velocity vector, |v| = vfl andits direction φ0.

Table 1. List of the Parameters Obtained from the Fit of theData Shown in Figure 7c by Means of Eq 4

aging (s) τ0 (s) μ vfl (Å/s)

silica filler 691 71 ± 17 1.1 ± 0.3 55.9 ± 0.358 ± 5 μ = 1.5 55.9 ± 0.3

1929 179 ± 13 1.1 ± 0.1 27.37 ± 0.04155 ± 5 μ = 1.5 27.36 ± 0.04

carbon black filler 221 40.0 ± 0.5 2.0 ± 0.2 28.0 ± 0.141 ± 2 μ = 1.5 27.9 ± 0.2

Figure 10. Homodyne correlations (50 frames) obtained for the silicafilled sample (aging = 430 s) at different φ values. Solid lines cor-respond to the fit with eq 5. The black dashed line is the fit with eq 6a.

Figure 11. Variation with q of the relaxation time τ0(max) determinedby means of eq 5 over 100 frames for the carbon black filled sample(filled circles) and 50 frames for the silica one (open triangles). Theinset shows the variation of the exponent μ with q for the silica filledsample.



particles quoted above, the shear flow is generated externally.In the present case, the shear flow of filler particles would beinduced by the tensile strain applied in the perpendiculardirection. It is likely that this effect results from the lateraldeformation of the solid sample. The general form of thehomodyne correlation function taking into account the shearvelocity vsh writes as

51

τ β ττ

α τα τ

α τα τ

= + −μ⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

g aqv

qv

qvqv

( ) exp 2sin( cos( ) )

( cos( ) )

sin( sin( ) )( sin( ) )

s

sh(2)

0

sh s

sh s

2

sh

sh s

2

(6)

where αs is the angle between the velocity vector vsh and q .When αs is equal to 0, which is obtained in the case of Figure10 for φ = 180°, eq 6 simplifies to

τ β ττ

ττ

= + −μ⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥⎡⎣⎢

⎤⎦⎥g a

qvqv

( ) exp 2sin( )

( )sh(2)

0

sh

sh

2

(6a)

Equation 6a has already been used by Fluerasu et al.52 for theanalysis of the homodyne XPCS measurements mentionedabove. Lhermitte51 has shown that, when divided by the fittedshear terms, the experimental correlation functions G(τ) − 1obtained for each αs angle collapse into a single curve.In eqs 6 and 6a, vsh is the shear velocity corresponding to the

difference in the velocity of particles moving in a box of sizeL/2, L = 20 μm being the size of the X-ray beam. vsh is relatedto a shear rate γ by means of γ = 2vsh/L. Equation 5 will be usedfor describing the correlations in the direction of q∥ (φ = 266°).It can be assumed that τ0 is the same in eqs 5 and 6a and

corresponds to the maximum value of the “apparent τ0”determined by fitting with eq 5 the correlations obtained fordifferent angles φ. Because, generally, the variation of theapparent τ0 displays an upper and a lower plateau, the value ofτ0 (τ0(max)) introduced in eq 6a is a mean value. Similarly, thevalue of the shear velocity vsh will be a mean value of resultsobtained from the fit of the correlations for which the apparentτ0 values are the shortest (i.e., αs = 0). The dashed line inFigure 10 corresponds to the fit of the data with eq 6a usingτ0 = 76 s. The value of vsh (about 4 Å/s) reported in Figure 10corresponds to a very small shear rate (4 × 10−5 s−1).In a dilute dispersion of colloidal particles at equilibrium,

constituting an ergodic system, the characteristic time τ0 scalesas q−2. It is related to the constant of diffusion D0 due toBrownian motion of the particles in the surrounding mediumby means of τ0 = 1/(D0q

2). In the case of the strained filledelastomers investigated here, τ0 scales as q

−1 (Figure 11). Theinset in this figure shows that μ fluctuates between 1.5 and 2.The mean value is close to 1.8. For the correlation fitted to eq 6a,shown in Figure 10, the value of μ is slightly smaller (μ = 1.28 ±0.08) but still larger than 1. The compressed-exponential shapeand the q−1 dependence of the characteristic time τ0 arefingerprints of slow relaxations in jammed systems.14,15 Thesefeatures were already reported for filled elastomers.5,19,20 Theq−1 dependence of the characteristic time τ0 rules out diffusivemotions and suggests a mechanism characterized by aninhomogeneous distribution of stresses leading to an inhomoge-neous distribution of velocities characterized by v = (qτ0)

−1.14,55

This velocity, describing local ballistic motions between theparticles, will be named vball in the following. It is clear that vball

is not a uniform velocity as that measured in the heterodynearrangement.

3.5. Comparison between Homodyne and Hetero-dyne Relaxation Times. Figure 12 shows that the relaxation

times deduced from the heterodyne correlations are significantlysmaller than those obtained in homodyne correlations. Becausehomodyne and heterodyne measurements are performedalternately, the aging times are different. τ0 was shown to increasewith aging time.34

This feature is actually observed in Figure 12b for the hetero-dyne correlations measured at two different aging times (229and 716 s). However, the values of τ0 obtained from thehomodyne correlations measured after 432 s aging, at differentangles φ, are systematically larger than the heterodyne ones.It follows that the damping of the oscillations deduced from thefit of the heterodyne correlations with eq 4 is stronger thanexpected from the homodyne correlations. It is then necessaryto examine the origin of this mismatch. Among others, one mayquote the width Δq and Δφ of the q and φ domains in whichthe correlations are measured. In addition, as it will be shownbelow, for the carbon black filled sample, the modulus of thevelocity vector and its direction change with aging. Rewritingeq 4 in order to include such effects as well as shear motionsand to obtain the homodyne−heterodyne damping agreementis currently in progress.

4. AGING BEHAVIOR OF THE DYNAMICS OF FILLERPARTICLES IN STRAINED SAMPLES4.1. Aging Features in Heterodyne Correlations.

Figure 13a indicates a power law decrease of the particlevelocity with aging for both fillers:

∝ −v t mfl a

f (7)

Figure 12. Variation of the time constant τ0 with the angle φ in thecase of the silica (a) and the carbon black (b) filled sample.



The exponents mf, however, are different. For the silica filledsample, mf is close to 1. For carbon black filled sample, it isclose to 2. A t−1 (mf =1) behavior of the particle velocity hasbeen observed for a different series of carbon black filledsamples after release of a 100% tensile strain.5,6 In the presentcase, the strain is maintained constant at 60% elongation, butthe exponent is the same for the silica sample. For the strainedcarbon black filled sample, the decrease of the particle velocityis faster (mf is close to 2) than it was during recovery after aninitial strain. As already underlined, the major difference bet-ween the two samples is the filler−matrix interaction. Carbonblack filled elastomers are known for their strong polymer−filler interaction which can be traced back to the disorder-induced localization of polymer chains on the rough (fractal)and chemical heterogeneous surfaces of the carbon blackparticles.56 It follows that the movement of the carbon blackparticles reflects that of the cross-linked elastomer chains in thecase of a purely elastic deformation or result from breaking offiller−matrix bonds (debonding) in plastic deformations. Thehydroxylated silica particles interact very weakly with the chainsand form an H-bonded network spanning through the cross-linked polymer network. Thus, the cross-linked silica filledsystem could be similar to a colloidal gel resulting from theinterpenetration of the silica aggregates and the cross-linkedpolymer chains networks.The flow velocity evidenced for both samples under strain,

results from the collective transfer of particles in order for thesample to accommodate the new length and if possible, toreach a new equilibrium state. Because the carbon blackparticles are attached to the polymer network, their flow resultsfrom the tensile deformation (elastic and plastic) of the wholesystem. At increasing elongations, the elastomer chains aresignificantly stretched. They become strongly immobilized57

and/or tend to form a “glassy” environment for the carbonblack particles.58 The small velocity attained by the carbonblack particles after less than 600 s aging and its nearly constantvalue after 1000 s aging would agree with the above intuitivedescription. Furthermore, as shown in Figure 13b and alreadymentioned,34 a change in the direction of the velocity vector isobserved. This feature reveals a modification of the direction ofstress in the sample in which the dynamics is becoming slow.For the sample filled with hydroxylated silica, the direction

of the velocity vector remains nearly parallel to that of thestretch (Figure 13b). The velocity of the particles results fromthe deformation of the silica network in a uniaxially strainedviscoelastic medium (the cross-linked polymer). An exponentclose to 1, characteristic of universal pictures for aging, was alsoobserved in strained colloidal gels.55 This observation, whichwill be discussed in more detail in the last paragraph, would agreewith the gel-like character of the silica filled sample suggestedabove.

4.2. Aging Features in Homodyne Correlations.Figure 14 shows the evolution, during aging, of the relaxation

time τ0 measured in homodyne conditions for the silica filledsample.In each series of homodyne measurements, correlations were

determined by means of sequences of 50 frames (delay time =105 s) for the first two series and 100 frames (delay time =210 s) for the last homodyne series. Figure 14 shows that theincrease of τ0 is different within each homodyne domain. Thesefeatures suggest beam damage as the sample is no longer pro-tected by the compacted silica used as the static scatterer inheterodyne measurements. These damages would induce anunexpected increase of τ0, i.e., a slowing down of the silicaparticles mobility. Shinohara et al.20 report radiation damage(structural and dynamic) in filled elastomers during homodyneXPCS measurements by irradiation of X-rays (10.5 keV) over900 s. These authors, therefore, limited the irradiation time atthe same sample position to 600 s.The physicochemical effects of moderate X-rays energy

(7.448 keV) used in the present work on filled elastomers are infact not well documented. Exposure of polymers to soft X-rays(0.315 keV) causes radiation damages in the form of the loss ofmass and changes to the chemical structure of the polymers.59

Recently, Maiti et al.60 observed a hardening of stretched filledrubber after γ irradiation (Co-60 source, 1.4 MeV). Electron-beam-induced damage was reported by Urushihara et al.61

Figure 13. Evolution of the modulus vfl (a) and of the direction φ0 (b)of the velocity vector with aging time ta for the silica and the carbonblack filled samples under strain (60% elongation).

Figure 14. Evolution of the relation time τ0 measured at the angle φ0(τ0(max)) as a function of the aging time for the stretched silica filledsample.



These authors performed field-emission scanning electronmicroscopy (FE-SEM) in order to observe filler displacementsduring tensile deformation of nanosilica filled rubbers. Theymention the hardening of the surface by a long-term irradiation.When the total scan time was reduced, surface hardening wasminimized and disappeared after releasing the applied strain.The increase of τ0 observed here reveals also a hardening

effect during a given series of homodyne measurement. At thebeginning of the next homodyne series, the time constants areunaffected because the sample position was changed. In the caseof the carbon black filled sample, the same test of beam damagecannot be performed as a result of the larger τ0 values and a tooshort duration of the aging measurements. It could be assumedthat damage results from an increase of the polymer cross-linking degree or a change in the polymer−filler interaction asin the case of γ-irradiation of EPDM−silica nanocomposites.62

In such conditions, the silica filled sample would be intuitivelymore sensitive than the carbon black one. At this stage, however,it remains difficult to propose a pertinent explanation to thesefeatures.It will now be assumed that the correlations obtained from

the first 50 or 100 frames in each homodyne series for the silicafilled sample are not affected by beam damages. The variationof τ0 with the angle φ for these series is plotted in Figure 15a.

This figure indicates that the angle corresponding to the minimumand the maximum of τ0 does not change significantly during aging.Figure 14 also shows that the relaxation time τ0(max) increaseswith aging time as t0.45.For the carbon black filled sample, the variation of the

homodyne relaxation time τ0 with the angle φ (Figure 15b) isdifferent from what is observed for the silica filled sample(Figure 15a). The position of the minimum assumed to be the

direction of shear, changes between 235° (ta = 432 s) andnearly 180° (ta = 2467 s), i.e., in the direction perpendicular tothe stretch, as for the silica filled sample. However, theheterodyne measurements performed just before indicate thatthe direction of the flow velocity φ0 (Figure 13b) is close to180°. Because no heterodyne measurements were performedafter this homodyne series, it is difficult to decide whether flowand shear velocities become parallel or whether the direction ofthe flow velocity changed again. Figure 15b also shows that thevariation of τ0 with the angle φ is weak in the case of the dataresulting from the first 300 frames (frames 731 to 1030) of thesecond homodyne series corresponding to an aging time of2047 s. Under such circumstances, the determination of a shearvelocity, as it becomes very small, from the fit of the correlationdata with eq 6 is hardly possible. Also, the maximum value isvery close to that obtained for frames 931 to 1230 (aging =2467 s).The variation of vball (corresponding to the relaxation times

plotted in Figure 14 for the silica sample) with aging time isshown in Figure 16. This figure also shows the variation of vsh.

For both samples, the shear velocity vsh is much larger than theinterparticle velocity vball. For the silica sample, vsh and vballdisplay a power law decrease with aging with exponents msh andmball equal respectively to 0.60 and 0.45. For the carbon blacksample, the values of vball are about 10 times smaller than forsilica. They are characterized by a slower power law decreasewith mball = 0.20. It may be noticed that for vsh the slope(−0.58) of the line drawn between the two points is close tothat obtained for the silica sample (−0.60).

4.3. Analysis of the Aging Exponents. The exponentsmball and msh describing the aging of vball and vsh, respectively(indicated in Figure 16), as well mfl are collected in Table 2.Most papers dealing with slow aging dynamics are devoted

to the behavior of the ballistic movements for which theexponents are generally ranging between 0.4 and 1. However,

Figure 15. Variation of the relaxation time τ0 as a function of the angleφ at different aging time for the silica (a) and the carbon black (b)filled samples. The curves plotted with blue symbols are the onesshown in Figure 12.

Figure 16. Evolution of the interparticles velocity vball and the shearvelocity vsh during aging under strain of the silica and the carbon blacksamples.

Table 2. List of Exponents Characterizing the Power LawDecrease of the Force ([F(t) − F∞] for the Silica Sample),the Particle Velocity vfl, the Interparticle Ballistic Velocity,vball, and the Shear Velocity vsh

exponent force α − 1 flow mfl ballistic mball shear msh

silica 0.17 ± 0.01 1.04 ± 0.02 0.45 ± 0.02 0.60 ± 0.01carbon black log 1.97 ± 0.05 0.20 ± 0.02 (0.58)



a superlinear behavior characterized by an exponent equal to1.8 was reported by XPCS in the case of aging of aqueous sus-pensions of laponite.15 After rejuvenation, the exponent becameclose to 1,16 i.e., a value similar to that reported earlier bydynamic light scattering experiments.11 Sublinear aging wasshown to characterize lamellar gels for which the aging exponentwas equal to 0.77,14 close to the value equal to 0.88 predictedby Kob and Barrat for a Lennard-Jones glass63 or to 4/5 in theBouchaud−Pitard model.64 More recently, Shinohara et al.19,20

obtained an exponent equal to 0.5 in the case of aging (aftermixing) of silica particles in unvulcanized rubber. This value isclose to mball reported in Table 2 for the silica sample but largerthan that obtained for the carbon black sample. At this stage, itis difficult to go deeper into a discussion about the physicalmeaning of the different exponents obtained in our XPCS andHD-XPCS experiments. It may be noticed, however, that thevalues obtained for the silica sample measurements areconsistent with the ones reported in the literature.On the contrary, for the carbon black sample, the exponents

mball and mfl seem to be “out of range”. Moreover, the directionof the flow and the shear velocity vector varies with aging time.As reported in section 3.1 and in Table 2, the mechanical agingis also different from that of the silica sample. The logarithmicdecrease of the force during the application of the tensile strainsuggested local plastic deformations yielding filler−filler orfiller−matrix cracks and, therefore, modifications in thedynamics of the filler particles. Interestingly, Cipelletti et al.14

remark that a ballistic dynamics could also result from over-damped motions of the strength of the stress sources. It followsthat the XPCS results obtained for the carbon black samplecould be interpreted in a different way. Also, possible hetero-geneous dynamic features could be examined.

5. SUMMARY AND CONCLUSIONWe have combined tensile force and XPCS (in homodyne andheterodyne configuration) measurements. The goal was toinvestigate the mechanism of aging of carbon black and silicafilled elastomers under strain. The first system is characterizedby a strong particle−matrix interaction. In the second one,hydroxylated silica interacts very weakly with the matrix butdisplays a hydrogen-bonded network throughout the matrix.The analysis of homodyne correlations allowed us to show,

for the first time, the existence of a shear effect induced bystretching in both samples. For the silica sample, the agingexponents obtained for the shear and the ballistic movementsare close to the ones reported in the literature for colloidal gelsor soft glasses. Additionally, our measurements allowed us toanalyze the effects of beam damage. It follows that, in general,aging exponents deduced from XPCS homodyne measure-ments should be considered with some care. For the carbonblack sample, our homodyne results show that the mobility ofthe filler particles is greatly reduced as compared to that of thesilica ones dispersed in the same cross-linked elastomer. Thisresult attributed to the strong filler−matrix interaction agreeswith the interpretation of the behavior of the tensile forceduring aging.The use of heterodyne XPCS permitting to avoid the

problem of beam damage in filled elastomers yields a series ofnew relevant information about the dynamics of filler particlesin a strained sample. These informations are deduced from thebehavior of the flow velocity of the particles during aging, therelaxation times being difficult to analyze. It is shown that theaging behavior of this velocity is not the same in the strained

silica sample and in the carbon black one. The value of thepower law exponent obtained for the flow velocity suggeststhat the strained silica filled system behaves as colloidal gels orsoft glasses (aqueous suspensions of laponite, for example), inagreement with the results deduced from the homodynecorrelations. In the case of the strained carbon black sample, itis possible that the relaxation of the stress involves microcracksin a polymer which would be below its glass transitiontemperature.The comparison of the aging behavior of the tensile force

measured simultaneously with the XPCS measurements deservesa few comments. For the silica sample the particle velocity islarger than for the carbon black one. Its decrease with aging isslower for the first one than for the second one. For the tensileforce, the opposite is observed since the decrease of the force isfaster in the silica sample than in the carbon black one. Theseobservations indicate that the silica particles are able to moveeasily in a viscoelastic medium which is more viscous than elastic.These movements would be responsible for the fast decrease ofthe stress measured macroscopically by the tensile force. As forthe carbon black sample, the slow decrease of the force could berelated to the rapid decrease of the mobility of the carbon blackparticles moving very slowly in a nearly solid elastic system as aresult of the strong particle−matrix interaction.To conclude, the combination of HD-XPCS and tensile

measurements described in the present article appears as apowerful method for relating the macroscopic mechanicalbehavior to the nanoscale dynamics in filled elastomers.

■ AUTHOR INFORMATIONCorresponding Author*Tel +33 476635880; Fax +33 476635495; e-mail [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSUse of the APS was supported by the DOE, Office of BasisEnergy Sciences, under Contract W-31-109-Eng-38. Theauthors thank Rene Jurk (IFP Dresden) for the preparationof filled elastomers and Philippe Beys (LiPhy-CNRS-UJF) forhis help in adapting the software “coherent” to the laboratorycomputing facilities.

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XPCS Investigation of the Dynamics of Filler Particles in Stretched Filled Elastomers