XPCS Investigation of the Dynamics of Filler Particles in Stretched

Article pubs.acs.org/Macromolecules

XPCS Investigation of the Dynamics of Filler Particles in Stretched Filled Elastomers Françoise Ehrburger-Dolle,*,† Isabelle Morfin,† Françoise Bley,‡ Frédéric Livet,‡ Gert Heinrich,§ Sven Richter,§ Luc Piché,∥ 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’Hères, France § Leibniz-Institut für Polymerforschung Dresden, 010169 Dresden, Germany ∥ Physics Department, McGill University, Montreal, Quebec H3A 2T8, Canada ‡

ABSTRACT: The complexity of the mechanical behavior of filled elastomers can be partly attributed to the fact that the duration of an applied strain plays a crucial role. In order to bring new insights into this still incompletely solved problem, we look for relationships between the macroscopic mechanical relaxation and the relaxation of the filler particles at the nanoto mesoscale. To this end, X-ray photon correlation spectroscopy (XPCS) in homodyne and heterodyne configurations combined with tensile stress relaxation is employed. The paper is devoted to the study of the role of the filler−filler and the filler−matrix interactions in a cross-linked elastomer on the aging mechanisms under strain. The fillers investigated are carbon black, as an example of strong filler−matrix interactions, and hydroxylated silica for which the filler−filler interaction is strong (H-bonds). Homodyne XPCS correlation reveals features of jammed systems (compressed exponential and ballistic motion) for both systems. The exponents characterizing the aging of the homodyne relaxation times are not the same in the carbon black and in the silica filled samples. For both systems, the decrease of the particle velocity determined by heterodyne detection with aging time follows a power law. The silica sample is characterized by a slow decrease of the velocity during aging. For the carbon black sample, the velocity remains small and decreases faster than for the silica sample. The reverse is observed for the behavior of the tensile force. The first reason concerns the possibility to relate mechanical relaxation (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 natural in the physics of disordered out-of-equilibrium systems like gels or glasses.7 Dynamic light scattering (DLS) combined with rheology was used to relate internal dynamics and elasticity in fractal colloidal gels8 local dynamics and nonlinear rheology of soft colloidal paste9 or ultraslow dynamics in aging of a gel composed of multilamellar vesicles10 or in a suspension of clay laponites.11,12 This methodology allowed researchers to find universal features in the fluid-to-solid transitions, the jamming state, or the sol-to-gel transition in colloidal suspensions.13,14 For several years, XPCS has been used to complement DLS when investigating the aging behavior of colloidal gels,15−18 silica particles dispersed in unvulcanized rubber,19,20 or the dynamics of colloidal particles in ice21 or that of magnetic particles dispersed in water and submitted to a magnetic field.22,23 The combination of XPCS and rheological properties is also reported in the case of gel-forming silica suspension in decalin24,25 or in

1. INTRODUCTION Over the past several years, X-ray photon correlation spectroscopy (XPCS), originally named intensity fluctuation spectroscopy (IFS),1 has been shown to be an important technique for investigating slow dynamics in soft and hard condensed matter systems.2−4 It provides a tool complementary to dynamic light scattering for the study of small-scale high-q dynamics. Even for length scales accessible to light scattering, it can be used for opaque samples. A few years ago, we developed a heterodyne technique (HD-XPCS) for XPCS measurements5,6 which yields information about the phase shift in the scattering signal. This allows us to measure the velocity of filler particles during the relaxation after an initial tensile deformation of un-cross- and cross-linked carbon black filled elastomers. It was shown that the velocities were larger for the un-cross-linked sample than for the cross-linked one. In both cases, however, the velocity decreased with the aging time as 1/t. This first series of measurements proved that HD-XPCS was very well suited for investigating the mechanisms of aging of filled elastomers after release of a strain. Therefore, we decided to perform extended measurements in which tensile force and XPCS measurements on strained samples would be combined. Such measurements were expected to be particularly interesting for at least two reasons. © 2012 American Chemical Society

Received: July 3, 2012 Revised: September 5, 2012 Published: October 16, 2012 8691

dx.doi.org/10.1021/ma3013674 | Macromolecules 2012, 45, 8691−8701

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peroxide in a two-roll mill for 5 min at 50 °C. Afterward, the mixtures were cured in molds at 160 °C under pressure during 10 min. The thickness of the samples was about 1 mm. The volume fraction of filler corresponding to 40 phr is close to 0.18 for the carbon black filled sample and to 0.16 for the silica one, i.e., in both cases, above the “percolation” threshold. In our measurements, the 1 mm thick plates are punched out to the classical dumbbell shape (width = 4 mm). 2.2. Force Measurements. An apparatus that permits simultaneous stress and scattering measurements has been developed.34 The setup is sized as to fit in a vacuum can with X-ray aperture windows and mounted on a translational stage on the IMMY/XOR-CAT (8-ID) beamline at APS (Argonne, IL). The sample placed perpendicular to the beam axis was subjected to a rapid vertical extension step which was then held constant. A strain gauge and a digitizing unit were used to measure the resulting evolution of the stress on the sample every 2 s. By using a symmetric stretch, the instrument was designed so that there is a nearly fixed point on the sample for the X-ray beam to pass through. This setup permitted measurements of the same section of material during the entire range of strain conditions applied during the course of a particular experiment. A complete description 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 stress relaxation which is the time-dependent reduction in stress, i.e., the force needed to keep the sample under constant deformation. 2.3. XPCS Measurements. XPCS measurements were performed on the IMMY/XOR-CAT (8-ID) beamline at APS in similar experimental conditions as for the previous series.5,6 The X-ray wavelength λ was equal to 1.663 Å (7.448 keV). The 20 μm × 20 μm beam size was selected by means of carefully polished slits placed 0.64 m before the sample. Guard slits were added, 0.16 m before the sample, in order to limit the scattering from slit diffraction in the SAXS region. The incident beam intensity I0 (∼109 photons s−1) was measured by a diode located in the beamstop. A heterodyne signal was obtained from a powder of fume silica (Aerosil 200), compacted in one of the two holes (diameter and depth 1 mm) drilled in a plate placed immediately upstream the filled elastomer sample. The total thickness of material in the scattering volume was kept close to 2 mm, and interferences from scattering in this volume could be observed. The requirement for this reference sample was that it occupies the same coherence volume as the specimen to be investigated. The second hole was empty in order to permit homodyne measurements. The sample holder was mounted on an x−z stage, perpendicular to the beam. By translating the sample holder, measurements could be made either directly on the fluctuating sample alone or on the hybrid sample (sample + reference), thus allowing for an easy comparison of homodyne and heterodyne results. In the latter case the beam intensity was reduced by a factor of 13 owing to the absorption in the reference. The sample-to-detector distance was 2.8 m. A direct-illumination deep-depletion CCD (PI 1152 × 1242, 22 μm resolution) camera was used as an area detector. Figure 1 shows a typical speckle pattern. The beam center and beamstop were placed in the upper right corner. Reading of each frame takes about 2 s. At each elongation step, the duration of X-ray illumination was 100 ms per frame in the homodyne and the heterodyne setup for the carbon black filled sample. The heterodyne (het) and homodyne (hom) sequences were organized as follows: 100 frames-het, 100 frames-hom, 500 frames-het, 500 frames-hom. The total duration of the measurement, including measuring transmission and dark, was 2805 s. For the silica filled sample, the exposure time was 500 ms in the heterodyne setup and 100 ms in the homodyne one. The sequences (total duration equal to 4126 s) were organized 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 provided by the software “coherent” (Matthew A. Borthwick and Peter Falus, 22 Nov 2003 MIT, 1996)39 available at beamline 8-ID at APS. This software is designed to permit the determination of the static SAXS curve I(q) normalized by the transmission and the incident beam intensity, by means of the application Sqphi. The experimental XPCS correlation functions G(τ) is obtained by means of a multiple tau correlation calculation in the application g2qphi. In order to determine

polymer melts26,27 and in several other systems recently reviewed.3 The common feature of all the above quoted experiments uses signatures in the relaxation, such as ballistic movements and particular exponents in the power law aging, to try and put a given complex out of equilibrium system into a given class of materials (e.g., gels, glasses, jammed systems, etc.).7,28 As a consequence, it becomes possible to learn more about which are the controlling parameters (particle−particle or particle matrix interactions, mechanical history, and so on) and, eventually, to extend the underlying theories. The second reason for carrying out a new series of HDXPCS measurements is the similarity of some of our results on filled elastomers [refs 5, 6 and unpublished results] with those reported on jammed systems. Robertson and Wang29,30 gave experimental evidence suggesting the existence of an analogy between dynamic strain-induced nonlinearity in the modulus of filled rubbers, the physics of the glass transition of glass-forming materials, and the jamming transition of vibrated granular materials. Actually, the mechanical behavior of filled elastomers and its time involvement is an active research area since more than 50 years. Bhattacharya et al. [ref 31 and references therein] reviewed several “classical” models attempting to take into account the complexity of the problem. These authors also reported an extensive study of the nonlinear, time-dependent mechanical behavior of a cross-linked carbon black filled elastomer. They concluded that none of the existing models were able to capture the complexity of these systems, and they underlined the need for the introduction of nonlinear stress relaxation in the models. Combination of transmission electron microscopy, mechanical measurements, and small-angle scattering (neutrons and X-rays) recently highlighted the role of the polymer−filler interfacial interaction32 and that of the filler dispersion33 on the mechanical reinforcement in model nanocomposites. Our goal was to investigate the role of these parameters 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 of the filler volume fraction (below and above the percolation threshold) and the role of filler−filler and filler−matrix interactions on the dynamics and tensile properties of a filled elastomer (un-cross-linked or cross-linked). HD-XPCS measurements were performed on strained samples while the stress relaxation was simultaneously measured. This experimental approach is particularly suitable for systems for which the duration of the application of a strain (the “waiting time”) plays a crucial role. Preliminary homodyne and heterodyne XPCS results34 have shown that the aging behavior under 20% tensile strain of the cross-linked elastomer filled with carbon black was different from the one filled with hydroxylated silica. The present paper is aiming to describe the experimental results obtained for the same cross-linked elastomer filled either with carbon black or with hydroxylated silica stretched at 60% elongation. A particular attention will be paid to the methods used for the analysis of the mechanical and the XPCS (homodyne and heterodyne) data.

2. MATERIALS AND EXPERIMENTAL METHODS 2.1. Samples. The present study involves two model samples consisting of an elastomer (ethylene propylene diene monomer, EPDM, rubber) filled with carbon black (N330) or hydoxylated pyrogenic silica (Aerosil 200, Degussa). The morphological characteristics of both fillers have been previously described in detail.35−37 The filled elastomers (40 parts of rubbers i.e., 40 phr) were prepared by mixing 40 g of filler with 100 g of rubber and 3 g of dicumyl 8692


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with time as follows: |ΔF(t )| ∝ t −α Δt

(1)

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

Figure 1. Typical speckle pattern obtained by the direct illumination CCD camera. Each image is partitioned into 30 angular sectors of 5.65° (angles φ vary between 140° and 306° counterclockwise) and 30 q-domains ranging from 2.7 × 10−3 to 3.2 × 10−2 Å−1. The direction of the strain is vertical and corresponds to φ = 270°. the correlation functions in heterodyne and homodyne configurations, the delay per tau level (dpl) is fixed at 150. Because of the anisotropy of the correlation functions, images are partitioned as shown in Figure 1. Fitting of the experimental curves is achieved by nonlinear regression procedures by means of the Marquardt−Levenberg algorithm (SigmaPlot 10.0).

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

3. ANALYSIS OF THE EXPERIMENTAL DATA 3.1. Force Measurements. The elongation of the samples was increased by steps of 20% from 0 to 60%, as shown in Figure 2. For the 60% step investigated here, the duration was

F(t ) = At (−α + 1) + F∞

(2a)

• for the carbon black filled sample F(t ) = A log(t ) + F1

(2b)

The result obtained for the silica sample (Figure 3) indicates that the force reaches an asymptotic limit F∞ = 2.413 N. On the contrary to the strained silica sample, the force measured for the carbon black sample does not tend to an asymptotic limit. It is slowly decreasing from F1 = 3.19 N (when t = 1 s). This conclusion, however, results from observations during less than 3000 s. For filled elastomers, stress relaxation is generally associated with reorientation of the polymer network under strain, with the rearrangement of the chain entanglements and with the breaking of bonds between chains, between filler particles or between chains and filler particles (debonding). Gent40 reports that the stress relaxation generally decreases with a logarithmic behavior, as observed here in Figure 3, for the carbon black reinforced samples. A logarithmic decrease of the stress has been also reported in the case of tensile deformation of semicrystalline polymers.41 Theoretical models and simulations have shown that logarithmic relaxation at the macroscopic level is often associated with strongly interacting elements at the microscopic scale.42,43 Very recently, Amir et al.44 have observed that, in many cases, the relaxation of very different systems in nature is logarithmic. These authors propose a generic model for slow relaxations 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 0.17 were reported for cross-linked polymer networks mechanically deformed (stretch, shear, or creep) by Curro and Pincus45 and

Figure 2. Mechanical history of the hydroxylated silica and the carbon black filled elastomers. The curves in red correspond to the stress relaxation at 60% elongation investigated in this article. The arrows indicate the origin of the time scale for aging (ta = 0).

4126 s for the silica filled sample and 2805 s for the carbon black one. A detailed analysis of the force (stress) relaxation at a given constant deformation measured for all samples and all elongations reached from below (“up”) or from above (“down”) will be presented elsewhere [manuscript in preparation]. In brief, it appears that, in all cases for which noise was not a factor, the experimental rate of change of the force |ΔF(t)|/Δt decreases 8693


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later by Heinrich and Vilgis.46 This apparent power law behavior is better represented by the empirical Chasset and Thirion equation47,48 into which eq 2a can easily be transformed. The value of the exponents ranging between 0.05 and 0.20 were related to the degree of cross-linking. Ng and McKinley49 have shown that the shear modulus of gluten gels at finite strain decreases as t−n with n = 0.175 ± 0.005. Thin films of an aqueous suspension of laponite previously subjected to a creep flow field undergo a time-dependent enhancement of the elastic modulus G′ ∝ t0.22.50 An exponent equal to 0.4 was reported by Guo et al.24,25 for the growth of the storage modulus of gelforming colloidal suspensions at large waiting times. These examples suggest that the stress relaxation of the strained silica filled sample is typical of a gel-like viscoelastic material. The force tends to an asymptotic limit which may correspond to a new stable arrangement of the H-bonded silica network in the stretched sample. On the contrary, the logarithmic relaxation of the strained carbon black sample does not reveal a limiting value for the force with the same time domain. This system could relax under strain as an elastoplastic solid and/or according to viscoelastic damage models.31 3.2. Static Scattering Measurements. Figure 4 shows the SAXS curves deduced from the speckle patterns by means

a butterfly pattern when stretched at 52%. This anisotropy appeared between 7 × 10−4 and about 3 × 10−3 Å−1, i.e., in a q-domain that was not reached in the present experiment. In the intermediate q-domain, the shape of the SAXS curves is the same as previously described. For the carbon black filled sample, the curve I(q) scales as q−p with p = 3.76. This feature corresponds to the scattering of the surface of the primary particles characterized by a surface fractal dimension DS = 6 − p = 2.24.35 For the silica filled sample, the curve I(q) scales as q−m with m = 2.26. This power law, characterized by an exponent which absolute value is smaller than 3, describes the scattering of fractal silica aggregates with Dm = 2.26.35,37 Figure 4b indicates that aging of the samples under strain does not involve any structural change at mesoscopic length scales (approximately 230 to 21 nm). The SAXS intensity curves obtained for the two samples in homodyne and heterodyne configuration are compared in Figure 5. These curves yield an experimental estimation of the

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

mixing coefficient x(q) = Is(q)/[Is(q) + Ir(q)] where Is(q) is the intensity scattered by the sample (in the homodyne configuration) and Ir(q) the intensity scattered by the compacted Aerosil 200 used as reference in the heterodyne configuration (Figure 6). It may be noticed that, above about 8 × 10−3 Å−1, for the silica (Aerosil 200) filled sample x(q) remains constant. At smaller q values, the scattering of the silica dispersed in the elastomer increases more than in the compacted state because fractal aggregates are less interpenetrated at a large scale than at a shorter one.35,37 3.3. XPCS in Heterodyne Regime. Theoretical backgrounds and experimental conditions of X-ray heterodyning were discussed in details previously.5,6,51 Owing to an expected relative velocity v ⃗ between the filler particles in the strained sample and the immobile silica particles in the reference, the correlation function acquires a phase factor of exp(iq⃗·vτ⃗ ). We can define ω =→ q⎯ ·v ⃗ = qv cos(φ − φ0) (3)

Figure 4. Comparison between the SAXS curves obtained for the silica (open triangles) and the carbon black (closed circles) filled samples at 60% elongation: (a) for different angles φ and (b) for different aging times.

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

where φ is the angle given by q⃗ and φ0 is the angle for the direction v.⃗ In this case the heterodyne correlation function can be written as5 8694


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Figure 6. Variation with q of the mixing coefficient x in the case of the carbon black and the silica filled sample. (2) ghet (τ ) = 1 + A + B[g(1)(τ /τ0)]2

+ C cos(ωτ ) Re[g(1)(τ /τ0)]

(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 our experimental setup. The exponent μ introduced in the damping term results from the assumption of a possible compressed exponential behavior observed in jammed systems.14,15 For the suspension of colloidal latex particles in glycerol investigated earlier,5 μ was equal to 1 and ω equal to zero as expected for a dilute system at equilibrium. The heterodyne correlation function (eq 4) implies that the reference sample does not display any dynamic component. Thus, it was necessary to determine its homodyne correlation over a large period of time, equal at least to the delay time involved in the heterodyne measurements. The experimental homodyne correlations G(τ) obtained for the reference sample for two q-values plotted in Figure 7a indicate that our setup is stable over at least 1000 s. Figure 7b shows the correlation curves obtained for the silica filled sample at φ = 208° for different q-values. These curves are deduced from 50 frames, i.e., 122 s, among 500 frames obtained in the third series of heterodyne measurements. The value of aging indicated in these figures corresponds to the mean value of the aging time for the first and the last frame considered for the correlations. Each block of φ, q was fit to eq 4. Figures 8 and 9 show that eq 3 is valid, and a single |v|⃗ and φ0 explains all blocks. The inset in Figure 8 shows that, in the range of q investigated, the value of the exponent μ fluctuates between 0.5 and 1.5 with quite a large statistical error primarily because the delay window is not broad enough to precisely determine the damping of the oscillation. The two-times correlation matrices in ref 51 show that the velocity varies with aging under strain. Thus, the delay window must be narrow in order to avoid spurious effects in the correlations. On the other hand, it is possible to determine quite precisely the velocity within a delay window that needs not to be larger than half a period. This possibility will be used to determine the variation of the velocity as a function of the aging time under strain. In such conditions, information about the damping of the oscillations (characterized by μ and τ0) is lost. Nevertheless, Figure 7c clearly shows that, even in a narrow time window, the amplitude of the oscillations decreases much faster in carbon black filled sample than in the silica one. This feature appears as well for

Figure 7. (a) Homodyne correlations obtained for the reference sample. (b) Examples of heterodyne correlations obtained for the silica filled sample at different q values at an angle φ = 208° and (c) for the carbon 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 4 in 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 the correlation curves partly shown in Figure 7b. The inset shows the fluctuations of the exponent μ with q.

similar values or different values of ω (in the case of the silica filled sample). At large q values, the damping of the oscillation amplitude becomes strong enough to eventually make difficult a fit with eq 4 over the whole delay window, as already reported.34 8695


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Figure 9. Constancy of the modulus of the velocity vector, |v⃗| = vfl and its direction φ0.

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

The above comments based on the aspect of the curves plotted in Figure 7b are confirmed by the values of the parameters τ0 and μ deduced from the fit with eq 4 given in Table 1. The strong Table 1. List of the Parameters Obtained from the Fit of the Data Shown in Figure 7c by Means of Eq 4 aging (s) silica filler

691 1929

carbon black filler

221

τ0 (s)

μ

± ± ± ± ± ±

1.1 ± 0.3 μ = 1.5 1.1 ± 0.1 μ = 1.5 2.0 ± 0.2 μ = 1.5

71 58 179 155 40.0 41

17 5 13 5 0.5 2

vfl (Å/s) 55.9 55.9 27.37 27.36 28.0 27.9

± ± ± ± ± ±

0.3 0.3 0.04 0.04 0.1 0.2

damping of the oscillations in the strained carbon black filled sample is characterized by a shorter relaxation time than for the hydroxylated silica one, and the exponent μ remains close to 2 for the former. Now, the influence of the value of μ on the quality of 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 the fitting equation but fixed to 1.5 in both cases (dashed lines in Figure 7b). As shown in Table 1, the only significant effect is the expected lowering of τ0 for the silica filled sample. The value of the particle velocity vfl remains the same in all cases. It follows that the obvious difference in the damping of oscillations having nearly the same frequency results from the difference in τ0 which is much smaller for the strained carbon black sample than for the silica one. 3.4. XPCS in Homodyne Regime. The homodyne correlation function is written as follows: ⎡ ⎛ τ ⎞ μ⎤ (2) ghom (τ ) = a + β exp⎢ −2⎜ ⎟ ⎥ ⎢⎣ ⎝ τ0 ⎠ ⎥⎦

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

Anisotropic dynamics has been reported, for example, in ferroglasses under an applied magnetic field.23 In this case, the magnetic field induces a structural anisotropy evidenced by SAXS in the same q-domain. In the q-domain investigated here, the SAXS curves show no structural anisotropy induced by elongation of the sample. It follows that, from this point of view, our homodyne correlations are supposed to be isotropic. Cipelletti et al.14 have observed in concentrated emulsions relaxation times that are larger for q perpendicular to centrifugation acceleration than for q parallel. This anisotropy was attributed to the internal stress as the sample is loaded and centrifuged. Anisotropic correlations have also been reported in XPCS experiments for probing the diffusive dynamics of PMMA colloidal particles in a shear flow.52−54 These authors have observed that the correlation curves indicate a faster decorrelation (i.e., a smaller apparent τ0) in the direction of the shear flow (q∥) than in the direction perpendicular (q⊥). In our case (Figure 10) it appears that decorrelation is faster for φ = 180°, i.e., in the direction perpendicular to the stretch, than in the direction nearly parallel to the stretch (φ = 266°). The comparison between the two homodyne situations suggests the existence of a shear flow in the stretched samples perpendicular to the stretching direction. In the case of the microfluidic experiments performed on solutions of PMMA colloidal

(5)

where a, supposed to be equal to 1, will be considered as a free parameter in the nonlinear regression. It appears that, in most cases, 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 heterodyne correlation function. It was also shown that τ0 was isotropic and scaling as q−2 as expected for diffusion. In the present case, the homodyne correlations change with the angle φ (Figure 10) and the characteristic time τ0 scales as q−1 (Figure 11). 8696


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particles quoted above, the shear flow is generated externally. In the present case, the shear flow of filler particles would be induced by the tensile strain applied in the perpendicular direction. It is likely that this effect results from the lateral deformation of the solid sample. The general form of the homodyne correlation function taking into account the shear velocity vsh writes as51

is not a uniform velocity as that measured in the heterodyne arrangement. 3.5. Comparison between Homodyne and Heterodyne Relaxation Times. Figure 12 shows that the relaxation

⎡ ⎛ τ ⎞ μ⎤⎡ sin(qv cos(α )τ ) ⎤2 sh s (2) gsh (τ ) = a + β exp⎢ −2⎜ ⎟ ⎥⎢ ⎥ ⎢⎣ ⎝ τ0 ⎠ ⎥⎦⎣ (qvsh cos(αs)τ ) ⎦ ⎡ sin(qvsh sin(αs)τ ) ⎤2 ⎢ ⎥ ⎣ (qvsh sin(αs)τ ) ⎦

(6)

where αs is the angle between the velocity vector vs⃗ h and q⃗. When αs is equal to 0, which is obtained in the case of Figure 10 for φ = 180°, eq 6 simplifies to (2) gsh (τ )

⎡ ⎛ τ ⎞ μ⎤⎡ sin(qv τ ) ⎤2 sh = a + β exp⎢ −2⎜ ⎟ ⎥⎢ ⎥ ⎢⎣ ⎝ τ0 ⎠ ⎥⎦⎣ (qvshτ ) ⎦

(6a)

Equation 6a has already been used by Fluerasu et al.52 for the analysis of the homodyne XPCS measurements mentioned above. Lhermitte51 has shown that, when divided by the fitted shear terms, the experimental correlation functions G(τ) − 1 obtained 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 size L/2, L = 20 μm being the size of the X-ray beam. vsh is related to a shear rate γ by means of γ = 2vsh/L. Equation 5 will be used for 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 for different angles φ. Because, generally, the variation of the apparent τ0 displays an upper and a lower plateau, the value of τ0 (τ0(max)) introduced in eq 6a is a mean value. Similarly, the value of the shear velocity vsh will be a mean value of results obtained from the fit of the correlations for which the apparent τ0 values are the shortest (i.e., αs = 0). The dashed line in Figure 10 corresponds to the fit of the data with eq 6a using τ0 = 76 s. The value of vsh (about 4 Å/s) reported in Figure 10 corresponds 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 scales as q−2. It is related to the constant of diffusion D0 due to Brownian motion of the particles in the surrounding medium by means of τ0 = 1/(D0q2). In the case of the strained filled elastomers investigated here, τ0 scales as q−1 (Figure 11). The inset 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 shape and the q−1 dependence of the characteristic time τ0 are fingerprints of slow relaxations in jammed systems.14,15 These features were already reported for filled elastomers.5,19,20 The q−1 dependence of the characteristic time τ0 rules out diffusive motions and suggests a mechanism characterized by an inhomogeneous distribution of stresses leading to an inhomogeneous distribution of velocities characterized by v = (qτ0)−1.14,55 This velocity, describing local ballistic motions between the particles, will be named vball in the following. It is clear that vball

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

times deduced from the heterodyne correlations are significantly smaller than those obtained in homodyne correlations. Because homodyne and heterodyne measurements are performed alternately, the aging times are different. τ0 was shown to increase with aging time.34 This feature is actually observed in Figure 12b for the heterodyne correlations measured at two different aging times (229 and 716 s). However, the values of τ0 obtained from the homodyne correlations measured after 432 s aging, at different angles φ, are systematically larger than the heterodyne ones. It follows that the damping of the oscillations deduced from the fit of the heterodyne correlations with eq 4 is stronger than expected from the homodyne correlations. It is then necessary to examine the origin of this mismatch. Among others, one may quote the width Δq and Δφ of the q and φ domains in which the correlations are measured. In addition, as it will be shown below, for the carbon black filled sample, the modulus of the velocity vector and its direction change with aging. Rewriting eq 4 in order to include such effects as well as shear motions and to obtain the homodyne−heterodyne damping agreement is currently in progress.

4. AGING BEHAVIOR OF THE DYNAMICS OF FILLER PARTICLES IN STRAINED SAMPLES 4.1. Aging Features in Heterodyne Correlations. Figure 13a indicates a power law decrease of the particle velocity with aging for both fillers: vfl ∝ ta−mf 8697

(7) dx.doi.org/10.1021/ma3013674 | Macromolecules 2012, 45, 8691−8701

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and/or tend to form a “glassy” environment for the carbon black particles.58 The small velocity attained by the carbon black particles after less than 600 s aging and its nearly constant value after 1000 s aging would agree with the above intuitive description. Furthermore, as shown in Figure 13b and already mentioned,34 a change in the direction of the velocity vector is observed. This feature reveals a modification of the direction of stress 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 the stretch (Figure 13b). The velocity of the particles results from the deformation of the silica network in a uniaxially strained viscoelastic medium (the cross-linked polymer). An exponent close to 1, characteristic of universal pictures for aging, was also observed in strained colloidal gels.55 This observation, which will be discussed in more detail in the last paragraph, would agree with the gel-like character of the silica filled sample suggested above. 4.2. Aging Features in Homodyne Correlations. Figure 14 shows the evolution, during aging, of the relaxation

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 carbon black filled samples under strain (60% elongation).

The exponents mf, however, are different. For the silica filled sample, mf is close to 1. For carbon black filled sample, it is close to 2. A t−1 (mf =1) behavior of the particle velocity has been observed for a different series of carbon black filled samples after release of a 100% tensile strain.5,6 In the present case, the strain is maintained constant at 60% elongation, but the exponent is the same for the silica sample. For the strained carbon black filled sample, the decrease of the particle velocity is faster (mf is close to 2) than it was during recovery after an initial strain. As already underlined, the major difference between the two samples is the filler−matrix interaction. Carbon black filled elastomers are known for their strong polymer− filler interaction which can be traced back to the disorderinduced localization of polymer chains on the rough (fractal) and chemical heterogeneous surfaces of the carbon black particles.56 It follows that the movement of the carbon black particles reflects that of the cross-linked elastomer chains in the case of a purely elastic deformation or result from breaking of filler−matrix bonds (debonding) in plastic deformations. The hydroxylated silica particles interact very weakly with the chains and form an H-bonded network spanning through the crosslinked polymer network. Thus, the cross-linked silica filled system could be similar to a colloidal gel resulting from the interpenetration of the silica aggregates and the cross-linked polymer chains networks. The flow velocity evidenced for both samples under strain, results from the collective transfer of particles in order for the sample to accommodate the new length and if possible, to reach a new equilibrium state. Because the carbon black particles are attached to the polymer network, their flow results from the tensile deformation (elastic and plastic) of the whole system. At increasing elongations, the elastomer chains are significantly stretched. They become strongly immobilized57

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 filled sample.

time τ0 measured in homodyne conditions for the silica filled sample. 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 the increase of τ0 is different within each homodyne domain. These features suggest beam damage as the sample is no longer protected by the compacted silica used as the static scatterer in heterodyne measurements. These damages would induce an unexpected increase of τ0, i.e., a slowing down of the silica particles mobility. Shinohara et al.20 report radiation damage (structural and dynamic) in filled elastomers during homodyne XPCS measurements by irradiation of X-rays (10.5 keV) over 900 s. These authors, therefore, limited the irradiation time at the 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 in fact not well documented. Exposure of polymers to soft X-rays (0.315 keV) causes radiation damages in the form of the loss of mass and changes to the chemical structure of the polymers.59 Recently, Maiti et al.60 observed a hardening of stretched filled rubber after γ irradiation (Co-60 source, 1.4 MeV). Electronbeam-induced damage was reported by Urushihara et al.61 8698


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These authors performed field-emission scanning electron microscopy (FE-SEM) in order to observe filler displacements during tensile deformation of nanosilica filled rubbers. They mention the hardening of the surface by a long-term irradiation. When the total scan time was reduced, surface hardening was minimized 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 the beginning of the next homodyne series, the time constants are unaffected because the sample position was changed. In the case of the carbon black filled sample, the same test of beam damage cannot be performed as a result of the larger τ0 values and a too short duration of the aging measurements. It could be assumed that damage results from an increase of the polymer crosslinking degree or a change in the polymer−filler interaction as in the case of γ-irradiation of EPDM−silica nanocomposites.62 In such conditions, the silica filled sample would be intuitively more sensitive than the carbon black one. At this stage, however, it remains difficult to propose a pertinent explanation to these features. It will now be assumed that the correlations obtained from the first 50 or 100 frames in each homodyne series for the silica filled sample are not affected by beam damages. The variation of τ0 with the angle φ for these series is plotted in Figure 15a.

direction of shear, changes between 235° (ta = 432 s) and nearly 180° (ta = 2467 s), i.e., in the direction perpendicular to the stretch, as for the silica filled sample. However, the heterodyne measurements performed just before indicate that the direction of the flow velocity φ0 (Figure 13b) is close to 180°. Because no heterodyne measurements were performed after this homodyne series, it is difficult to decide whether flow and shear velocities become parallel or whether the direction of the flow velocity changed again. Figure 15b also shows that the variation of τ0 with the angle φ is weak in the case of the data resulting from the first 300 frames (frames 731 to 1030) of the second homodyne series corresponding to an aging time of 2047 s. Under such circumstances, the determination of a shear velocity, as it becomes very small, from the fit of the correlation data with eq 6 is hardly possible. Also, the maximum value is very 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 is shown in Figure 16. This figure also shows the variation of vsh.

Figure 16. Evolution of the interparticles velocity vball and the shear velocity vsh during aging under strain of the silica and the carbon black samples.

For both samples, the shear velocity vsh is much larger than the interparticle velocity vball. For the silica sample, vsh and vball display a power law decrease with aging with exponents msh and mball equal respectively to 0.60 and 0.45. For the carbon black sample, the values of vball are about 10 times smaller than for silica. They are characterized by a slower power law decrease with mball = 0.20. It may be noticed that for vsh the slope (−0.58) of the line drawn between the two points is close to that obtained for the silica sample (−0.60). 4.3. Analysis of the Aging Exponents. The exponents mball 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 the exponents are generally ranging between 0.4 and 1. However,

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

This figure indicates that the angle corresponding to the minimum and the maximum of τ0 does not change significantly during aging. Figure 14 also shows that the relaxation time τ0(max) increases with 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) is different from what is observed for the silica filled sample (Figure 15a). The position of the minimum assumed to be the

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

8699

exponent

force α − 1

flow mfl

ballistic mball

shear msh

silica carbon black

0.17 ± 0.01 log

1.04 ± 0.02 1.97 ± 0.05

0.45 ± 0.02 0.20 ± 0.02

0.60 ± 0.01 (0.58)


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silica sample and in the carbon black one. The value of the power law exponent obtained for the flow velocity suggests that the strained silica filled system behaves as colloidal gels or soft glasses (aqueous suspensions of laponite, for example), in agreement with the results deduced from the homodyne correlations. In the case of the strained carbon black sample, it is possible that the relaxation of the stress involves microcracks in a polymer which would be below its glass transition temperature. The comparison of the aging behavior of the tensile force measured simultaneously with the XPCS measurements deserves a few comments. For the silica sample the particle velocity is larger than for the carbon black one. Its decrease with aging is slower for the first one than for the second one. For the tensile force, the opposite is observed since the decrease of the force is faster in the silica sample than in the carbon black one. These observations indicate that the silica particles are able to move easily in a viscoelastic medium which is more viscous than elastic. These movements would be responsible for the fast decrease of the stress measured macroscopically by the tensile force. As for the carbon black sample, the slow decrease of the force could be related to the rapid decrease of the mobility of the carbon black particles moving very slowly in a nearly solid elastic system as a result of the strong particle−matrix interaction. To conclude, the combination of HD-XPCS and tensile measurements described in the present article appears as a powerful method for relating the macroscopic mechanical behavior to the nanoscale dynamics in filled elastomers.

a superlinear behavior characterized by an exponent equal to 1.8 was reported by XPCS in the case of aging of aqueous suspensions of laponite.15 After rejuvenation, the exponent became close to 1,16 i.e., a value similar to that reported earlier by dynamic light scattering experiments.11 Sublinear aging was shown to characterize lamellar gels for which the aging exponent was equal to 0.77,14 close to the value equal to 0.88 predicted by Kob and Barrat for a Lennard-Jones glass63 or to 4/5 in the Bouchaud−Pitard model.64 More recently, Shinohara et al.19,20 obtained an exponent equal to 0.5 in the case of aging (after mixing) of silica particles in unvulcanized rubber. This value is close to mball reported in Table 2 for the silica sample but larger than that obtained for the carbon black sample. At this stage, it is difficult to go deeper into a discussion about the physical meaning of the different exponents obtained in our XPCS and HD-XPCS experiments. It may be noticed, however, that the values obtained for the silica sample measurements are consistent 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 direction of the flow and the shear velocity vector varies with aging time. As reported in section 3.1 and in Table 2, the mechanical aging is also different from that of the silica sample. The logarithmic decrease of the force during the application of the tensile strain suggested local plastic deformations yielding filler−filler or filler−matrix cracks and, therefore, modifications in the dynamics of the filler particles. Interestingly, Cipelletti et al.14 remark that a ballistic dynamics could also result from overdamped motions of the strength of the stress sources. It follows that the XPCS results obtained for the carbon black sample could be interpreted in a different way. Also, possible heterogeneous dynamic features could be examined.

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AUTHOR INFORMATION

Corresponding Author

*Tel +33 476635880; Fax +33 476635495; e-mail francoise. [email protected].

5. SUMMARY AND CONCLUSION We have combined tensile force and XPCS (in homodyne and heterodyne configuration) measurements. The goal was to investigate the mechanism of aging of carbon black and silica filled elastomers under strain. The first system is characterized by a strong particle−matrix interaction. In the second one, hydroxylated silica interacts very weakly with the matrix but displays 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 by stretching in both samples. For the silica sample, the aging exponents obtained for the shear and the ballistic movements are close to the ones reported in the literature for colloidal gels or soft glasses. Additionally, our measurements allowed us to analyze the effects of beam damage. It follows that, in general, aging exponents deduced from XPCS homodyne measurements should be considered with some care. For the carbon black sample, our homodyne results show that the mobility of the filler particles is greatly reduced as compared to that of the silica ones dispersed in the same cross-linked elastomer. This result attributed to the strong filler−matrix interaction agrees with the interpretation of the behavior of the tensile force during aging. The use of heterodyne XPCS permitting to avoid the problem of beam damage in filled elastomers yields a series of new relevant information about the dynamics of filler particles in a strained sample. These informations are deduced from the behavior of the flow velocity of the particles during aging, the relaxation times being difficult to analyze. It is shown that the aging behavior of this velocity is not the same in the strained

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Use of the APS was supported by the DOE, Office of Basis Energy Sciences, under Contract W-31-109-Eng-38. The authors thank René Jurk (IFP Dresden) for the preparation of filled elastomers and Philippe Beys (LiPhy-CNRS-UJF) for his help in adapting the software “coherent” to the laboratory computing facilities.

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REFERENCES

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XPCS Investigation of the Dynamics of Filler Particles in Stretched

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