Multiscale Approach to Dynamic-Mechanical Analysis of Unfilled

Article pubs.acs.org/Macromolecules

Multiscale Approach to Dynamic-Mechanical Analysis of Unfilled Rubbers Marina Saphiannikova,*,† Vladimir Toshchevikov,† Igor Gazuz,† Frank Petry,‡ Stephan Westermann,‡ and Gert Heinrich† †

Leibniz-Institut für Polymerforschung Dresden e.V., Hohe Strasse 6, D-01069 Dresden, Germany Goodyear Innovation Center Luxembourg, Avenue Gordon Smith, L-7750 Colmar-Berg, Luxembourg

‡

ABSTRACT: A versatile multiscale theoretical approach for the viscoelasticity of the homogeneous rubber matrix has been established taking into account relaxation processes at different time and length scales as well as nonpolymeric relaxation processes at extremely high frequencies. It allows to fit and describe the dynamic moduli of unfilled S-SBR rubbers over 16 frequency decades with a limited set of parameters (relaxation times, scaling exponents) which have a clear physical meaning and obey the relations motivated by the statistical− physical theory of polymer melts and polymer networks.

1. INTRODUCTION The understanding of the relationship between the dynamicmechanical properties of polymers and their molecular structure has considerably improved in the past decades with the development of molecular dynamics theories. Besides the academic interest of molecular theories relating molecular dynamics to the macroscopic viscoelastic behavior, the accurate description of the structure/property relationships is of the highest importance for the polymer and rubber industry, as the processing as well as final properties of polymer products are directly governed by their dynamic-mechanical behavior. Nowadays, the linear dynamic-mechanical properties of entangled polymer melts can be represented reasonably well using a multiscale approach which combines different relaxation regimes: glassy regime at very high frequencies, followed by Rouse regime at the intermediate frequencies and reptation regime at low frequencies.1−3 It is even possible to predict the linear viscoelastic shear modulus of monodisperse entangled polymer melt using some molecular estimates from the molecular dynamics simulations of its unentangled analogue.3 The frequency-dependent complex shear modulus G*(ω) = G′(ω) + iG″(ω) is usually calculated as the Fourier transform of the time-dependent shear relaxation modulus G(t) which in the frame of multiscale approach can be written as the superposition of shear moduli for different relaxation regimes. So, in the frame of the original reptation model of Doi and Edwards1 the total shear relaxation modulus for an entangled polymer melt contains contributions from the glassy, Rouselike, and reptation relaxation processes:3 G(t ) = Gglass(t ) + G Rouse(t ) + Grept(t ) © XXXX American Chemical Society

The shear relaxation modulus in the Rouse model exhibits a well-known power-law asymptotic behavior with the exponent −1/2 between the shortest τ0 and the longest τR Rouse modes:1,4 G Rouse(t ) ∝ (t /τ0)−1/2

for τ0 < t < τR

(2)

The power-law asymptotic behavior is an intrinsic feature of the polymer dynamics and statistics as has been shown by de Gennes in his book “Scaling Concepts in Polymer Physics”.5 For example, the power-law behavior in the Rouse model arises due to a power-law distribution of the relaxation times for sufficiently long chains (see Appendix A).1 Further, it has been established that the lower frequency part of transition zone nicely obeys the Rouse-like power law with G′(ω) ∝ ω1/2, while the higher frequency part of the transition zone exhibits a stronger power-law behavior with G′(ω) ∝ ωγ with γ ≈ 3/4; e.g., see section 8.7.2 of ref 4. The exponent γ ≈ 3/4 indicates a semiflexible chain behavior on the lengths shorter than the Rouse segment. However, a number of authors6,7 point out that one should be cautious when interpreting results from this higher frequency zone as we will discuss in the Experimental Section. In any case, many simple scaling laws and thus multiscale approaches emerge naturally from the statistical theories of polymer melts. Over decades they showed themselves as a powerful tool when molecular parameters are intended to be Received: June 4, 2014 Revised: June 17, 2014

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dx.doi.org/10.1021/ma501159u | Macromolecules XXXX, XXX, XXX−XXX

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matrix followed by a numerical procedure of finding its eigenvalues which are directly related to the network relaxation times.19,20 It was also proposed that the power law features can be a signature of a broad distribution of relaxation times and multiscale dynamical processes.21 So, for moderately crosslinked polymer networks a stretched exponential law was derived, with an exponent of 1/3, which results in G′(ω) ∝ ω2 and G″(ω) ∝ ω independently on the cross-link density.22 In the present study we develop for the first time a theoretical multiscale approach which allows to model the dynamic moduli of randomly cross-linked polymer networks across a broad frequency domain of up to 16 decades. This approach simultaneously takes into account contributions from the glassy, Rouse-like, and long-time relaxation regimes. Comparing the form of the long-time relaxation spectrum with different theoretical approaches, we conclude that this spectrum is caused by slow reptation dynamics of dangling chains. The proposed approach allows us to interpret experimental data based on structural characteristics of polymer materials as the length of the Kuhn segment, the numbers of Kuhn segments in an entangled fragment, in the active network strands between junctions, and in dangling chains, and the volume fractions of active chains and dangling material.

extracted from the experimental data on dynamic-mechanical behavior of unentangled and entangled polymer melts. On the other hand, to the best of our knowledge, a physically motivated multiscale approach similar to that used for the polymer melts has not yet been applied to the crosslinked networks. It is expected that cross-linking of polymer chains in the entangled melt should not affect the high- and intermediate-frequency regimes, i.e., the glassy and Rouse relaxation spectra. The network chains between cross-links however cannot reptate; they can only perform slight sliding motions along the reptation tube. The corresponding stress relaxation modulus in the case of entangled monodisperse chains between the network junctions has been calculated by Edwards et al.8 using a rigorous statistical-mechanical approach. The authors predicted a power-law frequency behavior G′(ω) ∝ G″(ω) ∝ ωa with an exponent a = 1/2 at intermediate frequencies. Note that at low frequencies G′ for the polymer networks tends to the plateau modulus Geq. Therefore, here and below the symbol ∝ in asymptotic relations for the storage modulus of polymer networks denotes behavior of the purely dynamic part of the storage modulus, G′ − Geq. At even smaller frequencies, near the low-frequency plateau Geq, Gotlib and coworkers predicted the appearance of collective interchain relaxation modes of the regular polymer network which result in the power-law frequency behavior G′(ω) ∝ ωa with an exponent a = 3/2.9,10 Such low-frequency scaling behavior of the storage modulus around the low-frequency plateau corresponds to a collective motion of chains incorporated into a single network structure, which does not contain dangling chains. This scaling behavior was observed for poly[oxy(methylsilylene)] networks which can demonstrate both isotropic and nematic states.11,12 In randomly cross-linked networks usually extremely longtime relaxation processes are observed,13 the structural origin of which is still a matter of discussion. To explain such a long-time relaxation spectrum in randomly cross-linked networks, several approaches have been proposed. Calculations of Curro and Pincus in the framework of the dangling chain picture predict an appearance of extremely long terminal time τmax depending exponentially on the average length of dangling chains.14 On the other side, Heinrich and Vilgis proposed that the topological constraint release of network strands can also result in an extremely large terminal relaxation time.15 These authors assumed that a characteristic time of the constraint release obeys a very similar exponential law as was used by Curro and Pincus.14 The law was originally proposed for a melt of branched star molecules16 and assumes the dangling chain picture. With both approaches, the empirical Chasset−Thirion law13 can be explained which shows that the stress relaxation modulus obtained for cross-linked unfilled elastomers in the regime of finite deformations can be well fitted by the power law −m

G(t ) = G∞[1 + (t /τmax ) ]

2. EXPERIMENTAL SECTION 2.1. Materials. A solution-polymerized styrene butadiene rubber (S-SBR) was provided by Lanxess (Leverkusen, Germany). It has a styrene content of 25% and a vinyl content of 50% (Buna VSL50250); the density ρ = 0.93 g/cm3. An exact composition of the S-SBR copolymer is given in Table 1.

Table 1. S-SBR Copolymer Composition styrene 25%

cis-PB 8%

trans-PB 17%

vinyl-PB 50%

The weight-average molar mass and other polymer parameters of the unvulcanized rubber were measured by size exclusion chromatography using an Agilent 1200 instrument equipped with a Wyatt miniDAWN TREOS and an Optilab rEX detector. Two Agilent PLgel Mixed-C 5 μm columns were used for the chromatographic separation which was conducted at ambient temperature. The resulting weightand number-averaged molar masses are MW = 2.69 × 105 g/mol and MN = 1.84 × 105 g/mol, respectively, with a polydispersity index PI = 1.462. Three cross-linked samples were produced from the unvulcanized S-SBR using different amount of sulfur. The curing package consisted of 1.0, 1.5, and 2.75 phr sulfur and 1.2, 2.3, and 3.3 phr CBS (N-cyclohexyl-2-benzothiazolesulfenamide), respectively. Table 2 shows the corresponding cross-link density of the samples, determined via fitting the stress−strain dependences with the extended tube model of rubber elasticity23 (see section 3).

Table 2. The Samples

(3)

Exponents m varying between 0.066 and 0.382 depending on the molecular weight and the degree of cross-linking were reported.17 The approaches14,15 predicted a linear dependence of the exponent m with the cross-link density. In addition to these approaches, a nontrivial long-time relaxation spectrum in polymer networks can be caused by the heterogeneous network topology and random connectivity of networks strands.18 Such a kind of relaxation spectrum can be investigated by constructing a generalized Rouse connectivity

sample

sulfur content [phr]

cross-link density [nm−3]

Tg [°C]

SBR950 SBR953 SBR956

1.0 1.5 2.75

0.027 0.037 0.059

−16.3 −14.3 −11.0

2.2. Stress−Strain Measurements. The stress−strain dependences for the S-SBR samples have been measured in the tensile mode using the materials testing machine (Zwick 1456, Z010, Ulm, Germany) at the testing velocity of 50 mm/min and at room temperature. The measurements were done according to DIN53504/ S2/50 with the optical recording of sample elongation. The samples were 25 mm long and had the cross-section of about 2 mm × 4 mm. B


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Figure 1. Master curves and typical power-type frequency behavior of the shear storage modulus G′ (left) and loss modulus G″ (right) at the reference temperature Tref = 25 °C for the samples SBR950, SBR953, and SBR956. The degree of cross-linking is increasing from the top to the bottom. procedure is applied:24,25 a large number of experimental curves are measured with a small temperature step, usually 5 °C. This provides a good overlapping of experimental curves and is equivalent to the first procedure in the case of thermorheologically simple materials when the time−temperature superpositioning is valid. The use of the master curves is based on the time−temperature superposition principle, which states that with decreasing temperature the shape of the relaxation curves remains the same but the characteristic relaxation time increases (in the vicinity of the rubber−glass transition region, the relaxation times diverge). Such a scenario is based on the picture theoretically supported e.g. by the mode-coupling theory26 that the glass transition is a purely kinetic phenomenon and is not associated with structural changes (e.g., phase transitions) in the material. On the other side, there are a number of indications that the TTS is not fulfilled in the entire glass transition zone. This was first discovered by Plazek6,27,28 and later investigated in detail in the viscoelastic studies of Roland and co-workers7,29 as well as in the optomechanical studies of Inoue et al.30,31 The data obtained for thermoplastic polymers indicate slightly different temperature dependences for the large-scale chain modes (reptation) and high-frequency local segmental motion. Being aware of this complication, we have nevertheless constructed master curves to get a better overview and to prepare the data for the following analysis in the frame of multiscale approach. Presently, to our knowledge, there exists no other alternative to extract multiple molecular parameters without constructing the master curves. Besides, as we will show below, the mastering appears to be quite accurate as all extracted parameters have reasonable values.

For every sample three measurements were done. Although the elongation at break differs from measurement to measurement for the same sample, the stress−strain dependences are found to be very close up to 400% of elongation. 2.3. Linear Dynamic-Mechanical Measurements. Linear dynamic-mechanical measurements were performed in the tensile mode with an Eplexor 2000N dynamic measurement system (Gabo Qualimeter, Ahlden, Germany). For the measurement of the complex tensile modulus E* as a function of frequency f, a static load of 1% prestrain was applied and then the samples were oscillated to a dynamic load of 0.2% strain. The complex shear modulus was determined as G*( f) = E*( f)/3 assuming the sample isotropy and incompressibility. The complex modulus was measured in a combined temperature−frequency sweep at temperatures from −60 to 120 °C and frequencies from f = 0.5 to f = 50 Hz. Using the time−temperature superposition (TTS) principle, the master curves were produced at the reference temperature of 25 °C from the data at temperatures between (Tg − 10 °C) and 120 °C, where Tg is the glass transition temperature, which depends on the sulfur content as measured by the DSC method (see Table 2). Here we note that to create master curves for un-crosslinked melts one usually uses a few experimental curves measured with temperature steps of about 10−15 °C and covering up to 4 decades of frequency. However, it is not possible to measure the dynamicmechanical response of cross-linked rubbers at frequencies above 50 Hz due to the extremely high stiffness of the sample. The artifact-free experimental window by the rubber testing devices is usually limited to 2 decades of frequency. To compensate for this limitation, a different C


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To construct the master curves, both the horizontal shift factor, aT, and the vertical shift factor, bT, were applied in the double logarithmic scale to the frequency dependences of the storage and loss moduli at varying temperature. The vertical shift for dynamic moduli of polymers is usually related with the prefactor ρkBT, where ρ is the mass density of the polymer.1,4 However, at T > Tg one can neglect the change of the mass density as compared to the factor kBT.32 At the same time, at 10 deg below the glass transition temperature the relative change of ρkBT can be estimated as ∼10 K/300 K, i.e. 3%. Hence, the influence of the vertical shift on the master curves is negligible below the glass transition temperature. Thus, in a whole temperature region we apply in a good approximation the shift factor calculated as bT = Tref/T. The horizontal shift factor was then applied to the quantities bTG′ and bTG″. Note that without the vertical shifting it was impossible to get master curves in the frequency domain of the plateau modulus for G′ using only the horizontal shift since the plateau modulus rises with the increase of temperature. On the other side, application of the vertical shift factor bT = Tref/T allowed us to construct smooth master curves for both G′ and G″ simultaneously (see Figure 1). For the three samples SBR950, SBR953, and SBR956, we show the mastered linear storage and loss moduli in Figure 1. For all the samples, the power laws with the exponents 3/4 and 1/2 at high- and intermediate-frequency regions are observed, respectively. The plateau value Geq at low frequencies can be extracted from a fitting procedure in a low-frequency domain using the Chasset−Thirion law (see eq 3).13 We have found that this fitting procedure provides the value Geq which coincides with the minimal value of G′ inside an experimental error. Furthermore, the dynamic moduli obey a power law with the same exponent at lower frequencies: G′(ω) ∝ G″(ω) ∝ ωα. It proves that the values Geq were determined correctly. The exponent α depends on the sulfur content: it systematically increases from 0.13 to 0.31 with the increase of sulfur content. The final long-time relaxation time decreases strongly with increasing sulfur content and thus is also cross-link density dependent.

Figure 2. Static stress−strain curves for the S-SBR rubbers crosslinked with different amount of sulfur: 1 phr (SBR950), 1.5 phr (SBR953), and 2.75 phr (SBR956). Experimental data (symbols) are fitted with the help of eq 4 (solid lines).

eq 4 reveals that the entanglement modulus Ge does not depend on the cross-link density. The average value over all samples is found to be Ge = 0.323 ± 0.016 MPa, which corresponds to the number density of entanglement strands νe = 0.156 nm−3. The number of Kuhn segments Ne in an entanglement strand has been estimated from the values νekBT = 2Ge = 0.646 MPa and ckBT = 7.7 MPa, where c is the number density of Kuhn segments (see Table 4). Thus, we obtain Ne = ckBT/(νekBT) = 12. The cross-link modulus Gc and hence the number density of network strands νc increase linearly with the sulfur content (see Table 3). The number of Kuhn segments Nc in a network Table 3. Parameters of the S-SBR Rubbers with Different Sulfur Contents

3. FITTING THE STATIC STRESS−STRAIN CURVES Structural parameters of S-SBR rubbersthe cross-link density and the density of entanglementshave been obtained from the fitting of stress−strain dependences using the extended tube model (ETM) of the rubber elasticity.23 The ETM has been top ranked33 as the best constitutive model for rubbers that gives excellent quantitative agreement with experiments. This advantage of the ETM with respect to a wide variety of hyperelastic models is attributed to the fact that ETM realistically describes the effects of the tube deformation and of the finite extensibility of the network subchains on the stress−strain response of the network. The engineering stress σeng in the ETM is given by23 σeng

Gc (MPa)

a

Gqs (MPa)

νc (nm−3)

Nc

1.0 1.5 2.75

0.112 ± 0.002 0.152 ± 0.001 0.243 ± 0.006

0.0096 ± 0.0005 0.0155 ± 0.0003 0.026 ± 0.001

0.433 0.470 0.553

0.027 0.037 0.059

69 51 32

strand has been estimated from the value of Gc = νckBT as follows: Nc =

ckBT Gc

(5)

Nc decreases with the cross-link density as shown in Table 3. The values of the quasi-static shear moduli for the samples can be calculated as

⎧ ⎫ 1−a a ⎬ = Gc(λ − λ 2)⎨ − 2 1 − a(I1 − 3) ⎭ ⎩ 1 − a(I1 − 3)

+ 2Ge(λ−1/2 − λ−2)

sulfur content (phr)

Gqs = Gc(1 − 2a) + Ge

(4)

(6)

One can see from Table 3 that an increase of the degree of cross-linking leads to the increase of the quasi-static moduli of the rubbers.

where Gc = νckBT is the cross-link modulus, νc is the number density of mechanically active network strands between the junctions, Ge = 0.5νekBT is the entanglement modulus, νe is the number density of chain fragments between entanglements (entanglement strand), and parameter a takes into account the finite extensibility of polymer chains between the network knots. I1 in eq 4 is the first scalar invariant of the Cauchy− Green deformation tensor; it is equal to I1 = λ2 + 2/λ in the case of uniaxial deformation of incompressible solid.34 Here λ = 1 + ε is the elongation ratio and ε is the strain. Figure 2 shows the dependences of engineering stress σeng on the optically acquired deformation ε for S-SBR rubbers with 1, 1.5, and 2.75 phr sulfur. The fit of these stress−strain curves with the help of

4. MODELING OF DYNAMIC-MECHANICAL BEHAVIOR USING A MULTISCALE APPROACH To model the dynamic-mechanical behavior of the S-SBR rubbers, we developed a multiscale theoretical approach based on the logarithmic spectral density of the relaxation times H(τ). The shear relaxation modulus G(t) for a polymer network is related to H(τ) as follows:35 G(t ) = Geq + D

∫0

∞

d lnτ H(τ ) exp( −t /τ )

(7)


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After the Fourier transformation one obtains the frequencydependent dynamic moduli35

∫0

G′(ω) = Geq + G″(ω) =

∫0

∞

d lnτ H(τ )

∞

d lnτ H(τ )

(ωτ )2 1 + (ωτ )2

ωτ 1 + (ωτ )2

(8)

Here ω = 2πf is the angular frequency. It is well-known that a power law H(τ) ∝ τ−a with the exponent −a transforms to a power law in the moduli G′(ω) ∝ G″(ω) ∝ ωa with the exponent a (see, e.g., p 183 of ref 20 and Chapter 4 of ref 35). In Appendix B we calculate the relative errors for the power law G′(ω) ∝ G″(ω) ∝ ωa if the spectral density function H(τ) obeys the power-type behavior only in a limited time domain τ1 ≤ τ ≤ τ2 (τ1 ≪ τ2). We show that the moduli in a very good approximation obey the power law G′(ω) ∝ G″(ω) ∝ ωa in a broad frequency domain τ2−1 ≪ ω ≪ τ1−1. Using the inverse relation for the power laws between the moduli and H(τ), we propose the piecewise-power-law ansatz for the spectral density function: ⎧τ β , ⎪ ⎪ ⎪(τ /τ0)−3/4 , H(τ ) ∝ ⎨ ⎪(τ /τ )−1/2 , 0 ⎪ ⎪(τ /τ *)−α , ⎩

τ ≤ τb τb ≤ τ ≤ τ0 τ0 ≤ τ ≤ τ * τ * ≤ τ ≤ τmax

(9)

In eq 9, we introduced the characteristic times τb, τ0, τ*, and τmax, which separate the different power-law regions; their meaning will become clear from the following discussion. Obviously, the piecewise-power-law ansatz ensures the experimentally observed form of the moduli in the four frequency regions (see section 2.3). This can be also clearly seen from Figure 3, which illustrates both the spectral density function H(τ) and the dynamic moduli calculated on the base of H(τ) using eqs 8 and 9 with the prefactors extracted below (see eq 18). In order to determine the corresponding prefactors, we shall consider the asymptotic behavior for different scales of motions, as provided by molecular theories, and require the function H(τ) to be continuous. Let us start from the intermediate frequency domain, in which the dynamic moduli exhibit the power law behavior with the exponent 1/2, typical for the Rouse-like regime. In this regime the spectral density function obeys an asymptotic behavior:35 HRouse(τ ) ≃

⎛ τ ⎞−1/2 1 νekBT ⎜ ⎟ 2 ⎝ τe ⎠

Figure 3. Logarithmic spectral density H(τ) as a continuous, piecewise-power-law function and the storage and loss moduli calculated on the base of H(τ) using eqs 8 and 18. The following parameters were used: α = 0.2, β = 0.1, τb = 10−6 s, τ0 = 10−3 s, τ* = 10−1 s, τmax = 106 s, and ckBT = 7.7 MPa.

Here Ne/n is the number of Gaussian springs comprising the entanglement strand; it is calculated as a ratio of Ne of Kuhn segments in an entangled strand to the number n of Kuhn segments in a Gaussian spring. Note that the Gaussian spring does not coincide with the Kuhn segment since the end-to-end statistics of a rigid Kuhn segment is not a Gaussian one. As it was shown,36 the chain fragment composed from three Kuhn segments obeys the Gaussian statistics in a good approximation. Thus, in the following we take n = 3. Using eq 11, we can rewrite eq 10 as

(10)

−1/2 ckBT ⎛ τ ⎞ HRouse(τ ) = ⎜ ⎟ 3π ⎝ τ0 ⎠

where νe is the number density of entanglement strands and τe is the Rouse time of an entanglement strand. In Appendix A we derive the spectral density function for the Rouse model starting from a classical equation, which includes a summation over normal modes for the model. τe can be expressed via the shortest relaxation time τ0 of a Gaussian spring in the Rouselike entanglement strand as follows:1,35 τe =

2 4 ⎛ Ne ⎞ ⎜ ⎟ τ 0 π2 ⎝ n ⎠

(12)

where c = Neνe is the number density of Kuhn segments. Further, in the frequency domain, which is above the Rouselike regime, the moduli increase more rapidly, exhibiting the power law behavior with the exponent 3/4. This effect can be interpreted as being due to the bending rigidity of the polymer chains4 that manifests itself on times smaller and comparable with the relaxation time of a Kuhn segment. In this study we will assume that the crossover between the Rouse-like regime

(11) E


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and the material constant ν′ that is found to be close to 0.6.4,40 The relaxation time τα is defined by eq 13 from ref 40 and is roughly the Rouse time for a subchain between two entanglements:

and the bending regime takes place at τ0, which means that the equality HRouse(τ0) = Hbend(τ0) should be satisfied. This results in the following form of the logarithmic spectral density in the regime of bending rigidity: Hbend(τ ) =

−3/4 ckBT ⎛ τ ⎞ ⎜ ⎟ 3π ⎝ τ0 ⎠

τα = (13)

(15)

From eq 14 it immediately follows that a logarithmic spectral density function in the dangling chain regime is given by

At even higher frequencies, τ < τb, eq 13 does not hold anymore, as the dynamics in this regime is characterized by small high-frequency vibrations of chain fragments comparable with the size of monomers (glassy regime). In this regime the relaxation modulus is known to obey the Kohlrausch− Williams−Watts time behavior G(t) ∼ exp[−(t/τb)β] with 0 < β < 1.35,37−39 At small times one can expand the exponent into a series: G(t) ∼ 1 − (t/τb)β. Thus, the relaxation in the high-frequency regime (0 < τ < τb) can be approximated by the power-law function H(τ) ∼ τβ, the exponent β being a phenomenological parameter of our approach. Now, let us turn to the low-frequency decay with the crosslink density dependent exponent α. As discussed in the Introduction, a physically motivated explanation of this decay in randomly cross-linked networks is still a controversial issue. Nevertheless, the sliding motion of entangled network strands between junctions provides the asymptotic behavior:8 G′ ∝ G″ ∝ ω1/2 and thus cannot describe our experimental data with an exponent α < 1/2 for the moduli. Furthermore, random connectivity of network strands between junctions leads to the following asymptotic behavior for the moduli in the framework of phantom Gaussian chains: G′ ∝ G″ ∝ ωd/2, where d is a spectral dimension of the network structure.19,20 Obviously, the dimension should be higher than unity, d > 1; for randomly cross-linked networks the index d can be even approximated as d ∼ 4.19 Thus, the random connectivity of network strands is also not able to provide an exponent α < 1/2. Reptation of free chains in a (possible) sol fraction leads to the entanglement plateau modulus for G′ and to a maximum in a low-frequency domain for G″. It is not the case for our experimental data. Thus, we conclude that the entangled dangling chain picture proposed by Curro and Pincus14 provides for our systems a plausible interpretation: it leads to the asymptotic behavior G′ ∝ G″ ∝ ωα, the index α being a function of the degree of crosslinking. We note that the above-mentioned mechanisms (sliding motions of network strands between junctions, collective motions of network strands incorporated into a single network structure, random connectivity of network strands, reptation of free chains in a sol fraction) can also slightly contribute to the mechanical moduli. However, these contributions can be neglected as compared to the main contribution, which is related to the constrained dynamics of entangled dangling chains and provides an index α < 1/2 for the dynamic moduli. We will show that the Curro and Pincus picture allows us to extract a number of reasonable microscopic parameters. Below we will use an exact numerical result obtained by Curro et al. for the diffusion of dangling chains in the presence of topological constraints.40 Using eqs 7 and 27− 31 from the study of Curro et al.,40 it can be shown that dangling chains contribute to the shear relaxation modulus as Gdang (t ) = νdgkBTA(α , v′)(t /τα)−α

4 π 1/2 τ0Ne 2 3 ν′3/2

Hdang(τ ) = νdgkBT

A(α , v′) (τ /τα)−α Γ(α)

(16)

If we define that a crossover between the Rouse-like regime and the dangling chain regime takes place at the characteristic relaxation time τ*, it will mean that the equality HRouse(τ*) = Hdang(τ*) should be satisfied. This provides the following expression for the relative volume fraction of dangling material, fdg, in a randomly cross-linked network: fdg ≡

νdg νe

=

α 2 1/2 − α Γ(α) ⎛ 3ν′3/2 ⎞ ⎛ τ0Ne ⎞ ⎟ ⎜ 1/2 ⎟ ⎜ 3πA(α , v′) ⎝ 4π ⎠ ⎝ τ * ⎠

(17)

Summarizing the results for the four relaxation regimes, we can finally write the complete logarithmic spectral density function as follows: ⎧(τ /τ )−3/4 (τ /τ )β , b ⎪ b 0 ⎪(τ /τ )−3/4 , ck T ⎪ 0 H (τ ) = B ⎨ 3π ⎪(τ /τ )−1/2 , 0 ⎪ ⎪(τ */τ )−1/2 (τ /τ *)−α , ⎩ 0

τ ≤ τb τb ≤ τ ≤ τ0 τ0 ≤ τ ≤ τ * τ * ≤ τ ≤ τmax (18)

With the multiscale theoretical approach, as described by eqs 8 and 18, we fitted the master curves for the storage and loss moduli for the S-SBR rubber samples with different cross-link density (see Figure 4). The parameters ckBT, α, β, τb, τ0, τ*, and τmax were first predefined manually. Two relaxation exponents α and β were taken as slopes of the very low and the very high frequency regions, respectively. Further, as one can see from Figure 4, the frequency dependences of the storage and loss moduli cross each other at three frequency points. The value of τb = ωb−1 was taken from the highest frequency cross-point at ωb. The parameter ckBT was taken as a value of G′ = G″ at the middle frequency cross-point ω0, from which also a value of the τ0 = ω0−1 was extracted. The value of τmax was taken from the lowest frequency cross-point at ωmax. Finally, a value of τ* was extracted from the frequency point between ωmax and ω0, at which G″(ω) has the highest curvature. We found that ckBT and β were not changing with the cross-link density. Therefore, we fixed ckBT and β and used the manually adjusted values of the time parameters τb, τ0, τ*, and τmax and of the long-time relaxation exponent α as the initial values for the nonlinear fitting routine of the Matlab software. Final values of the fitting parameters for the three rubber samples are summarized in Table 4. The plots in Figure 4 show the relaxation part of the storage modulus, G′ − Geq, and the loss modulus, G″, as functions of angular frequency. Here Geq is the plateau modulus which corresponds to the limiting value of the storage modulus at low frequencies: Geq = G′(ω → 0). The value Geq is given in Table

(14)

where νdg is the number density of entanglement strands in the dangling chains. The coefficient A is given by eqs 30 and 31a from ref 40. Its magnitude depends on values of the exponent α F


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The following relation correlates the half-wavelength L of the bending mode with the corresponding relaxation time:4,41 τ (L ) ≃

ζsls 2 2π 4kBT

(L /ls)4

(19)

where ls is the Kuhn segment length and ζs is the Kuhn segment friction coefficient. The smallest length scale, at which the bending rigidity comes into play, contains two rigid monomers, 2lm, which can bend around the joining group; in our approach this length scale corresponds to the relaxation time τb. The largest length scale, at which the bending rigidity is still noticeable, corresponds to the relaxation time τ0 of a Gaussian spring, the latter being about 3 times the length of Kuhn segment ls. This implies that ⎛ 3l ⎞4 τ0 ≃⎜ s⎟ τb ⎝ 2lm ⎠

(20)

The values of the relaxation time τb and the ratio (τ0/τb)1/4 as a function of the cross-link density ν are presented in Figure 5a,b. First, we can see from Figure 5a that τb slightly increases with the degree of cross-linking. This result displays the increase of the monomeric friction coefficient ζmon with increasing cross-link density due to the increase of Tg (see Table 2). A similar finding was published earlier by Marzocca et al.42 on different unfilled sulfur cured natural rubber vulcanizates. Here, the increase of ζmon was correlated with the crosslink density and the network structure formed. It is instructive to rescale the frequency axis with the ratio s = τx/τ950, where τx is the value of τb for the sample SBRx and τ950 is the value of τb for the sample SBR950. s = 1.0, 1.5, and 3.0 for the samples SBR950, SBR953, and SBR956, respectively. Such rescaling leads to the dependences of G′ and G″ which coincide in the high-frequency range, as can be seen from Figure 6, especially from the two insets. This clearly shows that the physical processes occurring at short times are identical for all three studied samples and thus are cross-link density-independent. Therefore, the ratio (τ0/τb)1/4 (see Figure 5b) is independent of the cross-link density, which is in accordance with eq 20. Thus, the latter relation can be used to extract the ratio of the Kuhn and the monomer lengths, which is found to be ls/lm = (2/3)(τ0/τb)1/4 ≈ 3.5. The ratio ls/lm is known to change for flexible hydrocarbon polymers from 2.5 for poly(ethylene oxide) to 5 for polystyrene.43 Thus, the multiscale approach proposed by us allows to extract a realistic value of this parameter which indicates that the mastering at high frequencies was done really well and deviation from the thermorheological simplicity in our systems if exists is only slight. In order to compare this result with an actual value for the SSBR copolymer, we first need to calculate the average copolymer parameters from the composition (see Table 1). A number of monomer parameters for corresponding homogeneous polymers can be found in ref 44; they are summarized in Table 5. To get the monomer parameters for the actual S-SBR

Figure 4. Fits of the master curves for the storage and loss moduli for the samples SBR950, SBR953, and SBR956 with the multiscale approach given by eqs 8 and 18.

4. We note that the value Geq differs from the quasi-static shear modulus Gqs presented in Table 3. The reason is that the quasistatic modulus was measured during several seconds, and in this time region the long-scale relaxation is not yet finished according to Figure 1. Thus, the quasi-static modulus includes contributions both of mechanically active network strands between junctions and of nonrelaxed dangling chains, which contribute in this time regime as mechanically active chains into the quantity Gc (see Table 3). On the other side, the plateau modulus Geq includes only contributions from mechanically active network strands between network junctions, when the dangling chains have been fully relaxed. Now we turn to the relations between the fitting parameters in Table 4, which follow from the physical processes underlying the power laws in eq 18. First, let us consider the region τb < t < τ0 dominated by the bending rigidity of the polymer. Table 4. Fit Parameters (See Figure 4) sample

ckBT [MPa]

α

β

τb [s]

τ0 [s]

τ* [s]

τmax [s]

Geq [MPa]


7.7 7.7 7.7

0.1 0.2 0.3

0.15 0.15 0.15

1.2 × 10−7 1.71 × 10−7 3.35 × 10−7

8.82 × 10−5 1.29 × 10−4 2.58 × 10−4

4.68 × 10−2 4.48 × 10−2 1.58 × 10−1

2.1 × 104 2.4 × 104 3.18 × 104

0.217 0.283 0.377

G


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Figure 5. Fitted parameters and relations between them as functions of the cross-link density.

Figure 6. Storage and loss moduli for the samples SBR950, SBR953, and SBR956 with the frequency rescaling. Two insets show a very good overlapping of the curves in the high-frequency range.

From the last equation we obtain Ms ≈ 300 g/mol, which is in close agreement with the value given in Table 5, Ms = 353 g/ mol, obtained from the mathematical analysis of the chemical structure of S-SBR. It is known that the dangling chain picture provides correct scaling dependences for the exponent α ∼ νc and the terminal relaxation time ln(τmax) ∼ 1/νc as functions of the cross-link density νc.14,40 We can indeed observe both tendencies for our systems as shown in Figure 5c,d. Here we normalize the value of τmax on the corresponding value of τ* to eliminate a small shift to larger values observed for all relaxation times upon cross-linking due to the increase of Tg (see Table 2). The next step is to check whether the dangling chain picture provides plausible numerical values of material parameters such as the relative fraction fdg of dangling chains and the average number of Kuhn segments in dangling chains Ndg. According to eq 17, fdg is a complex function of multiple parameters: τ0, τ*, α, and the material constant v′ as well as the number of Kuhn segments Ne in the entanglement strand. The latter was estimated from the fitting of stress−strain curves to be about 12 (see section 3). The value of ν′ was found in refs 4 and 40 to be

Table 5. Monomer Parameters Ms [ g/mol] m0 [ g/mol] ls [ nm] lm [ nm]

styrene

cis-PB

trans-PB

vinyl-PB

S-SBR

720 52 1.77 0.255

114 13.5 0.93 0.441

117 13.5 1.12 0.515

289 27 1.37 0.255

353.5 30 1.4 0.314

copolymer, the weighted average has been calculated using Tables 1 and 5. The average values obtained in this way are presented in the last column of Table 5. Using these data, we obtain for the length ratio ls/lm the value of 4.4, which is in reasonable agreement with the value of 3.5 estimated from the ratio (τ0/τb)1/4. Further, using the number density of Kuhn segments c and the mass density of the S-SBR ρ = 0.93 g/cm3, we can independently calculate the molar mass of the Kuhn segment Ms: Ms =

ρRT ckBT

(21) H


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close to 0.5−0.6. Using eqs 30 and 30a from ref 40, we estimated the average value of parameter A(α,v′) and the relative fraction of dangling chains at ν′ = 0.6 at different values of the exponent α. The latter values can be also used to estimate a number of Kuhn segments in the average dangling chain as Ndg = Ne/(αv′).4,40 The dangling parameters are provided in Table 6.

To obtain the fraction of dangling material, it has to be taken into account that the first or last ndg monomers in the chain are unreacted and that there are two dangling ends per chain45 fdg =

Np − 1

∑

r(1 − r )ndg (2ndg + 1)

ndg = 0

(23)

Then, the fraction of elastically active material in network strands fstr can be extracted from the normalization condition

Table 6. Dangling Parameters for ν′ = 0.6 sample

α

A

fdg

Ndg


0.1 0.2 0.3

1.054 1.285 1.61

0.50 0.21 0.09

206 100 66

fsol + fdg + fstr = 1

(24)

Note that the same result for fstr can be obtained independently using a rigorous formalism presented in ref 45. Figure 7a shows all three fractions as a function of reduced extent of reaction r/rc. Interestingly, the fraction of dangling chains in a randomly cross-linked network has a maximum at r/ rc ≈ 2 where it reaches quite a high value of 0.60. We found this value to be independent of the length of primary chains. One can see from Table 6 that the volume fraction of dangling chains fdg estimated from the fitting procedure always satisfies the condition which follows from rigorous molecular theories for randomly cross-linked polymer networks: fdg < 0.6. Furthermore, it is usually assumed that the length of dangling chains is comparable with the length of network strands. Contrary to this intuitive assumption, the theory45 predicts that the length of dangling chains can be as long as 1.5 of the length of the network strands at low extents of reaction (see Figure 7b). This is due to the fact that at least one branching point is necessary to obtain dangling chains, and it cuts the primary chains in two equal parts, whose lengths is 1/2 of the primary chain on average. At the same time, to have a network strand, it is necessary to have at least two branching points which cut the primary chains in three equal parts, so that the network strands have on average 1/3 of the primary chain length. Thus, the dangling chains are about 1.5 times longer as compared to the network strands between junctions. Comparing the average numbers of Kuhn segment in network strands, Nc (see Table 3), and in dangling chains, Ndg (Table 5), we obtain the ratio Ndg/Nc ≈ 3 for the sample SBR950 and Ndg/Nc ≈ 2 for the samples SBR953 and SBR956. These ratios are in a qualitative agreement with theoretical estimations but, however, slightly exceed the maximal theoretical value 1.5. Presumably, a sol fraction should be also considered. It is not high but contains long primary chains with potentially extremely slow reptation motions in the surrounding them network structure. We conclude by noting that the multiscale approach presented above allows us to fit precisely the frequency

One can see that both the volume fraction of dangling chains, fdg, and their average length, Ndg, decrease with increasing of the cross-link density. This result is intuitively understandable for randomly cross-linked networks as in our case. In the next section we compare the obtained values with numerically exact theories devoted to molecular structure of randomly crosslinked polymers.

5. DISCUSSION The plausibility of the values obtained in the frame of the dangling chain picture can be established by comparing them with the estimates provided by molecular theories describing the random cross-linking process. A rigorous consideration of network defects in randomly cross-linked networks has been done in studies.45,46 Here we will restrict our consideration to main defects such as dangling chains and uncross-linked chains comprising the sol fraction and assume for simplicity that the primary chains are of the same length. Following the theory,45 the average lengths of dangling chains and network strands can be calculated as a function of the extent of reaction r, which has a meaning of the reacted fraction of cross-linkable monomers. It is known that the vulcanization process is a type of gelation for which the mean-field theory works well due to a high degree Np of cross-linkable monomers in primary chains. The meanfield theory prediction for the gel point is rc = 1/Np.4 So, it is convenient to normalize the extent of reaction on this value as a precise number of cross-linkable monomers in a primary chain is an unknown parameter. For the same reason, the average lengths of dangling chains and networks strands have been normalized on Np. The sol fraction is given by45 fsol = (1 − r )Np

1 Np

(22)

Figure 7. Fractions of primary chains (sol), dangling chains, and network strands as a function of reduced extent of reaction r/rc calculated using eqs 22−24. I


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APPENDIX B. SPECTRAL DENSITY FUNCTION WITH THE POWER-TYPE TIME BEHAVIOR Below we consider the frequency behavior of the storage and loss moduli for spectral density function H(τ), which obeys a power-type behavior in a long time domain:

dependences of the storage and loss moduli simultaneously across a broad frequency domain (16 decades). The fitting parameters provide the molecular characteristics which are reasonable for the studied rubber materials and are in a good agreement with rigorous theoretical calculations. These results demonstrate a great potential strength of the proposed theoretical multiscale approach which can be used in the future for investigation of the molecular mobility and structure of other rubbery materials which are of the great importance for industry and everyday life.

H (τ ) = A τ − a ,

τ1 ≤ τ ≤ τ2

(B.1)

where 0 < a < 1 and τ1 ≪ τ2. The integral in eq 8 can be represented as a sum of integrals from 0 to τ1, from τ1 to τ2 and from τ2 to ∞. In the second region the function H(τ) is given by eq B.1, and the integrals from τ1 to τ2 can be well approximated by the integrals from 0 to ∞ at τ1 ≪ τ2. Thus, the dynamic part of the storage modulus, Ĝ ′, and the loss modulus are approximated by1

6. CONCLUSIONS A versatile theoretical multiscale approach was elaborated to describe the linear dynamic moduli of the unfilled rubbers (see section 4). It combines ideas developed in the statistical− physical theory of polymers for different frequency regimes: (1) nonpolymeric relaxation processes at extremely high frequencies, (2) the bending modes of the polymer chain at high frequencies, (3) the Rouse relaxation processes at intermediate frequencies, and finally (4) reptation dynamics of dangling chains at low frequencies. Only after taking into account this richness in relaxation processes on various time and length scales, an excellent description of dynamic moduli of unfilled rubbers was achieved. The values of the material parameters extracted from our fits correlate well with the actual values for the S-SBR copolymer used in the samples.

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Ĝ ′(ω) = G′(ω) − Geq ≅ = Aωa G″(ω) ≅

π 2 sin(απ /2)

∫0

∞

d lnτ Aτ −a

∫0

∞

d lnτ Aτ −a

(ωτ )2 1 + (ωτ )2

ωτ π = Aωa 2 cos(απ /2) 1 + (ωτ )2

The errors of this approximation are now given by ΔĜ′(ω) ≅ (

τ1

∞

∫0 +∫τ

)d lnτ[H(τ ) − Aτ −a]

(ωτ )2 1 + (ωτ )2

)d lnτ[H(τ ) − Aτ −a]

ωτ 1 + (ωτ )2

2

ΔG″(ω) ≅ (

τ1

∫0 +∫τ

∞

2

(B.3)

APPENDIX A. SPECTRAL DENSITY FUNCTION FOR THE ROUSE MODEL

The relative errors can be shown to be small enough in the frequency domain τ2−1 ≪ ω ≪ τ1−1. The integrals with the terms Aτ−a in the right-hand sides of eqs B.3 can be calculated analytically. The integral from 0 to τ1 with the term H(τ) can be estimated using the change of variables x = ωτ and using the first term in the Taylor expansion in the vicinity of the limit ωτ1 ≪ 1. The integral from τ2 to ∞ with the term H(τ) can be estimated using the change of variables λ = (ωτ)−1 and using the first term in the Taylor expansion in the vicinity of the limit (ωτ2)−1 ≪ 1. As a result, the relative errors of the approximation given by eq B.2 are estimated as

The relaxation modulus in the framework of the Rouse model reads as1 N

G(t ) = νkT ∑ exp( −t /τp) (A.1)

p=1

where ν is the number density of polymer chains, each of which is composed from N Gaussian springs, and τp are the stress relaxation times of the model. The relaxation times vary between their minimal and maximal values: τmin = τN and τmax = τ1. In the time domain τmin < τp < τmax the relaxation times are well approximated as τp = τmax/p2.1 Thus, in the time domain τmin < t < τmax the sum in eq A.1 is well approximated by the integral: G(t ) ≅ νkT

∫1

N

dp exp( −tp2 /τmax )

|ΔĜ ′(ω)| 2 sin(απ /2) 1−a ≅ (ωτ1)2 − a ̂ π 1+a G′(ω) −a a − 1 + (ωτ2) a

ε1 =

|ΔG″(ω)| 2 cos(απ /2) a ≅ (ωτ1)1 − a π G″(ω) a−1 a + (ωτ2)−1 − a a+1

ε2 = (A.2)

Now, introducing a new variable of integration τ = τmax/p2, eq A.2 can be rewritten in the following form: G(t ) ≅ νkT

∫τ

τmax

dτ

min

1 2τ

τmax exp( −t /τ ) τ

τ 1 νkT max 2 τ

(B.4)

The first terms in the brackets are caused by the contributions of the short relaxation times at τ < τ1. One can see that the relative contribution of the short times is quite small in the frequency domain ω ≪ τ1−1, since ε1 ∝ (ωτ1)2−a ≪ 1 and ε2 ∝ (ωτ1)1−a ≪ 1 at ωτ1 ≪ 1. For instance, for a = 3/4 and at frequency ωτ1 = 0.1 the relative contributions of the short times are estimated as ε1 ≈ 0.5% and ε2 ≈ 5%. Obviously, at ωτ1 < 0.1 the errors are much smaller. The second terms in the brackets in eqs B.4 result from the long times at τ > τ2. One can see that the relative contribution of the long times is also quite small at frequency domain ω ≫ τ2−1, since ε1 ∝ (ωτ2)−a ≪ 1 and ε2 ∝ (ωτ2)−1−a ≪ 1 at ωτ2 ≫ 1. For instance, for a = 3/4

(A.3)

Comparing the last equation with eq 7, we conclude that at τmin < τ < τmax the spectral density function for the Rouse model is approximated as H (τ ) ≅

(B.2)

(A.4) J


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and at frequency ωτ2 = 10 the relative contributions of the long times are estimated ε1 ≈ 3.5% and ε2 ≈ 0.2%. Obviously, at ωτ2 > 10 the errors are much smaller. Thus, in a long frequency domain τ2−1 ≪ ω ≪ τ1−1 the inverse relation of the power laws takes place in a good approximation: H(τ) ∝ τ−a leads to G′ ∝ G″ ∝ ωa. Note, however, that in the vicinities of the boundaries of the domain (i.e., at ω ∼ τ1−1 and at ω ∼ τ2−1) the contribution of the short and long times can be comparable with the main contribution given by eqs B.2 and the graduate crossover between different frequency regimes can be observed, as it can be seen in Figures 1 and 4.

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(23) Kaliske, M.; Heinrich, G. Rubber Chem. Technol. 1999, 72, 602− 632. (24) Stöckelhuber, K. W.; Svistkov, A. S.; Pelevin, A. G.; Heinrich, G. Macromolecules 2011, 44, 4366−4381. (25) Rooj, S.; Das, A.; Stöckelhuber, K. W.; Wang, D.-Y.; Galiatsatos, V.; Heinrich, G. Soft Matter 2011, 9, 3798−3808. (26) Götze, W.; Sjogren, L. Rep. Prog. Phys. 1992, 55, 241−376. (27) Plazek, D. J. J. Phys. Chem. 1965, 69, 3480−3487. (28) Plazek, D. J. J. Rheol. 1996, 40, 987−1014. (29) Santangelo, P. G.; Roland, C. M. Macromoleucles 2006, 31, 3715−3719. (30) Inoue, T.; Okamoto, H.; Osaki, K. Macromolecules 1991, 24, 5670−5675. (31) Inoue, T.; Matsumoto, A.; Nakamura, K. Macromolecules 2013, 46, 6104−6109. (32) Dealy, J. M.; Larson, R. G. Structure and Rheology of Molten Polymers; Hansa Publications: Cincinnati, 2006. (33) Marckmann, G.; Verron, E. Rubber Chem. Technol. 2006, 79, 835−858. (34) Graessley, W. W. Polymer Liquids and Networks: Dynamics and Rheology; Taylor & Francis Group: London, 2008. (35) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley & Sons: New York, 1980. (36) Toshchevikov, V. P.; Heinrich, G.; Gotlib, Y. Y. Macromol. Theory Simul. 2010, 19, 195−209. (37) Williams, G.; Watts, D. C. Trans. Faraday Soc. 1970, 66, 80−85. (38) Shore, J. E.; Zwanzig, R. J. Chem. Phys. 1975, 63, 5445−5458. (39) Gotlib, Y. Y.; Toshchevikov, V. P. Polym. Sci., Ser. A 2001, 43, 525−534. (40) Curro, J. G.; Pearson, D. S.; Helfand, E. Macromolecules 1985, 18, 1157−1162. (41) Morse, D. C. Macromolecules 1998, 31, 7044−7067. (42) Marzocca, A. J.; Steren, C. A.; Raimondo, R. B.; Cerveny, S. Polym. Int. 2004, 53, 646−655. (43) Chatenay, D.; Cocco, S.; Monasson, R.; Thieffry, D.; Dalibard, J. Lecture Notes of the les Houches Summer School 2004; Elsevier: Amsterdam, 2005. (44) Huneke, B.; Klüppel, M. Kautsch. Gummi Kunstst. 2006, 59, 242−250. (45) Lang, M.; Göritz, D.; Kreitmeier, S. Macromolecules 2003, 36, 4646−4658. (46) Chasse, W.; Lang, M.; Sommer, J.-U.; Saalwächter, K. Macromolecules 2012, 45, 899−912.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge a technical support from T. Götze and K. Scheibe (Leibniz-Institut für Polymerforschung Dresden e.V.). M.S. and V.T. thank Prof. J.-U. Sommer and Dr. K.W. Stöckelhuber for inspiring discussions. S.W. and F.P. are grateful for the outstanding collaboration and support by Dr. F. Schmitz (Goodyear Innovation Center Luxembourg). We also thank the Goodyear Tire & Rubber Company for the permission to publish this paper.

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Multiscale Approach to Dynamic-Mechanical Analysis of Unfilled

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