Filled elastomers are systems of very great practical importance because of their unique properties. Nonreinforced polymer matrices generally do not exhibit mechanical properties suitable for practical purposes, being too soft and fragile.1 On the contrary, elastomers filled with carbon black or silica particles have a shear modulus much (up to a few 100 times) higher than that of the pure elastomer, exhibit a high dissipative efficiency, and are extremely resistant to both fracture and abrasion, which makes them essential for damping materials, shock absorbers or tyres.1–10 Another important feature of filled elastomers is their nonlinear behavior, in the nondestructive regime. When submitted to oscillatory deformations at a frequency ω of the order typically 1 Hz and of amplitudes γ of order a few percent or more, the elastic modulus G′(ω,γ) decreases to values much smaller than the value in the linear regime: this is the so-called Payne effect. In some of the systems studied by Payne, the modulus G′(ω,γ) drops from a few 107 Pa to a few 106 Pa.9 Payne showed that in these systems, the loss modulus G″(ω,γ) exhibits a peak up to a few 106 Pa at deformations of typical amplitude 1%,8 which is a key feature of strongly reinforced systems. In less reinforced systems, that is, for smaller filler volume fractions, the peak is less pronounced. The corresponding data are plotted in Figure 1.
Another remarkable property is the so-called Mullins effect, which refers to the drop of elastic modulus (or stress softening) observed after a large amplitude stress–strain cycle. This results in the well-known strongly nonlinear, banana-shaped, stress–strain cycles in filled rubbers.5, 11, 12. An important feature is that the elastic modulus may recover the initial value—at least partially–on a long-time scale. In this article, we shall consider the drop and subsequent recovery of the modulus after large amplitude oscillatory stress–strain cycles, which involves essentially the same physical mechanisms as the proper Mullins effect. Thus, we will call this phenomenon Mullins effect as well for the sake of simplicity.
All these features are of primary importance in, for example, tyres applications, as regards rolling resistance, grip, and durability.10 To be able to control and tune the properties of these systems, understanding this unique behavior is thus an issue of major importance. Filled elastomers might also be considered as paradigms of nanocomposites. Predicting their properties is therefore a hard test of theories aimed at describing the physics of polymers structured at the nanoscale.
We recently proposed that the essential feature for predicting these effects is the presence of a glassy layer around the fillers13–15 (Fig. 2), which explains both the linear and the nonlinear behavior.16, 17 Indeed, it has long been proposed that the polymer matrix in the vicinity of the filler is glassy.18–24 This picture is consistent with experimental studies, which demonstrated that the glass transition temperature Tg in polymer thin films in contact with a solid substrate differs from that in the bulk, with a sign of the effect depending on the polymer–substrate interaction.25–34. The link between glass transition in thin films and physical properties of nanocomposites have also been emphasized by many authors.35–43 Recently, a mesoscale model has been proposed to deal with large scale elastic and plastic properties of filled elastomers.17, 44–46 This model takes into account the existence of glassy layers around the fillers. When these glassy layers overlap, thus forming glassy bridges (Fig. 2), the reinforcement can be very strong, as it is the case in Payne's systems. The dynamical state of these glassy bridges is characterized by a relaxation time or life-time. This life-time is assumed to depend on the distance between the particles representing the fillers, the polymer–filler interaction, the local value of stress, and also the history. Macroscopic deformations result in an increase of the local stress in between the fillers, which leads to a reduction of the local Tg and an acceleration of the local dynamics. When the stress is released after local plastic deformation, physical aging takes place. The life-time of the glassy bridge increases again. These features result in a complex history dependence of the dynamical state of the glassy bridges. In a permanent regime of sinusoidal deformation, the storage and dissipative moduli depend on the steady-state distribution of glassy bridge life-times in the system. This mesoscale model was able to account for (1) reinforcement when a large cluster of long-lived glassy springs was present; (2) the Payne effect which was interpreted as the consequence of the typical life-time of glassy bridges becoming shorter than the period of the deformation; (3) the Mullins effect which is the consequence of the aging of the distribution of the glassy bridge life-times back to the initial distribution.
The purpose of this article is to study in more detail the plastic behavior and the recovery of the mechanical properties of filled elastomers after a first series of deformations and to compare the predictions of our model to known experimental results. The typical situations considered here will be the following ones. (1) The elastic and dissipative moduli G′(γ0,ω) and G″(γ0,ω) are measured as a function of the deformation amplitude γ0 of the initial systems, by applying deformations of the type γ(t) = γ0 cos(ωt) for a few periods. This number of period is typically a few tens at least to allow the samples to reach the permanent regime, characterized in particular by the distribution of breaking times of glassy bridges. (2) At some point during this applied periodic deformation, the system is maintained in a fixed deformation state at the maximum γ0 of the deformation during a long waiting time tw. Then, the linear properties around this new reference state are measured by applying small amplitude deformations. (3) Starting from this deformed state, the stress is relaxed to zero for a long time by allowing the shape of the system to relax. The final state is usually different from the initial state (before applying any deformation) and characterizes the plastic deformation of the sample. Then, the mechanical properties of the system in this new state are studied. The aim of this study is to show how the permanent kinetics of breaking and rebirth of glassy bridges can account for experimental results regarding the plastic and recovery behavior of filled elastomers in these kind of experiments.47 The article is organized as follows. In “Physical mechanisms of reinforcement and nonlinear properties of filled elastomers” section, we describe the physical mechanisms we propose to be relevant for the reinforcement and nonlinear behavior. In “Description of the Numerical modelling” section, we describe the basic features of the numerical model and the way in which the relevant physical quantities can be translated into the input parameters of the numerical simulations. The results are presented and discussed in “Results and Discussion” section.
PHYSICAL MECHANISMS OF REINFORCEMENT ANDNONLINEAR PROPERTIES OF FILLED ELASTOMERS
Shift of the Glass Transition Temperature in the Vicinity of the Fillers
The model is based on the presence of a glassy layer around the fillers when the interaction between the matrix and the fillers is sufficiently strong. The presence of this glassy layer has been demonstrated in refs.13–15. It has the same physical origin as the Tg shifs measured in thin polymer films.28, 29, 35. It has been proposed that the Tg shift at the distance z from an interface is of the form:31, 34
where Tg is the bulk glass transition temperature of the pure rubber, the exponent ν ≈ 0.88, and the length β depends on the matrix–filler interaction. For strong interactions, it is of the order 1 nm. It follows from Eq. (1) that at temperature T, the fillers are surrounded by a glassy layer of thickness eg:13–15
For the sake of simplicity, we assume from now on that the Tg shift at a distance z is given by
and that the glassy layer thickness as a function of T can be calculated accordingly.
Let us discuss typical values of the parameter β in the regime of interest. In a film with strong substrate–polymer interactions, the increase of Tg can be as large as 50 K at a distance of 10 nm.25–27 This corresponds to a value of β of 1.5 nm. For β ∼ 1 nm and Tg ≈ 200 K, we obtain an increase of Tg of 100 K for z = 2 nm, which is a typical interparticle distance for strongly reinforced elastomers as we shall see later.
The glass transition temperature as a function of the distance to the filler–matrix interface is exemplified in Figure 3, in units of the particle diameter.
When glassy layers overlap, the macroscopic shear modulus G′ is related to the shear modulus of the glassy polymer G′g through geometrical effects. Indeed, a macroscopic deformation ε is amplified locally in between fillers by a factor λ of the order of the ratio between the filler diameter and the distance between two neighboring fillers. In a plane normal to the direction of elongation, the stress is supported by glassy bridges which represent an area fraction Σ < 1 (Fig. 4). Both λ and Σ values depend on the considered systems. In the small deformation regime, an estimate of the macroscopic modulus within a glassy bridge is thus given by
Assuming that the fillers are spherical particles of the order 10 nm in diameters with typical interparticle distance of a few nanometers, we deduce that λ is of order a few units and Σ is of order a few 10−2, depending on the ratio between the glassy layer thickness and the nearest neighbor distance. As G′g ∼ 109 Pa, a macroscopic shear modulus of about 108 Pa is obtained, which corresponds to very strong reinforcement.
Effect of An Applied Stress: Yielding of Glassy Bridges
As shown above, when a strain is applied to a strongly reinforced sample, the stress is concentrated in the glassy bridges. The local stress σ results in a lowering of the local glass transition of the polymer given by ref.16:
The first term in the right-hand side of eq (5) represents the effect of the interaction between the fillers and the matrix as described in the earlier subsection. The second term is the decrease of Tg due to the local stress, which is the plasticizing effect of an applied stress. The parameter K depends on the polymer; it is known from macroscopic experiments. It relates the yield stress σy to the temperature T and the polymer glass transition temperature Tg by σy = K(Tg − T). K is of the order 106 Pa K−1 typically.48
Let us discuss the effect of a macroscopic deformation ε of a few percent. The local stress σ is then of order σ ∼ λ εG′g ∼ 108 Pa. With K ∼ 106 Pa K−1,48 a local Tg reduction of 100 K is obtained. This is comparable with the increase of Tg due to interfacial effects described in the earlier subsection. Therefore, glassy bridges yield, which results in a lowering of the shear modulus, for macroscopic deformations of order a few %. This is indeed comparable with the Payne effect,8 which is a sharp decrease of the elastic modulus in this range of deformation.
Experimentally, it has been observed that the typical strain amplitude at which the nonlinearity occurs depends much more on the filler amount than on the Tg of the elastomer matrix (or more precisely, on the quantity T − Tg).12 This is actually what is observed in our simulations. It was shown in ref.17 that the critical deformation for the appearance of the Payne effect depends only weakly on temperature. This is due to the fact that the Tg variation as a function of the local stress is strong. Also, as Tg is reduced, glassy bridges which no longer contribute to the elastic modulus in the linear regime do not contribute to the yield behavior any longer either. Both effects contribute to reduce the dependence of the critical deformation on Tg when compared with what might be expected from eq (5).
Life Time of Glassy Bridges: Aging
Glassy bridges are not permanent. Indeed, as we will discuss, they break under applied strain. Within a glassy bridge in between two neighboring particles, at equilibrium, we assume that the polymer has locally the dominant relaxation time τα given by the William-Landel-Ferry (WLF) law of the corresponding polymer,49 modified by the Tg shift due to interfacial effects and the local stress σ. We assume that the breaking times of glassy bridges are comparable with the local dominant relaxation times τα of the glassy bridges. The breaking time is thus given by
where Tg(z,σ) is given by eq (5), τg = 100 s (the relaxation time at Tg) and T is the temperature. C1 and C2 are the WLF parameters of the considered polymer.49 The breaking time of a glassy bridge between two neighboring fillers is plotted as a function of the distance z between the fillers, for different values of the interaction parameter β (in units of the filler diameter) in Figure 5 (in the absence of stress). For β ≈ 0.06, the breaking time becomes longer than experimental time scales at distances about 0.4 (in units of the filler diameter). The WLF law used here corresponds to that of polyisoprene with the parameters Tg = 213 K, C1 = 12.8, and C2 = 34 K.50
Equation (6) gives the equilibrium value of the breaking time, which is obtained when the distance z, the local stress σ, and the temperature T have been maintained fixed for a long time. In general, the breaking time depends on the history of the glassy bridge and is denoted by τα(t). To describe the corresponding evolution, we assume that, at any time, a glassy bridge has a probability for breaking per unit time, dP/dt, given by
where α is a number of order 1, but smaller than 1 (see discussion below). We will assume α = 0.3 – 0.4 in the following. When a glassy bridge breaks, the local stress σ is relaxed and drops to a much smaller value, which is the rubbery contribution. Immediately after breaking, we assume that τα relaxes to a value , where denotes the local deformation rate. In practice, typical deformation rates in our simulations will be of order . The local breaking time τα(t) undergoes a subsequent evolution, analogous to an aging process.51 Thus the evolution of the breaking time of a glassy bridge, τα(t), is given by
By definition, the time τα is bounded by the time τWLF(T − Tg(z,σ)) given by eq (6). Equations (7)–(9) describe the evolution of the local breaking time. Thus, we assume that, as the local stress increases, the breaking time decreases to a smaller value, and then increases progressively again when the stress decreases, up to the value set by the local glass transition temperature, which depends on the local stress and the distance between neighboring fillers. When breaking occurs, we assume that the breaking time drops to the smallest value τmin, before increasing slowly again. The fact that α has to be chosen smaller than 1 is directly related to the aging dynamics [eq (8)]. Values for α larger than or equal to 1 would not allow the system to age, which has no physical meaning. Thus, we assume that the breaking time is comparable with, but larger than, the α-relaxation time, which is itself comparable with the aging time (under permanent conditions).
Just after breaking, a glassy bridge can experience subsequent frequent ruptures, if the instantaneous value of the WLF time τWLF(T − Tg(z,σ)) is small when compared with an experimental time scale ≃ . This occurs for instance during an oscillatory shear experiment, if the local stress is above the local yield stress σy(z) = K(Tg(z) − T). The glassy bridge may then break and rebuild permanently. On the other hand, when local stresses are kept at a low level, for example, if the global stress is released after a first deformation, the WLF time τWLF(T − Tg(z,σ)) is comparable or even larger than ≃ . The probability of rupture [eq (7)] is small, rupture events are unfrequent, and a glassy bridge progressively rebuilds between the considered fillers.
DESCRIPTION OF THE NUMERICAL MODELING
The physical mechanisms discussed above are introduced in a dissipative particle dynamics numerical model. The implementation of the model into a numerical code is described in detail in ref.17. The model is an extension of a model that we proposed for describing the mesoscale behavior of soft thermoplastic or reinforced elastomers in the high-temperature regime when glassy layers do not overlap.44–46
Basic ingredients of the model are permanent elasticity, disorder, and excluded volume effects. Solid filler particles are represented by hard spheres randomly distributed in space, with a given volume fraction (all considered deformations are at constant volume). Examples of interparticle distance distributions are shown in Figure 6.45 The degrees of freedom are the centers of mass of the fillers (Fig. 2). The diameter d of the fillers sets the unit-length scale. The equations of the dynamics are noninertial and include a source of dissipation in the form of a hydrodynamic friction term. The equation of motion for particle i is thus:
where is the elastic force between two neighboring particles and is the sum of the contribution of the glassy bridges and of the rubbery matrix. is the hardcore repulsion, and is the friction force between two neighboring particles, respectively. In this section, we discuss how the physical parameters of the model are translated into parameters of the simulation to solve the model numerically.
In addition to excluded volume interactions, two neighboring fillers interact with two distinct forces schematized in Figure 7 and described by two types of springs: (1) permanent springs of modulus k∞ = 1. In our dimensionless system, this spring constant sets the unit of modulus. To be specific, we assume that it corresponds to a value of 5 × 105 Pa in the following. All the results will therefore be expressed in units of MPa; (2) springs corresponding to glassy bridges. These springs have finite life-times: they break and rebuild permanently (Fig. 7). The glassy bridge modulus ≈ 109 Pa is typically three orders of magnitude larger than the rubbery modulus. However, due to the geometric effects mentioned in “Shift of the Glass Transition Temperature in the Vicinity of the Fillers” section, when translated in terms of spring stiffness, this quantity has to be rescaled by the product λΣ which we choose to be 10−1. The glassy spring constant is thus set to k0 ≈ 100. Therefore, the force due to the rubbery matrix is given by
l0 is the equilibrium length of the springs, is the vector joining the centers of the two particles, and is the unit vector joining the two particles. The force due to the glassy bridges reads
where and is the vector position of filler i. The reference state corresponds to the relative positions of the fillers when the last glassy bridge breaking occurred. When breaking takes place at time t (Fig. 7), eq (12) cancels, and subsequent relative displacement at time t′ > t are calculated according to the new reference state . The life-time of glassy bridges are the results of a dynamical evolution calculated according to eqs (6)–(9). This dynamical evolution depends on the local glass transition temperature [eq (6)], which allows for calculating time scales in physical units (seconds). We thus take the local history into account in a simple way.
Local Tg Between Neighboring Particles
The values to use in the simulations for the parameters β and K, as defined by eqs (3) and (5), are discussed here. In the simulations, the distance z is taken to be half the distance between particle surfaces.
The dimensionless value of β depends on the size of the fillers. The diameter d of fillers sets the unit length. Assuming a filler diameter of 10 nm, a value of β ≈ 1nm results in a dimensionless value of 0.1. In the simulations, this value corresponds to strong interactions between the fillers and the matrix. The local glass transition temperature is plotted as a function of the distance to a filler particle for different values of the interaction parameter β in Figure 3. The thickness of the glassy layer around the fillers may thus be obtained from this plot at a given temperature and for a given interaction parameter.
Yield Behavior of Glassy Bridges
The value of the constant K to use in the simulations for describing the yielding of glassy bridges will be denoted Knum. When a stress is present in a glassy bridge, the constant Knum must allow for calculating the corresponding Tg shift (in Kelvin). Therefore Knum must satisfy the following equation:
where the superscript phys denotes real physical quantities, and the superscript num denotes quantities used in the simulations. For an applied deformation ε, the local numerical stress is σnum = k0ε, whereas the local physical stress is given by σphys ≈ G′gλε. The model proposed here is not a micromechanical model. The stress is computed at the scale of the simulation box while only forces are computed at the local scale. Thus, when solving the model, the local Tg shift is computed with the norm of the local interaction force between two neighboring fillers. This is appropriate as only deformations at constant volume (shear or unixial extension) are considered.
Given that the elastic constant of the numerical simulation is k0 ≈ G′gΣλ/G′rubber, Knum is given by
Assuming Σ = 0.05, G′rubber = 5 × 105 Pa and Kphys = 106 Pa K−1, we obtain Knum ≈ 0.1 K−1. For these systems, the Payne effect is expected for deformations εc such that k0εc/Knum ≈ 100 K. Thus, we obtain εc ≈ 10%. We will take Knum = 0.3 K−1 in this manuscript.
In the numerical simulations, viscous dissipation is accounted for by a friction coefficient ζnum, such that the viscous force between two neighboring particles 1 and 2 is written as
The relevant value of the friction coefficient ζnum has to be determined according to the physics of the considered systems. In ref.17, we have shown that the main contribution to the dissipation is due to the shearing of the polymer in between the fillers after the glassy bridges have yielded. After yielding, the viscous stress is , where η is the viscosity of the yielded polymer, which depends on the local strain rate . When considering real physical quantities with physical dimensions, the contribution of the dissipative force in eq (15) to the stress is , where d is the filler diameter or typical distance between fillers. In the numerical simulations, the stress is measured in units of the elastic modulus G′rubber of the pure rubber matrix above Tg. The expression for the friction coefficient ζnum in the numerical simulations states that the viscous stress (expressed in units of G′rubber) is the product of the friction coefficient ζnum by the shear rate . This relation thus reads: , where we have used the fact that the filler diameter d sets the unit length scale. Note that the friction coefficient ζnum has the dimension of a time and is expressed in seconds. The quantity is typically of order 107 Pa. We assume that Σ ≈ 10−2, , and G′rubber = 5.0 × 105 Pa. Thus, it results that ζnum, which will be denoted by ζ from now on, is of order a few seconds. Specifically, the value ζ = 4 s has been chosen in the simulations.
Parameters of the Simulations
The above discussion allows for determining the relations between the parameters of the model (β,K,ζ,k0,k∞) and the physical quantities and for determining the values to use in the simulations according to the regimes of interest. The time scale is set by the WLF law of the polymer matrix. Therefore, the time unit of the simulations are expressed in seconds and represent real physical time scales. The parameters and the corresponding values used in the simulations are summarized in Table 1. Note that some care should be taken to translate the volume fraction Φ used in the simulations to that in real physical systems reinforced with carbon black or silica aggregates of complex structures. In real systems, a change of behavior is often observed around a so-called percolation threshold, whose precise location depends on the considered system. In real systems, the distribution of distances between fillers may be expected to be broader than in the simulations, with fillers being able to come very close to each other (a few nanometers typically) even at a quite low volume fraction (of the order 7% in some cases12). We shall concentrate in what follows in strongly reinforced systems, with a high volume fraction (40% typically).
Table 1. Parameters of the Model
RESULTS AND DISCUSSION
To illustrate the Payne and Mullins effects, we will consider in the following a reinforced system, with a volume fraction Φ = 40%, a moderate polymer–matrix interaction β = 0.04,17 and a relatively low temperature T = 263 K. The WLF parameters have been taken to be those of polyisoprene with Tg = 213 K. The time step used is dt = 0.001 s and the breaking time cutoff τmin = 0.01 s. The other parameters are given in Table 1.
Stress curves during an oscillatory shear are shown in Figure 8 for different shear amplitudes. The evolutions of the storage and loss moduli as a function of the shear amplitude are shown in Figure 9. The modulus is measured after 10 shear cycles so that a steady state is probed. At small deformation amplitudes, the stress is practically in phase with the strain. In this regime, glassy bridges do not yield, and the response of the system is elastic, with a storage modulus G′(ω) larger than the loss modulus G″(ω). G′(ω) is controlled by the glassy springs having a life-time larger than the period Tperiod = 2π/ω ≃ 6.3 s.
On increasing the strain amplitude, the phase difference between the stress and the strain increases. A maximum for the loss modulus and a drop of the storage modulus for a strain amplitude of order a few percents are observed in Figure 9.
For deformations larger than 10%, the elastic modulus is brought back to a value, typically a few MPa. This drop is reminiscent of the Payne effect observed in carbon black elastomers by Payne.8 In both our systems and the experimental systems studied by Payne, this nonlinear effect occurs for relatively low deformation amplitudes (a few percents). We interpret the Payne effect in our systems as resulting from the progressive breaking of the glassy springs under the effect of the local stress. This breaking corresponds physically to the stress-induced yielding of the glassy layers surrounding the fillers. In the model, the value of G′ is directly related to the fraction of glassy bridges having a life-time larger than the characteristic time scale of the shear cycle Tperiod. The larger the amplitude of deformation, the larger is the fraction of glassy bridges which have broken, and the lower is the elastic modulus. It is also important to note that the Payne effect is accompanied by a significant dissipation, as measured by the loss modulus G″. For deformations comparable or larger than a few percents, the loss modulus G″ is larger than the storage modulus G′. In this regime, dissipation is determined by the breaking and rebirth of glassy springs, in a timescale comparable with the period Tperiod, and we interpret the existence of a maximum of the loss modulus in the model by this process. Note that a pronounced maximum of the loss modulus was also observed by Payne.8
Short-Time Recovery of the Mechanical Behavior
After the systems have lost their storage modulus after a small deformation, they recover their initial elastic behavior after a relatively short waiting time. We now study this recovery of the elastic response. We first consider the evolution of the systems after a first oscillatory strain of amplitude γ0. After approximately 10 cycles, the strain is maintained fixed at the value γ0 corresponding to the maximum of the cycles (Fig. 10). The evolution of the shear stress in this deformed state, for different values of γ0 is shown in Figure 11. The shear stress starts to decrease sharply at early times, t ≤ 5 s. Then it relaxes slowly toward a nonvanishing value, which increases with γ0. During this relaxation, the glassy springs that were broken during the first oscillatory deformation rebuild progressively, as shown in Figures 12–14. The life-time of the initially broken glassy springs increases with time. Note, however, that a substantial fraction of the glassy springs have not been broken during the first deformation. These latter do not break during the subsequent stress relaxation and keep their long life-times. Immediately after the deformed system is put at rest, a fraction of the glassy springs have broken, while the rubber springs, which are permanent, are stretched significantly. The glassy springs that were not broken during the first deformation are stretched in the deformed state and develop high local stresses. As aging takes place, new glassy bridges rebuild progressively in a new configuration. These new bridges allow the stress to relax in a configuration which might be very different from the initial one.
To probe the recovery of the elastic behavior of the model systems, the elastic modulus is measured in the deformed state, after various aging times tw. The small amplitude storage modulus is measured with oscillatory shear cycles of low amplitude γmax = 0.006. Only the three first cycles are considered, corresponding to a duration t = 3 × 2π/ω ≃ 19 s (Fig. 15). The resulting storage moduli for different values of the strain γ0 are shown in Figure 16. The most striking feature is that even for a waiting time tw = 1 s, the storage modulus is quite close to the low-amplitude modulus of the reference undeformed state, whose value is reported in Figure 9. For instance, for the deformation γ0 = 0.06, the storage modulus measured after tw = 1 s is G′ ≃ 37 MPa, which is quite close to the value G′ ≃ 40 MPa reported in Figure 9 in the absence of previous deformation. Also note that this value G′ ≃ 37 MPa is already larger than the storage modulus G′≃ 20 MPa measured for a shear amplitude γmax = 0.06 in Figure 9. The same holds for the various considered shear deformations γ0. Even for γ0 = 0.36, the small amplitude storage modulus is G′ ≃ 20 MPa, which is in a way larger than the initial Payne value measured for γmax = 0.36 and G′ ≃ 5 MPa (Fig. 9). Thus, the different systems analyzed are able to recover the elastic behavior of the undeformed initial state in a relatively short time. In particular, for deformation amplitudes γmax smaller than typically 10%, the systems acquire small strain mechanical properties, which are very close to those in the initial state. For larger deformation amplitudes γ0, the systems are somewhat less stiff than in the initial state. However, they develop larger storage moduli than a system submitted to a shear deformation γ0. This is due to the progressive rebuilding of glassy springs after a first deformation, which contributes to increase the storage modulus with time.
Plastic and Recovery Behavior
We now study the recovery of the mechanical behavior after long waiting times, up to tw = 105 s. After waiting tw = 104 or 105 s in the deformed state, we let the system reach a new reference state in the following way. We allow the stress σxy to relax to zero by progressively deforming the system with an elementary shear deformation dγxy = −Cσxy, where C is a constant taken here to be 10−3 MPa−1. The other components of the strain tensor are kept at the constant value characterizing the initial undeformed state. Each incremental deformation is followed by a relaxation at constant shape during 10 time steps. In this way, after a long time, the system reaches a state with zero stress and a permanent residual deformation γf (Fig. 17). The strain evolution in systems preliminary aged in a deformed state with different values of the deformation γ0 is shown in Figure 18. The strain initially drops steeply down to a finite value before decreasing slowly. Although the relaxation is only partial, a final residual strain γf at long times (here 8000 s), which characterizes the plastic deformation of the sample, may be defined. The value of the residual strain increases with the initial deformation γ0, and at a fixed value of γ0, it increases with the waiting time tw in the deformed state. Note that this residual strain is quite large: it can reach values of a few tenths of percents. Such a plastic behavior has also been observed in the absence of glassy springs,45 but the reported values of the residual strain were quite smaller than those observed here. This is the consequence of the presence of glassy springs. Indeed, when submitted to large amplitude deformations, the distribution of glassy bridge life-times shifts to smaller times. Then, when the deformation is maintained fixed, these life-times increase again (rebirth of glassy bridges) to reach a new permanent distribution, which is not different from the initial one (before any deformation was applied) when the deformation amplitude γ0 is smaller than ∼ 10%. However, glassy bridges have rebuilt in a very different microscopic configuration (Fig. 19), leading to a residual deformation γf when the stress is released. Indeed, the rubbery network might be stretched when compared with the initial undeformed state, but the corresponding contribution to the stress is compensated by the new configuration of the glassy bridges, which are much stiffer. We thus predict an enhancement of the plastic behavior of filled elastomers in the strongly reinforced regime when compared with a regime in which no glassy bridges would be present, for example, at higher temperature or lower filler–matrix interaction.
Figure 20 shows the stress relaxation (same data as in Fig. 18) plotted in the stress–strain plane. The stress starts to relax steeply in a relatively short time. During this initial stress relaxation, the strain has practically not relaxed and keeps a value close to γ0. Later on, both stress and strain relax toward a state of vanishing stress. This state corresponds to a nonzero residual value of the strain, as explained above. The distribution of breaking times for the different values of γ0 is given in Figure 21.
Relaxation curves as shown in Figure 20 are indeed typical of thermoplastic elastomers,52 in which the buckling process illustrated in Figure 19 eventually leads to a large amplitude plasticity. However, plasticity is present as well and has been studied in filled elastomers submitted to repeated large amplitude sollicitation, even though typical amplitudes of residual deformations (often called “permanent set” in the context of filled rubbers) are smaller in this case.53
We analyze now the mechanical and recovery behavior of the model systems in the new deformed state, corresponding to various preliminary deformation amplitudes γ0. In these systems, the strain has relaxed to the residual value shown in Figure 18. Then, the systems were oscillatorily sheared with different shear amplitudes γmax (Fig. 22). The corresponding storage moduli, measured after 10 cycles as previously, are shown in Figure 23 as a function of the shear amplitude γmax. The Payne curve shown in Figure 9, which corresponds to the absence of past deformation, is also represented in Figure 23. Clearly, all the systems considered exhibit a Payne effect, that is, a drop of the modulus as γmax increases. For an initial deformation γ0 smaller than 10%, the values of the storage moduli in the new state are practically undistinguishable from those measured in the initial undeformed state. Hence, the mechanical properties in the deformed state have almost completely recovered: both states are almost undistinguishable from a mechanical view point. When the initial deformation γ0 is larger than about 10%, the amplitude of the subsequent Payne effect decreases. However, for subsequent deformations γmax larger than 10%, the different curves merge: the large amplitude deformation behavior does not depend much of the history of the considered samples.
The loss moduli are plotted in Figure 24 as a function of the strain amplitude, in systems initially deformed at different amplitudes. The loss moduli of a reference system which has not undergone preliminary deformations is also shown for comparison. All the systems considered show a peak of dissipation for an amplitude of a few percents. In the case of moderate preliminary deformations γ0 < 0.1, the values of the loss moduli are quite close to that measured in the initial reference system. Hence the dissipative properties in the deformed state are the same than those in the undeformed reference state. In systems that have been preliminarily deformed at larger amplitudes, both the small amplitude and peak values of the loss modulus are somewhat smaller than the values in the undeformed reference state.
Recently, Sternstein and coworkers have studied the influence of the mechanical history on the mechanical behavior of filled elastomers.47 In particular, they have considered the effect of a finite static strain on the storage and loss moduli of their filled systems. They have shown that the application of a static strain has no effect on the storage and loss moduli of their samples. However, note that the amplitude of static strain considered was at most 10%. Also, the samples have been allowed to equilibrate several hours after the application of a static strain. They have shown that modulus recovery requires thousands of seconds. These experimental observations may be interpreted with our results. Just after a first deformation, some glassy bridges have been broken, and the system has lost part of its stiffness. As the deformation is maintained fixed, the broken glassy bridges will rebuild slowly. The distribution of breaking times will evolve slowly toward the equilibrium one. Such an aging process is shown in Figures 12–14.
In particular, the number of glassy springs having a breaking time larger than a characteristic time scale (the period Tperiod = 2π/ω of a shear cycle) will increase with time. Even if the distribution of breaking times has still not converged toward the equilibrium one, the number of glassy springs having a breaking time larger than, for example, 1 s may become comparable with the corresponding number in the equilibrium state. As the corresponding glassy bridges mainly contribute to the stiffness of the system, the storage modulus may have recovered its value in the undeformed reference state. This is what we observe for long deformations of the order of 10%, if we let the system age during a few tens of seconds. This is also what is observed in the experiments of Sternstein and coworkers.47 For static deformations larger than 10%, we observe only a partial recovery in the model systems. Full recovery would necessitate longer waiting times so that the population of long-lived glassy springs can be reconstructed. Experimentally, it would be interesting to study the effects on the mechanical properties of preliminary strains of amplitudes larger than 10%. In particular, the ability of our model to explain and/or predict the recovery effects described by Sternstein and coworkers47 is a strong argument in favor of the explanation of reinforcement as a consequence of the presence of glassy layers around the fillers, as opposed to other explanations for example, entanglement effects in the vicinity of the fillers.54 Note that Roland and coworkers55, 56 did not observe the presence of such layers by DSC. However, the associated DSC signal is expected to be quite weak and might be spread over a large temperature range. Moreover, the particles used in refs.55 and56 were 100 nm diameter, which is large, reducing even further the effect.
We have described a model for the reinforcement of filled elastomers, which should also be valid for more general nanocomposites with hard inclusions dispersed in an elastomeric matrix such as some thermoplastic elastomers. It allows for describing the behavior of these systems both in the linear and nonlinear (nondestructive) regimes, which are the so-called Payne and Mullins effects. This model is based on the presence of glassy layers around the fillers. The presence of overlapping glassy layers results in high level of stress between fillers, with finite life-times depending on the history, on temperature, on the distance between fillers, and on the local stress in the material. We show how the dynamics of yield and rebirth of glassy bridges account for the nonlinear behavior and dissipative properties of filled elastomers. The storage modulus is determined by the fraction of glassy bridges with life-time comparable with or larger than the period of the oscillatory deformation, and the loss modulus is dominated by the fraction of glassy bridges with life-times comparable with the deformation period. The populations of glassy bridges also depend on the amplitude of the deformation.
We have studied the behavior after large deformations, for which the storage modulus has dropped significantly. Putting the system at rest after a first large deformation, we have considered the evolution of the storage and loss moduli with time. We have shown that for initial deformations smaller than typically 10%, the deformed systems have elastic properties almost equal to those in the reference undeformed state. The Payne curves characterizing the undeformed and the deformed states are identical. For small deformations, we have also shown that the recovery of the initial elastic properties requires relatively short waiting times, of the order of a few seconds. For larger deformations, the recovery is only partial. However, the large strain behavior of the undeformed and deformed systems is the same. Also, preliminarily deformed systems develop a small strain modulus, which is much larger than the Payne modulus corresponding to the same deformation amplitude. All these results allow to interpret the experiments of Sternstein and coworkers, who have found no effect on the storage and loss moduli of a static strain smaller than 10% in filled elastomers. We interpret these experimental results in the following way: after a first deformation, a fraction of the glassy bridges have been broken because of the high levels of stress. When the deformed system is put at rest, the broken bridges undergo an aging process and start to rebuild slowly. Hence, the distribution of breaking times converges toward the equilibrium one, the fraction of glassy springs having a breaking time larger than a characteristic time scale increases with time, and thus the system stiffens and recovers the elastic behavior of the initial undeformed state. The simulations show that this recovery process may be very short if the initial deformation is not too large, that is, smaller than 10% typically. Finally, we have shown that the model systems show plastic behaviors characterized by large values of the residual strain. This plasticity is enhanced when compared with situations where glassy springs would not be present.