Unique plastic and recovery behavior of nanofilled elastomers and thermoplastic elastomers (Payne and Mullins effects)

Authors

  • Samy Merabia,

    Corresponding author
    1. Laboratoire de Physique de la Matière Condensée et Nanostructures, Université Lyon I/CNRS, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cédex, France
    • Laboratoire de Physique de la Matière Condensée et Nanostructures, Université Lyon I/CNRS, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cédex, France
    Search for more papers by this author
  • Paul Sotta,

    1. Laboratoire Polymères et Matériaux Avancés, CNRS/Rhodia, 85 avenue des Frères Perret, F-69192 Saint-Fons, France
    Search for more papers by this author
  • Didier R. Long

    Corresponding author
    1. Laboratoire Polymères et Matériaux Avancés, CNRS/Rhodia, 85 avenue des Frères Perret, F-69192 Saint-Fons, France
    • Laboratoire Polymères et Matériaux Avancés, CNRS/Rhodia, 85 avenue des Frères Perret, F-69192 Saint-Fons, France
    Search for more papers by this author

Abstract

We have proposed recently that the mechanical properties of nano-filled elastomers are governed by the kinetics of rupture and re-birth of glassy bridges which link neighboring nanoparticles and allow for building large rigid clusters of finite life-times. The latter depend on parameters such as the temperature, the nanoparticle-matrix interaction, and the distance between neighboring fillers. Most importantly these life-times depend on the history of deformation of the samples. We show that this death and re-birth process allows for predicting unusual non-linear and plastic behavior for these systems. We study in particular the behavior after large deformation amplitude cycles. At some point we put the systems at rest under large deformation, and let the stress relax in this new deformed state. During this relaxation process the life-time of glassy bridges increases progressively, even for large deformation states. The systems thus acquire a new reference state, which corresponds to a plastic deformation. The stretching energy of the polymer strands of the rubbery matrix is larger than in the initial undeformed state, but this effect is compensated by a new configuration of glassy bridges, which are much stiffer. For plastic deformations of less than about 10%, the new system acquires mechanical properties around this new reference state which are very close to those of the initial system. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1495–1508, 2010

INTRODUCTION

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.

Figure 1.

The loss modulus G″ (in Pa) measured in carbon black-filled butyl rubbers as a function of the deformation amplitude and three different filler volume fractions: □: Φ = 38.6%; ○: Φ = 33.6%; •: Φ = 23.2%.8

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.

Figure 2.

At low volume fractions and/or at high temperatures, the glassy layers around filler particles (light gray dashed circles) do not overlap. At lower temperature, they do overlap and build glassy bridges between fillers (plain white circles).

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

equation image(1)

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

equation image(2)

For the sake of simplicity, we assume from now on that the Tg shift at a distance z is given by

equation image(3)

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.

Figure 3.

The local glass transition temperature Tg(z) as a function of the distance z to the surface of filler particles, in units of the particle diameter, as given by eq (1), for different values of the matrix–filler interaction parameter β: β = 0.02, 0.04, 0.06, and 0.08 (from bottom to top curve). When Tg(z) is equal to the temperature T of the experiment, the distance z is equal to the glassy layer thickness eg. The dashed line is the Tg of the pure rubber. The glassy layer thickness eg at T = 293 K for β = 0.08 is indicated.

When glassy layers overlap, the macroscopic shear modulus G′ is related to the shear modulus of the glassy polymer Gg 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

equation image(4)
Figure 4.

The macroscopic stress is supported by the glassy polymer fraction which bridges two neighboring filler aggregates. Aggregates of about 100 nm made of primary particles of 10 nm are schematically represented here. Aggregates are surrounded by a glassy layer which is roughly represented. The fraction of glassy polymer in a section normal to the applied stress is Σ ∼ 1 %. The macroscopic deformation is amplified in between fillers by a factor typically λ ∼ 10, which results in a macroscopic modulus G′ of order 107 to 108 Pa.

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 Gg ∼ 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:

equation image(5)

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(TgT). 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 εGg ∼ 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 TTg).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

equation image(6)

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

Figure 5.

Breaking time (or life-time) of glassy bridges as a function of the distance z to the particle surface (in units of the particle diameter) for different values of the matrix–filler interaction parameter β: β = 0.02, 0.04, 0.06, and 0.08. The life-time is given by eq (6).

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

equation image(7)

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 equation image, where equation image denotes the local deformation rate. In practice, typical deformation rates in our simulations will be of order equation image. 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

equation image(8)

if

equation image(9)

By definition, the time τα is bounded by the time τWLF(TTg(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(TTg(z,σ)) is small when compared with an experimental time scale ≃ equation image. 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(TTg(z,σ)) is comparable or even larger than ≃ equation image. 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:

equation image(10)

where equation image is the elastic force between two neighboring particles and is the sum of the contribution of the glassy bridges and of the rubbery matrix. equation image is the hardcore repulsion, and equation image 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.

Figure 6.

The probability density P(ri) for a given bond (spring) i to have a length ri in the initial equilibrium state for different values of the volume fraction and the connectivity (number of springs per bead): plain curve: Φ = 0.40, n = 12; Δ: Φ = 0.20, n = 12; ∇: Φ = 0.20, n = 8.5. The dotted curve is the distribution in the system Φ = 0.40, n = 12, at a shear deformation of amplitude γmax = 2.0. In the concentrated systems, the two maxima correspond to the shells of first and second neighbours, respectively.

Elastic Forces

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

equation image(11)

l0 is the equilibrium length of the springs, equation image is the vector joining the centers of the two particles, and equation image is the unit vector joining the two particles. The force due to the glassy bridges reads

equation image(12)

where equation image and equation image is the vector position of filler i. The reference state equation image 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 equation image. 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.

Figure 7.

Modeling of a filled elastomer in the strong reinforcement regime. Two neighboring filler particles interact via two distinct forces, which correspond to (a) permanent springs of stiffness k = 1 representing the rubber matrix contribution, and (b) nonpermanent springs due to the glassy bridges of much larger stiffness k0 ∼ 100, which have finite life-times depending on the local history, the local stress at earlier times, and the local glass transition temperature. When the local stress increases it can lead to a breaking of a glassy bridge and the fillers take a new reference relative position Rmath image.

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.

Filler–Matrix Interaction

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:

equation image(13)

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 σphysGgλε. 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 k0GgΣλ/Grubber, Knum is given by

equation image(14)

Assuming Σ = 0.05, Grubber = 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.

Friction

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

equation image(15)

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 equation image, where η is the viscosity of the yielded polymer, which depends on the local strain rate equation image. When considering real physical quantities with physical dimensions, the contribution of the dissipative force in eq (15) to the stress is equation image, 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 Grubber 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 Grubber) is the product of the friction coefficient ζnum by the shear rate equation image. This relation thus reads: equation image, 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 equation image is typically of order 107 Pa. We assume that Σ ≈ 10−2, equation image, and Grubber = 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
inline image

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.

Payne Effect

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.

Figure 8.

Shear stress during sinusoidal cycles of shear amplitudes γmax = 0.6, 0.06, 0.025 and 0.006, and pulsation ω = 1 s−1. The dashed curve is in phase with the shear strain γ(t) = γ0 cos(ωt). Note for the larger shear amplitude considered, the stress contains harmonics nω, with n ≥ 2. The parameters are Φ = 0.4, β = 0.04, K = 0.3, and the temperature is T = 263 K. The WLF parameters are those of PI with Tg = 213 K.

Figure 9.

Storage (○) and loss (□) moduli as a function of the shear amplitude γmax, measured in oscillatory shear at a pulsation ω = 1 s−1. tan δ (⋄) vs γmax (magnified by a factor 5) is also plotted. Same parameters as in Figure 8.

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.

Figure 10.

Relaxation of the shear stress of systems maintained in a deformed state (with a fixed value of the shear strain γ0) after an oscillatory deformation of amplitude γ0 has been applied during n cycles (n = 10 typically).

Figure 11.

Relaxation of the shear stress of systems maintained in a deformed state, with various fixed value of the shear strain γ0: γ0 = 0.5, 0.36, 0.1, and 0.06. Before maintaining the strain fixed, the different systems were oscillatory sheared with an amplitude γ0 during 10 cycles with a pulsation ω = 1 s−1 (Fig. 10). Same parameters as in Figure 8.

Figure 12.

Distributions of relaxation times for systems aged during different values of time tw in a deformed state with a fixed strain γ0 = 0.06. The dashed curve corresponds to the initial distribution before shearing.

Figure 13.

Distributions of relaxation times for systems aged during different values of time tw in a deformed state with a fixed strain γ0 = 0.12. The dashed curve corresponds to the initial distribution before shearing.

Figure 14.

Distributions of relaxation times for systems aged during different values of time tw in a deformed state with a fixed strain γ0 = 0.36. The dashed curve corresponds to the initial distribution before shearing.

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.

Figure 15.

Measurement of the small amplitude storage and loss moduli of systems having aged in a deformed state for different values of the fixed strain γ0. The moduli depend on the aging time tw spent in the deformed state.

Figure 16.

Small amplitude storage modulus of systems having aged in a deformed state, under different values of the fixed strain γ0, as a function of the aging time tw spent in the deformed state (see Fig. 15). From top to bottom: γ0 = 0.06, 0.12 and 0.36. The small amplitude storage modulus was measured using oscillatory shear cycles (shear amplitude 0.005, pulsation ω = 1 s−1. Only the first three cycles have been used to measure the storage modulus, which corresponds to a time span of ≃20 s. Note that after this period of time, the storage modulus measured is significantly larger than the modulus corresponding to the Payne effect (compare with Figure 9). Same parameters as in Figure 8.

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 17.

Evolution of the strain during the stress relaxation of systems which were preliminary allowed to age during a time tw = 104 or 105 s at different fixed values of the strain γ0. At long times, the system relaxes to a state of zero stress and deformation γf < γ0, which defines the plastic deformation of the system (see, e.g., ref.45).

Figure 18.

Evolution of the strain during the stress relaxation of systems which were preliminary allowed to age during a time tw = 104 or 105 s at different, fixed values of the strain γ0 (Fig. 17). For each value of γ0, we have considered two values of the aging time tw = 104 and tw = 105 s. At a given value of γ0, the curve corresponding to tw = 104 s is below the curve corresponding to 105 s. The different values of γ0 are, from top to bottom, γ0 = 1.2; γ0 = 0.5; γ0 = 0.36; γ0 = 0.25; and γ0 = 0.12.

Figure 19.

Schematics showing the distribution of glassy bridges in the system. Only glassy bridges between close-by particles have a long relaxation time, that is, are effective. Effective glassy bridges are indicated by thick black lines. Elastomeric springs are indicated in light grey. After shearing, glassy bridges are rebuilt in a different configuration. For instance, particle 1 interacts (via glassy bridges) with particles 4, 5, and 6 in the initial state and with particles 2, 3, and 6 in the final state; particle 6 has undergone a buckling in which its local environment has changed.

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.

Figure 20.

Relaxation of the shear stress (same as in Fig. 18) plotted in the shear stress/shear strain plane.

Figure 21.

Distribution of relaxation times in systems deformed at different values of γ0 during 105 s, after having relaxed their shear stresses (Fig. 17). From top to bottom, the values of γ0 are as follows: γ0 = 0.12, 0.36, 0.5, and 1.2. The dashed curve corresponds to the initial distribution prior to any deformation.

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.

Figure 22.

Measurement of the storage and loss moduli at different shear amplitudes, γmax, in systems aged during tw = 105 s at different fixed values of γ0 and in which the stress was subsequently relaxed (as described in Fig. 17).

Figure 23.

Storage modulus as a function of the shear amplitude γmax for systems which have been preliminary deformed during tw = 105 s at different fixed values of γ0: γ0 = 0.12 (⋄)); γ0 = 0.25 (Δ); γ0 = 0.36 (∇); γ0 = 0.5 (×), and γ0 = 1.2 (☆) (Fig. 22). The curve denoted γ0 = 0 (□) refers to the Payne curve shown in Fig. 9, obtained in the absence of past deformation. Just after the long deformation at γ0, and prior to the second oscillatory deformation, the shear stress of the different systems has been relaxed, as displayed in Figure 20. Same parameters as in Figure 8.

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.

Figure 24.

Loss modulus as a function of the shear amplitude γmax for systems which have been preliminary deformed during tw = 105 s at different fixed values of γ0: γ0 = 0.12 (⋄)); γ0 = 0.25 (Δ); γ0 = 0.36 (∇); γ0 = 0.5 (×), and γ0 = 1.2 (☆) (Fig. 22). The curve denoted γ0 = 0 (□) refers to the Payne curve shown in Fig. 9, obtained in the absence of past deformation. Just after the long deformation at γ0, and prior to the second oscillatory deformation, the shear stress of the different systems has been relaxed, as displayed in Figure 20. Same parameters as in Figure 8.

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.

CONCLUSIONS

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.