Effect of Filler Morphology and Distribution State on the Linear and

Article pubs.acs.org/Macromolecules

Effect of Filler Morphology and Distribution State on the Linear and Nonlinear Mechanical Behavior of Nanofilled Elastomers Mathieu Tauban, Jean-Yves Delannoy, Paul Sotta, and Didier R. Long* Laboratoire Polyméres et Matériaux Avancés, CNRS/Solvay UMR 5268, 87 avenue des Fréres Perret, F-69192 Saint-Fons, France S Supporting Information *

ABSTRACT: We extend the model proposed by Merabia et al. [2008] regarding reinforcement mechanisms of filled elastomers. This model is based on the presence of glassy layers around filler particles which may bridge neighboring particles. The model was solved by Merabia et al. for spherical particles only and for a single dispersion state. However, experiments show that mechanical properties depend crucially on the complex shape of fillers as well as on their spatial distribution. We consider extensively both aspects in this article. We show that the distribution state of the fillers is key for controlling the reinforcement at high temperature. For a given distribution state, we show a strong effect of filler morphology on reinforcement. Distances between fillers are smaller with fractal aggregates, which leads to stronger reinforcement. Our model opens the path for the development of systems with tailored properties by tuning the filler distribution state and morphology. amplitude 1%.12 In carbon black-filled materials, this behavior is associated with a drop of the electrical conductivity.3 These features are of primary importance in tire applications, as regards to rolling resistance, grip, and durability.13 Over the past decades, many studies have been focused on the effect of the filler volume fraction. The strength of filler− matrix interactions has been investigated for carbon black and silica aggregates.5,7 Conversely, studies regarding the effect of the filler morphology and distribution state are scarce.15−17 From an experimental point of view, it is difficult to discriminate the effect of filler distribution and morphology from the effect of other parameters such as surface chemistry, filler matrix interactions, or even compounding process. Berriot et al.14,18 developed model filled rubbers by incorporating spherical silica particles in an ethyl acrylate matrix. The spherical particles were highly monodisperse, and the synthesis route allowed tuning the distribution state of the fillers.18 They studied the evolution of the elastic modulus G′ under an oscillatory shear of frequency 1 Hz of increasing amplitude above the glass transition temperature in the reinforcement regime (Tg + 50 K). The elastic modulus of the sample with a good distribution state remained mostly constant as a function of strain, while it dropped by about 20% in the sample with a nonhomogeneous distribution state.15 This demonstrates that the amplitude of the Payne effect is related to the distribution state of the filler. Scotti et al.19 studied the role of particle morphology on the filler reinforcing effect. They observed a relation between the

I. INTRODUCTION Composite materials based on rubber and nanoparticles (NPs) are of a great practical relevance because they exhibit significantly improved performances in comparison to pure polymer matrices.1 Adding NPs in a rubbery polymer matrix, even at a relatively small volume fraction, has considerable impact on electrical conductivity,2−4 mechanical strength,5 and stiffness.6 Elastomers filled with carbon black or silica particles may have an elastic modulus up to a few hundred times higher than that of the pure elastomer, exhibit high dissipative efficiency, and may be extremely resistant to tear and wear. These characteristics have been widely used for materials such as tires, shock absorbers, and impact modifiers.1−3,7−13 The reinforcement R = G′(ϕ)/G′ may be defined as the ratio of the filled elastomer modulus over that of the pure matrix at the same temperature. In the strongly reinforced systems studied by Payne,12 the maximum reinforcement peaked up to 200. In industrial samples, this quantity can reach 30 in the systems studied by Wang.5 For model systems with welldistributed spherical particles, R exhibits a narrow peak of magnitude about 100 followed by a step decrease as temperature increases.14 Another important feature of these materials is their nonlinear behavior in the nondestructive regime. Under oscillatory shear strain at a frequency of the order typically 1 Hz and of increasing amplitude of the order of a few percent or more, the elastic modulus drops down to values much smaller than the linear modulus. This is the so-called Payne effect. In some of the systems studied by Payne, the elastic modulus G′ dropped from a few 107 Pa to a few 106 Pa.3 This drop of elastic modulus is associated with a peak in the loss modulus G″ that exhibits a peak up to a few 106 Pa at deformations of typical © XXXX American Chemical Society

Received: May 10, 2017 Revised: August 13, 2017

A

DOI: 10.1021/acs.macromol.7b00974 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 1. Simulated SEM images of systems filled with spherical particles of diameter D = 1 at a volume fraction ϕ = 0.2 with a varying distribution state parameter rc as indicated. The numerical method for controlling the distribution state is described in section III.A, and the procedure for obtaining such images is based on a ray tracing method as described in ref 27.

Figure 2. Simulated TEM images of thickness 1 D of systems homogeneously filled with aggregates of various number of primary particles as indicated. In each system, the envelope size of the fillers is fixed to 1 D, which can be mapped to 60 nm. The method for obtaining such simulated TEM images is based on ray tracing and considering a Beer−Lambert absorption model as described in ref 27.

providing a good dissipation power only in a limited frequency range. To contribute to this aim, we extend the theoretical Glassy Bridge Reinforcement Model (GBRM)20−22 and consider the effects of both complex filler morphologies and spatial distribution on reinforcement. It was proposed over the past 15 years14,20,23 that a thin layer of polymer surrounding nanoparticles exhibit mechanical properties different from those of the bulk. This change of properties may be expressed by a shift of the glass transition temperature that occurs in the vicinity of filler surface.23−25 Neighboring filler particles may be connected by polymer glassy bridges to build a large scale skeleton that reinforces the nanocomposite. Under applied strain, these glassy bridges may yield. The dynamics of yield and rebirth of glassy bridges allows for explaining the Payne effect and the recovery properties (Mullins effect) of filled elastomers.21,22 This model allows to

elastic modulus in the linear regime and the aspect ratio of the fillers. Fillers with higher aspect ratios and/or tenuous structures were related to higher elastic modulus in the linear regime and a higher Payne effect amplitude. It was also found that the dependence on the aspect ratio was amplified by increasing the volume fraction of the fillers over 20 phr. This effect was attributed to the formation of a continuous percolative network of silica nanoparticles connected by thin polymer films and invoked the larger interface polymer/filler for the high-aspect-ratio fillers, resulting in a higher interaction strength between the fillers and the matrix. The dependence of the Payne effect amplitude on the distribution state of the filler has not been investigated in a systematic way so far. Understanding the effect of filler morphology would allow new perspectives for the development of fillers having both the ability to reinforce the matrix while B


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the filler surface and the matrix and β ∼ 1 nm −2 nm20 in a system with strong interactions. This latter value will be used in our simulations. For simplicity we shall assume from now on that ν = 1. II.B. Effect of an Applied Stress. When a reinforced system is submitted to an applied strain, the stress is locally concentrated in the confined polymer between filler particles. The local yield stress σy of a glassy bridge is proportional to the difference between the local Tg and the temperature T. The yield stress can be written σy = K(Tg(z) − T), where Tg(z) is the local glass transition temperature computed in the previous section and K a parameter that depends on the polymer. This parameter is of the order 106 Pa/K typically.28 The relation can be rewritten to express the local Tg as a function of the confinement z and of the local stress σ:

give a semiquantitative prediction of the mechanical behavior of filled elastomers in terms of reinforcement and Payne effect,20 plasticity and recovery behavior,21 or dissipation in the nonlinear, nondestructive regime.22 Up to now, fillers have been described as spherical particles distributed randomly in space. More complex aggregates have not been considered. The specific effect of the spatial distributions has not been studied, though it is key for the mechanical properties. It is essential to consider fillers as more complex objects with a multiscale structure as investigated in ref 26. Hence, the aim of this article is to extend the GBR model to the case of complex aggregates and to study in detail the effect of the spatial distribution of the fillers on the reinforcement properties. The new developments allow finely tuning the distribution state of the fillers and the filler morphology in the systems and studying reinforcement properties of systems close to real systems such as those used for making tires. In order to be specific and identify as clearly as possible the cause of the various effects we study, we will consider filler aggregates of identical overall sizes, but of different morphologies and dispersion states. We choose this representative size to be 60 nm. We will vary the diameter d and number np of primary particles contained in an aggregate. The typical fractal dimension df will be close to 2. Thus, a typical aggregate of diameter D = 60 nm shall contain a number N of order (D/d)df of primary particles. We will vary np from np = 1 (spherical particles of diameter 60 nm) up to 40. The latter value corresponds to roughly fractal aggregates made of primary particles of diameter about 10 nm. Therefore, we shall first consider spherical particles with various dispersion states as illustrated in Figure 1. We will explain later in the article how the dispersion state is controlled by the parameter rc. Then, we will consider the effect of the morphology and study the reinforcement properties of systems such as thoses depicted in Figure 2. The number np is the number of primary particles per aggregate of overall size 60 nm. The article is organized as follow. The physical model based on earlier works,20−22 and its new extension is described in section II. The numerical implementation of the model is presented in section III. Results are presented and discussed in section IV. The effects of the filler distribution state and of the filler morphology are successively studied.

⎛ β⎞ σ Tg(z , σ ) = Tg ⎜1 + ⎟ − ⎝ ⎠ z K

Let us introduce a local dominant relaxation time τα(r) that can be defined for any position r of the volume Vf occupied by the polymer matrix. We assume that the local dominant relaxation time τα(r) of the polymer is bounded by the relaxation time at equilibrium τWLF(r) given by the William−Landel−Ferry (WLF) law of the corresponding polymer modified by the local Tg(r) due to interfacial effects. Thus, the local relaxation time at equilibrium is given by ⎛ τ (r) ⎞ C1(T − Tg(r, σ )) log⎜⎜ WLF ⎟⎟ = − C2 + (T − Tg(r, σ )) ⎝ τg ⎠

Tg

⎛β⎞ ⎜ ⎟ ⎝z⎠

∂τα (r , t ) = 1 ∂t

(4)

while it is bounded by the equilibrium value τWLF(r, σ) discussed earlier:

1/ ν

=

(3)

where τg is the relaxation time at Tg of the polymer. C1 and C2 are the WLF parameters of the considered polymer. The local Tg in the absence of stress at position r in the polymer matrix is expressed as a function of the distance z between position r and the nearest filler surface and is given by eq 1, which we approximate by Tg(z) = Tg(1 + β/z). Let us consider an elastomer with a glass transition temperature in the bulk Tg = 213 K (case of polyisoprene) and an interaction parameter β = 1.8 nm. At a distance z = 5 nm (equivalent to a surface to surface distance of 10 nm between fillers) and at temperature T = Tg + 60 K = 273 K, the local relaxation time of the polymer is of order τWLF(z) ∼ 107 s, which may be obtained only after a long aging time, whereas the equilibrium relaxation time of the same polymer in the bulk at the same temperature is τWLF(∞) ∼ 10−2 s. Before reaching its equilibrium value, the local relaxation time, denoted τα(r,t), depends on the local history. The polymer undergoes aging, during which the relaxation time follows the Struik law20 given by τα ∝ twμ where tw is the waiting time since the last relaxation and μ ≈ 1 the aging exponent. Thus, the evolution of the local relaxation time τα(r,t) is given by

II. PHYSICAL MODEL In this section we recall the basic ingredients of the physical model originally described in refs 20−22. II.A. Effect of Polymer Confinement. As supported by a recent study,25 the key assumption of the GBR model is that the increase ΔTg of glass transition temperature Tg of the polymer confined between filler surfaces is at the origin of the reinforcement and nonlinear properties of filled elastomers. This increase can be expressed as a function of the distance z from the filler surface using the relation derived by Long and Lequeux in 2001:24 ΔTg

(2)

(1)

τα(r, t ) ≤ τWLF(r, σ )

where the exponent ν ≈ 0.88 is the critical exponent for the 3D correlation length in percolation. The parameter β accounts for the strength of polymer matrix interactions. The typical value is β ∼ 0.1 nm25 in a system with a moderate interaction between

(5)

As specified in eq 2, the equilibrium relaxation time τWLF(r) decreases as the local stress increases or the particles are brought away from each other. To describe aging in the C


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Macromolecules simulations, we assume that the local relaxation time τα(r,t) follows any decrease of τWLF(r) and then increases again progressively according to eq 4 as the stress decreases or particles get closer to each other. According to the definition of the relaxation time τα, at any time, the polymer has a probability for relaxing per unit time dP/dt given by dP = α

dt τα(t )

(6)

where α is a number of order 1 (but smaller than 1) that allows local aging of the polymer.20,25 Figure 3. Degrees of freedom in the simulation. Randomly generated aggregates inserted in the simulation box are defined by a position and orientation with respect to cell frame. Interactions (springs, soft- and hard-sphere repulsion) act at the scale of primary particles. The equations of motion are solved at the scale of rigid body aggregates. Blue arrows affixed to aggregates picture aggregation orientation. Thick lines denote pairwise interactions between primary particules, as detailed in the zoomed right-hand side cartoon. Spring interactions may be glassy (blue) or rubbery (gray) depending on the distance, local stress, and history of the particular bond.

III. IMPLEMENTATION OF THE MODEL To solve the model, we have developed a mesoscale simulation tool where the degrees of freedom are the centers of mass and orientations of the filler aggregates distributed in the matrix. A filler is defined as a rigid set of np spherical primary particles. The rubbery elasticity of the matrix is accounted for by harmonic springs connecting primary particles. The rubbery matrix also exerts a hydrodynamic friction on the aggregates. In section II we have considered the relaxation time of the matrix τα(r) for any position in the matrix. To simplify this picture, we only consider the relaxation time of the polymer confined in between fillers. A rigid bridge will be modeled by a harmonic spring with specific relaxation time (the spring can break) and strength. While the equations of motion are solved at the scale of the fillers, the forces are defined between primary particles that belong to distinct aggregates. Forces between primary particles are of two kinds. The first category corresponds to both the entropic contribution of the elastomer matrix and to the (nonentropic) contribution of glassy bridges between primary particles. The second one corresponds to repulsion forces between primary particles. III.A. Hard- and Soft-Sphere Repulsion. The hard-sphere force is approximated by the repulsive part of a Lennard-Jones potential of the form VRep = εrij−12, where rij is the center-toij center distance between particles i and j, as depicted in Figure 3. This potential is set to zero for distances rij > (di + dj)/2, where di (dj) is the diameter of particle i (j), so that the resulting force cancels for nonoverlapping particles. We also consider an other interaction that acts only during the equilibration of a system. To obtain a distribution of nonoverlapping complex fillers in a reasonable computational time, the systems are constructed using a modified Lubachevsky−Stillinger procedure.29 Complex fillers are randomly distributed in space (random position and random orientation) at a very small volume fraction (ϕ ∼ 10−5). A repulsion potential is applied between primary particles that belong to distinct aggregates, and the forces are relaxed by solving the equation of motions for the position and orientation of the aggregates until the forces cancel. The repulsion potential is set to zero for distances rij − (di + dj)/2 > rc, where rc is called the screening parameter. When the forces are relaxed, the system undergoes a homothetic compression leading to an increase of the volume fraction and followed by the relaxation of the repulsion forces. This procedure is repeated until a given volume fraction ϕ is reached. When the system is equilibrated at a volume fraction ϕ, the long-ranged repulsion is removed, and only the hard-sphere repulsion remains active during the subsequent simulations. More details regarding the applied potential are given in the Appendix (Supporting Information).

In this way, the distribution state of particles can be tuned by changing the minimum distance of approach (screening parameter rc). III.B. Elasticity of the Matrix. In order to model the elasticity of the matrix, a permanent network of springs is connected between primary particles. We approximate the elastic behavior of the rubbery matrix by harmonic springs. Let us discuss how to map the strength of a harmonic spring on the elastic modulus of the matrix element it represents. The size of a primary particle is not a constant of our model. The harmonic spring between two particles represents the local force the matrix would sustain. Let us consider a small volume element in the polymer comprised between two filler surfaces. The initial length of the matrix element between fillers is h0. Because of filler relative displacement, the length h0 becomes h, which corresponds to a local deformation ϵl = (h − h0)/h0. The polymer element supports a local stress in the direction of deformation: σ = G ϵl

(7)

with G the elastic modulus of the polymer element. The equivalent force may be estimated by considering a disk S of diameter d perpendicular to the line joining the center of particles i and j and located in between. The disk area is πd2/4 ≈ d2 (see Figure 4). Thus, the equivalent local force reads f = d 2Gϵl = d 2G

h − h0 h0

(8)

The local force in eq 8 is similar to the one of a harmonic spring connecting particles i and j with a strength k: k=

d 2G h0

(9)

with a length at rest h0. Therefore, eq 9 shows that the strength of a spring needs to be defined locally in a simulation. As a result, the local force between two primary particles i and j is calculated as f ijRE(t ) = −kij∞(rij(t ) − rij(0)) D

(10) DOI: 10.1021/acs.macromol.7b00974 Macromolecules XXXX, XXX, XXX−XXX

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Figure 4. Mapping between a polymer matrix element and an equivalent spring connecting two filler particles i and j.

where rij is the center-to-center vector from particle i to particle j taken at time t or at equilibrium for t = 0. The strength is a pairwise constant and reads kij∞ =

(di + dj)2 4h0

Gr′

(11)

In the dimensionless model, the elastic modulus of the matrix G′r sets the unit of modulus and assumes the value 1, corresponding in physical units to about 106 Pa. The beads are connected on average with a connectivity of nc = 10 springs per bead. Since the number of springs depends on the choice of nc and on the number of primary particles of a filler aggregate, the numerical stress computed in the system shall be normalized accordingly (see Appendix, Supporting Information). III.C. Elasticity of Glassy Bridges. In addition to the permanent spring representing the matrix, two neighboring particles interact with a second harmonic spring corresponding to a rigid bridge. These rigid springs have finite lifetimes. They break and rebuild permanently when the distance and/or local stress between the particles is sufficiently small. This network represents the contribution of the rigid bridges between fillers. Its strength depends on temperature and on the morphology of the systems. The fraction of the polymer matrix affected by interfacial effects such that the local Tg > T is denoted Σ(T). Let us define Λi(u) = |ri − u| − ai the distance between a position u within the matrix in the simulation domain, and the surface of the primary particle i whose center is ri. Let Λ(u) = min(Λi(u)) be the shortest distance between position u and the surface of all primary particles of the domain (see Figure 5). The local Tg at position u is given by eq 1, that is ⎛ β ⎞ Tg(u) = Tg ⎜1 + ⎟ Λ (u ) ⎠ ⎝

Figure 5. Estimate of the local glass transition temperature at any position u using eq 12. Gray regions correspond to glassy domains, with Tg(u) > T.

k 0 = Σ(T )

1 V

⎛ ⎛

∫V du Θ⎜⎝Tg⎜⎝1 +

k∞

The reference state rij(tl) and the reference surface-to-surface distance hij correspond to configuration of the system when the last bridge breaking occurred at an earlier time tl. III.D. Equation of the Motion. The viscous dissipation is accounted for by a friction coefficient ζnum such that the viscous force between two neighboring particles i and j reads

Fij = −ζpnum(vi − vj)

(15)

with vi and vj the velocities of particles i and j. Since we perform the dynamics at the scale of the aggregates, elementary friction forces acting on primary particles of an aggregate A are summed and attached to the center of mass of the aggregate, which leads to the total force FA and torque TA:

(13)

FA = −ζAnum ,Trans(vA − v∞)

From its definition, Σ(T) is a decreasing function of T and goes from 1 − ϕ for T ≤ TBulk to zero for T ≫ TBulk g g . Hence, the above discussion allows us to estimate the strength of the rigid bridges that connect particles as G′ ∼ ΣGg′. Translating this behavior into the strength of the spring connecting two primary particles, the local strength of a rigid bridge reads

num TA = −ζA,Rot (ωA − ω∞)

f

(14)

Fij = −k 0(rij(t ) − rij(tl))

(12)

⎞ β ⎞ ⎟ − T⎟ Λ(u ) ⎠ ⎠

Gr′

Finally, the force due to the glassy bridges reads

If T − Tg(u) < 0, the position u is considered to be glassy. Using the Heaviside step function30 Θ(x), the fraction of glassy polymer in the matrix may be written Σ (T ) =

Gg′

(16)

with vA the velocity and ωA the angular velocity of aggregate A and v∞ the local mean field velocity and ω∞ the local mean field angular velocity surrounding aggregate A. The numerical translational and rotational friction parameters are related to E


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Macromolecules num the friction on a single primary particle by ζnum and A,Trans = npζp num 2 num ζA,Rot = Rg ζp . The dissipation is calibrated to represent the dissipative power of the rubbery matrix as discussed in ref 20. More details on how the motion of the aggregates is calculated are given in the Appendix (Supporting Information). III.E. Parameters of the Simulations. The parameters and the corresponding values used in the simulations are summarized in Table 1. We consider in what follows systems at a volume fraction ϕ = 20 vol %.

Table 1. Parameters of the Model (β, K′, ζ, k0, k∞) physical value

parameter number of fillers number of primary particles filler external diameter primary particle diameter connectivity of springs rubbery modulus glassy modulus rigid bridge strength

60 nm 60−15 nm

glass transition temperature of the pure rubber temperature of the experiment oscillatory shear typical frequency oscillatory shear typical deformation dissipative modulus at large deformation amplitude coefficient β coefficient K coefficient α

106 Pa 109 Pa Σ(T) 109 Pa 213 K T ω γ0 106 Pa 1.8 nm 106 Pa/K

value in the simulation 1000 1000−40000 D = 1.0 d = 1.0−0.2 n = 10 k∞ = 1

Figure 6. Structure factor of spherical filler particles dispersed at a volume fraction of ϕ = 0.20 computed from the pair distribution function using eq 17. A varying screening parameter rc has been used for constructing systems with a different dispersion state. Noticeable structure factors are for rc = 0.400D (+) and for rc = 0.100D (■) since they repectively reproduce the characteristics of the experimental structure factors of a sample with a homogeneous microstructure and a sample with an aggregation filler particles. The insert reproduces the experimental structure factor of the systems studied by Berriot et al. from ref 18.

k0 = 1000Σ(T) Tg = 213 K T = Tg...Tg + 100 K ω = 2π rad/s γ0 = 0.005−0.1 ζ = 30 s

large objects, that is, the presence of large aggregates. The second distinctive feature of the structure factor is the position of the first peak corresponding to the distance between first neighbors. By using the Percus−Yevick approximation, we find that this peak at about qR ≈ 2.4 corresponds to a nearest surface-to-surface distance of about 0.36D in the homogeneously distributed sample. We reproduce this behavior in the simulations by setting the screening parameter rc = 0.400D. In the experimental aggregated sample, the peak at about qR ≈ 3.4−3.5 corresponds to a first approach distance of about 0.1D, which we approximately reproduce by setting a screening parameter rc = 0.1D. Thus, realistic filler dispersion states can be reproduced by setting the range of a repulsive interaction between particles while constructing the system. Let us emphasize again that in this section particles are simple spheres with no aggregation. We will then consider systems with a repulsive length rc = 0.4D as well dispersed and systems with a repulsive length rc = 0.1D as badly dispersed. The dispersion states obtained with different rc values are illustrated in Figure 1. For all results shown in this section, the interaction parameter is β = 0.030D, which corresponds to β = 1.8 nm in physical units for fillers with D = 60 nm. This corresponds to strong filler−matrix interactions. The volume fraction is ϕ = 0.20. WLF parameters are those of polyisoprene, with Tg = 213 K. The yield parameter is K = 0.01. A.2. Reinforcement. The elastic modulus G′ measured at small shear amplitude (γ = 5 × 10−3) and angular frequency ω = 6.28 rad/s is plotted in Figure 7 as a function of temperature for systems of spherical particles (D = 60 nm) at a volume fraction ϕ = 0.20 built using a screening parameter rc varying from 0.4D (24 nm) down to 0.06D (3.6 nm). The curves in Figure 7 show that a modulus of around 107 Pa can be obtained over a large temperature range as soon as the dispersion state of the fillers is sufficiently inhomogeneous. The

0.030 0.01 K−1 0.4

IV. RESULTS AND DISCUSSION IV.A. Effect of Filler Dispersion. As discussed in section III, the dispersion state of our systems is obtained by using a modified Lubachevsky−Stillinger algorithm29 with an adjustable repulsion range, or minimum distance of approach, rc. In this section, we show that this numerical approach allows us to recreate realistic filler distribution states. We shall then study the impact of the dispersion state on the reinforcement and on the Payne effect in nanofilled elastomers. A.1. Realistic Dispersion States. The structure factor of a distribution of spherical particles is calculated as31 S(q) = 1 + 4πN

∫0

∞

(g (r ) − 1)

sin(qr ) 2 r dr qr

(17)

N is the number of particles per unit volume, and g(r) is the pair distribution function for the center of mass of the primary particles of the fillers. The structure factors computed in our simulations for samples with different repulsion screening parameter rc are compared to experimental structure factors of two samples with different dispersion states in Figure 6. Berriot and co-workers obtained different dispersion states from the same colloidal solution of grafted spherical silica particles (D = 40 nm) in acrylate monomers using slightly different solvent transfer protocols as described in ref 18. For the experimental aggregated system in the inset of Figure 6, the structure factor S(q) shows an upward turn as the scattering vector q tends to zero. This indicates the presence of F


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Figure 7. Variation of the elastic modulus (left) and reinforcement (defined as R = G′/G′rubber) (right) as a function of temperature for systems of spherical particles constructed with different screening parameters rc. In the graph on the left, the Ref curve models the behavior of the unfilled matrix.

Figure 8. Simulated elastic modulus G′ (in MPa) (left) and dissipative modulus G″ (in MPa) (right), measured in oscillatory shear at angular frequency ω = 6.28 rad s−1, as a function of the shear amplitude γ at temperature T = 263 K, for different values of the screening parameter rc between 0.4D and 0.1D. The parameters are ϕ = 0.2, β = 0.030, and K = 0.01 and WLF parameters of PI with Tg = 213 K.

well-dispersed samples with rc ≤ 0.4D exhibit a sharp drop of the elastic modulus in a temperature range between Tg and Tg + 50 K. Dispersion states with intermediate values of the distribution parameter (rc = 0.3D, rc = 0.2D) result in a drop of the modulus over a broader temperature range, up to Tg + 80 K. Badly dispersed samples (with rc ≤ 0.1D) show a smoother decrease of the elastic modulus. For Tg + 120 K, the rigid bridge network gives a contribution to the macroscopic elastic modulus that is still quite strong (macroscopic modulus of 107 Pa). These data may alternatively be plotted in terms of the reinforcement ratio R = G′/Grubber ′ . The reference elastic modulus of the pure polymer matrix is plotted in Figure 7. The corresponding heuristic formula is given in ref 25. The glassy modulus assumes the value G′g = 109 Pa and the rubbery modulus Gr′ = 106 Pa. The reinforcement curves show a marked peak at a temperature a few tens of kelvin above the matrix Tg. The magnitude of this peak depends only slightly on the dispersion state of the fillers. The effect of the latter is more pronounced in the high temperature regime (above Tg + 50 K). The reinforcement for a well-distributed sample reaches its high temperature limit (close to 3) at Tg + 50 K, whereas for more heterogeneous microstructures this limit is not reached in the considered temperature range, up to Tg + 100 K. A.3. Payne Effect. The simulated elastic modulus G′ obtained in our simulations is plotted in Figure 8 as a function of the oscillatory shear amplitude at temperature T = 263 K = Tg + 50 K for various screening parameters between rc = 0.400D (good distribution state of the fillers) and rc = 0.100D (bad distribution state of the fillers). The elastic modulus is measured in permanent regime, after typically two cycles in the

transient regime. The low shear amplitude modulus is about G0′ = 13 MPa for a bad dispersion state and only G′ = 3 MPa for a good dispersion state of the fillers. Hence, the elastic modulus at low shear amplitude indeed strongly depends on the dispersion state of the filler (see also Figure 7). According to our model and for all the systems, the larger the shear amplitude, the larger the fraction of rigid bridges which have yielded under stress and the lower the elastic modulus. As shown in Figure 8, the shear amplitude at which the Payne effect occurs depends on the dispersion state of the fillers. The system with a bad dispersion state (rc = 0.1D) exhibits a drop of the elastic modulus at strains about γ = 0.1 and systems with a better dispersion state (rc = 0.3D) show a drop of the elastic modulus for smaller strains γ ≈ 0.01. This observation indicates that the drop of the elastic modulus is indeed due to the melting of rigid bridges due to the local stress in the confined elastomer between the fillers. Hence, rigid bridges with high Tg shifts (and hence small z) leads to a Payne effect for higher shear amplitude. In a filled elastomer, the local strain εl may be amplified as compared to the macroscopic strain. It reads typically εl ∼ λεm, with λ ∼ d/z where z is the surface to surface distance between the fillers. If the strain was locally amplified in an essentially affine way, the local stress would be σ = (d/z)G′εm and the polymer confined in between close fillers (with small distance z) would support a higher stress than the polymer between more distant fillers. As a result, rigid bridges between very close particles would yield first. In fact, we observe in our simulations that deformations tend to localize in the softer regions (low density of rigid bridges). As a consequence, glassy bridges between very close particles tend to yield later. G


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Figure 9. Left: relative drop of elastic modulus G′/G0′ simulated in our work in oscillatory shear at a pulsation ω = 6.28 rad s−1 as a function of the shear amplitude γ at temperature T = 263 K, for various screening parameter rc between 0.4D and 0.1D. The parameters are ϕ = 0.2, β = 0.030 and K = 0.01, and WLF parameters of PI with Tg = 213 K. Right: relative drop of elastic modulus G′/G0′ as measured by Montes and co-workers15 in a homogeneously distributed (▼) and partially aggregated (■) of grafted silica particles (ϕ ≈ 0.20) in acrylate monomers. Measurement performed at Tg + 50 K.

The data from Figure 8 can be used to predict the relative drop of elastic modulus G′/G0′ with the strain amplitude. G0′ is the low strain elastic modulus measured at γ = 10−3. The elastic modulus is essentially independent of the strain amplitude for the system with a good dispersion state, while it decreases significantly for the system with a bad dispersion state. It appears that the nonlinear behavior of filled elastomers depends on the relative position of the particles in the matrix. The relative drop of elastic modulus in our results is compared to the experimental results of ref 15 in Figure 9. In this reference, the authors used the same materials as those in ref 23 but with different dispersion states. Then they compare the relative amplitude for the Payne effect of the systems with different dispersion states at a temperature T = Tg + 50 K. We obtain the same characteristics in our simulation. The quality of the dispersion state has a strong impact on the amplitude of the Payne effect. A good dispersion state leads to a small Payne amplitude and a bad dispersion state leads to a large Payne effect amplitude. The loss modulus G″ as a function of the shear amplitude is also reported in Figure 8. The systems which displays a significant loss of elastic modulus with the strain amplitude also exhibits a dissipation peak at intermediate strains. Note that the dissipation at low strain (in the linear regime) is not a monotonous function of the dispersion state. This will be discussed in the next section. A.4. Discussion. For the studied dispersion states, the elastic modulus exhibits a smooth decrease with temperature up to a characteristic temperature Tc where it undergoes a rather sharp drop down to the high temperature regime value. This behavior is not observed in experimental results where a slow, progressive decrease is usually observed. In fact, according to the model, the temperature Tc at which the high temperature value of the reinforcement is reached is determined by the melting temperature of the most rigid glassy bridges in the system. These glassy bridges correspond to the closest interparticle distances. The distributions of the surface-tosurface distances between a particle and its 10 first neighbors are reported in Figure 10 for different screening parameters rc. For badly reinforced samples with rc ≲ 0.6D, the distance distribution drops down to zero abruptly, within an distance interval smaller than 0.01D, as r decreases down to the value of the screening parameter rc chosen during the system preparation: as a consequence, there is a large population of rigid units having the relaxation time τ(rc) associated with a

Figure 10. Distribution of the distances within the tenth first neighbors in systems with a varying screening parameter rc. We believe that small distances leads to stronger glassy bridges (bridges with a higher local Tg) that drive the mechanical behavior of filled elastomers in the reinforcement regime. In our simulation, the screening parameter allows to finely define the smaller distances achievable in the systems.

shift of glass transition ΔTg/Tg = 2β/z. The sharp decrease of the elastic modulus at a characteristic temperature Tc = Tg(1 + 2β/z) may be thus associated with the large population of such glassy bridges. Figure 10 shows that for a more heterogeneous system the closest approach distance is more broadly distributed on a distance range of about 0.3D. As a result, these systems exhibit a broader decrease of the elastic modulus as a function of temperature. In order for our model to be more representative of the experimental phenomenology, it would be possible to consider a sufficiently representative set of systems with various dispersion states. We consider here the loss modulus in the linear regime. It can be seen from Figure 8 that the relative position of G″0 does not vary monotonously with the distribution state of the fillers. We observe that G0″ ≈ 1.5 MPa for the system with rc = 0.4D, G0″ ≈ 3 MPa for the system with rc = 0.3D, G0″ ≈ 2.5 MPa for the system with rc = 0.2D, and G0″ ≈ 3.2 MPa for the system with rc = 0.1D. The glassy bridges between fillers can be seen as Maxwell elements made of a purely elastic spring of strength k and a purely viscous damper of viscosity η = kτ. From the distribution of relaxation time p(τ) for the rigid units, the loss modulus as a function of the shear frequency ω may be approximated by H


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Figure 11. Distribution of relaxation times n(τ > τ0) after an oscillatory shear of various amplitudes γ from 0.001 to 0.5 (from top to bottom, respectively, the value of the shear amplitude γ for the curves are the same of the markers of Figure 8) at T = 263 K for systems of a varying distribution state: (a) rc = 0.400D, (b) rc = 0.300D, (c) rc = 0.200D, and (d) rc = 0.100D.

G″(ω) ∝

∫0

∞

p(τ )

ωτ dτ 1 + ω 2τ 2

In our simulations, G″ exhibits a maximum for deformation range between 0.01 and 0.1 for systems with a bad dispersion state. The position of this maximum depends on the dispersion state of the systems. The less reinforced system with a good dispersion state of the fillers exhibit only a small decrease of the loss modulus with the strain amplitude. In the reinforced systems studied by Payne, a maximum for the loss modulus was also observed, more pronounced in more strongly reinforced systems.3 Therefore, we predict here that the dispersion state of the filler is key for tailoring the dissipative properties of filled elastomers. A bad dispersion state results in stronger reinforcement, that is, in a higher modulus, in the linear domain, but it correlatively induces a more pronounced dissipation at intermediate strain amplitudes. IV.B. Effect of Filler Morphology. In this section, we consider the specific effect of the filler morphology. The morphological parameters are described, and the effects on reinforcement and Payne effect are studied. We used the building procedure described in section III with the parameter rc = 0.4D, which gives a good distribution state. The volume fraction is fixed at ϕ = 0.2. The interaction parameter is still fixed at β = 0.030. WLF parameters are those of polyisoprene with Tg = 213 K. In all the systems we only consider monodisperse populations of fillers in terms of number of primary particles. B.1. Morphological Parameters. In what follows, we consider fillers with a varying number np of primary particles and with an overall diameter D ≈ 1, where the unit length scale corresponds to 60 nm. The gyration radius D = 2Rg is calculated as

(18)

According to eq 18, the loss modulus G″ is proportional to the total number of rigid units and is mostly affected by rigid units with relaxation time of the order of magnitude as the experimental relaxation time τexp = 2π/ω. The number of rigid units n(τ > τ0) with a relaxation time τ larger than a reference relaxation time τ0 is plotted in Figure 11 as a function of the reference relaxation time τ0 for different shear amplitudes. The quantity n(τ > τ0) can be expressed as a function of the distribution of relaxation time p(τ): n(τ > τ0) = n0(1 −

∫0

τ0

p (τ ) d τ )

(19)

The derivatives of the curves in Figure 11 are thus directly related to the distribution of relaxation time p(τ). Figure 11a shows that at the considered temperature T = 263 K more than 90% of the rigid units in the well-dispersed system (rc = 0.4D) have already melted. The slope of the curves at τ0 = τexp decreases slightly with the strain amplitude, thus inducing a small decrease of the loss modulus with the strain amplitude as shown in Figure 8. Compared to the well-dispersed sample (rc = 0.4D) the system with rc = 0.3D has a larger number of rigid units, especially at low strain amplitude. Moreover, the slope of the curve associated with a small shear amplitude γ = 0.001 in Figure 11b for the system with rc = 0.3D exhibits a maximum for τ0 = τexp. Thus, the loss modulus at small shear strain amplitude is about 2 times larger in the system with rc = 0.3D than the system with rc = 0.4D. The maximum for the slope of the curve is brought back to smaller reference relaxation time with the shear strain amplitude, and the total number of rigid units that has not already melted is subsequently diminished. When the total number of rigid units and the slope of the distribution are sufficiently high, the loss modulus exhibits a peak, for intermediary strains as shown in Figure 8 (right).

Rg2 =

1 np

np

∑ |ri − R|2 i

(20)

Aggregates are randomly generated using a modified Diffusion Limited Aggregation (DLA) algorithm.32 We then apply a scale I


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Macromolecules Table 2. Morphological Parameters and Representative Filler Morphologies Used in This Studya

a Sizes are scaled so that the average diameter of the filler population ⟨D⟩ ≈ 1. d is the diameter of primary particles, and dreal is the corresponding physical diameter (given that D = 60 nm).

dimension of the aggregates has been estimated around Df = 1.97. Figure 13 gives a visual representation of systems made with filler aggregates with a varying number of primary particles np. Each system contains 1000 filler aggregates. Since aggregates containing 40 primary particles are less dense than aggregates of the same overall size but containing a smaller number of (bigger) primary particles, the latter systems are slightly larger for a fixed volume fraction ϕ = 0.2. Indeed, the density of fillers depends on the number of primary particles according to a scaling law ϕf = np1−3/Df. The spring network is generated in order to connect neighboring primary particles which do not belong to the same aggregate. The elastic strength of the rubbery spring network is set to 1, the unit of elastic modulus in our system. The elastic strength k0 of glassy springs is set to a value that depends on the temperature and on the considered system:

factor which depends on the number of primary particles per aggregate such that the average envelope diameter ⟨D⟩ = 1. Morphological parameters of the 1000 fillers used in the simulations are reported in Table 2. The diameter of the primary particles varies from 1 D down to 0.2 D, which corresponds to a primary particle size of 60 nm down to 12 nm for a filler of diameter D = 60 nm. An estimate of the fractal dimension Df of the filler population and the size distribution of the 1000 aggregates in the studied systems are represented in Figure 12. The fractal

k0 =

Σ(T )Gg′ Gr′

k∞

(21)

The fraction of glassy polymer Σ(T) is defined by eq 13. This quantity has been calculated as a function of temperature for all the systems presented here and is reported in Figure 14. The actual strength of a glassy spring joining particle i and j is defined locally and reads k0ij = d2k0/h0. B.2. Reinforcement. The storage modulus as a function of temperature is reported in Figure 15 for aggregates with overall diameter 60 nm, made of a varying number of primary particles from 1 to 40. The curves in Figure 15 display not only a shift but also a broadening of the glass transition for systems made

Figure 12. Fractal dimension Df of the filler population (left) and distributions p(D) of the envelope diameter D of the filler populations (right). The fractal dimension Df is defined by equation log(np) = Df log(D/d). The slope of the dashed line is Df = 1.97.

Figure 13. Visualization of the equilibrated systems at a volume fraction ϕ = 0.2 containing 1000 filler aggregates each, with a varying number of primary particles np. For a matter of clarity, the primary particles belonging to an aggregate are painted with the same color. Interaction between primary particles are depicted by the mean of a solid blue line. J


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primary particles, and over 120 K for aggregates of 20 primary particles. The loss modulus G″ and loss factor tan δ are reported in Figure 16 as a function of temperature. The loss modulus displays a peak whose position depends on the filler morphology. The peak is shifted from Tg + 40 K for a system of spherical particles up to Tg + 60 K for aggregates of 40 primary particles. The peak is followed by a slow decrease as temperature increases further. The loss factor tan δ exhibits a peak with a position and amplitude that strongly depend on the filler morphology. Systems with spherical particles exhibit a sharp peak with a maximum located around Tg + 40 K while more complex systems exhibit a shift and a spreading of the peak to higher temperature. Therefore, we predict that systems made of complex fillers shall dissipate more energy at high temperature than systems made with spherical particles. B.3. Payne Effect. Oscillatory shear at a constant angular frequency ω = 6.28 rad/s have been performed at varying shear amplitudes γ. We consider here the elastic modulus G′, loss modulus G″, and loss factor tan(δ) as functions of the strain amplitude. The numerical results are reported in Figure 17 for T = Tg + 50 K and in Figure 18 for T = Tg + 100 K. In the reinforcement regime, at T = Tg + 50 K, the curves for the elastic modulus G′ display a large drop for deformation amplitudes of order a few percent to about 10% (Figure 17). This drop corresponds to the Payne effect. The smaller the primary particles, the larger the linear elastic modulus G0′ , and the larger the drop of elastic modulus with the deformation. This drop is accompanied by an increase of the loss modulus G″, resulting in peak in the loss factor tan(δ) . The peak in the loss factor appears for higher strain and goes to higher amplitude when the size of the primary particles is reduced. The curves shown in Figure 17 illustrate the combined effects of complex fillers, that is, fillers with an increasing number of primary particles. While, at a given temperature, volume fraction, distribution state, and shear amplitude, complex fillers give a higher elastic modulus, they also produce a larger amplitude of the Payne effect and larger dissipation at intermediate strain amplitudes. At higher temperature, T = Tg + 100 K, the elastic modulus at low shear amplitude is lower for all the systems, as discussed in the previous section. Nevertheless, the strong nonlinear behavior associated with more complex morphologies is still present. In fact, as reported in Figure 18, the most complex systems with fillers made of 40 primary particles exhibit a drop

Figure 14. Evolution of Σ as a function of the temperature for systems at a volume fraction ϕ containing 1000 fillers with a various number np of primary particles. Σ(T) is computed using eq 13 considering β = 0.03 and Tg = 213 K.

with smaller primary particles. A strong reinforcement, with a modulus between 10 and 100 MPa, can be obtained over a broad range of temperature, as it is indeed observed experimentally in highly reinforced samples (see e.g. ref 33). Contrary to pure elastomers, the elastic modulus is a decreasing function of temperature that never reaches a constant regime for the systems made with the smaller primary particles (system of aggregates with 30 and 40 primary particles). The data in Figure 15 can be alternatively plotted in terms of the reinforcement R = G′/G′matrix. Reinforcement curves in Figure 15 exhibit two main characteristics. The reinforcement peak, located approximately at Tg + 30 K for all the systems, has an amplitude that strongly depends on the filler morphology. In our systems with a high interaction parameter β = 1.8 nm, the peak amplitude goes from 10 for spherical particles up to 102 for aggregates of 40 particles. We observe here the dramatic impact of the filler morphology on the amplitude of the reinforcement peak, at a fixed filler volume fraction, same spatial distribution and filler average envelope size. The second feature of the reinforcement curves is the temperature range over which reinforcement is not a constant function of temperature. This width clearly depends on the filler morphology. The reinforcement effect spans over 70 K for a system of spherical particles, over 80 K for aggregates of 5

Figure 15. Storage modulus G′ (left) and reinforcement R (right) in the linear regime as a function of the temperature, measured in oscillatory shear at a pulsasion ω = 2π/s. The parameters are β = 0.030 and K = 0.01, and the volume fraction of filler is Φ = 0.20. The WLF parameters are those of polyisoprene with Tg = 213 K. The systems are filled with fillers of diameter D = 1 with a varying number of primary particles np as indicated. The corresponding sizes of the primary particles are reported in Table 2. K


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Figure 16. Loss modulus G″ (left) and loss angle tan δ (right) in the linear regime as a function of the temperature. Data are those of Figure 15 The systems are filled with fillers of diameter D = 1 with a varying number of primary particles as indicated. The corresponding size of the primary particles are reported in Table 2.

Figure 17. Payne effect for the systems with different morphologies at a temperature of Tg + 70 K (283 K). We represent here the elastic modulus G′ (in MPa), the loss modulus G″ (MPa), the loss angle tan(δ), and the relative modulus G′/G′0 as a function of the shear amplitude. The parameters are β = 0.030 and K = 0.01, and the volume fraction of filler is Φ = 0.20. The WLF parameters are those of polyisoprene with Tg = 213 K. The systems are filled with fillers of diameter D = 1 with a varying number of primary particles, as indicated.

of modulus of about 2 whereas the system of spherical particles display a more linear behavior. The dissipation measured by G″ or tan(δ) is also larger for systems with smaller primary particles. Therefore, we demonstrate here that the amplitude of the Payne effect strongly depends on the morphology of the fillers. B.4. Discussion. The larger reinforcement peak and broader glass transition can be related to the distance distribution, which strongly depends on the filler morphology. Nearer neighbors are associated with shorter distances and then to stronger glassy bridges that control the mechanical behavior of reinforced elastomers. The distributions of distances of the ten first neighbors of each primary particle are represented in Figure 19. Aggregates with smaller primary particles lead to smaller distances. The screening parameter rc used for generating the distribution states of the systems strongly affects the interparticle distance distributions of the systems made of

spherical particles. Indeed, spherical particles have an average of five neighbors located at this specific distance. Although the same screening parameter have been used to generate more complex systems, they do not exhibit such characteristic first approach distance. For most systems, the surface to surface distances between primary particles are smaller than the parameter rc, and the smaller the primary particles, the closer they get from each other. This effect can be understood by considering the average distance L between the centers of mass of the aggregates. Considering that the aggregates are homogeneously distributed in space, L can be related to the volume fraction ϕ and to the aggregate volume V by ϕ = V/L3. For nonoverlapping primary particles, the aggregate volume is V = npπd3/6. The size d of primary particles is related to the aggregate size D and to the number of primary particles per aggregate by np = (d/D)Df. L


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Figure 18. Payne effect for the systems with different morphologies at a temperature of Tg + 100 K (313 K). We represent here the elastic modulus G′ (in MPa), the loss modulus G″ (MPa), the loss angle tan(δ), and the relative modulus G′/G′0 as a function of the shear amplitude. The parameters are β = 0.030 and K = 0.01, and the volume fraction of filler is Φ = 0.20. The WLF parameters are those of polyisoprene with Tg = 213 K. The systems are filled with fillers of diameter D = 1 with a varying number of primary particles, as indicated.

Figure 19. Distribution of the distances within the tenth first neighbors for fillers with a varying number of primary particles np. We believe that small distances leads to stronger glassy bridges (bridges with a higher local Tg) that drive the mechanical behavior of filled elastomers in the reinforcement regime. In our simulation, systems filled with smaller primary particles are associated with closer distances and, as a result, exhibit a reinforcement of higher amplitude in a broaden range of temperatures.

Figure 20. Statistical distance between the center of mass of the aggregates for a distribution of filler aggregates as a function of the number of primary particles for a varying volume fraction, from top to bottom: ϕ = 0.01, ϕ = 0.02, ϕ = 0.05, ϕ = 0.10, ϕ = 0.20, and ϕ = 0.30. Data are computed using eq 22 with a Hausdorf fractal dimension set to Df = 2.0 in this example.

volume fractions as the number of primary particles per aggregate increases. When the volume fraction is high enough, the primary particles get closer than rc since the system cannot find a configuration where all repulsion forces would be relaxed. In fact, increasing the number of primary particles or increasing the volume fraction has the same effect on the average distance between the centers of mass of the aggregates and therefore on the surface-to-surface distances between fillers. Changing the morphology of the fillers at a given volume fraction leads to an increase in the elastic modulus that would be equivalently achieved by increasing the volume fraction. A consequence of eq 22 is that the effect reported here for ϕ = 0.20 may vanish for a smaller volume fraction.

Then the average distance between the centers of mass of the aggregates is given by ⎛ πn 1 − 3Df −1 ⎞1/3 L p ⎟ =⎜ ⎜ D ⎝ 6ϕ ⎟⎠

(22)

The average distance L, calculated by eq 22, is reported in Figure 20 as a function of the number of primary particles per aggregates np for different volume fractions. When L/D becomes smaller than 1, aggregate envelopes need to overlap to achieve a given volume fraction. This happens at smaller M


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Macromolecules IV.C. Relevance of the Physical Model. The effect of confinement on polymer dynamics is still a debated issue (see e.g. the book Polymer Glasses,34 Chapters 5−9). As regard to the glassy layer thickness itself, Papon et al.22 have measured a relatively small value for the parameter β: 0.1−0.2 nm. This value is smaller than the one we consider here for strong reinforcement. Several remarks may be made at this regard. It has been proposed by Dequidt et al.25 that the effect of confinement is not simply a shift of the glass transition temperature, but a large broadening of the glass transition itself. As a result, the effects of confinement on the mechanical properties of confined polymers persist at temperatures much higher than the Tg shift itself. In Dequidt et al., depending on the strength of the polymer−substrate interaction, a Tg shift of e.g. 10 K for a 10 nm thick film could be obtained, which would correspond to β ≈ 0.2 nm. On the other hand, due to the broadening of the glass transition, the elastic modulus decreases slowly as temperature increases and is still 10−100 times larger than in he bulk at temperatures higher than Tg + 80 K. This slow decay of the elastic modulus as a function of temperature is a defining feature of strongly reinforced elastomers. It is comparable to results reported e.g. by Wang.5 This feature also indicates that the effect of confinement is more complex than a simple shift of the glass transition. By studying the polymer dynamics in filled elastomers by NMR and DSC, Papon et al.35 have shown that the state of the polymer in the vicinity of fillers cannot be described by a simple Tg shift. They distinguish three different regions around the fillers: (1) a glassy layer in the immediate vicinity, (3) a rubbery state far from the fillers, and (2) an state intermediate between the glassy and rubbery states, which they denote “immobilized polymer”. In the samples studied by Papon et al., this intermediate “immobilized polymer” layer may have a thickness of about 10 nm at Tg + 60 K and represent a volume fraction 5 times or more larger than the fraction of the glassy polymer itself. Our point of view is that these experimental results, associated with classical mechanical measurements,5 can be understood by the theoretical results of Dequidt et al.25 For the sake of simplicity, we consider here that the effect of confinement is a simple Tg shift. To obtain results comparable to the measured mechanical properties of reinforced elastomers, this leads to choose a relatively large value of the β parameter, since the broadening of the glass transition is not explicitly taken into account. A further extension of the present reinforcement model will be to implement a more complex function for the change of the dynamics in between the fillers, deduced from the simulations of Dequidt et al. for instance. Among other explanations for reinforcement, it has been proposed that stress may be transmitted by contacts between fillers.26 Note that direct contacts are unlikely in systems in which fillers are dispersed in the elastomer matrix by melt mixing. In this process, distinct fillers cannot come into direct contact because it should be very difficult to completely expel the confined polymer between them. However, a high local stress may be transmitted between two neighboring fillers in particular circumstances where local shear or buckling is not allowed as a consequence of multiple interactions between complex aggregates. Similarly, Witten et al.36 have proposed that the stress is essentially born by the complex reinforcing aggregates in close mechanical contact. The mechanical properties of such network close to the rigidity percolation has been studied by Kantor and Webman.37

Our point of view is that the slow decay of the elastic modulus with temperature in strongly reinforced elastomers cannot be reconciled with a purely geometrical description. This does not rule out the possibility that such rigidity percolation effects do not contribute to the mechanical response.37 Without considering glassy effects in the vicinity of the fillers, it is difficult to account for the specific temperature dependence of reinforcement. In principle, these purely geometrical effects could be studied with our model by suppressing the glassy contribution to the mechanical response and allowing the fillers to come in close contact in the preparation step of our samples.

V. CONCLUSIONS We have described the Glassy Bridge Reinforcement (GBR) model for the reinforcement of filled elastomers. In this paper, we have proposed an extension of the GBR model to take into account the filler morphologies and distribution states explicitly. The model allows for describing the mechanical behavior of filled elastomers in the linear and nonlinear regimes of deformation (Payne effect). It is based on the effect of the confinement of the polymer matrix between two fillers, which leads to the presence of glassy bridges. The presence of those rigid bridges results in a high level of stress between fillers, with finite lifetimes depending on the local stress, history, temperature, and distance between fillers. We have extended this model in order to fine-tune the dispersion state of the fillers in the matrix. The obtained dispersion states have been characterized by calculating structure factors, as measured experimentally by small-angle neutron or X-ray scattering. We have studied the effect of the dispersion state of the fillers on the reinforcement and on the Payne effect amplitude. The elastic modulus of systems filled with spherical particles with various dispersion states has been studied. The reinforcement strongly depends on the distribution state of the fillers. Indeed, a homogeneous distribution state leads to a rapid decrease of the elastic modulus with the temperature and as a consequence to a narrow reinforcement regime. Conversely, an inhomogeneous distribution state leads to a much slower decrease of the elastic modulus with temperature. The Payne effect amplitude is also related to the distribution state of the filler in the matrix. The elastic and loss moduli of the same systems have been studied as a function of strain amplitude. A homogeneous distribution state leads to a mostly constant elastic modulus with the oscillatory shear amplitude. As soon as the distribution state becomes inhomogeneous enough, the Payne effect amplitude increases drastically. Systems with a bad distribution state exhibit a loss of elastic modulus of about 80%. This drop of elastic modulus is associated with a peak of the loss modulus, corresponding to enhanced energy dissipation during mechanical loading. We conclude from this study that the dispersion state of the filler is a key parameter to control the elastic modulus in the linear regime G0′ (T). A bad dispersion state leads to a larger elastic modulus and to an enhanced energy dissipation. Our samples exhibit two distinct kinds of behavior: no Payne effect for welldispersed samples and strong Payne effect with badly dispersed samples. A smoother transition between the two regimes may be obtained by introducing some disorder in terms of closest approach distance between the particles. We have studied the effect of the filler morphology on the reinforcement and on the Payne effect amplitude. We have fixed the size of the filler envelope to a constant diameter (60 N


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Macromolecules nm) and changed the filler morphology in a well-controlled way by increasing the number of primary particles per aggregate. We observe that the reinforcement strongly depends on the morphology of the fillers for a given volume fraction and distribution state. Fillers with an increasing number of primary particles lead to a reinforcement of higher amplitude in a broader temperature range. This feature has been associated with the presence of smaller surface-to-surface distances in systems filled with complex aggregates.

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(13) Heinrich, G.; Kluppel, M.; Vilgis, T. A. Reinforcement of elastomers. Curr. Opin. Solid State Mater. Sci. 2002, 6 (3), 195−203. (14) Berriot, J.; Montes, H.; Lequeux, F.; Long, D.; Sotta, P. Gradient of glass transition temperature in filled elastomers. EPL (Europhysics Letters) 2003, 64 (1), 50. (15) Montes, H.; Chaussée, T.; Papon, A.; Lequeux, F.; Guy, L. Particles in model filled rubber: dispersion and mechanical properties. Eur. Phys. J. E: Soft Matter Biol. Phys. 2010, 31 (3), 263−268. (16) Papon, A.; Montes, H.; Lequeux, F.; Oberdisse, J.; Saalwächter, K.; Guy, L. Solid particles in an elastomer matrix: impact of colloid dispersion and polymer mobility modification on the mechanical properties. Soft Matter 2012, 8 (15), 4090−4096. (17) Baeza, G. P.; Genix, A. C.; Degrandcourt, C.; Gummel, J.; Mujtaba, A.; Saalwächter, K.; Thurn-Albrecht, T.; Couty, M.; Oberdisse, J. Studying twin samples provides evidence for a unique structure-determining parameter in simplifed industrial nanocomposites. ACS Macro Lett. 2014, 3 (5), 448−452. (18) Berriot, J.; Montes, H.; Martin, F.; Mauger, M.; PyckhoutHintzen, W.; Meier, G.; Frielinghaus, H. Reinforcement of model filled elastomers: Synthesis and characterization of the dispersion state by SANS measurements. Polymer 2003, 44 (17), 4909−4919. (19) Scotti, R.; Conzatti, L.; D’Arienzo, M.; Di Credico, B.; Giannini, L.; Hanel, T.; Stagnaro, P.; Susanna, A.; Tadiello, L.; Morazzoni, F. Shape controlled spherical (0D) and rod-like (1D) silica nanoparticles in silica/styrene butadiene rubber nanocomposites: Role of the particle morphology on the filler reinforcing effect. Polymer 2014, 55 (6), 1497−1506. (20) Merabia, S.; Sotta, P.; Long, D. R. A Microscopic Model for the Reinforcement and the Nonlinear Behavior of Filled Elastomers and Thermoplastic Elastomers (Payne and Mullins Effects). Macromolecules 2008, 41 (21), 8252−8266. (21) Merabia, S.; Sotta, P.; Long, D. R. Unique plastic and recovery behavior of nanofilled elastomers and thermoplastic elastomers (Payne and Mullins effects). J. Polym. Sci., Part B: Polym. Phys. 2010, 48 (13), 1495−1508. (22) Papon, A.; Merabia, S.; Guy, L.; Lequeux, F.; Montes, H.; Sotta, P.; Long, D. R. Unique Nonlinear Behavior of Nano-Filled Elastomers: From the Onset of Strain Softening to Large Amplitude Shear Deformations. Macromolecules 2012, 45 (6), 2891−2904. (23) Berriot, J.; Montes, H.; Lequeux, F.; Long, D.; Sotta, P. Evidence for the Shift of the Glass Transition near the Particles in Silica-Filled Elastomers. Macromolecules 2002, 35 (26), 9756−9762. (24) Long, D. R.; Lequeux, F. Heterogeneous dynamics at the glass transition in van der Waals liquids, in the bulk and in thin films. Eur. Phys. J. E: Soft Matter Biol. Phys. 2001, 4 (3), 371−387. (25) Dequidt, A.; Long, D. R.; Sotta, P.; Sanséau, O. Mechanical properties of thin confined polymer films close to the glass transition in the linear regime of deformation: Theory and simulations. Eur. Phys. J. E: Soft Matter Biol. Phys. 2012, 35 (7), 1−22. (26) Baeza, G. P; Genix, A.-C.; Degrandcourt, C.; Petitjean, L.; Gummel, J.; Couty, M.; Oberdisse, J. Multiscale Filler Structure in Simplified Industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM. Macromolecules 2013, 46 (1), 317−329. (27) Tauban, M. Impact of Filler Morphology and Distribution on the Mechanical Properties of Filled Elastomers: Theory and Simulations. Ph.D. Université Lyon 1, 2016. (28) Monnerie, L.; Halary, J. L.; Kausch, H.-H. Deformation, yield and fracture of amorphous polymers: Relation to the secondary transitions. Adv. Polym. Sci. 2005, 187, 215−372. (29) Lubachevsky, B. D.; Stillinger, F. H. Geometric properties of random disk packings. J. Stat. Phys. 1990, 60, 561−583. (30) Abramowitz, M., Stegun, I. A, Eds; Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover: New York, 1972. (31) Higgins, J. S.; Benoit, H. C.. Polymers and Neutron Scattering; Oxford Series on Neutron Scattering in Condensed Matter; Clarendon Press: 1994. (32) Witten, T. A.; Sander, L. M. Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon. Phys. Rev. Lett. 1981, 47 (19), 1400.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b00974. Appendix (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (D.R.L.). ORCID

Mathieu Tauban: 0000-0001-5874-916X Paul Sotta: 0000-0002-4378-0858 Didier R. Long: 0000-0002-3013-6852 Present Address

J.-Y.D.: Complex Assemblies of Soft Matter, UMI:3254 CNRS/ University of Pennsylvania, Solvay, Solvay Center of Research and Technology, 350 George Patterson Blvd, Bristol, PA 19007-3624. Notes

The authors declare no competing financial interest.

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REFERENCES

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Effect of Filler Morphology and Distribution State on the Linear and

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