Fillers Mullins effect

L. Mullins, Effect of fillers in rubber, Chap. 11 in The Chemistry and Physics of Rubber-Like Substances, ed. by L. Bateman, Wiley, New York, 1963. 

Natural rubber exhibits unique physical and chemical properties. Rubbers stress-strain behavior exhibits the Mullins effect and the Payne effect. It strain crystallizes. Under repeated tensile strain, many filler reinforced rubbers exhibit a reduction in stress after the initial extension, and this is the so-called Mullins Effect which is technically understood as stress decay or relaxation. The phenomenon is named after the British rubber scientist Leonard Mullins, working at MBL Group in Leyland, and can be applied for many purposes as an instantaneous and irreversible softening of the stress-strain curve that occurs whenever the load increases beyond... 

It was shown that the stress-induced orientational order is larger in a filled network than in an unfilled one [78]. Two effects explain this observation first, adsorption of network chains on filler particles leads to an increase of the effective crosslink density, and secondly, the microscopic deformation ratio differs from the macroscopic one, since part of the volume is occupied by solid filler particles. An important question for understanding the elastic properties of filled elastomeric systems, is to know to what extent the adsorption layer is affected by an external stress. Tong-time elastic relaxation and/or non-linearity in the elastic behaviour (Mullins effect, Payne effect) may be related to this question [79]. Just above the melting temperature Tm, it has been shown that local chain mobility in the adsorption layer decreases under stress, which may allow some elastic energy to be dissipated, (i.e., to relax). This may provide a mechanism for the reinforcement of filled PDMS networks [78]. 

So far the micro-mechanical origin of the Mullins effect is not totally understood [26, 36, 61]. Beside the action of the entropy elastic polymer network that is quite well understood on a molecular-statistical basis [24, 62], the impact of filler particles on stress-strain properties is of high importance. On the one hand the addition of hard filler particles leads to a stiffening of the rubber matrix that can be described by a hydrodynamic strain amplification factor [22, 63-65]. On the other, the constraints introduced into the system by filler-polymer bonds result in a decreased network entropy. Accordingly, the free energy that equals the negative entropy times the temperature increases linear with the effective number of network junctions [64-67]. A further effect is obtained from the formation of filler clusters or a... 

The above interpretations of the Mullins effect of stress softening ignore the important results of Haarwood et al. [73, 74], who showed that a plot of stress in second extension vs ratio between strain and pre-strain of natural rubber filled with a variety of carbon blacks yields a single master curve [60, 73]. This demonstrates that stress softening is related to hydrodynamic strain amplification due to the presence of the filler. Based on this observation a micro-mechanical model of stress softening has been developed by referring to hydrodynamic reinforcement of the rubber matrix by rigid filler... 

When a strip of a filler-reinforced rubber is extended, returned to the unstressed state and then re-extended, the second stress-strain curve is found to lie below the original one, at least up to the elongation of the first extension. This phenomenon, known as stress-softening, has been the subject of much study as well as controversy. It is frequently referred to as the Mullins Effect, although it was well known even before the extensive work of Mullins and collaborators. The subject was thoroughly reviewed by Mullins (181) in 1969 and no attempt will be made here to cover it in detail. Instead, only a brief summary will be given, along with some relevant observations not emphasized in the Mullins review. 

Another important effect of fillers is stress softening, or the Mullins effect. If a filled sample is stretched for the first time to 100%, the stress-strain curve27 will follow that illustrated in Figure 6-11. Now the strain is removed, and the sample is restretched to 200%. The stress in the second cycle is lower than that in the first up to 100%, after which it continues in a manner following the first cycle. If we repeat the stress-strain in a third cycle, we again see a softening up to 200% due to the previous strain history. This stress-softening effect was first discovered by Mullins, after whom it is named. 

If the tensile modulus for an unfilled elastomer is 5xl06 Pa, what value would it have if it were filled to / = 0.3 Suppose after several strain cycles, half of the filler is ineffective due to Mullins effect. What is its tensile modulus now ... 

Equation (1.5) has also been used to estimate the force at which a rubber molecule will become detached from a particle of a reinforcing filler (e.g., carbon black) when a filled rubber is deformed (Bueche, 1960, 1961). In this way, a general semiquantitative treatment has been achieved for stress-induced softening (Mullins effect) of filled mbbers (shown in Figure 1.5). 

Explanations of the Mullins effect have included failure of weak linkages, failure of network chains extending between adjacent filler particles, and polymer-filler bond failure, a form of dewetting. Brennan et al. (1969) and... 

Rapid decrease of E was observed with strain increase. Pure NR and composites with a low CNT content (1 wt%) showed a moderate increase of stress up to about 75% strain, a sort of plateau up to 300% strain and a final more evident increase above 300%. Said increase could be due to rubber crystallization and to CNT alignment. The stress-softening effect, known as Mullins effect, was observed at large strain and attributed to detachment of rubber molecules from the surface of filler particles. The presence of CNT bundles (at 5, 7, 10 wt%) was commented to bring about a decrease of the stress." ... 

The Mullins effect is characterized by stress softening. In order to demonstrate the presence of the Mullins effect in NR nanocomposites, three successive tensile cycles were performed for each sample. For the unfilled NR matrix, curves corresponding to the successive cycles were perfectly superposed, to a few per cent (6-8%) (Figure 14.22(a)).For nanocomposites, a significant decrease in G o can be observed between the first and the second cycle, for films reinforced with 30 wt% starch nanocrystals (Figure 14.22(b)).Similar to the Payne effect, the magnitude of the Mullins effect also increases with filler content. Furthermore, this increase was almost proportional to the filler content. 

Non-linear viscoelastic mechanical behaviour of a crosslinked sealant was interpreted as due to a Mullins effect. The Mullins effect was observed for a series of sealants under tensile and compression tests. The Mullins effect was partially removed after a mechanical test, when a long relaxation time was allowed, that is the modulus increased over time. Non-linear stress relaxation was observed for pre-strained filler sealants. Time-strain superposition was used to derive a model for the filled sealants. Relaxation over long periods demonstrates that the Mullins effect is caused by non-equilibrium with experimental conditions being faster than return to the initial state. If experiments were conducted over times of the order of a day there may be no Mullins effect. If a filled elastomer were only required to perform its function once per day then each response might be linear viscoelastic. 

A number of specialized elastomeric quantities have also been investigated. PDMS networks have been particularly useful in investigating the Mullins Effect, in which filler-reinforced elastomers exhibit a reduction... 

Examination of the stress as a function of time during retraction at constant strain rate following extension, and in particular for several successive cycles of elongation and retraction, reveals some further complications. The stresses during retraction are smaller than would be calculated on the basis of equation 8, and are smaller during extension in the second cycle than in the first. (This phenomenon, known as the Mullins effect, is particularly evident in filled rubbers, but appears also in the absenee of filler. )... 

Another softening phenomenon which manifests the dependence of the stress upon the entire history of deformation is the so-called Payne effect. Like the Mullins effect, this is a softening phenomena but it concerns the behavior of carbon blackfilled rubber subjected to oscillatory displacement. Strain dependence of the storage and loss moduli (Payne effect) at 70 °C and 10 Hz for a rubber compotmd with different concentration of carbon black filler [7] (Fig. 26). Indeed, the dynamic part of the stress response presents a rather strong nonlinear amplitude dependence, which is actually the Payne effect [8, 16, 43]. 

The Mullins effect, which can be considered as a hysteretic mechanism related to energy dissipated by the material during deformation, corresponds to a decrease in the number of elastically effective network chains. It results from chains that reach their limit of extensibility by strain amplification effects caused by the inclusion of undeformable filler particles [24,25]. Stress-softening in filled rubbers has been associated with the rupture properties and a quantitative relationship between total hysteresis (area between the first extension and the first release curves in the first extension cycle) and the enei-gy required for rupture has been derived [26,27]. 

There is considerable evidence that all the hysteresis effects observed in these materials and most of the viscoelastic behavior can be caused by the time dependent failure of the polymer on a molecular basis and are not due to internal viscosity [1,2]. At near equilibrium rates and small strains filled polymers exhibit the same type of hysteresis that many lowly filled, highly cross-linked rubbers demonstrate at large strains [1-8]. This phenomenon is called the "Mullins Effect" and has been attributed to micro-structural failure. Mullins postulated that a breakdown of particle-particle association and possibly also particle-polymer breakdown could account for the effect [3-5]. Later Bueche [7,8] proposed a molecular model for the Mullins Effect based on the assumption that the centers of the filler particles are displaced in an affine manner during deformation of the composite. Such deformations would cause a highly non-uniform strain and stress gradient in the polymer... 

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