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# How Do Active Chain Density and Cross-Link Length Influence the Mullins Softening in Filled and Unfilled Natural Rubbers?

Grégoire Delahaye, Jean-Benoît Le Cam, Julien Rosselgong, Sophie M. Guillaume, Benoît Ruellan, Isabelle Jeanneau

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## To cite this version:

Grégoire Delahaye, Jean-Benoît Le Cam, Julien Rosselgong, Sophie M. Guillaume, Benoît Ruellan, et al.. How Do Active Chain Density and Cross-Link Length Influence the Mullins Softening in Filled and Unfilled Natural Rubbers?. ACS Applied Polymer Materials, 2024, 6 (20), pp.12623-12635. 10.1021/acsapm.4c02147 . hal-04764047

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Submitted on 3 Nov 2024

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# How do active chain density and cross-link length influence the Mullins softening in filled and unfilled natural rubber?

G. Delahaye,†,‡,¶ J.-B. Le Cam,*,†,¶ J. Rosselgong,§ S. M. Guillaume,§ B.

Ruellan,‡,¶ and I. Jeanneau‡,¶

†Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F-35042 Rennes, France.

‡Contitech AVS France, Rennes Cedex, France.

¶ELAST-D3 joint research lab Continental/UR/CNRS. Campus de Beaulieu, Bat 10B, 35042 Rennes Cedex, France.

§Univ. Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes) - UMR 6226, F-35042 Rennes, France.

E-mail: jean-benoit.lecam@univ-rennes.fr

## Abstract

The present study addresses the effect of the active chain density and cross-link length on the Mullins effect in both carbon black (CB) filled and unfilled natural rubber (NR). A wide range of vulcanization system, i.e., sulphur (S) and accelerator (A) contents has been considered in order to ensure significant variations of active chain density, coupled with different content ratios (|[A]|/|[S]|) to change the sulphur cross-link length; it corresponds to conventional vulcanization (CV) for predominantly polysulphide cross-links, semi-efficient vulcanization (SEV) for mono, di, and poly-sulphide cross-links,
and efficient vulcanization (EV) for predominantly monosulphide cross-links. In addition, different vulcanization conditions, corresponding to various time and temperature, were investigated. Equilibrium swelling tests were performed in order to determine the active chain density by using the Flory-Rehner relationship. A correction was applied to account for the insoluble components. Uniaxial cyclic tension tests were carried out at increasing level of stretches. The results of these investigations showed that the active chain density has a significant effect for unfilled NR. The effect of the cross-link length was found negligible. As expected, the active chain density showed much lesser effect for filled NR. The vulcanization conditions showed no significant impact, for both temperature and time of vulcanization. Finally, a comparison between CB filled and unfilled NR is made, and a discussion about the physical explanation for both results is provided.

Introduction

Vulcanized rubber is used in many industrial applications for its specific mechanical proper-
ties. For instance, carbon black (CB) filled natural rubber (NR) is widely used in the design
of antivibratory systems for its good damping and fatigue properties. Under cyclic loadings,
rubber exhibits a singular mechanical behavior: when a rubber specimen is stretched to a
given stretch and then retracted, a lesser stress is required to re-stretch the specimen to
the same stretch. First observed by Bouasse and Carrière in 1903, the softening effect was
then extensively studied by Mullins, which is why the phenomenon has been then referred
to as the "Mullins Effect".¹,² The Mullins effect includes different changes in the mechanical
response, that are summarized as follows:

* after a first cycle, a softening effect appears for stretches lower than or equal to the maximum stretch previously applied,

* the higher the stretch reached at the first cycle, the higher the softening effect,
* during the next cycle, the material stiffens as the current stretch is closed to the maximum one applied previously,

* when applying a higher stretch than the maximum stretch previously reached, the mechanical response follows the same path as a virgin material,

* the softening effect is maximal during the first cycle. The mechanical response stabilizes during the first ten cycles at the same stretch.

The Mullins effect has been observed in many filled rubbers such as NR,³ styrene butadiene rubber (SBR),⁴ ethylene propylene diene monomer (EPDM),⁵ nitrile butadiene rubber (NBR),⁶ silicone⁷,⁸ non-exhaustively. Furthermore, and more surprisingly, the Mullins effect has also been evidenced in unfilled rubbers such as NR by Harwood *et al.*⁹ and NBR by Kakavas.¹⁰

The interpretations reported in the literature apply mainly to filled rubbers. Three different interpretations are found: (i) the rubber matrix damaging such as disentanglement, changes in the active chain density, rubber chain scission, appearance of cavities, (ii) the filler network alteration such as loss of percolation, filler aggregate and agglomerate fracture and (iii) the modification of the rubber-filler interface such as bonds breakage at the filler surface, slipping of molecules, desorption/adsorption.⁴,¹¹ For further information on the different explanations summarized above, see for instance Blanchard and Parkinson¹² for breakage of weak bonds, Houwink¹³ and also Dannenberg and Brennan¹⁴ for the molecules slipping. Importantly, among the three interpretations previously mentioned, only the first one applies for the Mullins effect in unfilled NR. More generally, the fact that the Mullins effect can be observed in filled and unfilled rubbers questions the contribution of macromolecular network, especially the active chain density and the cross-link length.¹⁵ The active chain density is defined as the number of chain segments of a molecule between successive points of cross-linkage and is denoted $v_{ac}$ in the following.¹⁶ The literature has provided some initial answers about the effect of $v_{ac}$ and cross-link length on the Mullins effect. For filled SBR, Kraus *et al*.
conducted swelling tests in n-heptane after various cyclic tests.¹⁷ They estimated the Mullins effect by a ratio between area under stress-strain curves of virgin and softened styrene butadiene rubber (SBR). Their results indicated only small differences in the volume fraction of rubber at equilibrium, between virgin and softened specimens. Furthermore, these changes are of a relatively low order of magnitude in comparison to the softening, for a strain level of 250%. Thus, breakage in the SBR network is not sufficient to interpret the Mullins effect. Dannenberg and Brennan also conducted swelling tests on filled SBR. They evaluated the Mullins effect by a relationship based on the strain energy loss, such as Dannenberg and co-authors.¹⁴ Their results showed no changes in active chain density for softened SBR in comparison to virgin ones, and concluded that the Mullins effect is unrelated to cross-links breakage. Later, Roland studied an NR and several SBR, non-vulcanized with sulphur but with peroxide. Their materials were chosen to ensure variations in $v_{ac}$ and cross-link types. They concluded that the rubber microstructure had no effect on the softening since comparable results were obtained for each materials.¹⁸ However, from the six materials studied by Roland, it is difficult to extend the conclusions to the case of unfilled NR. More recently, in the study of Wan *et al.*, results from simulation showed a dependence of the softening effect on $v_{ac}$, for unfilled NR.¹⁹ Even more recently, the results of Lin *et al.* also showed an effect of $v_{ac}$ on the softening.²⁰ In view of the results presented in this literature review, it appears that only few unfilled NR have been studied, and that the authors have more often varied the microstructure for filled SBR. Finally, the authors were more concerned about the effect of the softening on the rubber network rather than the reverse situation, and the conclusion is that the Mullins effect has only minor effects on the network since it does not change significantly the $v_{ac}$. However, to the best of our knowledge, no information about a possible relationship between $v_{ac}$ and the softening was provided. This review highlights how the understanding of the macromolecular network - Mullins effect relationship has to be improved, especially in the case of unfilled NR.

To substantially advance the state of the art, a comprehensive characterization of the Mullins
effect in different formulations of the same NR, to cover a large range of $v_{ac}$ and cross-link length, both filled and unfilled NR, must be carried out. This includes some of the materials presented in Ref 21.

The present paper is composed as follows. Section 2 presents the experimental setup and the data-processing. Materials are described as well as the swelling procedure and quasi-static cyclic tests. The specimens used for both experiments are presented. The methodology used to evaluate the Mullins effect is precisely explained. In Section 3, results of the swelling measurements are given, and the mechanical response of the different NR tested are reported and discussed according to the active chain density and cross-link length. The discussion also compares the relative contribution of cross-link features and fillers to the Mullins effect. For the sake of clarity, the Mullins effect will be referred to as "Mullins softening" hereafter similarly to prior reports (see Refs 11,18). Concluding remarks and perspectives close the paper.

# Experimental section

## Materials of the study

The material considered here is a NR of Standard Malaysian Rubber (SMR) grade. This type of rubber, i.e., *cis*-1,4-polyisoprene is made of more than 98% of *cis*-configuration monomers. As a result, the macromolecules are highly stereoregular. Some components in the formulation of the materials studied are common to all of them: 10 phr, which is the part per hundred of rubber in weight, of zinc oxide is used. The same amount of plasticizer and anti-oxydants was also used, 2 and 3 phr, respectively. In order to evaluate the effect of $v_{ac}$ and cross-link length, different vulcanization systems and conditions have been investigated. In the present study, only sulphur vulcanization is studied, coupled with N-cyclohexylbenzothiazol-2-sulphenamide (CBS) accelerator. In order to change both the
cross-link length and $v_{ac}$, several vulcanization systems were defined. They were chosen
in agreement with the standard distinction proposed by Quirck,²² in which a distinction is
drawn between three principal types of materials: CV, SEV and EV systems. The difference
between them lies in the length of the cross-links formed, which is driven by the ratio between
the accelerator and the sulphur contents, denoted [A]/[S]. The smaller the ratio, the longer
the length of the cross-links formed. For instance, CV are mainly composed of polysuphide
cross-links while shorter cross-links such as mono-and-di-sulphide cross-links are obtained for
EV ones. It is noteworthy that cross-link length has an effect on the NR dynamics, shorter
cross-links being more restrictive than longer ones.²³

On a first hand, to vary the cross-link length, a total of five content ratios are defined; 0.2, 0.6, 1.5, 2.5 and 4. These ratios correspond to vulcanization system ranges from CV to EV, and were chosen to ensure sufficient variation in cross-link length. On a second hand, in order to vary $v_{ac}$, both accelerator and sulphur contents are changed while keeping a fixed content ratio. By keeping the ratio constant, it is possible to ensure a fairly identical mean length of the sulphur cross-links.²⁴ Additionnaly, in order to evaluate the contribution of fillers relatively to the cross-link features, NR has been filled with 20 phr of N330 High Abrasion Furnace (HAF) grade CB. This grade of CB is considered as a reinforcing filler.²⁵ Therefore, two 2 phr of plasticizer is added in order to facilitate the fillers dispersion.

Formulations are given in Table 1, presenting the designation of each formulation. The contents of accelerator, sulphur, their content ratio, and the content of CB are given. The "0 and 20" in the CB content indicates that both unfilled and 20 phr CB filled formulations exist.

## Vulcanization conditions

The specimens were obtained by a molding process. All the components were incorporated
into the internal mixer except those for the vulcanization system, which were added during
external mixing, in an open-roll mill, to ensure a temperature of the mix lower than 100°C
<table><caption>Table 1: Materials of the study. They are denoted $NR_B^A$, with A the content of sulphur [S] and B the [A]/[S] ratio. For the sake of clarity, designation are classified in accordance with the vulcanization system type: CV, SEV and EV in black, blue, and red, respectively. The formulation is given in phr. Vulcanization conditions are given: N°0 $t_{98}$ at 160°C, N°1 to $t_{80}$ at 160°C, N°2 to $t_{98+40\% of time}$ at 160°C, N°3 to $t_{98}$ at 150°C, N°4 to $t_{98}$ at 170°C.</caption><thead><tr><th>Designation</th><th>[A]</th><th>[S]</th><th>[A]/[S]</th><th>CB</th><th>Vulcanization condition N°</th><th>ν<sub>ac</sub></th></tr></thead><tbody><tr><td>NR<sup>2</sup><sub>0.2</sub></td><td>0.4</td><td>2</td><td>0.2</td><td>0 and 20</td><td>0</td><td>1.23</td></tr><tr><td>NR<sup>3</sup><sub>0.2</sub></td><td>0.6</td><td>3</td><td>0.2</td><td>0</td><td>0</td><td>1.63</td></tr><tr><td rowspan="4">NR<sup>4</sup><sub>0.2</sub></td><td rowspan="4">0.8</td><td rowspan="4">4</td><td rowspan="4">0.2</td><td>0 and 20</td><td>0</td><td>1.97</td></tr><tr><td rowspan="4">20</td><td>1</td><td>2.06</td></tr><tr><td>2</td><td>1.82</td></tr><tr><td>3</td><td>2.18</td></tr><tr><td>4</td><td>2.10</td></tr><tr><td>NR<sup>6</sup><sub>0.2</sub></td><td>1.2</td><td>6</td><td>0.2</td><td>20</td><td>0</td><td>2.88</td></tr><tr><td>NR<sup>2</sup><sub>0.6</sub></td><td>1.2</td><td>2</td><td>0.6</td><td>20</td><td>0</td><td>1.83</td></tr><tr><td>NR<sup>4</sup><sub>0.6</sub></td><td>2.4</td><td>4</td><td>0.6</td><td>20</td><td>0</td><td>2.91</td></tr><tr><td>NR<sup>6</sup><sub>0.6</sub></td><td>3.6</td><td>6</td><td>0.6</td><td>20</td><td>0</td><td>4</td></tr><tr><td>NR<sup>0.8</sup><sub>1.5</sub></td><td>1.2</td><td>0.8</td><td>1.5</td><td>20</td><td>0</td><td>1.23</td></tr><tr><td rowspan="4">NR<sup>1.2</sup><sub>1.5</sub></td><td rowspan="4">1.8</td><td rowspan="4">1.2</td><td rowspan="4">1.5</td><td>0 and 20</td><td>0</td><td>1.63</td></tr><tr><td rowspan="4">20</td><td>1</td><td>1.55</td></tr><tr><td>2</td><td>1.53</td></tr><tr><td>3</td><td>1.60</td></tr><tr><td>4</td><td>1.51</td></tr><tr><td>NR<sup>1.6</sup><sub>1.5</sub></td><td>2.4</td><td>1.6</td><td>1.5</td><td>20</td><td>0</td><td>1.92</td></tr><tr><td>NR<sup>0.8</sup><sub>2.5</sub></td><td>2</td><td>0.8</td><td>2.5</td><td>20</td><td>0</td><td>1.45</td></tr><tr><td>NR<sup>1.2</sup><sub>2.5</sub></td><td>3</td><td>1.2</td><td>2.5</td><td>0 and 20</td><td>0</td><td>1.79</td></tr><tr><td>NR<sup>1.6</sup><sub>2.5</sub></td><td>4</td><td>1.6</td><td>2.5</td><td>20</td><td>0</td><td>2.2</td></tr><tr><td>NR<sup>0.5</sup><sub>4</sub></td><td>2</td><td>0.5</td><td>4</td><td>0</td><td>0</td><td>1.2</td></tr><tr><td rowspan="4">NR<sup>0.8</sup><sub>4</sub></td><td rowspan="4">3.2</td><td rowspan="4">0.8</td><td rowspan="4">4</td><td>0 and 20</td><td>0</td><td>1.68</td></tr><tr><td rowspan="4">20</td><td>1</td><td>1.69</td></tr><tr><td>2</td><td>1.65</td></tr><tr><td>3</td><td>1.74</td></tr><tr><td>4</td><td>1.65</td></tr><tr><td>NR<sup>1.1</sup><sub>4</sub></td><td>4.4</td><td>1.1</td><td>4</td><td>0</td><td>0</td><td>2.3</td></tr></tbody></table>
in order to prevent the compound from early cross-linking reactions. The specimens were molded at a temperature of 160 °C, and at a time corresponding to $t_{98}$ (condition N°0 in Table 1), determined with a rheometer (Rubber Process Analyzer 1000, Alpha Technologies). In the rubber field, rheometry is used to monitor the vulcanization kinetics. In practice, $t_{98}$ corresponds to the time required to reach 98% of the maximal torque measured. This is related to the cross-linking process, its direct consequence is the stiffening of the compound. In addition, filled $NR_{0.2}^4$, $NR_{1.5}^{1.2}$ and $NR_4^{0.8}$ were also molded with four other conditions to vary both the temperature and the vulcanization time. First, the vulcanization time is changed: condition N°1 corresponds to an under-vulcanized material, while the condition N°2 corresponds to an over-cured one.²⁶ Then, a vulcanization at low and high temperature of 150 °C and 170 °C were defined as the conditions N°3 and 4, respectively. Time and temperature of vulcanization affect the length and the number of the cross-links formed. A too long vulcanization time is known to cause reversion. This reversion generally has an effect on the stereo-regularity of the macromolecules and can also induce chain scission.²⁷,²⁸ For vulcanization temperature, it can be stated that increasing the temperature, for a corresponding optimal time, decreases $v_{ac}$ and cross-link length.²⁹⁻³¹ These effects are further discussed in Results section.

**Remark:** CB structure has an effect on the mechanical behavior of NR, and the Mullins softening, see for instance Ref 32. This is not addressed in the present study.

## Evaluation of the active chain density

In this section, the procedure of the swelling tests is first detailed. Then, the data-processing as well as the assumptions used for the calculation are presented.
## Swelling tests procedure

The swelling experiments are made by using circular specimens cut with a punch in 2 mm thick sheet: the initial weight is 0.4 g, i.e., small specimens are preferred since they can swell up to several times their original volume. The reported values are the average of three samples.

The specimens of weight $w_i$ are immersed for 72 h in a closed vial of toluene, HPLC grade. Then, specimens are released, and a paper cloth is used to dry the solvent excess from their surface. The weight at swollen state ($w_s$) is then measured. In order to evaporate toluene trapped in the specimens, they are dried in a vacuum-oven until a constant weight is obtained, typically for 24 h at 60 °C. The specimens are finally weighed at the considered dry state ($w_d$). The weight is measured with a model XT-220A (Precisa scale) balance. These tests were performed at ambient temperature, without isolating the vial from the light. UV light is known to cause a chemical degradation of unfilled NR and could affect the swelling measurements.33 However, by looking at the swelling kinetics of Candau on similar materials, the degradation happened only at a relatively long time, around 300 h.34 Since it is considerably longer than the chosen immersion time, UV exposure is assumed to have no effect.

## Data-processing and assumptions

From swelling experiments, $v_{ac}$ can be calculated by using the following relation: $v_{ac} = \frac{\rho}{M_c}$, typically expressed in mol per unit volume. The term $\rho$ denotes the rubber density, and $M_c$ the molecular weight between cross-links. They are obtained by using the Flory-Rehner relationship with the assumption of a phantom network model (Eq. 1), i.e., cross-links junction points are considered to have a certain mobility.35–37 This model is known to be more realistic than the affine network one.24,38

$$M_c = - \frac{\rho v_{sol} v_p^{\frac{1}{3}} (1 - \frac{2}{\Phi})}{\log(1 - v_p) + v_p + \chi v_p^2} \quad (1)$$
with $\chi$ the Flory-Huggins interaction,³⁹,⁴⁰ $\Phi$ the functionality of the rubber network, $v_r$ and $v_{sol}$ the rubber volume fraction and the solvent volume fraction at equilibrium, respectively.

For the calculation of $v_{ac}$, several assumptions are made and detailed as follows. The functionality of the network $\Phi$ is taken as 4, which physically corresponds to a cross-link junction point joining on the average 4 rubber chains.⁴¹ For the Flory-Huggins interaction $\chi$, it is not defined as a constant, as it is the case in most studies. Indeed, since the parameter is dependent on $v_r$, it is calculated through the relation: $\chi = 0.427 + 0.112v_r^2$.²⁴ For $v_r$, it is equal to the inverse of the equilibrium swelling ratio (or degree of swelling) $Q$. The general assumption for the calculation of $Q$ is to consider the volume at swollen state as the sum of the solvent and the rubber volumes. In this case, taking the initial weight has no significance since it changes during the swelling due to the extraction of the soluble components in solvent. As a result, weight at the dry state is obviously lower than the one of the initial state, and is therefore chosen for the calculations. Soluble components can be un-reacted rubber chains²⁴ or adjuvants such as antioxidants. Importantly, fillers and activators have an effect on the swelling: they limit the fraction of solvent volume absorbed by the rubber.⁴²⁻⁴⁴ Since all of our materials are composed of 10 phr of ZnO, and 20 phr of N330 HAF CB for filled ones, their are subjected to this restriction phenomenon. The swelling restriction, which leads to an overestimation in the calculation of $v_{ac}$, must therefore be taken into account and corrected. As a result, a correction of the insoluble components in toluene is employed. Details and procedure are given in our recent study.²¹

## Procedure for uniaxial tests and specimen geometry

Uniaxial mechanical tests were carried out with a home-made biaxial testing machine (see Fig. 1 (a)). Specimens were cut with a water-jet cutting device from pure shear specimens. The useful part of the simple tension specimen have a length of 23 mm and a section of 8×2 mm². The extremities of the specimens are composed of cylindrical extension of Ø10 mm.
Tests are prescribed under displacement, specimens being mounted in the vertical direction, and stretched symmetrically. Sets of five mechanical cycles are applied at different levels from $\lambda=1.5$ to $\lambda=7.5$, with an increment of 1 (see Fig. 1 (b)). The loading rate is set to $\pm 250 \text{ mm/min}$, which corresponds to strain rate equal to $0.18 \text{ s}^{-1}$.

Such a procedure induces a non-negligible permanent stretch of the specimens. Indeed, the length of the unbuckled state is then increased. This induce a difference between the prescribed stretch and the actual stretch undergone by the specimen, as shown for instance in similar materials in the recent study of le Cam *et al.*$^{45}$ As a result, a correction of the permanent stretch at each cycle is applied. Once the force is greater than 0 N, the specimen is considered under tension, and the initial length $L_0$ is corrected. Note that the results will be reported in terms of prescribed stretch, for the sake of clarity.

<figure><img src="image_3.png" /><figcaption>Figure 1: (a) Overview of the experimental set-up used (b) stretch-time curve performed at $\pm 250 \text{ mm/min}$.</figcaption></figure>
## Evaluation of the Mullins softening

In 2010, Machado and coworkers proposed to quantify the Mullins softening from a ratio between the strain energies brought during the second ($W_{2L}$) and the first ($W_{1L}$) loading.⁸ It corresponds to the calculation of the area under the loading curves, given by Equation (2).

$$W_{iL} = \int_{1}^{\lambda_{max}} \pi d\lambda, i = 1, 2 \qquad (2)$$

In our case, since the cyclic tests are carried out at increasing levels of stretch, the calculation of $W_{1L}$ when a lower maximum stretch has already been prescribed, has to be modified otherwise it is strongly underestimated (see Refs 6,46). It is to be noted that $W_{2L}$ is not corrected. In practice, only the first load of $\lambda = 1.5$ could be estimated by using $W_{1L}$. Thus, the strain energy of the first load for the higher strain levels is calculated by fitting the monotonic tensile response, which corresponds to the non-accommodated state.

Corrected $W_{1L}$ is denoted $W_{1L}^*$ in the following. This prevents us from carrying out all the tests independently at each stretch level. For the sake of clarity, the difference between $W_{1L}$ and $W_{1L}^*$ is illustrated in Figure 2. In this Figure, two cyclic loadings (cycles #1 and #2) are considered at two increasing maximum stretch levels. In addition, the corresponding non-accommodated monotonic tensile loading is represented in dashed line. The strain energy of the loading of cycle #2 is under-estimated by using $W_{1L}$ (Fig. 2 (a)), since a previous cycle was applied. Consequently, using $W_{1L}^*$ gives a better estimation of the strain energy (Fig. 2 (b)). These quantities are illustrated by the colored areas.

Finally, the Mullins softening is evaluated by the ratio $C_W$, given in Equation (3). This ratio informs on the softening level. For instance, a ratio $C_W=1$ stands for no Mullins softening. As the ratio decreases, the Mullins softening becomes more pronounced, meaning that a lower quantity of strain energy is needed to reach the same prescribed stretch.
<figure><img src="image_4.png" /><figcaption>Figure 2: Schematic representation of two successive loading (cycles #1 and #2) at a given stretch, for a material already softened at lower stretch. The corresponding non-accommodated monotonic first load is given in dashed line. The determination of (a) $W_{1L}$ and (b) $W_{1L}^*$ are given with colored area.</figcaption></figure>

$$C_W = \frac{W_{2L}}{W_{1L}^*} \qquad (3)$$

# Results

## Effect of formulation and process conditions on active chain density

As above-mentioned, $\nu_{ac}$ is an intrinsic parameter that is calculated from swelling measurements. By changing the vulcanization conditions or/and system, this parameter varies. The effect of the sulphur [S] content, accelerator [A] content and the [A]/[S] ratio on $\nu_{ac}$ is summarized in the diagram proposed in Ref. 21 This diagram is presented in Figure 3, and enables us to gather all the information about the different vulcanization systems and their effect on $\nu_{ac}$. The X and Y axes represent the sulphur and accelerator contents, respectively. The dotted lines represent the different [A]/[S] ratios, which is a qualitative information on the length of the sulphur cross-links. As indicated by the red arrow in the middle of the diagram, the length of the cross-link decreases as the slope of the lines increases. Thus, the five cross-link length investigated in the present study are shown. Then, for a given [A]/[S],
the cross-link length is fixed and only $v_{ac}$ can be directly varied by sulphur and accelerator contents. This is illustrated by the red arrow below the $[A]/[S]=1$. For instance, for $[A]/[S]=0.2$ which corresponds to $NR_{0.2}^2$, $NR_{0.2}^3$, $NR_{0.2}^4$ and $NR_{0.2}^6$, increasing the sulphur and accelerator contents increases in consequence $v_{ac}$. This is the case for every other ratios, i.e., 0.6, 1.5, 2.5 and 4. The $v_{ac}$ of the NR formulations of the present study covers a range from 1 to $4 \times 10^4 \text{mol/cm}^3$. It is noteworthy that a value superior to $3 \times 10^4 \text{mol/cm}^3$ corresponds to very highly cross-linked NR, which are consequently very stiff.

<figure><img src="image_5.png" /><figcaption>Figure 3: Effect of the vulcanization system on $v_{ac}$ for filled and unfilled NR, vulcanized for the same conditions.</figcaption></figure>

A few details concerning the $NR_{1.5}^{0.8}$ and $NR_{0.2}^2$ is also given, as they provide some important information of the effect of the vulcanization. As shown in Table 1, these two materials have the same $v_{ac}$, even though they have a very different vulcanization system, i.e., cross-link length. In fact, this result highlights that for filled $NR_{0.2}^2$, within a low content of accelerator, a much higher content of sulphur is necessary to obtain such quantity of active chains. But, for filled $NR_{1.5}^{0.8}$, its significant content of accelerator is known to rapidly decomposed the polysulphide cross-links into shorter cross-links such as mono- and di-sulphide cross-links.²⁷ Consequently, this decomposition increased the overall number of cross-links. In this case,
it is not surprising that they can have the same $v_{ac}$, as they have very different cross-link
lengths.

For the effect of vulcanization time, filled $NR_{0.2}^4$ shows the largest variations of $v_{ac}$. A focus
on this NR and some details are given in the following. For filled $NR_{0.2}^4$, the lower the time
of vulcanization the higher the $v_{ac}$. At first sight, this result is surprising. The fact that
under-vulcanized filled $NR_{0.2}^4$ presents a higher $v_{ac}$ could be explained by the vulcanization
itself. Indeed, even though the specimens are removed early from the press, they are still
very hot: vulcanization can continue until the specimens have cooled sufficiently. Neverthe-
less, it is possible to avoid this effect by using a cooling of the part after vulcanization.⁴⁷
Following this idea, the optimally vulcanized filled $NR_{0.2}^4$ could already be in the beginning
of the reversion area. At this point, the polysulphide cross-links are likely to break. Two
competitive phenomena occurred: (i) the cross-links decomposition which tends to increase
the overall number of active chains and (ii) the formation of elastically inactive cross-links
such as cyclic polysulphide cross-links.⁴⁸ Since $v_{ac}$ decreased, one can assume that the second
phenomenon is predominant. For filled $NR_{1.5}^{1.2}$ and $NR_4^{0.5}$, vulcanization time showed only
a slight effect. This could be due to the fact that SEV and EV have a very wide plateau
before reversing.

For the vulcanization temperature, filled $NR_{1.5}^{1.2}$ and $NR_4^{0.5}$ exhibited the same trend: an
increase in temperature tends to decrease $v_{ac}$. However, for filled $NR_{0.2}^4$, it is unlikely that
a temperature of 160°C gives a lower $v_{ac}$ than a temperature of 170°C. This could be due to
process issue, and further demonstrates the high thermal sensitivity of long cross-link,³⁰,⁴⁹,⁵⁰
which could justify why CV are more complicated to manufacture.

## Effect of active chain density, cross-link length and fillers on mechanical response

A large number of materials have been studied in this work. Consequently, some of
them have been chosen to illustrate the changes induced by the vulcanization conditions,
vulcanization system (cross-link length and $v_{ac}$), and fillers on the mechanical response. The results not shown here are given in the SI section, see Figure S1 and S2. Figure 4 present the mechanical responses of the chosen materials. The results are presented in terms of the nominal stress $\pi$ expressed as a function of the prescribed stretch $\lambda$. For the sake of clarity, $v_{ac}$ as well as the vulcanization system type (CV, SEV and EV) are also given on the figures. Figures 4 (a-f) deals with the effect of both $v_{ac}$ and cross-link type on stiffness of unfilled NR. Then, Figure 4 (g-l) gives the effect of $v_{ac}$ and cross-link type for filled NR. The effect of adding CB fillers on the mechanical response is also discussed, but further elements are given in Appendix. This is explained in detail below.

Figs. 4 (a), (b) and (c) present the mechanical response of unfilled CV $NR_{0.2}^2$, $NR_{0.2}^3$ and $NR_{0.2}^4$. As these three unfilled NR have similar cross-link length, only $v_{ac}$ is varying from 1.15 to $2.02 \times 10^4 \text{mol/cm}^3$. As expected, increasing $v_{ac}$ has the effect of increasing the stiffness. For instance, for a prescribed stretch of $\lambda = 6.5$, the nominal stress is increased by a factor of 3 between unfilled $NR_{0.2}^2$ and $NR_{0.2}^4$, while at lower stretch, the nominal stress is increased by a factor 2. The evolution of the nominal stress $\pi$ as a function of $v_{ac}$ is shown in Figure S3. For prescribed stretch of 1.5 and 2.5, the nominal stress is linearly related to $v_{ac}$, while for 3.5 and 4.5, this relation changed. However, $v_{ac}$ and consequently stiffness cannot be increased "infinitely", as this sacrifices the deformability of the material. This is why the unfilled $NR_{0.2}^4$ broke for a prescribed stretch of $\lambda = 7.5$. To continue with Figure 4, the mechanical responses of unfilled EV systems are shown in (d), (e) and (f). This corresponds to the $NR_4^{0.5}$, $NR_4^{0.8}$ and $NR_4^{1.1}$; they presents a similar cross-link length, but different $v_{ac}$ ranging from 1.20 to $2.30 \times 10^4 \text{mol/cm}^3$. The increase in stiffness is again correlated with the increase in $v_{ac}$. The unfilled $NR_4^{1.1}$ broke for a prescribed stretch $\lambda = 6.5$, giving another illustration of the reduction of the elongation at break induced by an increase in $v_{ac}$.

For every unfilled NR presented in Figure 4, a hysteresis loop due to SIC is formed.⁵¹⁻⁵³ It was shown on similar materials that hysteresis loop is not due to intrinsic dissipation,
<figure><img src="image_6.png" /><figcaption>Figure 4: Effect of $\nu_{ac}$ and cross-link length on the mechanical response for (a-f) unfilled and (g-l) filled NR, plotted in terms of the nominal stress as a function of the prescribed stretch. Designation, vulcanization system type and $\nu_{ac}$ are given in the Figures. Two specimens broke (unfilled $NR_{0.2}^4$ and $NR_4^{1.1}$). The vertical line highlight the stretch at break.</figcaption></figure>
since there is no dissipation of mechanical energy into heat under adiabatic conditions.⁵²,⁵³
This means that there are no viscoelastic effects associated with the hysteresis of unfilled
NR.⁵⁴ Finally, as SIC only begins after a certain stretch threshold, around $\lambda_c = 4$, it is
logical that no hysteresis loop is observed for stretches below this threshold. It is perhaps
worth emphasising that for unfilled NR (i) the stretch threshold is independent of $\nu_{ac}$⁵⁵,⁵⁶
and (ii) SIC properties are related to both cross-link length and $\nu_{ac}$, see for instance Refs
51,56–60. The stress softening effect is also observed on the cyclic mechanical response for
each unfilled NR. This will be quantified in the following in order to characterize the effect
of $\nu_{ac}$ and cross-link length on the Mullins softening.

By comparing the mechanical responses in Figs. 4 (a) and (j) for $NR_{0.2}^2$, it can be concluded
that adding 20 phr of CB fillers - which is relatively low as CB filler can generally be added
up to 50 or 60 phr - causes a strong stiffening. Indeed, the stiffness is multiplied by a
factor of 3 for the highest stretch. The effect of CB filler on the mechanical response for
$NR_{1.5}^{1.2}$ and $NR_{2.5}^{1.2}$ also demonstrated a strong stiffening effect (Figure S4 in SI.) Adding
CB changed the NR microstructure: at the macromolecular scale, a rubber matrix coexists
with a filler network. Strong interactions between macromolecules and CB particles exists.²⁵
Consequently, CB filler network is recognized to reinforce the NR properties.⁶¹ For example,
CB changes the elongation at break and improve the tensile strength, depending on both
$\nu_{ac}$ and CB content.⁶² It should be noted that the tensile properties of highly cross-linked
unfilled NR are considerably enhanced by the addition of CB fillers.⁶³

Furthermore, CB filled NR also presents a hysteresis, which begins at around $\lambda = 3.5$.
Contrarily to unfilled NR, the physical phenomenon associated with the hysteresis loop is
not only due to SIC. In fact, this hysteresis is due to a competition between viscosity and
structural changes such as reorganization of the filler network and SIC.⁶⁴ Note that their
relative contribution can be quantified by measuring the surface temperature of the specimen
during tensile test. For more details, see Ref 65. It is also to be mentioned that SIC occurs
in CB-filled NR, at a certain stretch threshold. For instance, Candau found a $\lambda_c = 3$ and $\lambda_c = 2.5$ for NR filled with 20 and 50 phr of CB, respectively.³⁴ This is due to the fact that fillers induce a strain amplification, the higher the filler content the lower $\lambda_c$ value.⁶⁶ Indeed, in the bulk, the local stress is similar to those unfilled while in the vicinity of filler particles, stress is much higher.⁶⁷ Furthermore, crystallinity rate and maximum depends on the filler content.³⁴ Finally, a stress softening is also apparent each filled NR, and will be quantified in the following.

Thereafter, Figs. 4 presents the mechanical responses for filled NR$^{0.8}_{1.5}$ (g), for NR$^{1.2}_{1.5}$ (h) and for NR$^{1.6}_{1.5}$ (i), where only $\nu_{ac}$ is varied. As previously shown for unfilled NR, $\nu_{ac}$ also drives the stiffness for filled NR, with an increase in $\nu_{ac}$ leading to an increase in stiffness. Figs. 4 (j), (k) and (l) were added to investigate the effect of the cross-link length on the mechanical response. Results indicate that for a low $\nu_{ac}$, stiffness is very similar between CV and SEV ((g) and (j)), which suggests that cross-link length has no effect. For an $\nu_{ac}$ of around $1.60 \times 10^4 \text{ mol/cm}^3$, SEV and EV also present similar stiffness ((h) and (k)). But for high $\nu_{ac}$ ((i) and (l)), SEV is much stiffer than CV, indicating that short cross-link length increased the stiffness. Thus, the cross-link length effect depends on $\nu_{ac}$. CB filled NR having an $\nu_{ac}$ of $3 \times 10^4 \text{ mol/cm}^3$ showed catastrophic failures at low stretch. To conclude, both stiffness and deformability of NR can be completely tuned by CB fillers, $\nu_{ac}$ and cross-link length. This provides a wide range of parameters for controlling the mechanical response, and especially the Mullins softening.

## Mullins softening of unfilled NR

This section is dedicated to the evaluation of the Mullins softening for every unfilled NR. First of all, the softening effect is quantified by using $C_W$ (see Equation 3). Then, the $C_W$ values are related to $\nu_{ac}$. Results are given in Figure 5. This Figure maps the effect of $\nu_{ac}$ in the case of formulations having the longest cross-links (Fig. 5 (a)) and the shortest
cross-links (Fig. 5 (b)), for different prescribed stretches. First, it occurs that the Mullins
softening depends on the maximum prescribed stretch, whatever the formulation considered.
For stretch of $\lambda = 2.5$, the $C_W$ values are closed to 1, which is considered as negligible.
From $\lambda = 4.5$, the ratio started to decrease: the higher the stretch, the lower the ratio. In
view of the literature, this result confirms the fact that the maximum stretch applied drives
the Mullins softening. Consequently, $\nu_{ac}$ seems to have an effect. For each stretch presented
here, increasing $\nu_{ac}$ leads to a decrease of $C_W$. This effect suggests that the Mullins softening
depends on this parameter. Comparable results are obtained for the longest (Fig. 5 (a))
and shortest cross-link lengths (Fig. 5 (b)); $\nu_{ac}$ increases the Mullins softening, regardless of
the length of the sulphur cross-links. It is noteworthy that there is much more effect for $\nu_{ac}$
higher than $1.6 \times 10^4 \text{mol/cm}^3$.

In order to generalize $\nu_{ac}$ effect for every cross-link length, all the results obtained for the
8 formulations are gathered in Fig. 5 (c), which gives the coefficient $C_W$ as a function of
$\nu_{ac}$ and the stretch level. As expected, the Mullins softening is confirmed to depend on the
maximum applied stretch. Furthermore, for the higher stretches, the Mullins softening is
more pronounced: the trends for prescribed stretch of 7.5 and 2.5 are in fact very different.
This clearly highlights that $\nu_{ac}$ influences the physical phenomena responsible for the Mullins
softening. The ratio can vary up to 10% between the smallest and highest $\nu_{ac}$, for the highest
stretch. This highlights that the Mullins softening depends on the microstructure, and more
precisely on $\nu_{ac}$ for unfilled NR. **Concerning the effect of the cross-link, it is further addressed
in Discussion section. Consequently, it is proposed to model $C_W$ as a function of both the
stretch and $\nu_{ac}$.** The Y-axis interception decrease as $\lambda$ increases, and both $\lambda$ and $\nu_{ac}$ drives
the slope of the interpolated curves. So, it is proposed to link the softening via the empirical
relationship : $C_W = a\lambda(\nu_{ac} + 1) + b\nu_{ac} + c$, with a=-0.017, b=0.04 and c=1.062. This
indicates that an increase in $\nu_{ac}$ tends to reduce $C_W$, the magnitude of this reduction is
directly depending on $\lambda$ value.
<figure><img src="image_7.png" /><figcaption>Figure 5: Effect of $\nu_{ac}$ on the Mullins softening for NR with (a) long cross-links types and (b) short cross-links types. Dotted red lines correspond to zero Mullins softening, i.e., $C_W=1$. The effect of $\nu_{ac}$ on the Mullins softening for every vulcanization system, i.e., all the unfilled NR tested is presented in (c).</figcaption></figure>
## Mullins softening in CB filled NR

Since $v_{ac}$ was found to affect the Mullins softening for unfilled NR, it is of first importance to determine whether or not the microstructure still has an effect on the softening once the material is filled, particularly for modeling the mechanical response. First, the effect of the vulcanization system is addressed. Then, the effect of processing conditions is investigated.

## Vulcanization system

Figure 6 (a) presents $C_W$ as a function of $v_{ac}$, for filled NR, and for different stretch levels ranging from 2.5 to 7.5. For the filled NR with a high $v_{ac}$, the specimens broke due to the high stiffness and no $C_W$ values are given. First, this Figure indicates that the Mullins softening depends on the maximum applied stretch. It is in good agreement with the results from the literature, see for instance Ref 3. It is perhaps worth emphasising that (i) Mullins softening is sharply higher in CB filled NR than in unfilled one, for a given stretch and (ii) $C_W$ evolves linearly with $v_{ac}$, showing that the effect of $v_{ac}$ depends on whether the NR is filled or not. Thus, these first results contrast sharply with those obtained in the case of unfilled NR. However, the trend in the interpolated scatter-plots reveals a decrease of $C_W$ as $v_{ac}$ increases; but the effect of $v_{ac}$ is only slightly pronounced. In view of the linearity of the results, it is also important to notice that the cross-link length seems to only have minor effect on the Mullins softening. This result is consistent with the previous ones on unfilled NR. To continue, this result suggests that the Mullins softening is mainly driven by the filler effect as well as the maximal stretch applied. Thus, it is proposed to model $C_W$ as a function of these two parameters. In contrast to the unfilled NR, the y-intercept of the linear trend curves varies as a function of the prescribed stretch. It should be noted that although $v_{ac}$ demonstrated only a small effect, it was therefore chosen to model it. The following empirical relationship: $C_W = av_{ac} + b\lambda^c$, with a=-0.0158, b=1.3515 and c=-0.46. This relationship shows that an increase in $\lambda$ value tends to reduce $C_W$, the magnitude of this reduction is slightly depending on $v_{ac}$ value.
<figure><img src="image_8.png" /><figcaption>Figure 6: Effect of $\nu_{ac}$ on the Mullins softening for filled NR vulcanized with (a) condition N°0 in Table 1 and (b) condition N°0, 1, 2, 3, 4 and 5 in Table 1.</figcaption></figure>

## Vulcanization conditions

The results presented previously were obtained with vulcanization conditions corresponding to condition N°0 in Table 1, i.e., curing at 160°C during a time corresponding to $t_{98}$. However, to give a complete understanding of the changes in mechanical properties induced by vulcanization conditions, these parameters are varied accordingly to Table 1. Since industrial parts are formulated with fillers, the effect of the vulcanization conditions is only addressed for filled NR. Therefore, this sub-section aims at investigating in which extend the Mullins softening is affected by the vulcanization conditions, i.e., by microstructural changes induced by variation in the time and temperature of vulcanization. Thus, before presenting any data, few elements concerning time and temperature of vulcanization are recalled.

First, increasing the vulcanization temperature is known to reduce both $\nu_{ac}$ and cross-link length.²⁹,³⁰ Then, the vulcanization time strongly changes $\nu_{ac}$ as vulcanization process lies in cross-linking: increasing the time increases $\nu_{ac}$ up to a maximum.⁶⁸ But, once the optimum time has been reached, it is known that other major reactions take place, leading to important microstructural changes such as the formation of: (i) dangling (or pendant) cross-links (ii) cyclic cross-links (cross-linking of a same macromolecule) (iii) decomposition
of polysulphide cross-links into shorter cross-links and (iv) cross-links scission.⁴⁸,⁴⁹,⁶⁹,⁷⁰ These modifications are generally entered into competition. For instance, a CV system will be more affected by cyclic cross-links and decomposition of polysulphide cross-links into shorter cross-links, while an EV mainly promotes dangling cross-links. Furthermore, EV are also prone to rubber chains modifications such as macromolecular scission. To go further, *cis* isomeric structure can also change into *trans* structure, which directly affects the stereoregularity of the rubber network. For more details, see Refs 48,70.

<figure><img src="image_9.png" /><figcaption>Figure 7: Effect of $\nu_{ac}$ on the Mullins softening for filled NR$_{0.2}^4$, NR$_{1.5}^{1,2}$ and NR$_{4}^{0.8}$ vulcanized with different conditions which are (a) 150°C (N°3 in Table 1), (b) 170°C (N°4 in Table 1), (c) under-vulcanized at 160°C (N°1 in Table 1), and (d) over-vulcanized at 160°C (N°2 in Table 1).</figcaption></figure>
Figure 7 presents the effect of the vulcanization conditions on the Mullins softening, in the same diagram as those introduced previously. So, $C_W$ is given as a function of $v_{ac}$ and $\lambda$, for vulcanization temperature of 150 °C (Fig. 7 (a)) and 170 °C (Fig. 7 (b)), and also for under-vulcanization (Fig. 7 (c)) and reversion (Fig. 7 (d)). First of all, vulcanization temperature led to similar results than the ones obtained at 160 °C. Indeed, for both temperatures of 150 °C and 170 °C, the Mullins softening depends on $v_{ac}$, and also the maximal stretch applied. However, a vulcanization temperature of 170°C seems to be slightly more affected by $v_{ac}$, the trend indicating that an increase in $v_{ac}$ decreases $C_W$. It was previously concluded that cross-link length has no effect on Mullins softening. Since changing the temperature of vulcanization mainly resulted in changes in $v_{ac}$ and cross-link length, the results (Figs. 7 (a) and (b)) appear to be consistent with these possible structural changes. However, it is difficult to draw more conclusions on these results. For the results on the two different temperature of vulcanization, Mullins softening is linked to the stretch and $v_{ac}$. **This will be further addressed in Discussion.**

For a vulcanization time higher than the optimal time, the Mullins softening is no longer a function of $v_{ac}$. Indeed, $C_W$ varies slightly when increasing $v_{ac}$ for over-vulcanized materials. This is not the case for the under-vulcanized NR. In this sense, the different phenomenon induced by reversion seems to affect the Mullins softening; in fact they possibly compensate the effect of $v_{ac}$, or they may inhibit it. So, among phenomenon induced by reversion, and by considering previous results, changes of cross-link length is believed not to be the cause. Macromolecule scissions and/or changes in chain conformation, from *cis* to *trans* structures, could most probably be the reason. However, since reversion strongly depends on the vulcanization system type, it is complicated to discuss further about the results. It is recalled that only filled $NR_{0.2}^4$, which is mainly composed of long polysulphide cross-links, shows a strong decrease in $v_{ac}$ with over-vulcanization. **These results on vulcanization time and temperature are further discussed in Discussion section.**
Discussion

The results on the effect of $\nu_{ac}$ on the Mullins softening in unfilled NR raised some addi-
tional questions. It was shown that $C_W$ depends on the stretch, with softening effect strongly
increased for stretch higher than 4.5. As above-mentioned, this stretch value corresponds
to the onset of SIC for unfilled NR,⁵¹ and raises questions about the effect of SIC on the
Mullins softening. In addition, since an increase in $\nu_{ac}$ led to an increase in the softening
effect, one can wonder if increasing $\nu_{ac}$ also increased the SIC? However, the effect of $\nu_{ac}$ on
SIC remains unclear since contradictory results exist in the literature. Indeed, for Trabelsi et
al., increasing $\nu_{ac}$ from 0.8 to 1.88 ×10⁴ mol/cm³ decreased the maximum of crystallinity⁵¹
while for Tosaka et al., from 1.01 to 2.12 ×10⁴ mol/cm³, the maximum was found to increase.
Similarly, Chenal et al. discussed the possible existence of an optimum value of $\nu_{ac}$ maxi-
mizing the crystallinity rate.⁵⁶

To continue with the results of unfilled NR, the effect of the cross-link length remains unclear.
A COMPLETER.

The previous results on vulcanization time showed that too long time corresponding to
over-vulcanization could have an effect on Mullins softening. Consequently, an attempt to
compare relatively the values of $C_W$ for the different vulcanization times and temperatures
was carried out. Figure 6 (b) presents the results about the under- and over-vulcanized
plotted with those optimally vulcanized, and also results obtained for vulcanization at 150
and 170°C. It appears that the Mullins softening was comparable for the filled NR, regardless
of the vulcanization conditions. Thus, vulcanization time has little effect over the time range
investigated in this study.

In order to discuss on the results of both unfilled and filled NR, a 3D plot is proposed to
gather them. The X and Y axis referred to as the vulcanization system, *i.e.*, the sulphur
and accelerator contents, respectively. The Z axis represent the values of $C_W$ obtained for
unfilled NR ((a), (b) and (c)) and for the filled one ((d), (e) and (f)) in Figure 8. Each
subplot present the values for a given level of stretch, the latter being indicated in the top
right. It is decided to represent the $C_W$ values for $\lambda$ of 2.5, 4.5 and 7.5. The main analogy
between filled and unfilled NR is the fact that they both depend on the maximal stretch
applied. However, filled NR present a much greater extent of softening than unfilled one.
For their differences, the unfilled NR has very different values of $C_W$ for every stretch. But
for filled one, the $C_W$ values are almost independent of the vulcanization system.

<figure><img src="image_10.png" /><figcaption>Figure 8: Effect of the vulcanization system on the Mullins softening for every vulcanization system, for unfilled NR. The colorbar is common to all 4 subplots.</figcaption></figure>

An analogy is drawn between filled and unfilled NR. This analogy is limited since only the
vulcanization system was varied for unfilled NR. Consequently, results can only be discussed
for vulcanization conditions N°0. To this end, we decided to focus on the filled and unfilled
$NR_{0.2}^2$ and $NR_{0.2}^4$, which exemplify the effect of $\nu_{ac}$ on the Mullins softening. Furthermore,
they were chosen since they have the same [A]/[S] ratio, and very different $\nu_{ac}$. Figure 9
presents the Mullins softening as a function of the prescribed stretch for the four NR pre-
viously mentioned. The unfilled $NR_{0.2}^2$ has a smaller Mullins softening than $NR_{0.2}^4$, which
corresponds to higher values of $C_W$. Consequently, increasing $\nu_{ac}$ has the effect of amplifying
the Mullins effect. On the contrary, when the same quantity and type of fillers are added - 20 phr of N330 HAF CB - a comparable Mullins softening is finally obtained. This seems to indicate that in the case of filled NR, the $v_{ac}$ effect is overcome by the filler effect.

In perspective, these results raise additional fundamental questions: does the Mullins softening depend on $v_{ac}$ for higher content of fillers? And with another type of filler, would the result be the same? Since $v_{ac}$ has been shown to have a negligible effect on the Mullins softening for 20 phr CB-filled NR, it would be logical that a higher CB content would lead to the same conclusion. This would confirm that the origin of the Mullins effect is predominantly induced by fillers, either by phenomena due solely to the filler network or by interaction with the rubber matrix. The study of different filler types is necessary, as the interactions between the rubber matrix and the fillers may be different.

<figure><img src="image_11.png" /><figcaption>Figure 9: Effect of $v_{ac}$ on the Mullins softening for filled (diamond) and unfilled (circle) $NR_{0.2}^2$ (black) and $NR_{0.2}^4$ (green).</figcaption></figure>

Conclusions and perspectives

This study deals with the effect of the microstructure, i.e., active chain density and cross-link length, on the Mullins softening of NR. Vulcanization system and conditions have been
changed. Unfilled and filled formulations have been considered. Swelling tests were performed in order to estimate the active chain density of each material. Uniaxial cyclic tests were carried out at increasing stretch levels. The Mullins softening was evaluated from the cyclic tests by using the ratio of the strain energy of the second cycle over the first one. As expected, the cyclic stress-stretch responses are driven by the active chain density, the cross-link length and CB. The Mullins softening was found to depend on the active chain density for unfilled NR. Indeed, an increasing in active chain density led to an increase in the softening level. The sulphur cross-link length has no significant effect. However, the active chain density had only minor effect on Mullins softening for filled NR. It was found that the Mullins softening is mainly driven by the maximal stretch. For filled NR, the temperature and time of vulcanization has no significant effect. This study provides results of importance to better model the softening with physically motivated approaches, especially those inspired from the molecular network description.

## Acknowledgement

The authors thank the Contitech company for supporting this work and for fruitful discussions. The authors also thank the National Center for Scientific Research (MRCT-CNRS and MI-CNRS), and Rennes Métropole, and Région Bretagne for having supported this work financially. This publication is supported by the European Union through the European Regional Development Fund (ERDF), the Ministry of Higher Education and Research, the French region of Brittany and Rennes Métropole.

## Supporting Information Available

Supporting Information. The material supplied as SI contains 5 Figures, designated Figure S1, S2, S3 and S4. Figs. S1,S2,S4 correspond to mechanical responses. Fig. S3 shows the effect of the active chain density on stiffness for filled CV.
The following file is available free of charge.

• Supporting information: "SupportingInformation.pdf".

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