Comparison of Two Recent Non-Gaussian-Type Tube Models: Analysis of Mechanical Data on Unfilled Natural and Styrene-Butadiene Rubber Vulcanizates

Literature stress-strain data of filler-unfilled networks, observed in uni and biaxial deformation modes, were comparatively characterized using Hoei's finite extensibility tube model and Meissner and Matějka's primary finite extensibility tube model excluding their empirical correction term for the finite extensibility. As a result of this analysis, the former model was found to be able to more advantageously describe the stress-strain behaviors over the entire range of strain without needing the empirical correction term.     

Four Non-Gaussian-Type Tube Models of Rubber-Like Elasticity: Analysis of Tensile Data for Lightly and Highly Carbon-Black Filled Styrene-Butadiene Elastomers

Recently, using a new, non-Gaussian, tube-based entropy model for active-filler filled elastic networks with the polymer phase influenced by the filler loading, proposed by the author, a data analysis of literature tensile data for carbon-black filled styrene-butadiene rubber (SBR) networks of different loadings was carried out; the data-fittability and the particular influence of the filler loading were examined in detail. In this report, the data-fitting performances of three other non-filled network models, all tube-based and finite chain extensible, along with a common strain-amplification factor that is necessary to extend them to the analysis of the filled networks, as well as the model which was employed in the above study were compared. As tensile data for this analysis, those observed for the same unfilled SBR, the same lightly loaded SBR and one of the same highly loaded SBR networks, as in our previous study, were chosen.     

Characteristic Frequency of Carbon Black Filled Rubber

Due to the high electrical conductivity of carbon black (CB) particles, the presence of CB improves the conductivity of filled rubbers. The impedance spectra of CB filled rubber were simulated using an infinite resistor‐capacitor (RC) circuit by considering the tunneling conduction mechanism for the CB contact regions. The calculated results had a similar appearance to the experimental results, i.e., the Cole‐Cole plot was a semi‐circle and a peak appeared in a plot of the imaginary component of complex impedance with respect to the frequency from which the characteristic frequency was obtained. For a simple RC circuit in parallel connection, the logarithm of the characteristic frequency should be a linear function of the 1/3 power of the average number of primary particles per aggregate. A slight deviation from the line found in the simulations was attributed to the network contribution. Additionally, low CB loading had little effect on the characteristic frequency, in accordance with the experimental data, while high loading had a marked effect. Furthermore, the calculations showed that the characteristic frequency was affected not only by the distribution of individual CB aggregates, but also by the percentage of agglomerates at high CB loading.     

Simulation of Carbon Black Network in Filled Rubber

Carbon black (CB) is one of the most important fillers for rubber and plastics materials. How to describe the CB network is a fundamental problem for establishing relationships between the CB network and the mechanical properties of filled rubber. In view of the electrical conductivity of CB, an infinite circuit consisting of numerous contact resistors, interconnected with each other, is proposed to simulate the CB network in filled rubber; the resistances were determined by considering the tunneling conduction mechanism and a Gaussian distribution for the CB aggregate junction width. As an example, the electrical resistivity of CB (N330) filled natural rubber during uniaxial deformation was studied. It was found that the logarithm of resistivity was an approximately linear function of the extension ratio, and the resistivity increased with the increase of average number of primary particles per aggregates. Additionally, some published experimental points lie between the curves calculated for five primary particles and for seven primary particles per aggregate at extension ratios below 1.2. The calculations suggested that the average number of primary particles per aggregate for CB type N330 might be between five and seven.     

Analysis of Anomalous Swelling by a Rubber Elasticity Model for Entangled Networks

A complete equation of the swelling activity parameter (S) as a function of swelling deformation (λ_s ) is derived by using a non-Gaussian elastic network model, including a tube concept and the Flory–Rehner model, and by following McKenna's criterion that takes into account the disparity between the Flory–Huggins interaction parameters for cross-linked and uncross-linked polymers. However, only a part of tube at the network chain size scale is extended to that for a large-scale structure according to the “gel tensile blob” model for equilibrium swollen networks. This approach is basically best for the “ideal regular network + simple structured good-solvent” binary systems due to its model character. As a result, it reproduces well the literature data of S versus λ_s ² with a maximum/inflection measured for such actual systems.     

Effects of Carbon Black on Chain Mobility and Dynamic Mechanical Properties of Solution Polymerized Styrene-Butadiene Rubber

The structure of the bound rubber, the ¹H NMR (nuclear magnetic resonance) relaxation time, and the crosslink density of the physical network and the glass transition, were studied for solution polymerized styrene-butadiene rubber (SSBR) filled by carbon black, to investigate the effects of carbon black on the chain mobility and dynamic mechanical properties. It was found by ¹H NMR analysis that the rubber chains were adsorbed on the surface of carbon black to form physical crosslinks and restrict the mobility of the chains, especially for some high-mobility units such as chain ends. It was calculated, according to the molecular weight between adjacent crosslinks, that the main motion units of the tightly adsorbed chains appeared to be similar in size to the chain segments. The glass transition temperature (T _g) obtained by differential scanning calorimetry (DSC) could not be used to judge the effect of carbon black on chain mobility, while the appearance and change of the loss-tangent (tan δ) peak at high temperature in dynamic mechanical thermal spectrometry (DMTS) test showed that there were three chain states: free chains, loosely adsorbed chains, and tightly adsorbed chains. The dynamic rheology test showed that the unfilled SSBR compound had the rheological characteristics of entangled chain networks; however the nonlinear viscoelasticities of the filled SSBR were related to the gradual disentanglement of adsorbed chains and free chains. The peaks in tan δ vs. temperature curves implied that the motion unit size decreased with the increase of bound rubber content, and the modulus vs. temperature curve showed an apparently lower mobility of adsorbed chains than that of free chains through the very low dependence of modulus on temperature for the highly filled compounds. The extremely high tensile modulus of the vulcanizate with 63.6% carbon black at room temperature also implied that the adsorbed chains were in the glass state due to their restriction by the carbon black.     

Analysis of Tensile and Permanent Set Data for a Silica-Filled Silicone Elastomer via Four Tube Models of Rubber-Like Elasticity

Using four recent tube models of rubber-like elasticity, a literature set of tensile and permanent set data for a silica-filled silicone network before and after γ-ray irradiation was investigated. As the silicone elastomer contained ～40 phr of silica, for the stress–strain analysis the latest filler cluster (fractal) breakup mechanism was combined with each of the tube models as a filled model along with the corresponding nonfilled one. In the first part of this study the essential parameters that the models have were determined from fits to the stress–strain data. In the last part the permanent set analysis was conducted with the parameter values, using the modified Tobolsky model involving chain scission and cross-linking. This showed very good agreement with the experimental data via any of three filled models (one of the filled models was excluded due to its intractability). It was also confirmed, as would be expected, that each of them worked more favorably than a corresponding nonfilled model.