Guth equation

Thus, in the coarse carbon black-fiUed mbber, the modulus increases according to the Guth equation with filler content, because the contribution from the GH layer (1% of the diameter) will disappear... 

In this relation, 2C2 provides a correction for departure of the polymeric network from ideality, which results from chain entanglements and from the restricted extensibility of the elastomer strands. For filled vulcanizates, this equation can still be applied if it can be assumed that the major function of the dispersed phase is to increase the effective strain of the rubber matrix. In other words, because of the rigidity of the filler, the strain locally applied to the matrix may be larger than the measured overall strain. Various strain amplification functions have been proposed. Mullins and Tobin33), among others, suggested the use of the volume concentration factor of the Guth equation to estimate the effective strain U in the rubber matrix ... 

The initial modulus of the composite is correlated with the filler volume fraction through the Guth equation ... 

The viscosity of composites with silica content from 10 to 70 phr, prepared with the solution procedure with an amine as catalyst, was found to increase with the silica content, being however lower than the viscosity of composites containing nanostructured silica. This difference, attributed to the lower amount of silanols on the surface of in situ generated silica, could be also attributed to the lower structure of in situ silica, that means to the absence of occluded rubber. The /shape factor of the Guth equation (Equation (2.1)) was calculated to be 2.53. 

Anthony, Caston, and Guth obtained considerably better agreement between the experimental stress-strain curve for natural rubber similarly vulcanized and the theoretical equation over the range a = 1 to 4. KinelP found that the retractive force for vulcanized poly-chloroprene increased linearly with a — l/a up to a = 3.5. 

Many other empirical modifications of the Einstein equation have been made to predict actual viscosities. Since the modulus (M) is related to viscosity, these empirical equations, such as the Einstein-Guth-Gold (EGG) equation (8.3), have been used to predict changes in modulus when spherical fillers are added. 

As shown by the modified Einstein. Guth, and Gould equation,... 

The effect of polymer-filler interaction on solvent swelling and dynamic mechanical properties of the sol-gel-derived acrylic rubber (ACM)/silica, epoxi-dized natural rubber (ENR)/silica, and polyvinyl alcohol (PVA)/silica hybrid nanocomposites was described by Bandyopadhyay et al. [27]. Theoretical delineation of the reinforcing mechanism of polymer-layered silicate nanocomposites has been attempted by some authors while studying the micromechanics of the intercalated or exfoliated PNCs [28-31]. Wu et al. [32] verified the modulus reinforcement of rubber/clay nanocomposites using composite theories based on Guth, Halpin-Tsai, and the modified Halpin-Tsai equations. On introduction of a modulus reduction factor (MRF) for the platelet-like fillers, the predicted moduli were found to be closer to the experimental measurements. 

Since the polymer-filler interaction has direct consequence on the modulus, the derived function is subjected to validation by introducing the function in established models for determination of composite modulus. The IAF is introduced in the Guth-Gold, modified Guth-Gold, Halpin-Tsai and some variants of modified Halpin-Tsai equations to account for the contribution of the platelet-like filler to Young s modulus in PNCs. These equations have been plotted after the introduction of IAF into them. 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

In the presence of reinforcing fillers, the elasticity modulus of the elastomers increases in first approximation according to the Guth-Smallwood equation 1111171... 

Many attempts have been made to extend Equation 3 to higher concentrations. Of these, we have chosen the equation of Guth,... 

The kinetic theory of rubber elasticity was developed by Kuhn (1936-1942), Guth, James and Mark (1946), Flory (1944-1946), Gee (1946) and Treloar (1958). It leads, for Young s modulus at low strains, to the following equation ... 

An equation for the modulus of ideal rubber was derived from statistical theory that can be credited to several scientists, including Flory, and Guth and James (Sperling, 1986). A key assumption in derivation of the eqmtion is that the networks are Gaussian. 

Figure 8.11. Comparison of the prediction ofthe Guth-Gold equation with experimental data for N330-filled SBR. [Adapted, by pennission, from Wang M-J, Wolff S, Tan E-H, Rubb. Chem. Technol., 66, No.2, 1993, 178-95.]...

Einstein s viscosity equation modified by Guth and Gold pre-... 

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