Determination of the Plateau Modulus

If there is actually a flat plateau in the curve of either the storage modulus or the compliance, it is easy to obtain the value of the plateau modulus, but except for very high molecular weight, monodisperse, samples this is not the case, and most reported values are estimated from loss modulus data using the following relationship [41, p. 373]. 

The subscript FP signifies that only contributions from the flow and plateau regions should be taken into account in the integration. This equation is based on the fluctuation-dissipation theorem [42 ]. A relationship closely related to Eq. 5.12 makes use of complex compliance data. 

Equation 5.12 can be written in terms of the relaxation spectrum function H(t), which was defined by Eq. 4.19. 

Chompff and Prins [43] proposed letting a be the time at which H(t) is a minimum in the transition zone. And in practice, the upper limit is the value of InT above which the spectrum function is essentially zero. Janzen [44] has proposed the estimation of the plateau modulus as the value of G at = l/o.  

As shown in Fig. 5.7, for a narrow-MWD sample, there is a point of inflection in the plateau zone of the curve of storage modulus versus frequency, and Inoue et cd. [47] suggested the use of the value of (7 oi) at this point as an estimate of. Empirical relationships between the plateau modulus and the local maximum in the curve of either 7 (o) or /( ), have also been proposed [41, p. 376], as shown by Eqs. 5.15 and 5.16  

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A (rheology) [7]. Obviously the local freedom for segmental motion is larger than anticipated so far from rheology. The result casts some doubts on the determination of the plateau modulus from rheological data in terms of the reptation model. 

As can be seen, determination of the plateau modulus as a function of blend composition yields AGR, which is closely related to AN,. Thus, the free-energy change under steady-state shear is... 

For practical applications, the parameter Ge can also be determined from the value of the plateau modulus Gn°, since the relationship Ge l/2GN° applies in accordance with the tube model. This implies that the parameter Ge is not necessary a fit parameter but rather it is specified by the microstructure of the rubber used. Note that the fit value Ge=0.2 MPa obtained in Fig. 44 is in fair agreement with the above relation, since Gn° 0.58 MPa is found for uncross-linked NR-melts [171]. 

Figure 45a-c shows an adaptation of the developed model to uniaxial stress-strain data of a pre-conditioned S-SBR-sample filled with 40 phr N220. The fits are obtained for the third stretching cycles at various prestrains by referring to Eqs. (38), (44), and (47) with different but constant strain amplification factors X=Xmax for every pre-strain. For illustrating the fitting procedure, the adaptation is performed in three steps. Since the evaluation of the nominal stress contribution of the strained filler clusters by the integral in Eq. (47) requires the nominal stress aR>1 of the rubber matrix, this quantity is developed in the first step shown in Fig. 45a. It is obtained by demanding an intersection of the simulated curves according to Eqs. (38) and (44) with the measured ones at maximum strain of each strain cycle, where all fragile filler clusters are broken and hence the stress contribution of the strained filler clusters vanishes. The adapted polymer parameters are Gc=0.176 MPa and neITe= 100, independent of pre-strain. According to the considerations at the end of Sect. 5.2.2, the tube constraint modulus is kept fixed at the value Ge=0.2 MPa, which is determined by the plateau modulus Gn° 0.4 MPa [174, 175] of the uncross-linked S-SBR-melt (Ge=l/2GN°). The adapted amplification factors Xmax for the different pre-strains ( max=l> 1-5, 2, 2.5, 3) are listed in the insert of Fig. 45a.

Modulus of end-linked PDMS networks with/= 4 at 30 °C from S. K. Patel etal.. Macromolecules 25, 5241 (1992). The line has an intercept determined by the plateau modulus of a melt of high molar mass PDMS linear polymers-----------... 

The number density of network strands determines and the plateau modulus caused by inter-chain entanglements is Cg. This plateau modulus is understood on a molecular level by imagining surrounding chains confining each network strand to an effective tube. 

Various characteristic molecular weights can be defined from the molecular weight dependences of the rheological properties, including the molecular weight between entanglements. Mg, determined from the plateau modulus... 

The foregoing discussion shows that a direct connection exists between the various viscoelastic parameters, unperturbed chain dimensions, and p. Predictive capacities exist that show that the melt viscoelastic properties are determined by the joint interplay of the radius of gyration of a chain and its accompanying volume. Control of the plateau modulus, for example, can be exerted by manipulating /V, e.g., an increase in this value results in an increase in Gg,. 

The contribution from the first term (reptation branch) has the same order of magnitude as the contribution from the second term at very high frequencies. However, one has to take into account that, due to distribution of relaxation times, the limit value of the first term is reached at higher frequencies than the limit value of the second term. At lower frequencies the plateau value of the dynamic modulus is determined by the second term and coincides with expression (6.52). 

Figure 10.3 Mean molecular mass between chemical crosslinks and trapped chain entanglements Mc+e in a cured mixture of a poly(ethylene glycol) diacrylate (PEGDA) and 2-ethylhexyl acrylate (EHA) as a function of the EHA content [52]. Mc+e values were determined from (1/T2s)max and the plateau modulus (see Figure 10.2). A substantial difference in Mc+e value, as determined by these two methods at low crosslink density, is caused by the effect of network defects which decrease volume average network density determined by DMA (see Section 10.3). The molecular mass of PEGDA (Mn = 700 g/mol) is indicated by an arrow. The molecular mass of network chains in cured PEGDA is about three times smaller than that of the initial monomer. The molecular origin of this difference is discussed in Section 10.3...

The early attempts to interpret the dynamic mechanical behaviour in structural terms include that of Smith et al. where the plateau modulus was correlated with the fraction of non-crystalline material f, determined by NMR. Plots of the plateau compliances at —60 °C and —160 °C as a function of f suggested a modified Takay-anagi series model, with a constant amount of non-crystalline material in parallel with the simple series model. The modd showed good internal consistency, with values for the compliances of the non-ciystalline regions which were acceptable in physical terms. 

Uniaxial tensile tests were carried out to determine the stress-strain curves and document the damage growth on a computer-controlled Instron model 8516 servo hydraulic testing machine operating at a strain rate of 5% min . The macroscopic tensile yield stress was considered equal to the maximum stress on the loading curve. The Young s modulus was determined as the plateau value of a plot of the secant modulus as a function of the strain. 

The entanglement strand has entanglement molar mass M = N Mq. The entanglement strand effectively replaces the network strand in the determination of the modulus for networks made from long strands, and also determines the rubbery plateau modulus of high molar mass polymer -melts ... 

The central region of the curve, comprising the drop to the ionic plateau, the plateau modulus, and its drop-off towards the rubbery plateau, is determined by the form of Equation 1, K, EQ, Ef and b. 

The plateau modulus (Gp ) for starch-POClg gel was determined by mechanical oscillation measurements. Figure 16 represents the values of the storage moduli in the plateau region as a function of the reaction rate of starch-POClg gel. Gel formation takes several hours to reach completion this time requirement points to a very slow condensation during phosphate formation. 

Given the general agreement in the shape of the relaxation modulus, it is possible to determine the step length a of the primitive chain Though various ways are conceivable, a direct way is to use the plateau modulus... 

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