Graessley equation

The size of M i is determined from the Moony-Rivelin equation and F from the Graessley equation [12]. By dividing the value of molecular weight of a statistical segment (M i), we obtain the number of particles (statistical segments) (N, ) in one cluster. 

Thus if we know [tj] and [rj]e as a function of molecular weight we can plot the chain expansion factor as a function of concentration. A plot for polybutadiene from the work of Graessley is shown in Figure 5.21 and uses Equation (5.81) to describe the relationship between concentration and intrinsic viscosity. 

The Mark-Houwink equation for the linear polystyrene/THF system at 25 C was determined accurately by W. Graessley et al. We used their equation as follows ... 

The three papers just referred to share a further assumption, namely that a steady state is set up in the continuous reactor, so that all time derivatives in the kinetic equations may be equated to zero. Graessley (91) considered the unsteady state during the start-up of a continuous stirred reactor and found that Mw may in certain cases increase without bound instead of reaching a steady state this will occur if a branching parameter exceeds a critical value. His reaction scheme is restricted to mono-radicals, and the effect of loss of radicals from the reactor is not taken into account. If a steady state is set up, the MWD obtained is Beasley s, when termination by combination and branching by copolymerization of terminal double bonds are absent. Since there is reason (92) to doubt the validity of Beasley s conclusions, as discussed above, the same doubt must apply to this work of Graessley s. 

Graessley and co-workers have studied the rheological properties of solutions of branched PVAc in diethyl phthalate (178, 188), using polymer concentrations of 0.17, 0.225, and 0.35 g ml-1. At the lowest concentration, the low shear-rate viscosity was simply related to [17], so that it was lower for branched polymers the equation ... 

A theoretical value for M JMC can be calculated by equating Graessley s expression for Je° [Eq. (6-60)] with the Rouse expression ... 

It is observed that Bueche s equation in combination with the g1/2 rule explains the results of this work whereas in combination with g3/2 it does not. The correction to g for polydispersity brings closer the agreement between the data and the seventh power relation. The difference of a few percent between the expected and observed slopes of 7 and 6.4 may be attributed to an undercorrection for polydispersity in this regard according to Graessley s findings current theories do not sufficiently account for the reduction in viscosity with polydispersity, whether the Beasley or the Stockmayer molecular weight distribution is employed (29). 

It has been hypothesized by Graessley (30) and others (3, 31) that the g1/2 rule is applicable to star-shaped and slightly branched polymers whereas the g372 rule describes the behavior of highly branched "comb-shaped polymers for which [rj] B/ [y] l < 0.5. The results of this study indicate that the g1/2 rule in combination with Equation 11 adequately represents the random trifunctionally and tetrafunctionally branched... 

For the weakly entangled system, the steady-state modulus depends on the molecular weight of polymer as M 1, while for strongly entangled system, the steady-state modulus does not depend on the molecular weight of polymer, which is consistent with typical experimental data for concentrated polymer systems (Graessley 1974). The expression for the modulus is exactly the same as for the plateau value of the dynamic modulus (equations (6.52) and (6.58)) Expressions (9.42) lead to the following relation for the ratio of the normal stresses differences... 

Due to difficulties in measuring the zero-shear viscosity of such high molecular weight polymers, and thus deducing the monomeric friction coefficient from Graessley s uncorrelated drag model [43], the following equation adapted from the modified Rouse theory has been applied [8]. 

It should be mentioned that recently Dossin and Graessley used equations developed by Pearson and Graessley to derive asymptotic expressions for v, and in randomly crosslinked networks where the sol fraction is negligible. In this case, nearly all crosslinks will have two or more paths leading to the network. Under these conditions the expressions for structural parameters become relatively simple Values for v, and of end-linked PDMS networks have often been calculated... 

Graessley [17,18] has developed a statistical mechanical calculating device based on small network regions, the micronet, for estimating the change in chain conformation as a consequence of crosslinking. He found that the front factor depended only on the functionality (/) and not on the details of network topology, formation or crosslink distribution. His final equation is... 

I CR time Xcr for a sequence of Ncr of PtBS in the blend. This CR term is expressed in the discrete Rouse form with the eigenvalue ratio given by Equation (3.72). Because the local CR hopping of the PtBS chain is activated by the global motion of the PI chains, the onset time for the CR process, Xcr /CjNcR-ir should be determined by x pi of PI. Watanabe et al. (2011) utilized the Graessley model (Graessley, 1982) to relate xcr / /ncr-i and x h as... 

The molecular theories of Bueche and Graessley are similar in that both theories relate the pseudoplastic nature of polymer solutions as a function of a dimensionless Deborah number which represents a ratio of the response time of the polymer molecules in solution, X, to the time scale of the flow process (18). In simple shearing the time scale of the flow process is inversely proportional to the shear rate. Thus both equation (5), developed from the Bueche theory, and equation (14), developed from the Graessley theory, can be expressed in terms of the Deborah number,... 

Each of the flow curves obtained by the Haak viscometer were analyzed in the low shear rate range by using the theoretical models of both Bueche and Graessley. The shear stress, shear rate data taken from each flow curve were fitted to both equation (5) and equation (12) by linear regression. Best values for molecular response time and zero shear viscosity were then calculated using equations (6), (7), (15), and (16). These results are presented in Table 7. 

Here Tmom and Vmnr are microscopic parameters defined at the monomer scale. These equations can be derived either from a microscopic study at the scale of one chain portion between entanglements or more directly, by requiring that for N deoeasing to Ne, there must be a crossover toward Rouse behavior.A more detailed discussion is contained in recent papers by W. Graessley and co-worirers. ... 

Duiser and Staverman (67) and Graessley (70) have shown that the front factor depends on the functionality of the network. Representing the network functionality as /, equation (9.34) can be written... 

In terms of critical molecular weight, Graessley(33) has introduced a more sophisticated equation for on the basis of work by Doi and EdwardsD ) ... 

The concept of the Graessley theory that the steady-state concentration of entanglements is diminished during flow at high shear rates implies that the steady-state compliance Ry observed in recovery after cessation of steady-state flow will be larger than J and given by the equation ... 

For polymers with sharp molecular weight distribution, a terminal relaxation time T] can usually be determined experimentally from the flnal stages of stress relaxation either after sudden strain or after cessation of steady-state flow the latter kind of experiment weights the desired parameter more strongly as can be shown by equations 19 and 64 of Chapter 3, when expressed in terms of a discontinuous set of relaxation times rather than a continuous spectrum. Alternatively, it can be obtained from the constant Ag (the ratio of G /(tP at very low frequencies). Since the very narrow distribution of relaxation times in the terminal zone is close to a single terminal time t, which may be approximately identified with t , of the Graessley theory or of the Doi-Edwards theory (Section C3 of Chapter 10), equation 3 of Chapter 3 applies approximately and... 

For M M c, if there were really a single terminal relaxation time as implied by equation 14, combination of that equation with equation 34 of Chapter 3 would make the steady-state compliance the same as the plateau compliance 7 = 7/v = /G%. with J% given by equation 2. Actually, as shown by Graessley, the ratio Je/J% is the ratio of what may be termed the weight- and number-average relaxation times in the terminal zone ... 

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