The Monomeric Friction Coefficient

The friction coefficient has usually been calculated by applying equation 45 of Chapter 10 to the relaxation spectrum H, as follows  

Relaxation spectrum in the transition zone for the six methacrylate polymers of Fig. 12-1, similarly identified and reduced to 100 C, to illustrate calculation of fo- 

Since the retardation spectrum L provides in principle the same information as the relaxation spectrum H, it might be expected that fo could be calculated from L also, using the analog of equation 1, which cf. equation 24 of Chapter 9) is 

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Fig. 8. Self-diffusion coefficients of polyethylene chains as a function of molecular mass. The measurements were carried out at the same value of the monomeric friction coefficient. (Reprinted with permission from [48]. Copyright 1987 American Chemical Society, Washington)...

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

Setting the time scale, the monomeric friction coefficient is a basic quantity in all rheological measurements. This quantity is inferred indirectly in two ways ... 

Fig. 3.8 Temperature dependence of the monomeric friction coefficients for PEP and PE. The symbols present the NSE results (filled triangle PEP, filled circle PE). The solid lines display the respective rheological predictions extrapolations are shown as dashed lines (solid lines prediction [51],point dashed line prediction [34] for PEP)...

W. G. Miller, University of Minnesota, Minnesota In vlsco elastic studies the monomeric friction coefficient is used to describe motion. The same parameter Is used to look at translational diffusion of solvent and Its concentration dependence. Is there any relationship between this parameter and your three bond motion or Is the correlation length way too long ... 

In any circumstance where the whole displacement of a polymer chain is involved, it is important to characterise friction arising from the surroundings. This is achieved through the monomeric friction coefficient, , which can be determined from melt viscoelasticity measurement of the Newtonian viscosity, corresponding to the low frequency plateau in rj (co). f is calculated... 

At much higher temperatures, where CDCs are able to develop, one has a temperature dependence of ocdc> which involves both oy2 and the monomeric friction coefficient, f. In systems where the activation energy of f is high enough, its temperature dependence can lead to a steeper decrease of ctcdc with increasing temperature than for ay. For such polymers, at temperatures not much lower than Ta, CDCs can be the favoured micromechanism and a SDZ-CDC transition will take place. 

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]. 

The results have indicated that the monomeric friction coefficient is sensitive to the presence of the short chains, and is smaller, at a given temperature, than the homopolymer. This observed reduction in has been shown to be only attributed to an increase in the fractional free volume. Indeed, the temperature dependence of the monomeric friction coefficient is described by a WLF-type equation ... 

Figure 21. Logarithmic plot of the monomeric friction coefficient as a function of 1/(T-Tc ,) for PS2000 ( ) and PS2000/PS10(30 wt.-%) ( ).

All viscoelastic functions may be expressed in terms of a single reptation parameter (for example the plateau modulus or tube diameter) and the monomeric friction coefficient (or mobility factor in our terminology), in agreement with the above phenomenological presentation. 

The Doi-Edwards theory treats monodisperse linear chain liquids by a model which suppresses fluctuations and assumes a topologically invariant medium. Two parameters are required, the monomeric friction coefficient which characterizes the local dynamics and the primitive path step length a which characterizes the topology of the medium. The step length is related to the entanglement molecular weight of earlier theories, = cRqT/Gn, by Eqs. 1 and 37 ... 

The monomeric friction coefficient sets the time scale of all motions, e.g., the disengagement time ... 

The monomeric friction coefficient has been studied extensively. As in small molecule dynamics, 5o depends on many variables (polymer and solvent species, ccmcen-tration, temperature, etc.), but its independence of chain length beyond a certain length is well established. The value of can be estimated by a variety of methods. Die path step length a is virtually a universal fiiiKtion of the total length of chain backtxme per unit volume in the system and quite insensitive to all other variables. Furthermore, for a given polymer species, a where 0 is the volume fraction of polymer. The... 

The softening dispersions of entangled low-molecular-weight polymers are often modeled by the Rouse modes modified for undiluted polymers. From their very definition only involving the coordinates of a single chain, the Rouse modes are not intermolecularly coupled, and their relaxation times, t/ /, are proportional to the monomeric friction coefficient, fo that is. 

In the Coupling Model all primitive relaxation times, including roa and To/, are proportional to the monomeric friction coefficient, o- Hence from Eq. (5.32) the temperature dependence of rj depends on n/ and is given by... 

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