Monomeric friction

The density dependence of however, is not trivial since one must take into account the density dependence of the monomeric friction [41] given by the monomer reorientation rate W. If a scaling plot of the correct ratio DN/iyVl /N) is attempted, one again finds poor scaling in terms of the variable (Fig. 24(a)), and a good collapse of data if instead... 

Rouse motion has been best documented for PDMS [38-44], a polymer with little entanglement constraints, high flexibility and low monomeric friction. For this polymer NSE experiments were carried out at T = 100 °C to study both the self- and pair-correlation function. 

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. 

Table 4 Monomeric Friction Coefficients of Selected Polymers...

The Doi-Edwards roptation model thus predicts that the width of1 the modulus plateau varies as the square of the molecular weight, or, in comparing different polymers that have different Mr values, as (M/Mf)1. Another way of stating this is to say that the monomeric friction factor has been increased by the factor Furthermore, since in general =... 

The monomeric friction factors and i have a temperature dependence given by the Williams-Landel-Ferry (WLF) relation, equation (27) ... 

Here c is the polymer concentration by weight. < the density of the polymer, a an effective bond length or measure of the coil dimensions, and to the monomeric friction factor. The subscript zero indicates the pure polymer. Since 2 (H), the mean-square end-to-end chain separation, the viscosity will be directly proportional to the polymer concentration unless the plasticizer modifies the coil swelling. At high molecular weight the monomeric friction factor is increased by the factor (MIMf)" and M, is increased relative to the undiluted polymer [equation (55)]. Thus... 

Thus at a given temperature, the location of the transition zone of E(t)e on the time scale is determined by the monomeric friction factor, the height of the entanglement plateau by Af, and the width of the plateau by (Ml Mf)3. The time dependence of entanglement slippage. (0 nn describes the rate at which the entanglement plateau will drop to the equilibrium... 

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

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

Table 3.1 Monomeric friction coefficients from NSE experiments compared to rheological data from the compilation of Ferry [34] ...

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)...

Fig. 4.35 Right-hand side Monomeric friction coefficients derived from the viscosity measurements on PB [205]. The open and solid symbols denote results obtained from different molecular weights. Solid line is the result of a power-law fit. Dashed line is the Vogel-Fulcher parametrization following [205]. Left hand side Temperature dependence of the non-ergodicity parameter. The three symbols display results from three different independent experimental runs. Solid line is the result of a fit with (Eq. 4.37) (Reprinted with permission from [204]. Copyright 1990 The American Physical Society)...

Under the simplifying assumption that hydrodynamic interactions may be neglected, the only new parameter that controls the dynamics is a monomeric friction coefficient (Rouse model). Then the prediction for the rate Pj is given by ... 

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. 

It is not clear why this transition should occur at such a higher level of arm entanglement for polystyrene stars than for other star polymers. This observation is in direct conflict with the standard assumption that through a proper scaling of plateau modulus (Go) and monomeric friction coefficient (0 that rheological behavior should be dependent only on molecular topology and be independent of molecular chemical structure. This standard assumption was demonstrated to hold fairly well for the linear viscoelastic response of well-entangled monodisperse linear polyisoprene, polybutadiene, and polystyrene melts by McLeish and Milner [24]. 

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