Extrapolation length

Finally, we find the dependence of the twist angle on the z-coordinate in a twist cell with soft director anchoring at one boundary  

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However, in the case of large Kn, the no-slip approximation cannot be applied. This implies that the mean free path of the liquid is on the same length scale as the dimension of the system itself. In such a case, stress and displacement are discontinuous at the interface, so an additional parameter is required to characterize the boundary condition. A simple technique to model this is the one-dimensional slip length, which is the extrapolation length into the wall required to recover the no-slip condition, as shown in Fig. 1. If we consider... 

Indeed, the shear stress at the solid surface is txz=T (S 8z)z=q (where T (, is the melt viscosity and (8USz)z=0 the shear rate at the interface). If there is a finite slip velocity Vs at the interface, the shear stress at the solid surface can also be evaluated as txz=P Fs, where 3 is the friction coefficient between the fluid molecules in contact with the surface and the solid surface [139]. Introducing the extrapolation length b of the velocity profile to zero (b=Vs/(8vy8z)z=0, see Fig. 18), one obtains (3=r bA). Thus, any determination of b will yield (3, the friction coefficient between the surface and the fluid. This friction coefficient is a crucial characteristics of the interface it is obviously directly related to the molecular interactions between the fluid and the solid surface, and it connects these interactions at the molecular level to the rheological properties of the system. 

When the slip velocity is further increased, the Rouse friction [148] finally becomes dominant, for Vs>V ocN l. A linear friction regime is then recovered, with a constant extrapolation length, b much larger than b0 and comparable to what would be observed on an ideal surface without anchored chains [139]. 

Fig. 23. Evolution of the extrapolation length at low slip velocity, fr0, as a function of the surface density of grafted chains for the experiments reported in Fig. 22. The fact that b0 appears independent of o when y increases linearly with a indicates that in this range of surface densities, the surface layer has saturated the number of melt chains it can capture...

The Navier relation in Eq. (1), i.e., T y=Pvs, naturally introduces a length scale known as the extrapolation length b... 

The same Navier dynamic boundary condition Eq. (1) and the subsequent expression Eq. 3 for the extrapolation length b can also be written down for non-Newtonian and polymeric fluids, where r is the shear viscosity and 11 is the local viscosity at the interface. The expression Eq. (2b) for 3 is equally valid for poly-... 

Fig. 3. Interfacial slip of an entangled melt at a non-adsorbing perfectly smooth surface, where the dots represent an organic surface (e.g., obtained by a fluoropolymer coating), which invites little chain adsorption. Lack of polymer adsorption produces an enormous shear rate jiat the entanglement-free interface between the dots and the first layer of (thick) chains. y-x=vs/a is much greater than the shear rate y present in the entangled bulk. This yields an extrapolation length b, which is too large in comparison to the chain dimensions to be depicted here...

The data in Fig. 12 actually collapse onto a master curve when the wall stress o is rescaled by temperature T and the nominal shear rate y is normalized by a WLF factor aT [29]. Thus Eq. (6) for the critical stress oc is supported by the data in Fig. 12, where V does not change with T. Another feature of the transition is that the amplitude of the flow discontinuity does not vary with T. In other words, the extrapolation length bc, which is evaluated according to Eq. (4a) at the transition, is a constant with respect to T. Thus for a given surface, bc is more than just a material property such as the melt viscosity r. It essentially depends only... 

Mooney [71], a century after Navier s notion for interfacial slip [15], ignored the essence of the Navier extrapolation length b, and produced a different formula, which employs the slip velocity vs as the essential quantity to describe wall slip. Instead of evaluating b from Eq. (4a), which has been available since the last century [15,24], Mooney wrote down... 

The linear relation GC°=T observed in Fig. 12 is not sufficient evidence that would unambiguously support Eq. (6) and reveal the interfacial nature of the transition, because a bulk phenomenon may also produce such a temperature dependence. For instance, one might think of melt fracture and write down oc=Gyc that would be independent of Mw where yc would correspond to the critical effective strain for cohesive failure and modulus G would be proportional to kBT. Previous experimental studies [9,32] lack the required accuracy to detect any systematic dependence of oc on Mw and T. This has led to pioneers such as Tordella [9] to overlook the interfacial origin of spurt flow of LPE. It is in this sense that our discovery of an explicit molecular weight and temperature dependence of oc and of the extrapolation length bc is critical. The temperature dependence has been discussed in Sect. 7.1. We will focus on the Mw dependence of the transition characteristics. 

Molecular Weight Dependence of Constant-Stress Viscosity and Extrapolation Length... 

The molecular weight dependence of the extrapolation length b originates from the same molecular weight dependence of the constant stress viscosity q0. A recent experimental study shows... 

Mooney M (1931) Trans Soc Rheol 2 210. It was evident even at the time of Navier that the meaningful way to quantify any level of wall slip is to express it in terms of the extrapolation length b [24]. Equation (4a) clearly shows that any correction would enter as the ratio of b to a characteristic dimension of the flow apparatus, e.g., the diameter D of capillary dies. It is unfortunate that Mooney abandoned the notion of the extra polation length b in favor of the slip velocity vs... 

The slippage of the polymer melt can also be characterised by the extrapolation length b b = A logarithmic plot of b versus Vs is shown in Fig. 6.Three different regimes of... 

The consequences for the macroscopic experiments are clear however (he onset of strong slip falls now in a range accessible macroscopically. and as the extrapolation length in... 

It should be underlined that it was possible to obtain slip at low stress values, by considering the flow of a polydimethylsiloxane (PDMS) through a silica die with walls grafted by a fluorinated monolayer [11]. It can also be found in the experimental study published in 1993 by Migler et al. [12], which moreover validates the model prediction in term of extrapolation length, for sufficiently high slip velocities. 

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