The Extrapolation Length

A useful way to think about slip, and its effect on rheological measurements, is to define an extrapolation length b. The extrapolation length is the distance from the fluid-solid 

as is often assumed, the slip velocity is linear in the shear stress, and if the shear viscosity is shear-rate-independent, then the extrapolation length will also be shear-rate-independent. 

It is possible for the extrapolation length to be negative if, for example, there is a layer of fluid so tightly bound to the interface that it does not move, and the point of zero velocity is moved from the wall into the fluid by one or more molecular layers. 

A simple mechanism for yield in a crystalline solid is depicted in Fig. 1-21 (Crist 1993). Under an increasing shear strain, each row of atoms is displaced from its equilibrium position (a) with respect to the neighboring row. Below a critical strain, if the stress is removed, the atoms spring back to their original positions. However, since the arrangement of atoms is spatially periodic, if the deformation continues, each row of atoms will eventually find itself back in registry with its neighboring rows, with each atom simply dispaced by one 

A simple model for yield in disordered solids can be derived from the Eyring model, discussed in Section 1.5.1. If the enthalpy of a hop AH is very large, then flow is essentially impossible unless the applied shear stress is large, so that v a approaches AH in magnitude. Then the frcqueney of forward hops is much larger than that of reverse 

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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. 

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-... 

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... 

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. 

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 is also possible to characterize the polymer-wall slip by representing variations in the extrapolation length b as a function of the shp velocity. It should be recalled that b is defined as the ratio of the slip rate to the shear rate at the wall [9]. With the assumptions made in this study (power law, etc.) relation (2) can be used to determine the shear rate at the wall for flow regimes including shp, i.e. ... 

An attempt was made to determine the effect of wall material on friction with polymer melt slip, in terms of variations in the extrapolation length "b". 

Problem 1.5 Suppose you shear a polymer melt in a plane-Couette rheometer at a shear stress a of 0.1 MPa and measure the apparent shear rate = V/h, where V is the velocity of the moving plate (the other is stationary) and h is the gap between the plates. At a gap h of 2 mm, you measure an apparent shear rate of 1.5 sec- aXh — mm, you measure 2 sec- and at h = 0.5 mm. you measure 3 sec. What is the slip velocity V, and the true shear rate y at this shear stress What is the extrapolation length bl... 

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