ANEXOS

It is a fundamental assumption in many calculations involving polymer flow through capillary dies that the velocity of the polymer at the wall is zero [25,38,39]. However, many authors [25-28] have either demonstrated experimentally or discussed theoretically, the occurrence of non-zero wall velocities in capillary dies.

Constant wall slippage at the capillary wall has been investigated by a number of workers [25,40,41]. Mooney’s [40] paper of 1931, contains one of the earliest calculations of wall

slip in capillary rheometer dies. Mooney’s [40] original equation has been adapted by many authors including Lupton and Register [42] and Brydson [25]. The basis of the Mooney [40] equation has been presented by Brydson [25] in a general qualitative manner. Brydson explains [25] that if slip occurred at the wall a form of plug flow would superimpose on the normal flow patterns. Consequently if experiments were carried out on tubes of different dimensions, the flow curves plotted would not super-impose, because the calculated values of the shear rate would be in error [25].

Worth et al [41] have also investigated this method of calculating wall velocity, once again based on the Mooney equation adapted by Lupton and Register [42]. The Mooney equation expressed in the form used by Worth et al [41] may be written as follows

line of slope equal to the slip velocity. Worth et al [41] used the Mooney equation plots to show that the wall velocity increased with increasing wall shear stress. Both of the polymer melts ( low and high density polyethylenes) considered by Worth et al [41] exhibited wall slip at shear stresses above the critical shear stress for melt fracture.

Shaw [43] has also discussed the use of the Mooney [40] equation to correct capillary rheometer data for wall slip. Shaw [43] presents a compilation of experimental results of wall slip calculations from various authors.

All of the above workers have studied wall velocity in capillary flow by indirect measurements and by the use of the various forms of the Mooney [40] equation. However, a number of authors have attempted to measure wall velocity directly by a variety of

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experimental techniques. One such technique for the measurement of wall velocity has been developed by Maxwell and Galt [44]. This employed a small particle tracer technique, involving random distribution of small particles throughout a molten polymer, which recorded the individual motions of the particles when stress was applied to the polymer. The movement of the individual particles was recorded by a camera focusing through a microscope onto a transparent section of an extrusion die. The camera produced dashed traces as the focal area of the camera was illuminated by a test lamp beam periodically interrupted by a rotating timing wheel. The individual particle velocities were determined from the length of the traces. All polymers tested by Maxwell and Galt [44] were low density, branched polyethylenes. Results from this experimentation showed a mixture of stationary particles, which it was concluded were stuck to the wall, and moving particles which it was concluded were slipping along the wall. Thus the particles oscillated between two distinct velocities (one of which may be zero). Such movement of the particles, and therefore the polymer, was termed slip-stick. Maxwell and Galt [44] also demonstrated that the radial extent of the boundary annulus of slip- stick behaviour at the wall could be defined by the extent of the layer of zero velocity traces from the wall. It was concluded by Maxwell and Galt [44] that the flow in the die was laminar, due to the lack of particle rotation or spin. Such movement would not be susceptible to the simple analysis of continuous slip of Mooney [40] unless an average velocity is assumed.

Additional experimental techniques for measuring wall velocity have been investigated by Rhoades [45]. Rhoades [45] was particularly interested in the flow of polyborosiloxane during extrusion honing (section 2.3). A similar technique to Maxwell and Galt [44] was employed, where high speed photography was used to record the motion of particulate matter in the polymer viewed through a transparent die. However, Rhoades [45,46] found insufficient data points could be recorded for his theoretical model, due to turbulence within the polymer. Rhoades [45] also measured the hydraulic piston displacement of his

test equipment in order to estimate polymer wall velocity. However, this method may be described as indirect and gives measurement of the average polymer velocity rather than the specific velocity of polymer at the wall. Nevertheless, evidence was found of cycling of the piston velocity at such a frequency that it was considered likely to be the result of slip-stick [45]. Rhoades [46] suggested that the technique of hot film anemometry may be very useful in further studies of wall velocity of polyborosiloxanes through extrusion dies. The mechanism for wall slip or slip-stick has been discussed by a number of authors. Whorlow [28] stated very simply that if adhesion between the polymer and the die wall will withstand a shear stress up to some limit, local slippage will occur wherever the limit is exceeded, resulting in a non-zero wall velocity. Similarly Worth et al [41] stated that the wall slip may commence at a critical value, the value of which will depend on the particular interfacial situation, and the elastico-viscous nature of the fluid.

A number of authors [28,41] have discussed phenomena which may exaggerate wall slippage in capillary dies. Whorlow [28] states that a non-zero wall velocity will be obtained if a lubricating layer of low viscosity material occurs at the die wall, resulting in apparent slippage of the main body of the material. Worth et al [41] comment that the apparent manifestation of wall slip they recorded, in low density polyethylene, may not be true slip but a viscous heating effect caused by high shear rates near the die wall.

The mechanism proposed by Maxwell and Galt [44] extends the theories of continuous slip to slip-stick phenomena. Maxwell and Galt [44] proposed that the melt sticks to the die wall and is elastically deformed as a shear stress is applied by the bulk flow of the polymer. Once the deformation has resulted in a shear stress in excess of the adhesive force between the polymer and the die wall, the polymer slips, thus removing the elastic strain.

This process is cyclically repeated down the die resulting in an overall slip-stick effect at the wall [44].

Dealy [26] states that at flow rates above those at which slip stick occurs, a smooth extrudate may be observed. It is proposed that this results from the permanent detachment of the polymer from the die wall. Dealy [26] attached considerable importance to wall slippage as a mechanism for effects such as oscillating flow and melt fracture in capillary flow. These and other effects, considered to be closely linked with wall slippage are reviewed in the appropriate sections below.