Draw ratio, critical

Thus, fracture occurs by first straining the chains to a critical draw ratio X and storing mechanical energy G (X — 1). The chains relax by Rouse retraction and disentangle if the energy released is sufficient to relax them to the critically connected state corresponding to the percolation threshold. Since Xc (M/Mc) /, we expect the molecular weight dependence of fracture to behave approximately as... 

Isothermal draw resonance is found to be independent of the flow rate. It occurs at a critical value of draw ratio (i.e., the ratio of the strand speed at the take-up rolls to that at the spinneret exit). For fluids that are almost Newtonian, such as polyethylene terephthalate (PET) and polysiloxane, the critical draw ratio is about 20. For polymer melts such as HDPE, polyethylene low density (LDPE), polystyrene (PS), and PP, which are all both shear thinning and viscoelastic, the critical draw ratio value can be as low as 3 (27). The maximum-to-minimum diameter ratio decreases with decreasing draw ratio and decreasing draw-down length. 

The experimental and theoretical literature on instabilities in fiber spinning has been reviewed in detail by Jung and Hyun (28). The theoretical analysis began with the work of Pearson et al. (29-32), who examined the behavior of inelastic fluids under a variety of conditions using linear stability analysis for the governing equations. For Newtonian fluids, they found a critical draw ratio of 20.2. Shear thinning and shear thickening fluids... 

Fig. 14.12 Results of the linearized stability analysis for a White-Metzner-type fluid, indicating the dependence of the critical draw ratio on n and N. [Reprinted by permission from R. J. Fisher and M. M. Denn, A Theory of Isothermal Melt Spinning and Draw Resonance, AIChE J., 22, 236 (1976).]...

The effect of an orientation process on an isolated elliptic flaw is depicted in Figure 7A. Let < i and a2 be the permanent stretch (or draw) ratios (ratio of drawn to undrawn length) to which the master sheet is subjected in two orthogonal directions. If an elliptic flaw is originally at right angles to the ai direction (i.e.y p = ir/2) and a2 = 1, then R will increase with until a particular value of is reached at which R = 1. For larger values of i, R will then decrease but the major axis of the ellipse is now at ft = 0. The critical draw ratio at which the orientation P jumps from tt/2 to 0 and for which the ellipse is circular (R = 1) is described later. 

Figure 7. A. Schematic of an elliptic flaw initially at right angles to the draw direction. For a uniaxial draw, the draw ratio D = at. At a critical draw ratio D = I/R the ellipse becomes circular. For higher draw ratios (D > I/R), the ellipse forms with a 90° change in its orientation. B. Schematic of the effect of increasing draw ratio on an ellipse initially at some angle other than 90° to the draw direction. C. A general biaxial draw (al9 a2) acting on an ellipse at initial orientation pto the, major drawm direction (a. > as).

Physically, the flaws with R > 80° are more effectively blunted than are flaws at a lower angle to the draw direction. Above the critical draw ratio, they are oriented close to the draw direction (low R ) and account... 

Fracture by disentanglement occurs in a finite molecular weight range, McPercolation theory predicts that the critical draw ratio. 

Draw resonance occurs in processes where the extrudate is exposed to a free surface stretching flow, such as blown film extrusion, fiber spinning, and blow molding. It manifests itself in a regular cyclic variation of the dimensions of the extrudate. An extensive review [169] and an analysis [170] of draw resonance were done by Petrie and Denn. Draw resonance occurs above a certain critical draw ratio while the polymer is still in the molten state when it is taken up and rapidly quenched after take-up. 

Draw resonance will occur when the resistance to extensional deformation decreases as the stress level increases. The total amount of mass between die and take-up may vary with time because the take-up velocity is constant but not necessarily the extrudate dimensions. If the extrudate dimensions reduce just before the take-up, the extrudate dimensions above it have to increase. As the larger extrudate section is taken up, a thin extrudate section can form above it this can go on and on. Thus, a cyclic variation of the extrudate dimensions can occur. Draw resonance does not occur when the extrudate is solidified at the point of take-up because the extrudate dimensions at the take-up are then fixed [171, 172]. Isothermal draw resonance is found to be independent of the flow rate. The critical draw ratio for almost-Newtonian fluids such as nylon, polyester, polysiloxane, etc., is approximately 20. The critical draw ratio for strongly non-Newtonian fluids such as polyethylene, polypropylene, polystyrene, etc., can be as low as 3 [173]. The amplitude of the dimensional variation increases with draw ratio and drawdown length. 

Various workers have performed theoretical studies of the draw resonance problem by linear stability analysis. Pearson and Shah [174,175] studied inelastic fluids and predicted a critical draw ratio of 20.2 for Newtonian fluids. Fisher and Denn [176] confirmed the critical draw ratio for Newtonian fluids. Using a linearized stability... 

Draw resonance occurs in the extensional flow region of the spinning line after the extrusion of the filament. Draw resonance is the periodic fluctuation of the filament diameter at the take-up of the spinning line. Draw resonance manifests itself at a critical draw ratio as sustained periodic oscillations in spinline variables such as filament cross section and tension despite maintaining constant extrusion rate and take-up velocity. It can be recorded as a systematic deviation from unity in the ratio of maximum to minimum diameter of the drawn filament with increasing draw ratio. 

Draw resonance is a hydrodynamic instability and not a viscoelastic one, albeit the former can be affected by fluid viscoelasticity. Thus, even Newtonian fluids can exhibit this phenomenon [8]. It has been reported both theoretically and experimentally that a Newtonian liquid sustains in-phase oscillations in force and area at draw ratios of 20 [9, 10]. Hence, the critical draw ratio for Newtonian fluids is taken as 20. This is true for an isothermal spinning system. 

Figure 11.3. Critical draw ratio as a function of De = Xvo/L for a single-mode Phan-Thien/Tanner fluid with various values of e and Reprinted from Chang and Denn, Proc. 8th International Congress on Rheology, Naples, Italy, Vol. 3,1980, p. 9.

The steady-state solution for fiber spinning (Newtonian and isothermal case) was presented in Section 9.1.1, and it consists of Eqs. 9.26 and 9.28. Linearized (small disturbances) stability analysis involves (Fisher and Denn, 1976) the study of finite amplitude disturbances, and we do not present it. Rather, we present the results of such an analysis. The value of Dr = 20.21 is considered to be the critical draw ratio beyond which the flow becomes unstable. Figure 9.13 (Donnelly and Weinberger, 1975) shows experimental data that confirm the theory. More specifically, silicone oil (of viscosity equal to 100 Pa-s), which seems to be Newtonian, was extruded and the ratio of maximum to minimum filament diameters was plotted against the draw ratio. An instability appears at a draw ratio of about 17, or about 22 if we take into consideration about 14% die swell. The value of the critical draw ratio of 22 compares well with the theoretical value of 20.21. Pearson and Shah (1974) extended the analysis to a power-law fluid and included surface tension, gravitational. 

The energy equation, Eq. 9.39, should be incorporated into the model to account for temperature variation along the filament axial length. Pearson and Shah (1974) solved the system of equations subjected to a linearized analysis and found that the critical draw ratio depends, besides on the power-law index, on the dimensionless number, S ... 

Fisher and Denn (1976) presented the Unearized stability analysis for the isothermal viscoelastic case as an extension of the steady-state case presented in Section 9.1.3. The analysis showed (Fig. 9.14) that the critical draw ratio depends on the power-law index, n, and on the viscoelastic parameter a /", where a is defined in Eq. 9.83. Three regions are shown in Figure 9.14 stable, unstable, and unattainable. The lower boundary of the unattainable region is described by the relationship Dr = 1 At low values of the parameter... 

Experimentally, the critical draw ratio for various polymers is measured as the draw ratio at which the ratio of maximum to minimum filament diameters increases above 1. Figure 9.15 shows experimental data for PP, HDPE, and PS. The corresponding critical draw ratios are 2.7, 3.8, and 4.6. The power-law index of PP and PS is about 0.5, and thus the agreement between experimental data and theory is generally good. 

FIGURE 9.14 Critical draw ratio, Dr, as a function of the viscoelastic parameter a and the power-law index, n, for the fiberspinning process. (Reprinted by permission of the publisher from Fisher and Denn, 1976.)... 

Example 9.3. Critical Draw Ratio and Spinning Length... 

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