Polymer fluids calculations

The screw rotation analysis leads to the model equation for the extruder discharge rate. There are now two screw-rotation-driven velocities, and and a pressure-driven velocity, Pp that affect the rate. and transport the polymer fluid at right angles to one another. In order to calculate the net flow from screw rotation It Is necessary to resolve the two screw-rotation-driven velocities into one velocity, Vpi, that can be used to calculate the screw rotation-driven flow down the screw parallel to the screw axis (or centerline) as discussed in Chapter 1 and as depicted in Fig. 7.14. The resolved velocity will then be integrated over the screw channel area normal to the axis of the screw. 

Gibbs and DiMarzio [47] (GD) first developed a systematic statistical mechanical theory of glass formation in polymer fluids, based on experimental observations and on lattice model calculations by Meyer, Flory, Huggins, and... 

Figure 14. The fragility parameter v t = Ti/ Ti — Tg) calculated from the LCT as a function of the inverse number l/M of united atom groups in individual chains for constant pressure (P = 1 atm 0.101325 MPa) F-F and F-S polymer fluids. The single data point denoted by o refers to high molar mass F-S polymer fluid at a pressure of P = 240 atm (24.3 MPa).

Figure 18. Variation of the specific volume at the glass transition temperature Tg with the glass transition temperature Tg as calculated from the LCT for constant pressure P = 1 atm 0.101325 MPa) F-F and F-S polymer fluids. Both and Tg are normalized by the corresponding high molar mass limits (v or T ). (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Physical Chemistry B 109, 21285 (2005). Copyright 2005 American Chemical Society.)...

The calculated LCT configuration entropy for high molar mass E-S polymer fluids at pressures of P = 1 atm and P = 240 atm equals 0.1933 and 0.2147, respectively, and can be regarded, to a first approximation, as pressure independent. 

Solution 2.5b. The calculation of the uncertainty involves both an arithmetic function and a function of the product of the variables. The equation below relates the uncertainty of the viscosity with the difference in density between the steel ball and the polymer fluid as well as the measurement of time ... 

Luciani et al. (1998) critically examined the experimental methods used for the measurements of the interfacial coefficient in polymer blends as well as the theoretical models for its evaluation. A new working relation was derived that makes it possible to compute the interfacial tension from the chemical structure of two polymers. The calculations involve the determination of the dispersive, polar, and hydrogen-bonding parts of the solubility parameter from the tabulated group and bond contributions. The computed values for 46 blends were found to follow the experimental ones with a reasonable scatter of +/— 36 %. The authors mentioned also that since many experimental techniques have been developed for low-viscosity Newtonian fluids, most were irrelevant to industrial polymeric systems. For their studies, two were selected capillary breakup method and a newly developed method based on the retraction rate of deformed drop. 

As the polymer fluid density decreases the agreement between PRISM and simulations becomes quantitatively poorer, even if the simulated (k) is used in the PRISM calculation [29,30]. Such a reduction of quantitative accuracy of the RISM approach also occurs for small, rigid molecule fluids and is a well-understood and documented trend [6]. 

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

The viscosity of the polymer fluid can be calculated from Equations 8.32 and 8.34 ... 

As discussed above, the flow curves of polymer fluids can be obtained by Equations 8.18 and 8.38 (or 8.39), and the viscosities of the fluids can be calculated by Equation 8.41. While deriving these equations, one of the assumptions is that the flow pattern is constant along the pipe. However, in a real capillary flow, the polymer fluid exhibits different flow patterns in the entrance and exit regions of the pipe. For example, the pressure drops at the die entrance and exit regions are different from AP/Z. Therefore, corrections, e.g., Bagley correction, are needed to address the entrance and exit effects. Another assumption is that there is no slip at the wall. However, in a real flow, polymer fluid may slip at the wall and this reduces the shear rate near the wall. The Mooney analysis can be used to address the effect of the wall slip. In addition, the velocity profile shown in Figure 8.13 is a parabolic flow. However, the tme flow in the die orifice is not necessarily a simple parabolic flow, and hence Weissenberg-Rabinowitsch correction often is used to correct the shear rate at the wall for the non-parabohc velocity profile. 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

---