## The Extensional Flow of Polymers |

### Introduction

Polymers are now present in so many areas of society that they are frequently overlooked. Fields such as medicine, electronics, vehicle manufacturing and many others require precisely designed components with exacting mechanical standards. In order to manufacture such components, it is important to understand the flow properties of the molten polymer in great detail.

Extensional flow of polymers can have a very strong influence on the flow properties of the molten polymer. However, it is usually more difficult to characterise properties for extensional flow than for simple shear. As such, in many characterisations, extensional flow is simply overlooked.

It would therefore be useful to develop a simple apparatus to generate pure extensional flows in a controlled environment. The Multi-Pass Rheometer is well suited for this purpose. It is capable of delivering precise flows through a known geometry, and observing the flows using pressure transducers and optical birefringence.

The experimental results obtained can be compared with numerical simulations to test the validity of constitutive equations for extensional flow.

### Experimental apparatus

Schematic diagram of |
The Multi-Pass Rheometer has been modified to generate a stagnation point using cross-slot flow. The pistons move towards and away from each other to force the polymer through a cross-slot geometry. Nitrogen pressure is applied to the side arms in order to force the polymer back into the system so that multiple experiments can be carried out on the same sample. The flow through the cross slot is controlled by pistons capable of delivering flowrates between 1 and 1,000 mm3/s. In the cross slot apparatus, this corresponds with maximum extension rates between 0.2 and 200 s-1. The flow through the system can be observed by shining polarised light through quartz windows and recording optical birefringence with a camera. A heated silicon oil jacket controls the temperature. The system temperature and pressure are measured by transducers at the top and bottom of the central section. |

Photograph of MPR Apparatus set up for Cross-Slot Flow |

### Experimental Results

Some examples of birefringence patterns for cross-slot flow can be seen below.

Polydisperse polystyrene at a maximum extension rate of 4s-1 |

Monodisperse polystyrene at a maximum extension rate of 4s-1 |

Low Density Polyethylene at a maximum extension rate of 1s-1 |

### Optical Birefringence

Optical Birefringence generates a 'contour map' of stresses observed by taking advantage of the optical anisotropy of polymers under stress. To calculate whether lightness or darkness will be seen we need to know the Stress Optical Coefficient, the Principle Stress Difference, the path-length through which the light travels and the wavelength of the light.

We start with the definition of the Stress Optical Co-efficient:

... equation 1

where Δn = Birefringence, C = Stress Optical Coefficient (Pa-1), and (σ11 - σ22) = Principle Stress Difference (Pa).

We define m1 and m2 as the number of wavelengths that are present across the path-length when following axis 1 and 2 respectively.

where L = Path-length of light and λ1, λ2 = wavelengths of light along axes 1 and 2.

We now define m as the difference between the number of waves across the pathlength.

... equation 2

When m is a unit value, the two light waves will be in phase with each other and a black fringe will be observed. When m is 0.5, 1.5, 2.5 etc, a light fringe will be observed.

The refractive index is defined as:

... equation 3

where n = refractive index, c = speed of light in a vacuum, v = speed of light through the medium, λvac = wavelength of light in a vacuum, λ = wavelength of light through the medium.

The birefringence (the difference in refractive index between axis 1 and axis 2) is then defined as:

... equation 4

We can then substitute equation 4 into equation 2 to give:

... equation 5

This is sometimes expressed as a 'Retardation' (in units of length) where:

... equation 6

The Stress Optical Coefficient can then be substituted into equation 5 to give:

Thus if we know the Stress Optical Coefficient , the Principle Stress Difference, the path-length and the wavelength of the incident light, we can plot the presence or absence of a fringe.

### Simulations

The flow geometry has been numerically simulated using two simulation packages: Polyflow and Flowsolve. Polyflow is a commercial flow solving computer software package, which can incorporate a number of different constitutive equations. Flowsolve is a development software package from Leeds designed principally for the Pom-Pom and Rolie-Poly constitutive equations.

The images below show how results of simulations can be directly compared with experimental results. This data can be used to compare constitutive equations. In all cases below the experimental data is for polydisperse polystyrene melt at 180°C and a maximum extension rate of 4s-1.

Comparison of Integral Wagner simulation (Polyflow) with observed birefringence field. |

Comparison of Rolie Poly simulation (Flowsolve) with observed birefringence field. |

Comparison of Pom-Pom simulation (Polyflow) with observed birefringence field. |

Comparison of Rolie Poly simulation (Flowsolve) with observed birefringence field. |

### References

Mackley, M.R, Marshall, R.T.J, and Smeulders, J.B.A.F, "The Multipass Rheometer", Journal of. Rheology, **39**(6), 1293-1309, 1995.

Macosko, C.W., "Rheology: Principles, Measurements and Applications", Wiley, New York, pp 379-421, 1994.

Verbeeten, W.M.H., "Computational Polymer Melt Rheology", Universiteitsdrukkerij, Eindhoven, 2001.

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