Optical Rheology of a Polydimethylsiloxane Fluid in Jeffrey-Hamel Type Flow
Abstract
The rheo-optical behavior of a viscoelastic polydimethylsiloxane (PDMS) fluid was examined at room temperature for various pressure drops (flow rates) across a Jeffrey Hamel type converging wedge flow cell. The strain-rates were computed from local velocity measurements using laser Doppler anemometry (LDA), and the planar extensional flow behavior of the polymer melt was studied via birefringence measurements along the centerline of the flow cell. The linear relation between the stress and polarizability tensors were confirmed over a range of strain rates that extended well into the non-Newtonian region. The first normal stress difference (FNSD) was calculated from the local velocity measurements using a two-term Goddard-Miller model with a single Maxwell-type relaxation time constant of 0.0174 s computed from Rouse model and a zero-shear viscosity of 300 Pa.s. A linear stress-optical coefficient of 1.41 x 10-10 Pa-1 was obtained for PDMS in planar extensional flow at room temperature from the flow birefringence measurements and the first normal stress difference (FNSD) computed using the Goddard-Miller model. This compares well with values for PDMS in the range of 0.909 – 1.84 x 10-10 Pa-1 at room temperature as reported by various researchers
Keywords
Download Options
Introduction
Polymer macromolecules exhibit isotropic behavior when they are completely randomly distributed. However, flow deformation causes orientation of the macromolecules leading to anisotropy in the transport properties like birefringence [1-3]. Anisotropy to transmission of light by an optical medium produces birefringence, or differences in refractive indices in orthogonal directions. According to Flory [4], the degree of anisotropy in refractive index (birefringence) for a chain network is represented as
ΔN = (2π/9) (ν/V) [(n2 + 2)2/n] (αxx – αyy) (1)
Where ν/V represents the number of segments per unit volume (a segment is the portion of a macromolecular chain between two adjacent entanglement points), n is the refractive index of the isotropic non-ordered material, and (αxx – αyy) is the difference in the averaged polarizabilities of the chain along the x and y axes.
In the case of flexible polymer solutions and melts, the net optical anisotropy caused by flow can be obtained by measuring differences in refractive indices in the direction of the principal stresses. When the direction of propagation of the electric field vector of a polarized light beam coincides with the direction of one of the principal stresses of an optically anisotropic macromolecular fluid flowing through a transparent channel, the difference in birefringence in the other two directions is related to the difference in the corresponding principal stresses via the stress-optical law. According to this law, in a wide range of conditions involving not too large stresses, there is a linear relation between the components of the refractive index (polarizability) and stress tensors given by
ΔN = C Δσ (2)
where C is a material constant known as the stress-optical coefficient, ΔN is the difference in main refractive indices, and Δσ is the corresponding difference in the two principal stresses. The sign and magnitude of the stress-optical coefficient depends on the chemical structure of the polymer, which is governed by the polarizability of the bonds between the atoms of the polymer molecule and the direction of the bonds with respect to the polymer backbone.
Conclusion
We have shown that the linear stress-optical coefficients of an amorphous polymer melt or fluid can be estimated from flowinduced birefringence and laser Doppler anemometry measurements. Unlike many previous studies that measured stresses mechanically, the stresses can be evaluated by choosing an appropriate constitutive equation that best describes the rheological behavior of the polymer. The strain-rates can be computed from local velocity measurements using laser Doppler anemometry. The linear relation between the stress and polarizability tensors, which generally valid over a range of strain rates that extended well into the non-Newtonian region, can then be used to obtain the stress-optical coefficient.