2- Phase Polymer Flow
Moisés' research dealt with 2-phase polymer flows within the interdisciplinary µPP2 project. Polymer melts become highly birefringent upon flowing. Flow induced birefringence (FIB), that arises from the orientation and stretch of polymer chains under deformation, is a valuable technique for investigating new viscoelastic constituve equations. The Cambridge MultiPass Rheometer (MPR) is an excellent platform for experimental flow birefringence studies on small quantities of polymer (20 g). Optical observations of the stress field in flowing molten polymers are compared with the numerical predictions using the tube theory-based 'Pom-Pom' constitutive equations and the Lagrangian-Eulerian code 'FlowSolve'.
Flow induced birefringence in polymer melts
Flow induced birefringence is an optical characteristic exhibited by a molten polymer when it becomes anisotropic by the effect of orientation during flow (see, for example, Wales 1976). The state of polarisation of light is affected when monochromatic polarised light passes through a flowing molten polymer. As a consequence, the measured isochromatic fringes pattern is intimately linked with the stress field, according to the semi-empiral stress-optical rule (SOR):
which states that the difference in principal values of the refractive index tensor (birefringence, Dn) is proportional to the principal stress difference (PSD), being C the optical-stress coefficiente. The birefringence can be computed experimentally as:
where k is the fringe order, l the wavelength of monochromatic light, and d the depth of sample through which the light propagates. This equation was derived for a homogeneous material, having constant optical properties along the direction of light propagation.
Therefore, the experimental PSD can be calculated from the flow birefringence patterns by counting the fringe order, k=0, 1, 2, 3..., and assuming a typical value of the stress-optical coefficient. The stress-optical coefficient is essentially independent of deformation rate, molecular weight and molecular weight distribution, but dependent on the monomer unit identity, and mildly on temperature and optical wavelength.
The flow birefringence technique has been widely used as a complementary tool to rheological experiments to assess viscoelastic constitutive equations (see, for example, Baaijens 1994).
Numerical simulation using the 'Pom-Pom' model
Experimental flow birefringence results are compared with numerical predictions. In order to compute the flow field in polymer melts, the equations of mass and momentum conservation are solved together with a stress constitutive equation. The 'Pom-Pom' constitutive equation has proved successful in fitting shear, uniaxial and planar extension data simultaneously (Inkson et al. 1999). This new class of constitutive equation, developed by McLeish and Larson (1998), is based on the tube theory (Doi and Edwards 1996) for an idealised branched molecular geometry having two identical q-armed stars connected by a backbone segment, with different characteristic relaxation times for the stretch (ts) and orientation (tb) of the backbone. A single 'Pom-Pom' mode is thus characterised by two relaxation times, the number of arms (q) that corresponds to the maximum stretch possible and a modulus (g). In a multimode extension of the 'Pom-Pom' equations (Inkson et al. 1999) the polymer is modelled as a spectrum of up to 12 'Pom-Pom' modes, each mode with differing constitutive parameters. For a highly branched polymer, the slower modes with higher number of arms can be thought of as modelling the inside portion of the molecules, whilst the faster modes model the more easily relaxed ends.
The stress in the multimode 'Pom-Pom' model is given by:
where for each mode i there is a separate equation for the backbone stretch (l) and the backbone orientation tensor (S) given by:
where K is the applied strain tensor, n* is a parameter whose best value has been found to be 2/q (Blackwell et al. 2000) and A is a tensor related to the actual orientation tensor S.
Thus, there are four material dependent parameters required for each mode i. Orientation relaxation times (τb,i) and moduli (gi) are determined from linear dynamic measurements in simple shear (see, for example, Mackley et al. 1994). Two further parameters, stretch relaxation times (τs,i) and number of arms (qi) are obtained by fitting data from transient uniaxial extensional experiments.
The flow behaviour of polymer melts in the 'MultiPass' rheometer (MPR) can be simulated using 'FlowSolve', the Lagrangian-Eulerian finite element solver developed at the University of Leeds (Harlen et al. 1995; Bishko et al. 1999), using the 'Pom-Pom' constitutive model described.
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