The aim of this study is to develop a constitutive model for disperse blends applicable in complex flows and to cast this model in a finite element framework. As the number of droplets in realistic conditions is extremely large, it is computationally intractable to model all droplets individually. Droplet populations are modeled that have macroscopically averaged morphological properties. These properties are the droplet stretch ratio, the unstretched droplet radius, the orientation vector, and the number of droplets per unit volume. The evolution equations of these properties vary based on the morphological state transitions. The current model describes the morphology evolution in complex geometries, assuming Newtonian mixture constituents and monodisperse droplet populations. The numerical model has been validated for simple shear flow. Results are discussed for Poiseuille flow and the eccentric cylinder flow.

The blend morphology model developed by Wong et al. (Rheologica Acta, 2019), based on Peters et al. (J Rheol 45(3):659–689, 2001), is used to investigate the development of the polydispersity of the disperse polymer blend morphology in complex flow. First, the model is extended with additional morphological states. The extended model is tested for simple shear flow, where it is found that the droplet size distribution does not simply scale with the shear rate, because this scaling does not hold for coalescing droplets. Subsequently, the model is applied to Poiseuille flow, showing formation of distinct layers, which occurs in realistic pressure-driven flows. Finally, the model is applied on an eccentric cylinder flow, where histograms are made of the average droplet size throughout the domain. It is observed that outer cylinder rotation results in narrow distributions where the small droplets are relatively large, whereas inner cylinder rotation results in broad distributions where the small droplets are significantly smaller than in the case of outer cylinder rotation. Eccentricity seems to only have a minor effect if the maximum shear rate is held constant. The flow profile and history in combination with the maximum shear rate strongly determine how the polydisperse droplet size distribution develops.

The blend morphology model developed by Wong et al., based on Peters et al., is used to investigate the development of the disperse polymer blend morphology in twin-screw extruder flow. First, the model is written in a point-wise form suitable for using in conjunction with particle tracking. Particle tracking methods are used to generate trajectories along the flow field. Macroscopic droplet populations are placed along these trajectories and the velocity gradient tensor is extracted and applied on the point-wise blend morphology model. Very large morphology differences arise between trajectories that pass through the middle gap and those that do not. In the global distribution of (macroscopically averaged, monodisperse) droplet sizes, two distinct peaks appear due to these different trajectories. Given enough number of screw rotations, a droplet population can reach almost every position in the twin-screw extruder and travel along both types of particle trajectories. The effect of varying the gap size is that the largest droplets are unaffected, but the smallest droplets are smaller for a smaller gap size due to the higher maximum shear rate. The effect of varying the viscosity ratio on the global droplet size distribution is found to be nonlinear and is strongly determined by the Grace curve. The effect on polydisperse droplet populations is found to be that trajectories that do not pass through the gap evolve toward a single peak, whereas trajectories that do pass through the gap lead to a split into two peaks that ultimately rejoin as one peak. It is concluded that the initial position of a population in the twin-screw extruder has a very large effect on the developing transient blend morphology, though future work should be done on the importance of the initial position on the steady-state blend morphology after a very large number of screw rotations.

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