Numerical simulations of the polydisperse droplet size distribution of disperse blends in complex flow

Abstract

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.

Numerical Modeling of the Blend Morphology Evolution in Twin‐Screw Extruders

“…We describe a polydisperse droplet size distribution according to the method as described by Wong et al. [ 37 ] In the histogram that describes this polydisperse size distribution, the bin value multiplied by the bin width equals the volume fraction of droplets with $$ R_0$$‐values contained within this bin interval, compared to the full disperse phase. The total integral under the histogram equals 1.…”

“…We use the logarithmic variables $$ s = \text{log}(\beta )$$ and $$ v = \text{log}(R_0)$$, because in this way, $$ \beta$$ and $$ R_0$$ mathematically cannot become negative, of which the details are explained in our previous works. [ 36,37 ]…”

“…For simple shear flow, the Grace curve is given by $$\begin{align} \text{log}_{10}(\text{Ca}_{\text{crit}} ) &= -0.506 - 0.0994 \thinspace \thinspace \text{log}_{10}(\lambda _{\rm e} ) + 0.124(\text{log}_{10}(\lambda _{\rm e} ))^2 \nonumber\\ &\quad - \frac{0.115}{\text{log}_{10}(\lambda _{\rm e} ) - 0.6107}\end{align}$$and for planar extension, it is given by $$\begin{equation} \text{log}_{10}(\text{Ca}_{\text{crit}} ) = 0.0331(\text{log}_{10}(\lambda _{\rm e} ) - 0.5 )^2 - 0.699 \end{equation}$$For intermediate cases, we apply a linear interpolation as is described by Wong et al. [ 37 ] For these cases between pure shear and pure extension, we use the following equations to quantify the amount of shear and extension, based on the work of Hulsen [ 50 ] $$\begin{align} D_{\rm s} &= \sqrt { -\mathrm{det}{\left(\bm{L}+\bm{L}^T\right)} } \end{align}$$ …”

“…These evolution equations are slightly different than those in our previous works. [ 36,37 ] In those works, we described the morphology variables as continuous field variables and therefore, the left‐hand side consisted of a material time derivative with a local time derivative term and a convection term in an Eulerian frame of reference, so they did not contain the label $$ X$$. In this work, we describe the morphology variables in discrete particles that we follow along their trajectories, which eliminates the convection term and adds the label $$ X$$.…”

“…Based on Peters et al., [ 34 ] Wong et al. [ 36,37 ] have developed a numerical model to simulate the monodisperse and polydisperse blend morphology development in complex flow geometries. An important complex flow geometry is the twin‐screw extruder.…”