A hybrid lattice Boltzmann-molecular dynamics-immersed boundary method model for the simulation of composite foams

Abstract

Small fillers (e.g., carbon fibers) are commonly added to polymer foams to create composite foams that can improve foam properties such as thermal and electrical conductivity. Understanding the motion and orientation of fillers during the foaming process is crucial because these can affect the properties of composite foams significantly. In this work, a hybrid lattice Boltzmann method-molecular dynamics-immersed boundary method model is presented for simulating the foaming process of polymer composites. The LBM model resolves the foaming process, and the MD model accounts for filler dynamics. These two solvers are coupled by a direct forcing IBM. This solver can simulate composite foaming processes involving many bubbles and filler particles, including rigid and deformable 3D particles, and rigid, deformable, and fragile fibers. The solver relaxes most simplifying assumptions of earlier polymer composite models, allowing for a better understanding of filler motion and interaction with growing bubbles.

Appendix A: Conversion between dimensionless physical and lattice parameters

Dimensionless numbers are indicated with “*” superscript. Given a length scale $\delta x$, a timescale $\delta t$, a mass scale $\delta m$, and a substance scale $\delta n$, the following dimensionless numbers can be defined to convert lattice parameters to physical parameters, and vice versa.

$$\begin{aligned} \rho ^*= & {} \rho \frac{\delta x^3}{\delta m}\\ \nu ^*= & {} \rho \frac{\delta t}{\delta x^2}\\ q^*= & {} \rho \frac{\delta x^3}{\delta n}\\ D^*= & {} \rho \frac{\delta t}{\delta x^2}\\ \mathbf {g}^*= & {} \mathbf {g} \frac{\delta t^2}{\delta x}\\ L_f^*= & {} L_f \frac{1}{\delta x} \\ k_s^*= & {} k_s \frac{\delta t^2}{\delta m} \\ k_b^*= & {} k_b \frac{\delta t^2}{\delta m \delta x^2} \\ k_v^*= & {} k_b \frac{\delta t^2}{\delta m \delta x^2} \\ k_d^*= & {} k_d \frac{\delta t^2}{\delta m} \\ k_a^*= & {} k_a \frac{\delta t^2}{\delta m} \end{aligned}$$

where the dimensionless lattice parameters $\tau ^*$ and $\tau ^*_g$ are:

$$\begin{aligned} \tau ^*= & {} \frac{6\nu ^* + 1}{2} \\ \tau _g^*= & {} \frac{6D^* + 1}{2} \end{aligned}$$

Note that because of the relationship between pressure, density, and the speed of sound ($p = \rho c_s^2$), for a given pressure and density $\delta t$ and $\delta x$ cannot be changed independently. Refer to “Koerner, et al. Springer Science & Business Media, 2008” for more details.