Simulation of Polymeric Materials : Overview and Examples

Polymeric materials have multiscale properties ranging from nm to μm in spatial scale, so it is necessary to carefully consider which scale is the origin of phenomena and physical properties. Although many mechanisms have been clarified through the development of experimental and measurement techniques, simulation technology plays an important role in areas that cannot be captured by these approaches alone. In addition, simulation technology also plays a complementary role in the recent progress in the prediction of physical properties using data science.

In the following, we will introduce some of the representative simulation techniques for polymers, their cooperation, software, and so on. The methods are described in order from the smallest scale, as it is easier to understand if we focus on the spatial scale targeted by each method. Since the list of references for each method is endless, we basically refer the reader to the reference [1].

In Full Atomistic Molecular Dynamics (FAMD), one particle represents one atom (Figure 2 left), and the dynamics of each particle is calculated based on Newton's equation of motion. The force acting on each atom is given by a function and parameters, including the LJ potential mentioned above. FAMD can be used to evaluate the effect of changes in molecular structure, such as functional groups, on the conformation and dynamics of molecules. Examples are the density and elastic modulus of polymers in the bulk state, the free volume distribution and diffusion of gas molecules within it, and the oriented (crystalline) Structure. FAMD has become popular because of its applicability to a wide variety of applications. On the other hand, as long as general computers are used, the spatial scale that can be handled is about 10 nm, and the time scale is on the order of 100 ns even when using recent software and hardware (such as GPUs). This is insufficient to handle, for example, long-time relaxation phenomena of polymers.

Figure 2. Left: Molecular structure of Polyisoprene by FAMD, Right: CGMD image

Dissipative Particle Dynamics (DPD) [5] is one of the CGMDs, but its use of soft potentials that allow particle-particle interactions to slip through makes it suitable for phase separation and the evaluation of filler dispersion structures. Dynamics also includes hydrodynamic effects. In the Mean Field (MF) method, each component of a polymer is represented as a concentration (volume fraction) field to evaluate the dynamics of phase separation and even the equilibrium state. It can be coupled with diffusion and hydrodynamics of each component. In particular, Self-Consistent Field Theory (SCFT) can be used to take into account the shape of polymer chains. In these methods, the Flory-Huggins χ parameter is used as the interaction between components (particles). Methods for estimating this parameter using FAMD and quantum chemical calculations have been proposed [6].
Figure 4 (left) shows the phase-separation structure of polyelectrolyte and water calculated using DPD, and Figure 4 (right) shows the phase-separation (core/shell) structure of a three-component polymer system calculated using the mean-field method. Spatial scales of up to several hundred nm are targeted.

Figure 4. Left: Phase separation structure of polyelectrolyte by DPD, Right: Phase separation structure of three polymer components by mean-field method

Using the phase molecular structures obtained by the DPD and mean-field methods, Finite Element Method (FEM) calculations based on continuum models can be performed to evaluate physical properties such as average elastic modulus and thermal conductivity. The material properties of each component are input as parameters. Figure 6 (left) shows the strain energy distribution at the interface when deformation is applied to the phase-separated structure obtained in Figure 4 (right), and Figure 6 (right) shows the heat flux distribution when heat conduction calculations are performed on the nanofiller dispersion structure in the polymer obtained by DPD. The calculation in Fig. 6 is based on a structure obtained by some kind of simulation, but there are other ways to utilize the structure, such as using data from scattering tests or CT instead, or conducting numerical experiments on the relationship between physical properties and structure by creating a virtual structure.

Figure 6. Evaluation of mechanical properties by finite element method
(Left: Strain energy distribution in phase-separated structure, Right: Heat flux distribution in nanofiller dispersed structure)