Multiscale Modeling of Epoxies and Epoxy-Based Composites
287
Based on the above literature, interphase thickness and their corresponding
properties (i.e., interface stiffness) are key factors in creating a multiscale simulation
model [110]. The interphase thickness can be determined from density variation
near the fiber surface. At the interface of the polymer matrix and nanoparticles,
there is usually a layered mass distribution (as shown in Fig. 10b), from which
the interphase thickness can be calculated [22, 69]. Schadler [111] points out the
interfacial depth depends on the reinforcement particle size and the interaction
between the matrix and particle [54]. For example, the reported interface of DGEBF
epoxy nanocomposites is 0.64–0.69 nm, comparable to the 0.6–1.0 nm Al 2 O 3
particle size [57]. The interface of a DGEBF matrix is 0.29–0.34 nm, comparable
to the 0.7–1.4 nm CNT particle size [81]. The interphase of a DGEBF system is
2.2 nm and is comparable to the size of the silica particles 1.01 nm [22]. Depending
on the configuration of the simulated system, an interface can be as large as 250 nm,
which is half of the spacing between particles in some cases [111].
After determining the thickness and properties of the interphase region, material
properties predicted from MD or CG-MD can then be upscaled into FEM models
via a numerical homogenization method, i.e., representative volume element (RVE)
method. An RVE model can effectively account for the structure variation through
a large material size and bring small localized material properties predicted by
all-atom MD or CG-MD models into microscale and macroscale level continuum
models [53, 112]. The basic concept of an RVE is to discretize a cubic FEM matrix
domain into unit cells, and each unit cell is corresponding to a fiber/epoxy system.
Stochasticity is considered in the FEM model via a random sampling process, so that
each unit cell has different fiber fractions and degrees of cross-linking of epoxies.
The overall FEM model thus can be used to simulate a macroscale material with
various local material properties. Figure 11a shows a work conducted by Mortazavi
et al., where they used the RVE method to incorporate MD-simulated thermal
conductivity of graphene/epoxy nanocomposites to a FEM microscale composite
model with 10% volume fraction of unidirectional graphene fibers and predict its
anisotropic thermal conductivity. The macroscale thermal properties of graphene
reinforced epoxy composites were then predicted using a larger FEM model with
20 3 elements and randomly assigned thermal properties predicted from RVEs in
each element, as shown by different colors in Fig. 11b. The latter model predicted
isotropic thermal properties of the material and demonstrated that RVE method
can be used to achieve homogenization in nanocomposite modeling [55, 74]. Kim
et al. also used the RVE method combined with MD simulation and simulated
mechanical deformation of a SiC/epoxy nanocomposites [22, 56]. Subramanian et
al. used RVE method simulating the fracture of CNT/epoxy nanocomposites [53].
Other analytical-based homogenization methods are also used in many multiscale
simulation of polymer composites, but they are not yet commonly used for epoxy
systems [63].
287
Based on the above literature, interphase thickness and their corresponding
properties (i.e., interface stiffness) are key factors in creating a multiscale simulation
model [110]. The interphase thickness can be determined from density variation
near the fiber surface. At the interface of the polymer matrix and nanoparticles,
there is usually a layered mass distribution (as shown in Fig. 10b), from which
the interphase thickness can be calculated [22, 69]. Schadler [111] points out the
interfacial depth depends on the reinforcement particle size and the interaction
between the matrix and particle [54]. For example, the reported interface of DGEBF
epoxy nanocomposites is 0.64–0.69 nm, comparable to the 0.6–1.0 nm Al 2 O 3
particle size [57]. The interface of a DGEBF matrix is 0.29–0.34 nm, comparable
to the 0.7–1.4 nm CNT particle size [81]. The interphase of a DGEBF system is
2.2 nm and is comparable to the size of the silica particles 1.01 nm [22]. Depending
on the configuration of the simulated system, an interface can be as large as 250 nm,
which is half of the spacing between particles in some cases [111].
After determining the thickness and properties of the interphase region, material
properties predicted from MD or CG-MD can then be upscaled into FEM models
via a numerical homogenization method, i.e., representative volume element (RVE)
method. An RVE model can effectively account for the structure variation through
a large material size and bring small localized material properties predicted by
all-atom MD or CG-MD models into microscale and macroscale level continuum
models [53, 112]. The basic concept of an RVE is to discretize a cubic FEM matrix
domain into unit cells, and each unit cell is corresponding to a fiber/epoxy system.
Stochasticity is considered in the FEM model via a random sampling process, so that
each unit cell has different fiber fractions and degrees of cross-linking of epoxies.
The overall FEM model thus can be used to simulate a macroscale material with
various local material properties. Figure 11a shows a work conducted by Mortazavi
et al., where they used the RVE method to incorporate MD-simulated thermal
conductivity of graphene/epoxy nanocomposites to a FEM microscale composite
model with 10% volume fraction of unidirectional graphene fibers and predict its
anisotropic thermal conductivity. The macroscale thermal properties of graphene
reinforced epoxy composites were then predicted using a larger FEM model with
20 3 elements and randomly assigned thermal properties predicted from RVEs in
each element, as shown by different colors in Fig. 11b. The latter model predicted
isotropic thermal properties of the material and demonstrated that RVE method
can be used to achieve homogenization in nanocomposite modeling [55, 74]. Kim
et al. also used the RVE method combined with MD simulation and simulated
mechanical deformation of a SiC/epoxy nanocomposites [22, 56]. Subramanian et
al. used RVE method simulating the fracture of CNT/epoxy nanocomposites [53].
Other analytical-based homogenization methods are also used in many multiscale
simulation of polymer composites, but they are not yet commonly used for epoxy
systems [63].