Lattice spring model

Recently, some models (e.g., Halpin-Tsai, Mori- Tanaka, lattice spring model, and FEM) have been applied to estimate the thermo-mechanical properties [247, 248], Young s modulus[249], and reinforcement efficiency [247] of PNCs and the dependence of the materials modulus on the individual filler parameters (e.g., aspect ratio, shape, orientation, clustering) and on the modulus ratio of filler to polymer matrix. 

Using field-based models, it is more difficult to provide information about the chain conformation on the surface however, attempts have been made to understand phase separation and mechanical properties of composites. Shou et al. combined SCF/DFT techniques with lattice spring model (LSM) to study the effects of the spatial distribution and aspect ratio of particles (rods and spheres) on the mechanical properties of the composite. Buxton and Balazs combined TDGL theory for polymer blends with BD for nanorods in the simulations of nanocomposites. i A percolating network of nanorods was identified in the minority phase of a bicontinuous structure. Clancy and Gates developed a hybrid model for CNTs in a bulk poly(ethylene vinyl acetate) matrix. Molecular structures of... 

To study the dynamic behavior of the BZ gels, we numerically integrate Equations 3.1 - 3.3 in three dimensions using our recently developed gel lattice spring model (gLSM) [18, 19]. This method combines a finite element approach for the spatial discretization of the elastodynamic equations and a finite difference approximation for the reaction and diffusion terms. We briefly describe the 3D formulation of the gLSM below for more details of the model, we refer the reader to Ref [18]. 

Buxton G and Balazs A C (2002) Lattice spring model of filled polymers and nanocomposites, J Chem Phys 117 7649-7658. 

Mori-Tanaka model Kalpin-Tsai model Lattice-spring model Finite element method Equivalent continuum approach Seif-similar approach... 

The usual structure of off-lattice chain models is reminiscent of the Larson models the water and oil particles are represented by spheres (beads), and the amphiphiles by chains of spheres which are joined together by harmonic springs... 

Fig. 8a,b. Off-lattice representations of a three-functional star a Bead and Rod model b Bead and Spring model... 

The spring model suggests that the symmetry of the crystal lattice determines the different forms of the dielectric tensor that is, they are related to the seven types of crystalline solid (amorphous solids and most liquids are isotropic). This is summarized as follows ... 

To represent the molecular structure with reasonable accuracy as well as to reduce computational time, the coarse-grained, bead-spring model [Fig. 1.28(b)] was employed to approximate a PFPE molecule. This simplifies the detailed atomistic information while preserving the essence of the molecular internal structure [167]. The off-lattice MC technique with the bead-spring model was used to examine nanoscale PFPE lubricant film structures and stability with internal degrees of freedom [168],... 

So far, we have demonstrated that the MC simulation (lattice-based SRS model and off-lattice bead-spring model) results are in qualitative agreement with the experiments. A complementary approach is molecular dynamics (MD) simulation using the bead-spring model. Since MD study for PFPE is still the infant stage, we will discuss it only briefly. The equation of motion can be expressed in... 

Fig. 1.2 Typical coarse-grained models of a polymer chain (a) random flight model, (b) bead-spring model, (c) lattice model.

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