Size-dependent mechanical behavior of nanoscale polymer particles through coarse-grained molecular dynamics simulation
© Zhao et al.; licensee Springer. 2013
Received: 13 August 2013
Accepted: 5 December 2013
Published: 21 December 2013
Anisotropic conductive adhesives (ACAs) are promising materials used for producing ultra-thin liquid-crystal displays. Because the mechanical response of polymer particles can have a significant impact in the performance of ACAs, understanding of this apparent size effect is of fundamental importance in the electronics industry. The objective of this research is to use a coarse-grained molecular dynamics model to verify and gain physical insight into the observed size dependence effect in polymer particles. In agreement with experimental studies, the results of this study clearly indicate that there is a strong size effect in spherical polymer particles with diameters approaching the nanometer length scale. The results of the simulations also clearly indicate that the source for the increases in modulus is the increase in relative surface energy for decreasing particle sizes. Finally, the actual contact conditions at the surface of the polymer nanoparticles are shown to be similar to those predicted using Hertz and perfectly plastic contact theory. As ACA thicknesses are reduced in response to reductions in polymer particle size, it is expected that the overall compressive stiffness of the ACA will increase, thus influencing the manufacturing process.
Experimental research has been previously conducted to determine the mechanical response of micron-sized polymer particles by Zhang et al.[5–7]. They used a nanoindentation-based flat punch method to test the compressive response of polymer particles with diameters ranging from 2.6 to 25.1 μm. They observed that decreasing particle diameters resulted in increasing stiffness of the constituent polymer material. Although this type of size effect has been well-documented in crystalline, inorganic materials[8–14], it has not been carefully studied in organic, amorphous materials. The observed behavior of the polymer particles was explained by He et al.[5, 6] using a core-shell argument. That is, there exists a layer of polymer at the surface of the particles that has a molecular structure that differs from that found in the bulk polymer (toward the center of the particle). This surface layer has a constant thickness, regardless of the size of the particle. The presence of this surface layer has a diminishing influence on the overall mechanical response of the particle for increasing particle sizes. Although this explanation is plausible, it remains unverified. Because the mechanical response of the polymer particles can have a significant impact on the performance of ACAs, understanding of this apparent size effect is of fundamental importance in the electronics industry.
The objective of this research is to use a coarse-grained molecular dynamics model to verify and gain physical insight into the observed size-dependence effect in polymer particles. Three different types of analyses have been performed to accomplish the objective. First, simulations of the loading sequence of polymer nanoparticles under compression are presented. This is followed by a description of simulations of the unloading process, both of which serve to verify the previous experimental observations. Finally, a surface energy analysis is described where the surface energy is determined for different sizes of nanoparticles to provide physical insight into the size-dependence effect.
Spherical particle molecular models
Although polymer particles can be composed of a wide range of polymer chemistries, linear polyethylene (PE) was chosen as the model material for this study because of its simple conformational structure and the availability of coarse-grained (CG) potentials especially tuned for the surface tension. Zhao et al. previously demonstrated that the CG models are able to effectively capture the thermo-mechanical characteristics of PE in its glassy phase. Well-tuned CG models can be simulated with significantly less time than all-atom models and are especially advantageous for modeled molecular systems with large numbers of atoms. Because of the relatively large size of the simulated systems in this study, a CG modeling technique using LAMMPS molecular dynamic simulation code was adopted based on a semi-crystalline lattice method for generating entangled polymer structures[16–18].
Potential functions and corresponding parameters of coarse-grained method
kb = 6.96 (TT), kb = 6.16 (TM, MM)
r0 = 3.65 (TM), r0 = 3.64 (MM)
kθ = 1.09 (TMT), kθ = 1.19 (TMM, MMM)
θ0 = 175.5 (TMT), θ0 = 175 (TMM), θ0 = 173 (TMM)
ϵ = 0.469 (TT), ϵ = 0.444 (TM), ϵ = 0.42 (MM)
σ = 4.585 (TT), σ = 4.5455 (TM), σ = 4.506 (MM)
rc = 15 Å (truncation radius)
A = -583.81 (CT, CM)
rc = 10 Å (truncation radius)
Characteristics of coarse-grained linear polyethylene particles
Number of CG beads
Number of molecules
Loading step per 20 ps (pm)
Simulated compression loading
where h is the loading plate displacement from the initial contact position h0 (Figure 4b). It is important to note that although these parameters are not strains and stresses according to their classic tensoral definitions, they are used herein as simple scalar measures in a manner consistent with previous studies[5, 6]. Because the initial gap distance h0 is the same as the non-bonded cutoff radius between the rigid plate atoms and the CG beads (Table 1), any displacement of the plate toward the particle results in an axial loading via the repulsive energy potential; thus, there is no loading slack in the system when ϵ = 0. It was assumed that the distance between the particle surface and loading plate during the compression, hgap, was constant due to the repulsive energy potential.
In order to effectively evaluate the size effect in the polymer particles, a continuum model of a particle subjected to compressive loading between two flat plates was evaluated with finite element analysis (FEA). Because the size effect observed in polymer nanoparticles does not exist in the classical continuum modeling of materials, the response of the FEA model is independent of size effects and thus serves as an excellent control reference to compare the molecular modeling results with. Axisymmetric quadrilateral elements were used with the ANSYS finite element software package. Contact elements were placed between the surfaces of the sphere and the rigid plate. The Young's modulus and Poisson's ratio values determined from the bulk MD simulations of PE described in ‘Spherical particle molecular models’ section were used in the FEA model. Displacements were applied to the top surface of the model, and the nominal strains and nominal stresses were measured using Equations (1) and (2), respectively. It is important to note that elastic properties were used to simulate a large deformation of the material. Normally, a hyperelastic analysis would be appropriate for such an analysis; however, the linear approximation is sufficient for the current study as a simple baseline comparison to the MD models.
Figure 6b displays the particle nominal stresses as a function of particle diameter for different compressive strain levels. For compressive strains of 20%, a mild size effect is observed. At this strain, the nominal stress for the smallest particle is about 1.5 times that of the largest particle. When the compressive strain is increased to 30%, which is common for the micron-sized polymer particles used in ACAs, the nominal stress for the D5 particle is approximately three times that of D40 particle. The data in the Figure 6b also indicates that the particle nominal stresses for large particles approach that of the continuum elastic solution.
The size effect data shown in Figures 6 are consistent with the size effect observed experimentally. He et al. carried out experiments on micron-sized polystyrene-co-divinylbenzene (PS-DVB) particles. A nanoindentation-based flat punch method was used to determine the stress-strain behavior of particles in compression. The particle size varied from 2.6 to 25 μm. A strong size effect on the compressive stress strain curve was observed. As the particle size decreases, the mechanical response becomes stiffer.
Simulated compression unloading
Surface energy analysis
The results of the compression loading and unloading simulations clearly demonstrate the existence of a size effect in polymer particles. The MD models in this study can also be used to gain physical insight into the origin of the size effect. It is well known that crystalline[27–29] and amorphous materials[30–33] have molecular structures at the surface (or bi-material interface) that differ substantially than in the bulk. In fact, the CG potential used for the research described herein was developed specifically to accurately predict the bulk and surface structure of PE. For amorphous polymers, the above-cited references show that the mass density of the polymer is higher on the surface than in the bulk. This high-density layer has a thickness on the order of 1 nm.
The cause of the densification of polymer molecules on a surface is classically explained by the concept of surface tension. Segments of polymer molecules in the bulk have a relatively low energy state because of the balance of attractive short-range (e.g., covalent bonds) and long-range (e.g., van der Waals bonds) interactions in every direction. Segments of polymer molecules on a free surface (or a non-bonded bi-material interface of two dissimilar materials) do not have these strong attractive interactions in the direction normal to the surface and are thus pulled by the attractive forces in the opposite direction towards the bulk. As a result, there is a densification of the top layer of polymer molecules on a surface.
This densified surface layer of material has a constant thickness regardless of the size and geometry of the overall material structure. For polymer particles, this means that the surface layer will have the same finite thickness for any particle size. For decreasing particle sizes, the relative volume fraction of the densified material increases. Therefore, it follows that the smaller polymer particles studied herein are expected to have stiffer mechanical responses than the larger particles, as observed experimentally[5–7] and discussed in ‘Simulated compression loading’ and ‘Simulated compression unloading’ sections.
Contact radius during compressive loading
These two approaches are most valid for two extremes in material behavior: linear elasticity and perfect plasticity. However, polymer materials typically exhibit non-linear behavior that is between these two extremes, particularly the PE material considered herein. Therefore, it is important to determine the accuracy of these two simple approaches when applied to polymeric materials.
In agreement with experimental studies[5–7], the results of this study clearly indicate that there is a strong size effect in spherical polymer particles with diameters approaching the nanometer-length scale. As the particle diameter decreases from 40 to 5 nm, increases in elastic modulus are predicted from the molecular simulations. These increases in modulus are significant for compressive nominal strains below 30% and substantially large for strains greater than or equal to 30%. The results of the simulations also clearly indicate that the source of the increases in modulus is the increase in total energy at the surface of the particles, that is, the surface energy. As the particle diameter decreases, the relative surface energy (ratio of surface energy to equivalent bulk energy for the particle volume) increases. The increases in surface energy result from the increases in the mass density of the material at the surface. This local increase in mass density results in an overall increase in particle stiffness properties.
These results are of significant importance for two reasons. First, coated polymer particles used for electrical conduction in ACAs have a very strong size-dependent behavior. As particle sizes are reduced, they will have a stiffer response to the compressive forces, particularly for nominal compressive strains of at least 30%. Therefore, as ACA thicknesses are reduced in response to reductions in liquid-crystal display thicknesses, it is expected that the overall compressive stiffness of the ACA will increase, thus influencing the manufacturing process. Second, these results indicate the presence of very strong size-dependent effects in organic, amorphous nanostructures that have been well-documented for inorganic, crystalline nanostructures, such as nanowires and nanobelts. The size dependence is a direct result of the changes that occur in the structure of the polymer molecules on the particle surface.
This research was supported by the Research Council of Norway and our industrial partner Conpart AS (http://www.conpart.no) via the NANOMAT KMB Project MS2MP “From Molecular Structures to Mechanical Properties: Multiscale Modelling for Ugelstad Particles” (grant no. 187269), the Norwegian Metacenter for Computational Science (NOTUR), and the US-Norway Fulbright Foundation.
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