Mechanical Properties of Rubber Type Systems Predicted by DPD Simulations

Over the last decade, there has been a lot of scientific interest surrounding polymer-based nano-composites.[1] This was spurred by the realization that, by the controlled inclusion of nanosized particles with different characteristics inside a polymer matrix, one could substantially improve its properties or even create novel materials for entirely new applications, such as magnetoresponsive elastomers or organic photovoltaic cells. Apart from the technological advances, new fundamental questions and results have arisen from this research.


This application note is connected to what may be the oldest and most practical application of polymers incorporating solid particles, namely the mechanical reinforcement of rubber by fillers. In the early days, these fillers were essentially micrometer-sized carbon blacks but now, thanks to the introduction of new materials and production processes (i.e., surface-modified silica), the use of nano-particles has become more widespread, and many commercial rubber based products, like for example tires, nowadays contain nano-particles.


The basic principles of this case study may be summarized as follows. By incorporating a suitable amount of solid fillers within a cross-linked elastomer, one observes a large increase in its elastic modulus (typically by one or even two orders of magnitude), as well as an improved resistance to tear and abrasion. Unlike the original polymer network, the filled rubber responds non-linearly to deformation: the elastic component of its dynamic modulus decreases dramatically at moderate strains, and at the same time its dissipative component goes through a maximum (this is the so-called “Payne effect”)[2]. In this study we observe both the storage part of the dynamics shear modulus, G', and the Youngs modulus of the system, the latter clearly shows nonlinear behavior. In order to study such a nano-particle filled rubber, non- equilibrium Dissipative Particle Dynamics[3,4] simulations have been performed on a crosslinked rubber with 20 volume % filler content[5]. The filler was also highly cross-linked and chemically bonded to the rubber. A typical DPD system is shown in Figure 1.


Figure 1. Rubber matrix(not displayed) filled with silica particle(green) and bonds between particles and matrix(yellow).

A sinusoidal periodic shear was applied to the system in the range of 10-3 to 10-4 Hz in Tau units, with an amplitude around one, and the storage and loss component of the dynamic shear modulus G were calculated from the stress response of the system. Also the loss tangent(tan delta), the ratio between G´ and G´´ was calculated.


The loss tangent was found to be in the typical range for filled elastomer systems, which is between 0.05 and 0.4. Figure 2 displays the stress response of the system to an imposed shear. G' of the simulated system was 0.11 in reduced DPD units.


Figure 2. Imposed shear (blue), system stress response (red) and the fitted stress (green)

In order to showcase the non- linear response of the system, we have also performed longitudinal strain simulations on the same system, and found that at strains higher than 10 % the system starts to flow, this behavior is displayed in Figure 3.


Figure 3. Longitudinal stress strain response of particle filled rubber.

Contrary to the particle filled rubbers, unfilled rubbers or concentrated solutions of rubber based materials like styrene/ butadiene/ stytrene triblockcolpoymers should behave like ideal elastic networks[6]. These systems are of considerable industrial interest as melt adhesives. A basic understanding of rubber like block copolymers is crucial in the study of adhesive formulation. In particular, styrenic block copolymers (SBC) are a vital component in a wide range of hot melt adhesives. SBC’s have a characteristic structure that makes them ideally suited for many adhesive applications. As the name implies, SBC’s are made of blocks of distinct polymers that each contribute to the properties of the polymer. In this study the end blocks are made from 70 monomers of styrene, the middle block consist of 570 monomers of Isoprene. The styrene end-blocks are hard polymers that add strength to the additive, the mid-block combined with tackifiers and plasticizers produce the adhesive properties.These block copolymers arrange themselves into domains on a microscopic level. In this tiny segregated world, styrene end-blocks clump together with other end-blocks as well as they can, while the rubbery mid-blocks tangle together like rubber bands connected to bowling balls. Since each molecule of a tri-block polymer has styrene on each end, the styrenic domains “glue together” several molecules simultaneously. This is schematically shown in Figure 4, Figure 5 shows the resulting morphology of an equilibrated SIS- copolymer in a 50% Tetradecane solution from a DPD simulation. Tetradecane is a selective solvent for the middle block of the copolymer.


Figure 4. Micromicellar structure of SIS block copolymer schematicaly.

Figure 5. Micromicellar morphology of SIS triblock copolymer in Tetradecane solution. Styrene islands in red, isoprene segments in green, solvent not displayed.

As pointed out before, these polymer systems should behave like an ideal elastic network and hence the shear modulus should follow the Rouse – Mooney[6] behavior when plotted against the shear frequency. A power law with exponent 1/2 at high frequencies should be followed by a plateau zone at low frequencies.


In order to simulate this behavior, DPD simulations similar to the ones performed for the particle filled rubbers with varying periodic shear rates have been performed. These simulations have confirmed the ideal elastic behavior[7], as can be seen in Figure 6.


Figure 6. log- log plot of frequency dependence of G'. Ideal Rouse behavior in is represented by the red line

In our studies non equilibrium DPD simulations with applied periodic shear or strain have been performed on particle filled rubbers and rubber based adhesives. It has been proven, that both non linear behavior of the composite systems as well as ideal elastic behavior of the additive systems can be captured by DPD simulations and important insight can be gained on these commercially important systems.



  1. Kraus, G., Ed. Reinforcement of Elastomers; Wiley: New York, (1965)
  2. Payne, A. R.; Whittaker, R. E. Rubber Chem. Technol. , 44, 440-476(1971).
  3. P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett. 19, 155 (1992)
  4. R. D. Groot and P. B. Warren, J. Chem. Phys. 107, 4423 (1997).
  5. G. Raos, M. Moreno, and S. Elli, Macromolecules, 39, 6744 (2006)
  6. J.D. Ferry, Viscoelastic Properties of Polymers; Wiley: New York, (1980)
  7. Y.R. Sliozberg,J. W. Andzelm, J. K. Brennan, M. R. Vanlandingham, V. Pryamytzin, V. Ganesan, J. Pol. Sci. B, Polymer Physics, Vol. 48, 15–25 (2010)