Innovatives Supercomputing in Deutschland
inSiDE • Vol. 11 No. 1 • Spring 2013
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Structure and Dynamics at a Polymer-Solid Interface

Polymer based composite materials are indispensable in our modern world. We trust our life to them in high-technology applications like airplane wings and in mundane everyday technology like car tires. The aim behind mixing a hard solid and a polymer is always to improve selected properties, most often, and in the above mentioned applications, the mechanical properties like elasticity or yield stress. The composite material is meant to inherit the best of both worlds for its properties. But it not always does so, and if it does, why does it do so? The answer to this question has to lie in the properties of the interface between the solid and the polymer where both worlds meet. To improve our understanding of this interface is the aim of the focused research program SPP 1369, Polymer-Festkörper-Kontakte, funded by the German Science Foundation. The results here presented were obtained in one sub-project of this program and were made possible by a generous grant of computing time on the JUGENE computer of the Jülich Supercomputing Centre.

In engineering applications, these composite materials are typically described by a three-phase model, the polymer, the solid and a so-called interphase between the two with its own properties different from the other two phases. And it turns out that this interphase often has to be assumed to have an extension one to two orders of magnitude larger than the molecular constituents. When the polymeric material is composed of linear macromolecules, the typical extension of the random coil conformations these chains assume in a dense melt varies between a few nanometers (10-9 m) for short chains and about a hundred nanometers for ultra-high molecular weight polymers. To obtain a molecular understanding of the properties of polymer-solid composites, the structure of this interphase region from the size of the single atom to the size of the complete polymer has to be resolved and its dynamic behaviour, which underlies, e.g., the mechanical properties of this phase, has to be studied on all these scales as well.

There exists, of course, an uncountable variety of possible solid materials and polymer materials to mix into a composite, and the exact behaviour at the interface of the respective materials could well be influenced by the very specific chemistries meeting there. However, the large universality in the structural and dynamic behaviour of polymers encourages us to look into simplified model systems and extract from their behaviour qualitative insight which is independent of the specific model. When such a simplified model system can be studied in a chemically realistic way, we have the additional advantage that the computer simulation can predict the results of experiments performed on such a system and be validated this way. Both the airplane wing and the car tire have something in common: they employ a carbon based solid as filler for the polymer. Carbon is a highly versatile filler material, be it as carbon nanotubes or graphene sheets in the most advanced applications or as carbon black (soot) in a car tire. The polymer we studied is 1,4-polybutadiene (PB), one of the polymers making up the rubber material of car tires. This polymer had been studied by us before in its pure form, and a chemically realistic model capable of quantitative property predictions had been established. Our model system therefore consists of a melt of linear 1,4-polybutadiene chains confined by two graphite solids (cf Figure 1)

Figure 1: In the interior of the film the chains are assuming random coil conformations in the melt as exemplified by the green chain extracted from a simulation snapshot like the one shown in the inset. At the walls the chains can either adsorb lightly, like the gray chain on the right, or strongly, like the red chain on the left. This adsorption behaviour strongly influences the relaxation properties of the confined polymer film.

Both graphite and PB are apolar materials and inert, so we can perform a classical Molecular Dynamics (MD) simulation of an interacting N-particle system integrating Newton's equation of motion. When one simulates linear polymers, a choice of chain length, n, has to be made in addition. This choice is dictated by computational feasibility.

The longest relaxation time of a polymer chain behaves as tmax = t0 nx where the segmental time scale t0 increases from a few picoseconds at high temperature to 100 seconds at the glass transition temperature of the polymer (178 K for PB). The exponent x changes from x=2 for short chains below the so-called entanglement molecular weight to x~3 for long chains. For PB the entanglement molecular weight is around 30 repeat units, so we chose n=29 for our simulations. The integration time step in the MD algorithm, on the other hand, has to be around 1 femtosecond, sufficiently smaller than the time scales of the vibrational degrees of freedom in the model to be able to capture the dynamics on the atomic scale. Taking into account the extension of the chains and the expected extension of the interphase, we opted for simulation box sizes of 15 nm extension parallel to the walls and 10 nm or 20 nm extension between the walls, resulting in 1 - 2 105 particles in the simulation volume. The simulations for these system sizes were run in parallel with high efficiency on up to 2048 cores of the Juqueen (formerly Jugene) computer, which delivered a trajectory of 27 ns per day of CPU time for the smaller system. This allowed us to generate trajectories of up to one microsecond duration at lower temperatures. This limits the temperature range over which we can study our model system. We discuss these considerations in detail to illustrate that, especially when one has to study processes covering many decades in time, even the most modern supercomputers can not just be used for brute force solutions of the problem. The modeling has to be a compromise between what is desirable and what is doable. In our case this meant that we, for example, have no possibility to study the dynamics of long entangled chains at solid surfaces in a chemically realistic model and over a relevant temperature range.

Figure 2: The left part shows a close-up of the distribution of chain segments next to one of the walls. The layering in the segmental distribution is clearly visible, highlighted here by the choice of different colors for different layers. This layering is shown in the right part by the black squares. This figure in addition shows the layering occurring in the chain centre of mass positions by the red circles. Both densities are normalized by the bulk value.

A liquid, like a polymer melt, confined between hard solid walls always exhibits a layering next to the walls as is visible in the configuration snapshot on the left side of Figure 2 and as can be observed in the right part of Figure 2 for the segmental density shown by black squares. The atoms in a solid wall always attract the atoms of the liquid with Van der Waals forces and this leads to a strong layering, as for our case of graphite and PB. This behaviour is well known and it is present in all possible polymer solid composites, so it is one of the universal qualitative features we mentioned above. But additionally, a polymer melt has another structural length scale, the size of the polymer coil as measured, e.g., by its radius of gyration, Rg. When we look at the probability to find the centre of mass of a polymer coil at a distance z from the wall, we observe again a layering phenomenon as shown by the red circles in the right part of Figure 2. Instantaneously, polymer coils in the melt assume the shape of a soap bar, so they have three symmetry axis of different lengths. In the bulk, their average projection on all directions is equal, but close to a wall, the longest axis will always be oriented parallel to the wall. This breaks the rotational symmetry of the coil structure leading also to the breaking of the translational symmetry observable as a layering on the coil scale (for this polymer Rg~1.5 nm in the bulk) in Figure 2. These structural changes in the polymer solid interphase extend roughly over two molecular sizes into the adjacent polymer melt. They are also underlying all effects of the solid walls onto the dynamics in the interphase and therefore on the property modifications occurring in the composite material.

The segmental dynamics as well as the chain dynamics in a polymer melt are accessible to a range of experimental techniques, the most important of these being neutron scattering, nuclear magnetic resonance (NMR) techniques and dielectric spectroscopy (for the chain dynamics this is only applicable for polymers having a net dipole moment along the end-to-end vector of the chain). All these techniques can in principal also be used to gain information on the dynamics in the interphase, however only with increased experimental effort. From our discussion of the melt structure in the interphase it is clear that the dynamics in this region will be heterogeneous, i.e., depending on the distance to the wall, as well as anisotropic, i.e., depending on the direction of motion. For neutron scattering techniques, this means that the experimental signal depends on the direction of the momentum transfer with respect to the wall, for NMR and dielectric experiments it means that the signal depends on the direction of the applied magnetic or electric field, respectively. So, experimentally, one has to realize a well controlled geometry. Furthermore, it would be very desirable to specifically address relaxation processes which occur at a given distance from the wall, which is much harder to realize, as all these techniques are typically obtaining their signals as averages over the complete sample.

In the simulation it is, of course, easy to analyze the dynamics as a function of distance to the wall and as a function of the orientation of applied fields, the direction of momentum transfer or the direction of motion of the segments. We have discussed our findings so far in relation to neutron scattering experiments and NMR experiments in two publications [3,4]. Let us discuss the central physical result of these works looking at a quantity which is directly accessible in the simulation and which is a measure of the local directional mobility of the polymer segments – or the chains as a whole – the mean squared displacement (MSD). We will look into the MSD for the center of mass of chains in Figure 3 as its behaviour is a polymer specific result directly related to the layering in the centre of mass density shown in Figure 2.

Figure 3: Mean squared centre of mass displacement of polymer chains (left axis) compared to an adsorption autocorrelation function (magenta line and right axis) defined in the text. The blue lines are the displacements for a chain in a layer of thickness Rg at the wall, the red lines are for a chain in the centre of the film. Dashed lines are displacements parallel, full lines are displacements perpendicular to the walls.

The first thing to note is the heterogeneity of the centre of mass MSD. In the centre of the film – shown by the red curves which are basically indistinguishable – it is much faster than close to the wall. For this figure we analyzed the displacements in a region of width Rg next to the wall and in the centre of the film, respectively. At around 1 picosecond, there occurs a crossover from a vibration dominated regime with a slope around two of the MSD as a function of time, to a subdiffusive regime which is typical for the Rouse motion in dense polymer melts of short chains. In short, the centre of the film behaves like the polymer bulk. This is very different at the wall. The motion parallel to the walls (shown by the dashed blue line) is very similar to the motion in the centre of the film, however, the motion perpendicular to the walls is slowed down strongly. The flat regime between the short time motion and the Rouse motion is very reminiscent of what happens at the glass transition in a polymer melt, however, the data shown are taken at T=353 K, twice the glass transition temperature of this material. A glass transition can be described as a time scale separation between vibrational processes and relaxation processes which in a polymer melt is induced by packing effects and the presence of barriers impeding conformational changes of the chains [2]. The slowing down observable in Figure 3 is the indication of such a time scale separation, but it cannot be simply ascribed to the increased density at the wall shown in Figure 2 as the motion parallel to the walls is not slowed down. The time scale separation is induced by a new relaxation process introduced into the confined melt by the presence of the walls. This process is the desorption kinetics of the chains, and the magenta line in Figure 3 is an autocorrelation function which gives the probability that a monomer adsorbed at time zero is still adsorbed at time t. Clearly, the decay of this function and the motion of the whole chain perpendicular to the wall occur on the same time scale.

This new relaxation process will influence different experiments in different ways. For instance, it had been found by neutron spin echo spectroscopy [5] that the motion parallel to the walls is basically unaltered in a tube confinement, whereas secondary ion mass spectrometry [6] showed that the motion perpendicular to the walls is strongly slowed down and that this influence extends far into the polymer melt from the walls. This is in direct agreement with our findings from simulation. But it will also influence different experimental techniques aimed to determine the glass transition temperature in a confined melt in different ways, and this may by one of the reasons why the conclusions about the dependence of the glass transition temperature on film thickness drawn from different experiments have been so contradictory for almost 20 years now. We hope that further detailed analysis of our MD trajectories revealing the molecular motions occurring in the polymer interphase will help resolve this long-standing controversy in the future.

References

[1] S. G. Advani, Processing and Properties of Nanocomposites (World Scientific, Singapore, 2006)

[2] W. Paul, G. D. Smith, Structure and dynamics of amorphous polymers: computer simulations compared to experiment and theory, Rep. Prog. Phys. 67, 1117 (2004)

[3] L. Yelash, P. Virnau, K. Binder, W. Paul, Slow process in confined polymer melts: Layer exchange dynamics at a polymer solid interface, Phys. Rev. E 82, 050801 (2010)

[4] L. Yelash, P. Virnau, K. Binder, W. Paul, Three-step decay of time correlations at polymer-solid interfaces, Europhys. Lett. 98, 28006 (2012)

[5] J. Martin, M. Krutyeva, M. Monken- busch, A. Arbe, J. Allgeier, A. Radulescu, P. Falus, J. Maiz, C. Mijangos, J. Colmenero, D. Richter, Direct Observation of Confined Single Chain Dynamics by Neutron Scattering, Phys. Rev. Lett. 104, 197801 (2010)

[6] X. Zheng, M. H. Rafailovich, J. Sokolov, Y. Strzhemechny, S. A. Schwarz. B. B. Sauer, M. Rubinstein, Long-Range Effects on Polymer Diffusion Induced by a Bounding Interface, Phys. Rev. Lett. 79, 241 (1997).

• Kurt Binder (1)
• Wolfgang Paul (2)
• Mathieu Solar (2)
• Peter Virnau (1)
• Leonid Yelash (1)
(1) Institut für Physik, Johannes Guten- berg-Universität Mainz
(2) Institut für Physik, Martin Luther Universität, Halle


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