The investigation of the dynamical properties in physics is usually connected with the calculation of time correlation functions.35 In computer simulations such calculations can be performed relatively easily by saving corresponding coordinates of particles at every time step, although it requires significant use of the CPU time and memory of the computers. For this reason, in order to decrease the memory consumption, all films have been simulated for different time intervals, and with different time resolutions, depending on the length of each run. In what follows we will call these time intervals waiting times, although they are actually the observation times; the observables change during observation and the change is averaged over the interval. The segmental translational and orientational mobility will be measured over different time intervals. Therefore, the simulated results at fixed temperature, pressure, and for a chosen film or layer in Figures 5–9 will be built from few different curves.
4.1. Segmental Translational Dynamics
First the MSD for the non-deformed aPS capped film in different layers and at two temperatures is analyzed, Figure 5. Initially, the film was divided into five equal layers, with thickness for each layer. For simplicity we consider only the two layers that are in direct contact with the supporting substrates, and one layer in the middle part of the film. From now on, these layers will be called top, bottom, and middle layers, respectively. The two intermediate layers, between top and middle layer, and between middle and bottom layer, will not be shown.
We note that the glass-forming systems in bulk consist of dynamic, collectively rearranging regions (CRR), characterized by some characteristic length scale.36 The size of the CRR growths when approaching the glass-transition temperature from above. One may say that in a polymer film, the CRR length encompasses the entire film thickness, which would make a gradient in dynamics difficult to comprehend. The size of these CRRs is not investigated in this study, however, our results and results of other studies for films of comparable thickness23, 36 show that in thin polymer films a strongly attractive substrate drastically slowing down the segmental mobility. In this sense, the division of nanothin films into layers is justified.
Initially, particles move ballistically, meaning that they do not collide with any other particles, and the traveled distance is proportional to the time interval, giving a quadratic (slope = 2) increase in the MSD. In the ballistic regime , where is the average thermal velocity of particles; v ∼ 650 m · s−1 at 300 K and v ∼ 840 m · s−1 at 370 K. An increase in the temperature leads to an increase in the mobility of monomers, marked by an increase in the MSD at shorter times, Figure 5. When the temperature is below the glass transition, particles are temporally frozen and trapped in local cages made by surrounding united-atoms. Motion in a cage is exhibited by the plateau of the MSD, and is called β-relaxation. As can be seen from Figure 5, the cage size in different layers of film is different. The size of the cage (or actually the displacement within the cage) in the top/bottom and middle layers can be estimated from the level of the plateau; , and , for the top/bottom and middle layers, correspondingly. Knowing these values we can estimate the characteristic time for the ballistic motion from , which gives tblst = 0.09 ps and tblst = 0.14 ps inside the top/bottom layer, and middle layer, respectively. Note slightly higher values of the ballistic time in the middle layer which can be connected to a larger free volume there. The increase of the temperature causes an increase of the plateau value, meaning that the average size of the cage also increases, as shown in Figure 5. After some time particles start to escape from their cages, which is manifested by a further increase in the MSD. The out-of-cage escape is usually referred to as the α-relaxation process.15
During MD simulation particles can exchange between layers. Because of this the calculation of dynamical properties in layers is not so trivial. One way to solve this issue is just not to take into account particles which move outside the layer. However, in this study we accept the way which was also used by Baschnagel and Varnik,37 where the total number of particles inside a layer is defined at the initial time t0, and is kept during the calculation of dynamical properties. Obviously, with increasing time t the fraction of particles that leave the original layer increases. In this case, the calculation of local segmental mobility is made over particles which are still present in the original layer and particles that have already left it. Therefore, for a long simulation time the local dynamics in a chosen layer will be a mixture of local motions inside this layer plus local motions from the neighboring layers. So, in terms of the MSD we should define a limit after which the averaging over particles in a chosen layer has no meaning anymore. This limit might be estimated as the time which is needed for a particle to travel from the center of a layer to the neighboring layer in the direction perpendicular to the wall, i.e., to travel a distance of h/2 (dashed line in Figure 5). From this figure it is seen that for a sufficiently long time (more than 100 ns) the MSD does not reach meaning that the simulated particles are highly immobile. Another estimation can be produced from the displacement of the –CH– backbone united-atom (σCH = 0.4153 nm) on a half of diameter σCH/2 (also shown in Figure 5 by dashed line).
In analogy with the calculation of the structural properties, we compare local dynamics in different layers for the non-deformed and cyclicly sheared capped films. In Figure 6, the MSD is shown for the non-deformed state (solid lines) and cyclicly sheared state (dotted lines) capped films in the top (a) middle (b) and bottom (c) layers. In all figures, black and red lines correspond to films simulated at two temperatures, 300 and 370 K, respectively.
First let us discuss the MSD for the top layer, Figure 6a. The initial ballistic motion of particles inside both the non-deformed and cyclicly deformed film is the same, for both temperatures. Then, starting at t ∼ 1 ps the “cage effect” appears. It is clearly seen that the MSD for the cyclicly deformed film is shifted to lower values. Physically it means that motion of particles becomes more restricted after deformation. The average size of the cage is lower as compared to the non-deformed sample. The main cause of this is the rearrangement of atoms under influence of repeated shear that leads to better monomer packing. Due to these rearrangements the out-of-cage escape for the cyclicly deformed film happens at a later time.
The shift in the MSD to lower values for the deformed film can be explained assuming that polymer material in the top layer has become more aged and apparently the monomers continue to stay in their original cages. From the calculated velocity profiles, Figure 7, it is clearly seen that the monomers in the top three layers (up to about ≈1.2 nm) follow the displacement of the substrate upon shear deformation, meaning that monomers are rather weakly displaced with respect to each other, they continue to stay in their original cages.
From the velocity profiles, Figure 7, it is also clearly seen that the bottom layer is weakly deformed. Due to this one might expect that the polymer material in the bottom layers is even stronger aged than in the top layer. From Figure 6c it is clearly seen that the MSD in this layer is indeed shifted to lower values for the deformed films at two temperatures. Due to the aging the average cage size becomes smaller and out-of-cage escape starts later.
Figure 7 also shows that at low shear rate, , there is a strong difference between shear velocities of top and bottom parts. Such a difference causes high deformations in the middle layers. This effect is usually ascribed to the shear banding and is observable over different length and time scales.38 The shear-banding effect also appears at the highest shear rates, as can be seen in Figure 7. In this case, the shear banding occurs between the shear-affected 3rd top layer and the non-sheared 4th layer (arrows in Figure 7).
Due to the shear-banding effect we may expect that upon cyclic shear the material in the middle layer may become younger or rejuvenated. The rejuvenation will lead to the increase in the mobility of monomers, and this, in turn, will lead to an early out-of-cage escape. The simulation results reveal that after application of the cyclic-shear deformation the mobility of monomers in the middle layer is indeed increased, as is demonstrated in Figure 6b. From this figure it is seen that for the sheared middle layer the out-of-cage escape happens roughly at t ≈ 30 ps, much earlier than t ≈ 300 ps for the non-deformed middle layer. Based on this we may conclude that middle layers are rejuvenated after shear deformation. From Figure 6b it is also visible that the rejuvenation effect is temporal, depends on the simulation time and disappears after about 1 ns of relaxation.
After 1 ns simulation time the polymers in the middle layer also become aged, which causes a shift in the MSD to longer times. At long enough simulation times the translational mobility of monomers in the sheared sample approaches to the MSD values for the non-deformed sample, meaning that already after 1 ns the effect of rejuvenation is erased, and the sample reaches the same aged state as the non-sheared film.
4.2. Segmental Orientational Dynamics
Again, each capped film has been divided into five equal () layers and for each layer the orientational autocorrelation function P2(t) of the side vector has been calculated. As was already mentioned, during simulation the particles can move between neighboring layers. Obviously, it also causes the motion of the side-bond vectors to other layers. The problem with migration of these vectors has been solved almost in the same way as was done for the calculation of the MSD: vectors are inside a chosen layer, if the center of the side-group vector initially belongs to this layer.
Figure 8 presents the orientational autocorrelation function P2(t) of the side-group vector in the top (red lines), middle (blue lines), and bottom (black lines) layers for the non-deformed capped film at 300 K (solid lines) and 370 K (dotted lines) and at normal pressure 52 MPa. As can be seen from Figure 8 at short times (up to 0.2–0.3 ps) the P2(t) exhibit fast decay that is connected with the ballistic motion of side vectors. Then for longer times (up to about 300 ps), the cage effect occurs (see the MSD in Figure 5), which is reflected by the slow decay (β-relaxation) of the P2(t) autocorrelation function. Due to the presence of two supporting substrates the relaxation at two interfaces is very slow as compared to the middle layer. An opposite effect was observed in simulations made by Mansfield et al.42 of bulk atactic polypropylene with a free, not-supporting surface: the bulk polymer exhibits slow relaxation while the free surface shows much faster relaxation. Their results suggest that the role of a free interface is to increase the segmental mobility there, and, as a consequence, to reduce the average Tg value for the whole sample. The free interface behaves as a melt (i.e., it has a lower Tg) in contrast to the frozen glassy middle (bulk-like) layer. The simulations of the free-standing films made by Baljon et al.22 showed that the presence of the free interfaces affects α and β-processes differently; the β(α)-process is faster (slower) in the center of films and slower (faster) close to the free surface.
In Figure 8, it is shown that the relaxation of the side-bond vectors at high temperature (370 K, dashed lines) is much faster than at low (370 K, solid lines) temperature. In spite of this fast relaxation at elevated temperature, the aging effect is also clearly visible: with increasing simulation time the P2(t) curve is shifted to larger times.
Next we compare local relaxations in layers for the non-deformed and cyclicly sheared films. Figure 9 presents the orientational correlation function P2(t) of the side vectors in three layers of the capped film for the non-deformed (solid lines) and cyclicly sheared (dotted lines) capped films at 300 K (black lines) and 370 K (red lines) and at normal pressure 52 MPa.
The observed picture is in a qualitative agreement with the MSD measurements. The relaxation in the top and bottom layers, Figure 9a and c, of the non-deformed film is slower than in the middle layer, due to the confinement (i.e., strong interaction with substrate) and due to the aging. A rather weak shift of the P2(t) curve to larger relaxation times with increase of the simulation time is also observed.
As was shown previously, the application of cyclic shear does not rejuvenate polymers in the two interface layers. Instead, because of the rather long simulation time of 100 ns (Non-deformed state, Figure 1) and additionally 52 ns of simulation in the Reference state, Figure 1, the polymers are well aged. For the middle layer, Figure 9b, the physical picture is different as compared to the top and bottom layers, and is also in agreement with the MSD calculations. Application of the cyclic shear causes rejuvenation of polymers in the middle layer, although this effect is only temporal. The maximal effect of the rejuvenation is observed up to about t ≈ 1000 ps, where the difference between the segmental orientational mobility of sheared and non-sheared middle layers is rather large. For simulations longer than 1000 ps the effect of rejuvenation is almost erased, and a rather small difference between rejuvenated and non-sheared film is observed, Figure 9b.
The effect of aging and rejuvenation is also observed at high temperature, 370 K. At this temperature the decrease in the side-bond-vector orientation is much faster than at the low temperature, meaning that the united-atoms are more mobile.
The time relaxation of P2(t) cannot be fitted with a simple exponential function. Therefore, the stretched exponential function, the Kohlrausch–Williams–Watts (KWW) equation, is used to extract the relaxation times
Here τ is the characteristic relaxation time, and β is the stretching parameter, which varies between 0 and 1.
The correlation function P2(t) was calculated for different simulation times (from now, instead of “simulation time” we will, following ref. , use “waiting time,” tw. As was shown above, with increase of the waiting time the P2(t) curve is shifted to longer times, meaning that relaxation of the side-group vectors is slowed down due to aging. A similar observation has been made by Kob and Barrat,3 who performed MD simulations of aging effects in the Lennard–Jones binary mixture. They followed the relaxation of the system with the help of the two-time correlation function , where tw denotes the waiting time elapsed after the quench. They found that with increase of the waiting time the relaxation time also rapidly increases, meaning that the system loses its initial configuration slower. Additionally, it was also shown that the relaxation time changes with the waiting time as a power law, , with an aging exponent µ = 0.88.3
Using expression Equation 8, the relaxation times τ are measured for the whole film, top, middle, and bottom layers, at two temperatures and pressures, and plotted as a function of the waiting time tw, Figure 10. In this figure the scaling behavior with power law can be seen for the entire films and for different layers of these films. We found that the relaxation times of the whole non-sheared film are very close to each other at two temperatures, and at fixed tw, and follow a power law with increasing waiting time, with the average exponent µ = 0.53 ± 0.07. Immediately after the application of the cyclic-shear deformation the relaxation times are shifted to lower values. The main cause of this shift is the rejuvenation effect, which is temporal, and ends roughly at 1 ns. The relaxation of the whole cyclicly sheared film at pressure 52 MPa and temperature 300 K is very slow, with the aging exponent µ = 1.5 ± 0.2. At temperature 370 K the relaxation is much faster, and the aging exponent is µ = 0.44 ± 0.15. We note that the different scaling exponents µ are produced partly because of different definitions of the waiting time tw. Therefore the absolute values of the scaling exponents produced in this study are not compared with known results from literature. We point out that the aging well in the glassy state (at T = 300 K) is enhanced after the cyclic rejuvenation, and the scaling exponent after cyclic shear is higher than before. At T = 370 K the aging rates before and after shear are very similar to each other, and the scaling exponents in this case are identical within error bars.
Figure 10. (a) Relaxation time τ of P2(t) as a function of the total waiting time tw averaged over the whole capped films that are simulated at pressure 52 MPa, and at two temperatures, 300 and 370 K. Vertical dashed lines denote the end of stages according to Figure 1. (b) Relaxation time as a function of the waiting time for the whole film (squares), top (diamonds), middle (triangles), and bottom (cycles) layers at 52 MPa and 300 K. In both figures the filled and opened symbols correspond to the relaxation times for the non-deformed and sheared films, respectively. The solid lines are guides to the eyes and denote the power law with the exponent µ = 0.88.3 The size of the error bars is comparable with the size of the symbols.
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From the segmental translational and orientational mobility, Figures 6 and 9, it is clearly visible that segmental relaxation is different in different layers. Calculation of the relaxation time in layers showed, Figure 10b, that relaxation is much faster in the middle layer than in the top and bottom layers. After cyclic shear the relaxation of the middle layer is slightly shifted to larger relaxation times, while for the top and bottom layers the relaxation is slowed down, meaning that the aging effect is very strong at both interfaces.