Rejuvenation, Aging, and Confinement Effects in Atactic-Polystyrene Films Subjected to Oscillatory Shear

Abstract

Molecular-dynamics simulations of 5 nm-thick atactic-polystyrene films have been used to study the influence of cyclic-shear deformation on the stress–strain behavior and local segmental mobility. Upon cyclic yield the stress–strain behavior of the films slowly evolves towards a steady state which is characterized by a decrease of the maximum stress and by an enhanced dissipative process. Immediately after plastic deformation the storage modulus is decreased and the loss modulus is increased as compared with their initial values. Such changes in the viscoelastic moduli reflect the mechanical rejuvenation of a polymer glass. This mechanical rejuvenation of polymers is connected to the increase in the simulated segmental mobility, which is calculated for the entire film as well as in different layers.

1. Introduction

It is well known that polymer glasses have a broad spectrum of relaxation times that are connected with the structural relaxations of polymer chains and polymer segments.1 Additionally, there is an effect of the physical aging, first investigated by Struik,2 which strongly influences the segmental relaxation by shifting the relaxation times of polymer glasses to larger values.3 The physical aging in glassy polymers is much studied, but poorly understood, especially in thin polymer films.4–10

The effect of physical aging can be removed by heating a system above the glass transition, which is usually referred to as thermal rejuvenation. Struik2 suggested that mechanical deformation can disturb the physical-aging process, mainly by creating additional free volume in the perturbed glassy polymer. Due to the presence of the additional free volume, the mobility of polymer segments becomes enhanced, which causes the shift of the relaxation times to the smaller values. In this sense, the application of the plastic deformation causes the erasure of the aging history and is usually described as a rejuvenation effect. Some experimental results also show that dynamics can be accelerated by shearing a system at high deformation rates, whereas deformation at moderate shear rate accelerates the aging of the system and is referred to as “overaging.”11

The aging kinetics of polymer glasses has been experimentally studied by Govaert and co-workers12 by measuring the time evolution of the yield stress for both quenched and mechanically deformed polymer samples in the postyield regime. They found a significant decrease in yield stress for the mechanically pre-deformed polystyrene and polycarbonate glasses. Additional to this, Govaert and co-workers12 found that mechanically pre-deformed glass has the same rate of physical aging as thermally quenched glass. Due to this effect they concluded that the polymer glass can also be rejuvenated as a result of mechanical deformation.

The rejuvenation and aging effects have also been studied in computer simulations of non-crystallizing binary Lennard–Jones mixtures.13, 14 In terms of the energy landscape, it was shown that large strain deformation can rejuvenate a glass by bringing the system to shallower energy minima, whereas small-strain deformation overages the system by driving it into the deeper energy minima.14 In particular, Utz et al.13 have shown that after slow annealing a polymer glass occupies lower potential-energy sites, which reflects the aging process. This aging effect can be erased by applying plastic deformation in the yield regime, during which the system is brought into a higher-energy state. The molecular-dynamic (MD) simulation of bulk polystyrene and polycarbonate performed by Lyulin and Michels,15 where the influence of thermal and mechanical histories on the internal energy of these glasses have been studied, also derives a similar conclusion. It was found that the internal energy after an extension/recompression deformation cycle is significantly higher than that of the non-deformed glasses. Moreover, Lyulin and Michels also showed that the partitioning of the internal energy is different for the mechanically and thermally rejuvenated polymers: the thermal rejuvenation is for more than 80% due to weaker van der Waals interactions, while the difference after the mechanical rejuvenation is for about 40% due to increased torsion of the polycarbonate chains; this means that the mechanical rejuvenation is different from the thermal rejuvenation.

In most experimental studies, the dynamics of a polymer glass is probed during or immediately after the application of deformation. The dynamics of a mechanically deformed acrylate film was analyzed by Chateauminois and co-workers16 in terms of small-strain linear-viscoelastic measurements of the shear modulus. They found that immediately after the interruption of the plastic shear deformation, the linear viscoelastic shear modulus is drastically changed. Then, after the deformation in the yield regime, the viscoelastic shear modulus recovers back to its initial value, i.e., the value before the cyclic yield. This recovery process has been considered as an evidence of an enhanced rate of physical aging in the temporally mechanically rejuvenated glass.

Experiments and simulations have shown that segmental dynamics in glassy polymer films is different from that in a bulk.17, 18 NMR experiments17 on confined 2 nm polystyrene films revealed that the dynamically slowest styrene fragments are found in the vicinity of the confining surface, while faster relaxation appears toward the center of the film, with lower local density. Serghei and Kremer19 have suggested that the main reason for observation of the altered dynamics in thin polymer films might be the application of different annealing conditions and experimental techniques.

Molecular-dynamics simulation studies on capped polymer films have also shown that both the chain mobility20 and the chain relaxation times21 can be slowed down by few orders of magnitude in the vicinity of the surfaces. MD simulations of a free-standing film have shown an opposite effect: polymer segments near the free surfaces are much faster as compared to the middle part.22

The main goal of the present study is to investigate the mechanical properties of thin capped atactic-polystyrene films upon cyclic-shear deformation, and show the influence of the cyclic-shear deformation on the local segmental dynamics for the entire aPS film as well as in different layers. In analogy with the experiments in ref.,16 the mechanical properties have been analyzed in terms of small shear-strain linear-viscoelastic measurements of the shear modulus, and large shear-strain nonlinear plastic measurements of the apparent shear modulus. The storage and loss moduli during cyclic-shear deformations have been calculated and a qualitative comparison between simulations and experiments has been made. The segmental orientational and translational dynamics have been simulated in different film layers.

The paper is organized as follows. In Section 2, we give the description of the model of the capped aPS film together with the deformation protocol. The mechanical properties of capped films upon cyclic-shear deformation are discussed in Section 3. Local segmental dynamics in non-deformed and cyclicly sheared films is discussed in Section 4. Our conclusions are given in the last section of the paper.

2. Simulation Details

2.1. Simulated Model

The atactic-polystyrene (aPS) chains used in this study are modeled with the help of a united-atom representation.23 In this representation all hydrogen atoms are not presented explicitly, but, instead, they are collapsed onto the carbon atoms and the combined atoms are treated as effective particles, i.e., united atoms. The motivation of using this model is that computations are much faster for the united-atom model than for the all-atom model.24

Each simulated atactic-polystyrene chain consists of 80 (641 united atoms) monomers, with molecular weight  = 8.6 × 10³, which is below the aPS entanglement molecular weight M_e = 1.3 × 10⁴. The average gyration radius of these chains is found to be about 2 nm. The stereochemic configurations of the aromatic groups were generated in each chain at random so that the ratio of the number of meso to the number of racemic dyads was near unity. Additionally, each aPS chain has been prepared with different tacticity.

To study the mechanical behavior of aPS films under shear deformation we have prepared films capped by two identical substrates. Each simulated capped film consists of 16 chains (total 10 256 united atoms) and is placed inside a simulation box with fixed to 7 × 7 nm lateral dimensions. Periodic boundary conditions are implemented only in the X and Y directions. In Z direction, the periodic boundary conditions are not applied.

In simulated aPS films all interactions between united atoms are described by the following potential:

• ((1))

The potential U_aPS includes the contributions from the excluded-volume interactions (∼ε_ij) between united atoms that are on different chains or are separated by more than three covalent bonds; the stretching potential (∼k_l) between two neighboring atoms i and j that have shared valence electrons; the bending potential (∼k_θ) for all bond angles, including those in the phenyl rings; finally, the proper-torsion and improper-torsion potentials (∼k_ϕ). In this aPS model, the Coulomb interactions are not taken into account. For more details of the used force-field we refer to Ref. 25.

The leapfrog variant of the velocity Verlet algorithm has been used to integrate Newton's equations of motion with an integration time step Δt = 4 fs. The simulations of the capped films have been performed in NVT ensemble. The temperature in simulated systems is controlled with the help of the collisional thermostat.26

2.2. Polymer–Substrate Interactions

In order to simulate capped aPS films, completely smooth structureless substrates are introduced in the XY-plane of the simulation box. In simulating the structureless substrates, the interactions between substrate and polymer chains are described by the van der Waals interactions between implicit atoms of the substrate and polymer segments, and have the form of a truncated and shifted 9-3 Lennard-Jones (LJ) potential:

• ((2))

Here z denotes the distance from the monomer to the substrate and ε is the strength of the attraction to the substrate. z_min = 0.3 nm is the distance at the minimum of the potential and z_cut = 0.9 nm is the cut-off distance. This potential is similar to that used by Müller and MacDowell in simulations for a bead-spring model film,27 except that in their case the potential was not truncated or shifted. For a bead-spring model the wetting transition occurs at ε = 0.4 kcal · mol⁻¹ at 300 K,27 while in the present MD simulations the potential strength ε is set to 5.0 kcal · mol⁻¹, i.e., complete wetting of polymer segments has been simulated. This potential mimics the van der Waals interactions between the atoms of the substrate and polymer segments, and can be obtained by integrating the 12-6 LJ potential over a half space.28

The simulation of aPS films capped by two identical substrates has been performed with the help of the previously mentioned potential, Equation (2), together with an additional second potential U₂(x,y,z) introduced in ref.29 that reconstructs the structure of substrate and mimics an implicit crystalline lattice of the substrate atoms,

• ((3))

In this equation, α = 5 nm⁻¹, A = 0.3 kcal · mol⁻¹, B = C = 50, and C = 1; z denotes the distance from the monomer to the substrate, and x, y correspond to the lateral directions, where periodicity is built in. This potential was truncated at z_cut = 0.4 nm. The parameter a = 0.5 nm corresponds to the lattice parameter and is comparable to the size of the single phenyl ring. This choice of the lattice parameter allows us to have good physical junctions between polymer monomers and the substrate, and to prevent a sliding of monomers under applied shear deformation. If the lattice parameter is smaller than 0.5 nm the polymer monomers move freely, and, e.g., can slide with respect to the substrate when (even extreme computationally slow) shear is applied along the X-axis.

2.3. Preparation of Capped Films

In Ref. 23, we explained the simulation procedure (equilibration and preparation) for the supported films (with one free interface and one supporting interface), and of different thicknesses. In this paper, we provide details about the preparation of the capped films (confined by two identical substrates), which will be used for studying the mechanical response under shear.

As the initial point of the simulation of the capped film we used results earlier produced for supported 16-chains aPS film at low temperature, 300 K.23 This film has been placed inside the simulation box with lateral dimensions 7 nm × 7 nm in the X and Y directions. In the Z direction two identical confining substrates have been placed at z = 0 nm (bottom substrate) and z = 7 nm (top substrate). Both substrates are modeled with the help of the potential Equations (2) and (3), as discussed above.

After a short (about 500 ps) pre-equilibration the top substrate was moved down from z = 7 nm to z = 5 nm with compressing velocity 1 × 10⁻⁴ nm · ps⁻¹. After the compression the films were equilibrated for 2 ns. During this equilibration the normal pressure was monitored. At the end of the equilibration one film at normal pressure 52 MPa and temperature 300 K has been chosen, with a film thickness of 5 nm. Additionally, this film has been heated up to 370 K with heating velocity 0.01 K · ps⁻¹. Due to the confinement the pressure inside the film increased drastically. In order to decrease the internal pressure back to 52 MPa at 370 K the film thickness was increased by about 1%. Finally we obtained two capped films at pressure 52 MPa and two temperatures 300 and 370 K.

We should also note that the glass-transition temperature has not been calculated for these films. However, the simulation results in ref. 23 show that for the supported 16-chains aPS film of about 5 nm thickness the value of the glass-transition temperature is about T_g = 400 K. For the capped films the value of the T_g can be even higher.

2.4. Deformation Protocols

During simulations the stress was calculated with the help of the symmetric stress tensor τ^αβ using the virial expression:

• ((4))

here, α, β = x, y, z; angular brackets correspond to the ensemble averaging over all N particles, with momentum , mass m_i, position , and total force acting on the particle i. From this expression the external normal pressure was calculated as . The shear deformation was applied horizontally in the X-direction only to the upper substrate of the capped film, in a plane perpendicular to the Z-direction, and the shear stress was calculated as (henceforth we will drop the superscript xz). The shear deformation has been done at different shear velocities, and for different shear strains. The shear strain γ and shear rate are defined as and , respectively; here Δx is the displacement of the top substrate, is the film thickness, and v is the shear velocity.

In a typical rheological experiment, the sample is placed between two plates. While a bottom plate is fixed, the imposed time-dependent shear strain is applied to a top plate. Simultaneously, the time-dependent stress is measured. The time of one cycle, i.e., a frequency of oscillation ω, defines the time-scale of the experiment.

For an ideal elastic solid, the strain and the stress are in phase, while for an ideal viscous fluid there is a π/2 phase lag of strain with respect to stress. The polymer glass in its nature is amorphous, meaning that it has both solid and fluid properties and therefore some phase lag, will occur, and the shear stress is . At low strain amplitude the stress response of a viscoelastic material can be rewritten as ; here G′ is the storage and G″ is the loss components of complex shear modulus, G*, which is equal to .30, 31 The physical meaning of these moduli is simple: G′ defines the energy which is stored in the system upon applied deformation, and G″ defines the energy which is dissipated during deformation.

The chosen simulation protocol is similar to the procedure used by Chateauminois and co-workers16 in experiments. For the consistency with these experiments, we also use the same terminology in the current study. Initially, the film has been simulated for 100 ns in the absence of the shear deformation (non-deformed state, Figure 1). After that, 10 complete shear cycles have been performed in the linear regime up to γ = 2% of the shear strain (Reference state). Then, 10 shear cycles have been done at γ = 13% shear strain, which is above the yield peak (Cyclic-yield state). Finally, each film again has been cyclicly sheared in the linear regime. Despite the small deformation the film has some possibility for relaxation, and therefore this regime is referred to as the Recovery state. After that, each film has been allowed to relax for another 100 ns in the absence of the deformation, Sheared state in Figure 1. Cyclic-shear deformations in the Reference, Cyclic-yield and Recovery states, have been performed at computationally low () shear rate, which is still significantly larger than the frequencies (0.05–25 Hz) normally used in experiments.16, 32 The main purpose is to calculate the storage (G′) and loss (G″) components of the complex shear modulus G* during the Reference, Cyclic-yield and Recovery regimes. It is important to note that in experiments16, 32 the storage and loss moduli that are measured in the Reference and Recovery states, constitute the linear viscoelastic shear modulus, while in the Cyclic-yield state both moduli correspond to the strain- and time-dependent apparent shear modulus; they can be calculated from the in-phase and out-of-phase components of the lateral contact stiffness at the excitation frequency.16, 32

Figure 1. Schematic illustration of different stages of the aPS thin film simulation.

3. Simulation Results

3.1. Stress—Strain Behavior: Shear Modulus

In Figure 2, we show the typical stress–strain behavior, in a Lissajous representation, of the simulated aPS film in the Cyclic-yield state. In this representation, the stress–strain relationship has a hysteretic loop and is qualitatively similar to the experimental results,16, 33 see lower inset in Figure 2. During the plastic deformation both in simulations and in experiments the stress–strain behavior rapidly evolves from a nearly elliptic cycle (simulations: black line, Figure 2; experiments: bold line, lower inset in Figure 2) toward a slowly evolving state, which is characterized by a decrease of the maximum stress and an enhanced dissipative process. The latter is indicated by an increasing area of the hysteresis loop.

Figure 2. Stress–strain behavior in the Cyclic-yield (γ = 13%) state of the 16-chains atactic-polystyrene film during 1^st (black line), 5^th (red line), and 10^th (blue line) cyclic-shear deformation, performed at computationally low, , shear rate, temperature 300 K, and pressure 52 MPa. Upper inset: Shear-stress response as a function of time during 10 cycles in the Cyclic-yield state. Lower inset: Experimental results:16 the stress–strain behavior in the cyclic-yield state (γ = 6.4%) of an acrylate film during cyclic plastic deformation at shear rate . Bold line: 2^nd cycle; thin line: 60^th cycle.

In order to calculate storage and loss moduli, the time-dependent shear strain is imposed to a top substrate in both Reference and Recovery states; here γ₀ = γ is the strain amplitude (we henceforth drop the subscript 0). Simultaneously, we measure the time-dependent shear-stress response σ(t) during the cyclic-shear deformation, and plot it as a function of elapsed time, see upper inset in Figure 2. This shear-stress signal can be fitted by the equation ;31 here , where T is the period of one complete cycle, and G′ and G″ are fit coefficients. We should note here that this fit will adequately work only when the film is sheared in the linear regime, i.e., at low shear-strain amplitudes in the Reference state.

When the glassy polymer is subjected to the large strain amplitudes (i.e., Cyclic-yield), the overall shape of the shear-stress signal is periodic. However, at a closer look, see upper inset in Figure 2, the value of the stress signal at maximum of deformation is not constant and drifts. Additionally, there is no symmetry before and after the peak of deformation. Therefore, we may say that the stress signal consists of many frequencies, or harmonics, and the former can be analyzed in terms of the Fourier series.

The drift of the value of the stress signal at the maximum of deformation might be due to some kind of a Boltzmann “memory” effect, i.e., the Boltzmann superposition principle, which expresses the dependence of the mechanical response of linear viscoelastic material to the loading history. We also observed that this drift occurs when the strain amplitude is decreased from 13 to 2% after the Cyclic-yield regime. In order to take into account this effect, the stress signal σ(t) has been fitted by the empirical expression

• ((5))

where coefficient a defines the stress amplitude at t = 0 and b is the drift parameter: after the second and the third stage the system is not fully relaxed back, and by the end of the third (Cyclic-yield) stage this adds up to the stress a. The relaxation back to the original state takes place during the recovery stage, and accounts for the term bt.

In Figure 3a, the simulated storage G′ and dissipative G″ components of the complex shear modulus are presented as a function of the total number of shear cycles at contact pressure 52 MPa. In the Reference state, the storage and loss moduli are almost constant and do not change with number of cycles. The average values are G′ = 500 ± 11 MPa (upper dashed line, Figure 3a) and G″ = 28 ± 12 MPa (lower dashed line, Figure 3a). The average values of the storage and loss moduli in the Reference state are taken as the reference points, i.e., as the initial values of the moduli before the Cyclic-yield state.

Figure 3. (a) Storage G′ (filled symbols) and loss G″ (open symbols) components of the complex shear modulus as functions of the total number of cycles, simulated at temperature 300 K, contact pressure 52 MPa, and shear rate. The average values of the storage and loss moduli in the Reference state are G′ = 500 ± 11 MPa, G″ = 28 ± 12 MPa, as denoted by the dashed lines. (b) Experimental results for acrylate:16 Storage G′ (filled symbols) and loss G″ (opened symbols) components of the complex shear modulus as functions of total time of the experiment at shear rate. Dashed lines denote the initial values of the storage and loss components of the viscoelastic shear modulus before the Cyclic-yield regime, and correspond to G′ = 1 066 ± 9 MPa and G″ = 67 ± 3 MPa.

In the Cyclic-yield state, a strong drop in apparent G′ and a corresponding increase in apparent G″ are observed which reflect changes in the segmental glassy dynamics. Simulation results show that most of these changes occur within about 3 cycles, Figure 3a. Then, as the number of cycles in the yield regime increases, a rather small drift of G′ and G″ toward a steady state takes place.

In the Recovery state, immediately after interruption of the plastic deformation, the simulated storage modulus initially is decreased and the loss modulus is increased with respect to the Reference state, Figure 3a. Then, a recovery of the storage modulus back to the reference value is observed. No clearly visible recovery process is observed for the loss modulus. Despite fluctuations the value of the loss modulus rather weakly changes with the number of cycles, and the average value is G″ = 62 ± 10 MPa, which is about two times larger than the reference value, meaning that in the plastically sheared film the energy-dissipation mechanisms are enhanced.

The simulation results are in a qualitative agreement with the experimental results performed by Chateauminois and co-workers,16 which are also presented in Figure 3b. In the Reference state the experimental storage and loss components of the viscoelastic modulus (acrylate) are G′ = 1 066 ± 9 MPa and G″ = 67 ± 3 MPa. They are marked by the dashed lines in Figure 3b. The large value of the storage modulus (G′ = 1 066 ± 9 MPa) as compared to the simulated value (G′ = 500 ± 11 MPa) might be explained, first of all, by the difference in the polymer samples (polystyrene and acrylate), and secondly, by the difference in the film thickness (50 nm and 50 µm) and applied frequencies (10⁸ Hz and 1 Hz). Christopher et al.34 experimentally found that the elastic modulus decreases drastically as compared to the bulk value when the film thickness decreases. The large value of the loss modulus (G″ = 67 ± 3 MPa) as compared to the simulated value (G″ = 28 ± 12 MPa) can be also explained by the difference in the shear rates used in simulation and experiment: from the viscoelasticity theory it is seen that the loss modulus at higher frequency is smaller as compared to the lower-frequency values.31

As was mentioned above, in the Cyclic-yield state the storage and dissipative moduli represent the components of the apparent shear modulus. A strong drop in the apparent storage modulus and a corresponding increase in the apparent loss modulus is observed in experiments,16 which is similar to the simulation results. Most changes in the glassy dynamics happen within about 50 cycles (= 50 s), Figure 3b, and the slow drift of both moduli toward a steady state takes place.

In the experimental Recovery state, immediately after the cyclic yield, the storage modulus is decreased by about 15% and the loss modulus is increased by about 80% as compared to the initial (i.e., before yield) viscoelastic moduli, Figure 3b. Then, a progressive recovery of these linear viscoelastic properties is observed as a function of time. We should note here that the recovery process of the loss modulus is quite fast, while the storage modulus is not recovered neither after 240 s nor after additional 2000s of experiment. However, in experiment the initial viscoelastic properties were fully recovered when the film was heated above T_g and cooled down again. In the simulations, we have a somewhat opposite effect: the storage modulus almost recovers back, while the mean value of the loss modulus is twice larger than the reference value. The possible reason for this discrepancy could be, first of all, the computationally high shear rate () as compared to the low experimental value (). Also, from the point of view of simulations, the 50 µm-thin film used in experiment should be considered as a bulk as compared to the thin films on the nanoscale.

The large changes in the viscoelastic moduli upon application of the plastic deformation might be attributed to the mechanical rejuvenation, in sense that it reflects the enhanced mobility of plastically deformed polymer glass. A recovery process might be considered as an evidence of enhanced physical-aging rate of the mechanically rejuvenated glass.

3.2. Energy Partitioning in the Cyclic-Yield Regime

Figure 4 shows the evolution of the total internal energy (squares) during the 1^st and the 10^th cycle in the yield regime, at low simulated shear rate , temperature 300 K and contact pressure 52 MPa. As can be seen from this figure, upon the oscillatory shear deformation in the yield regime, the total internal energy regularly oscillates. These oscillations are connected to the corresponding oscillations of the shear-stress signal.

Figure 4. Various contributions to the total internal energy (squares) during the 1^st and the 10^th cycle of the deformation in the yield state, at low simulated shear rate , temperature 300 K and contact pressure 52 MPa. The contributions are given by excluded-volume (circles), stretching (crosses), bending (triangles), and torsion (diamonds) interactions.

The maximal change of the internal energy is observed at the maximal or minimal value of the shear-stress signal. Minima in the internal energy correspond to the moments when the shear strain is zero, but the film is at finite stress. It is clearly seen that after each subsequent half of the cycle, the film is brought to a higher-energy state as compared to the previous cycle. These results are in agreement with the simulation results for binary Lennard–Jones mixtures13, 14 and with simulations of bulk glassy polystyrene and polycarbonate.15 In all these previous studies, it was shown that due to the application of the plastic deformation the samples were brought into a new energy state with a higher energy as compared to the initial, non-deformed sample.

The time evolution of different contributions to the total internal energy (excluded-volume interactions, stretching of covalent bonds, bending, and dihedral angle torsions) during the cyclic yield is also shown in Figure 4. The overall change in the internal energy is mainly governed by the change in the non-bonded, excluded-volume interactions. Stretching, bending, and torsion contribute rather weakly to the total internal energy.

4. Layer-Resolved Segmental Dynamics

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.

Figure 5. Mean squared displacement in three layers (bottom (black lines), middle (blue lines), and top (red lines)) of the capped film for the non-deformed film at 300 K (solid lines) and 370 K (dotted lines) at normal pressure 52 MPa, Stage 1, Figure 1. The two horizontal dashed lines denote the squared half-width of the layer thickness (h/2)², and the squared radius (σ_CH/2)² of the united atom chosen to calculate the MSD. The numbers 2 and 0.5 show the theoretical slopes in the ballistic and sub-diffusive regimes, respectively.

Figure 6. MSD in the top (a) middle (b) and bottom (c) layers for the Non-deformed state (solid lines), and for the Sheared state (dotted lines), capped films. In all figures black and red lines correspond to simulated systems at two temperatures, 300 and 370 K, respectively.

Figure 7. Ratio of the averaged velocity of the polymer segments in different layers of simulated film to the actual velocity of the top substrate at normal pressure 52 MPa, temperature 300 K, and for high and low shear rates. The dashed line denotes the velocity profile in the case of the ideal affine shear deformation. The vertical arrows indicate the region of the shear-banding effect.

Figure 8. The orientational correlation function P₂(t) of the side vectors in three layers (bottom (black lines), middle (blue lines), and top (red lines)) of the capped film for the non-deformed capped film at 300 K (solid lines) and 370 K (dotted lines) and at normal pressure 52 MPa. Inset: the aPS monomer with the naming convention of the united atoms, shown together with the vector that is chosen to measure the order parameter P₂(t). Here o = ortho, m = meta, p = para.

Figure 9. The orientational correlation function P₂(t) of the side-group vectors in the top (a) middle (b) and bottom (c) layers for the Non-deformed (solid lines) and for the cyclicly Sheared (dotted lines) capped films. In all figures black and red lines correspond to simulated systems at two temperatures, 300 and 370 K, respectively.

4.1. Segmental Translational Dynamics

The most common approach to measure local segmental dynamics in any simulated system is to calculate the mean squared displacement (MSD) of chosen particles,

• ((6))

here is the radius vector of a given particle at time t₀, and is the distance traveled by a particle over the time interval t. Brackets denote an average over particles and over time t₀.

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 t_blst = 0.09 ps and t_blst = 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 (σ_CH = 0.4153 nm) on a half of diameter σ_CH/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 3^rd top layer and the non-sheared 4^th 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

In experiments polymer segmental relaxation is also studied with the help of different techniques such as NMR,39 dynamic light scattering,40 dielectric spectroscopy.41 In the MD simulations, the most common approach to study the polymer segmental relaxations is to follow the time-evolution of a specific vector that represents a chemical bond. For the polystyrene monomer various different vectors might be defined, see Figure 8. For the calculation of the segmental orientational dynamics we chose the side-group vector , that connects two united atoms, CHC. The orientation of this vector is measured by calculating the second-order Legendre polynomial function P₂(t)

• ((7))

where the position of the unit bond vector is measured at time t₀ and at time t + t₀. The brackets denote both a time averaging and the averaging over all vectors in the simulated system.

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 T_g value for the whole sample. The free interface behaves as a melt (i.e., it has a lower T_g) 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

• ((8))

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. [3], use “waiting time,” t_w. 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 t_w 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 t_w, 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 t_w, 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 t_w. 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 t_w 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.

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.

5. Conclusion

In the present study MD simulations have been performed in order to investigate the influence of the cyclic-shear deformation on the mechanical properties of atactic-polystyrene capped films both in the viscoelastic linear- and cyclic-yield regimes, and to study the local segmental dynamics in these films, simulated at two temperatures, and two external normal pressures.

The simulated results have been compared with the results obtained experimentally in ref. [16]. The experimental research has been focused on the nonlinear shear properties of glassy acrylate systems up to the yield point. Despite the difference in polymers and the time-scales covered by the simulations and experiments, we found that, to some extent, the mechanical properties of simulated thin films change in qualitative agreement with experiments.

The experimental results show that upon cyclic-shear deformation in the yield regime, the shear behavior of the acrylate glass evolves slowly toward a steady state, which is characterized by a decrease of the maximum stress and an enhanced dissipative process. In experiments this steady state is reached within 50 cycles, where the storage and loss components of the strain- and time-dependent apparent shear modulus rapidly decreases and rises, respectively. Similar results have been obtained in the present simulations. A strong drop of the simulated storage modulus and corresponding increase of the simulated loss modulus is observed, which reflect changes in the segmental glassy dynamics within about three cycles. Immediately after the application of the shear deformation in the yield regime, both experiments and simulations show that the storage modulus is decreased and the loss modulus is increased as compared to their initial value. Such changes in the viscoelastic moduli upon application of the plastic deformation might be attributed to the mechanical rejuvenation, in the sense that it reflects the enhanced mobility of plastically deformed polymer glass. After the cyclic-yield regime the recovery regime starts. This recovery process might be considered as evidence of enhanced physical-aging rate of the mechanically rejuvenated glass.

The evidence of the mechanical rejuvenation can be also observed by calculating the internal energy during the cyclic-shear simulation. Previous simulations of the glasses show that the plastic deformation brings a system to a higher energy state. A similar effect has also been found in the present study: upon cyclic-shear deformation, the total internal energy regularly rises and drops, and after each subsequent shear cycle, when the deformation is zero, the film is brought to a new state with a higher energy as compared to the previous cycle.

The local dynamics of polymer monomers has been studied by calculating the segmental translational and orientational dynamics for all simulated films, and also in the top, middle, and bottom layers of these films. We found that the segmental mobility of the monomers is different in different layers. Due to the strong polymer–substrate interactions α-relaxation at the top and bottom layers is much slower than in the middle layer. An increase of temperature causes a slight increase of the cage size, and a significant increase in the rate of α- and β-relaxation in the middle layers. The application of the cyclic-shear deformation causes significant acceleration of the α- and β-processes only in the middle layers, and can be interpreted as rejuvenation. In contrast to this, the top and bottom layers are not rejuvenated; instead, they are aged further. The effect of the rejuvenation in the middle layers is temporal and is erased by aging after a simulation time of 1 ns. After this time, the difference between the segmental dynamics of the sheared and non-sheared middle layers is rather small. We might conclude that the top and bottom layers compete with the middle layers and, on average, the polymer film is aged with time, as is manifested in the power-law increase of the relaxation time (averaged over the whole film) with increase of the waiting time. We should note here that our definition of waiting time is different from that in experiments, which already leads to a difference between aging exponents.

Acknowledgements

This study is a part of the research program of the Dutch Polymer Institute, project # 654. The work was also sponsored by the Stichting Nationale Computerfaciliteiten (National Computing Facilities Foundation, NCF) for the use of supercomputer facilities, with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scientific Research, NWO). We thank Dr. Antoine Chateauminois for helpful discussions and experimental data provided for this paper.