Molecular simulation of structural relaxation in ultrathin polymer films.

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Physical Chemistry Chemical Physics

View Article Online Molecular Simulation of Structural Relaxation in Ultrathin DOI: 10.1039/C3CP53555J

Qiyun Tang∗ and Wenbing Hu Department of Polymer Science and Engineering, State Key Laboratory of Coordination Chemistry, School of Chemistry and Chemical Engineering, Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

Nanjing University, 210093 Nanjing, China

Abstract Structural relaxation and physical aging in glassy polymer films have attracted much attention in the past two decades due to its strong correlation with the lifetime of polymer-based nano-devices. Currently, the observed physical aging in polymer films was explained on the basis of free volume diffusion model (FVDM) that has not yet been validated by simulations at the molecular level. Here we performed a Monte Carlo simulation by introducing the vacancy diffusion mechanism (similar to FVDM) to investigate the structural relaxation in ultrathin polymer films. The results show that the local average density of segments increases linearly with logarithm of time, similar to the fluorescence intensity and dielectric strength measurements in experiments. The responses of relaxation rates in ultrathin films to temperatures are consistent with that for glassy polymer thin films in experiments. The emergence of a peak of relaxation rates with decreasing temperatures can be attributed to the competition of two mechanisms (segment mobility and accurate initial vacancy concentration) from the molecular levels. Our results demonstrate that the simulations with vacancy diffusion model could indeed provide new insights from the molecular levels to understand the structural relaxation and physical aging in ultrathin polymer films.

∗

Author to whom correspondence should be addressed. E-mail: [email protected].

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Physical Chemistry Chemical Physics Accepted Manuscript

Polymer Films


I.

INTRODUCTION

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DOI: 10.1039/C3CP53555J

When polymers are quenched from high to low temperatures, T , these polymers will ation process [1–4]. If T is larger than the glass transition temperature, Tg , this relaxation behavior will be quickly damped in a short time (based on the WLF empirical relation [5], the relaxation time ranges from microseconds to hundreds of seconds by lowering the Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

temperature towards Tg ), which is usually hard to be detected by experiments [6]. While T is below Tg , the relaxation process will last to macroscopic time scales, such as hours or days (see Fig. 1) [3]. In this case, the relaxation process can be easily monitored by experiments with different methods, such as gas permeation [7–15], ellipsometry [16–20], and fluorescence measurements [21–27]. This long-term relaxation process will result in volume contraction and densification of polymers, leading to significant changes of physical properties in macroscopic time scales, such as gas permeability and mechanical, optical, and electrical properties, and is usually termed as physical aging [3, 4]. In recent years, much attention has been focused on the structural relaxation and physical aging in polymer free-standing and supported thin films due to its strong connection with the lifetime of polymer-based nano-devices. The study of physical aging in free-standing films could date back to early 1990’s when experiments demonstrated an accelerated physical aging behavior in gas separation polymer films with thickness reduced to hundreds of nanometers [7–15]. The factors such as collapse of underlying porous scaffold[8] and solvent loss[9] were soon excluded, meaning that structural relaxation of confined nonequilibrium glassy thin films played a key role in this accelerated physical aging behavior. For example, Huang et al. monitored the gas permeability of polymer thin films with thickness ranging from 62um to 400nm[13]. With proceeding time at 35◦ C, the films showed a pronounced steeper decrease in gas permeability for thinner films, indicating a clearly accelerated physical aging behavior when decreasing film thickness. The methods to determine the aging behavior employed in these experiments were the gas permeation measurements, which were demonstrated to be valid for high free volume polymers, such as polynorbornene[10], polyimide[9, 13, 15], polysulfone (PSF)[9, 13, 14], polyphenylene oxide (PPO)[13, 15], poly-1-trimethylsilyl-1-propyne (PTMSP)[11], and bisphenol-A benzophenone-dicarboxylic acid (BPA-BnzDCA)[12]. However, for low free volume polymers 2


relax from a non-equilibrium quenching state towards equilibrium with a structural relax-

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FIG. 1: Illustration of structural relaxation process.

such as poly(methyl methacrylate) (PMMA)[17], poly(styrene)(PS)[25], polyetherimide[28], and poly(isobutyl methacrylate)(PiBMA)[29], the gas permeation measurements showed little physical aging behaviors. Baker et al. [17] used oxygen permeation methods to study the aging behavior of free-standing PMMA films with thickness ranging from 800um to 190nm but no change of permeability was observed. Possible reason for these conflicts could be attributed to the low sensitivity of gas permeation methods to the small variation of free volume in low free volume polymers[17]. Due to these limitations, other methods such as ellipsometry[16–20] and fluorescence[21–27] measurements were employed to monitor the aging behaviors for these low free volume polymers. These procedures usually need substrates to support the sample to performing the measurements, leading to enormous investigations of structural relaxations and physical aging behaviors in supported polymer thin films. For example, Priestley et al. [25, 26] employed the fluorescence methods to characterize the physical aging at different regions in supported PMMA thin films. Due to the structural relaxation of chain molecules, labeled polymers was suppressed and densified to yield an increase in fluorescence intensity, which was applied to characterize the local physical aging in PMMA films. Results showed a smooth distribution of aging rate across the films and the strong influence of free surface and substrates on the local aging rate in films. Recent measurements performed by Napolitano et al. [6] with the dielectric spectroscopy show that the dielectric strength, related to the mean square dipole moment and also the density of films, shows a linear decrease with logarithm of time, indicating a linear increase of film density. These diversity relaxation behaviors emerged in free-standing and supported poly3


Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

DOI: 10.1039/C3CP53555J


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Article Online mer thin films require the understanding of structural relaxations and physical agingView from a DOI: 10.1039/C3CP53555J

molecular level, which needs theories and simulations to capture the main factors (such as so on) that influence these relaxation processes. In polymer thin films, free volume diffusion model (FVDM) has been widely employed in experiments to explain the observed physical aging phenomena. In 1943, Alfrey et al. proposed that the isothermal aging below Tg can be attributed to the diffusion of free volume Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

holes from interior of polymers into the surface[30]. This model was developed by Curro et al. [31] to quantitatively analyze the volume relaxation experiments of Kovacs et al. for poly(vinyl acetate)[32, 33]. The motion of free volume holes can be described by a diffusion equation:

where D = Dr exp[−B(f −1

∂f = ∇ · (D∇f ) (1) ∂t − fr−1 )] is the diffusion coefficient for free volume holes, in

which Dr and fr represent the diffusion constant and free volume fraction at the reference temperature respectively, and B is a material constant. This model was shown to semiquantitatively match the major phenomena observed for the aging of poly(vinyl acetate)[32]. In 2000, McCaig et al. extended this model by employing the lattice contraction mechanism to explain the observed gas permeation experiments of polymer thin films ranging from 0.25um − 33um[12, 34]. In this new dual model, f = fi − 4fLC − 4fD , where fi represented ∗ was the amount of vacancy lost due to the initial fractional free volume, 4fLC = fi − fLC

lattice contraction and 4fD = fi − fD represented the vacancy lost by diffusion. Results showed that for thicker films (l > 2.5um), aging was primarily due to the lattice contraction while for thinner films (l < 2.5um) aging was determined by free volume diffusion. Cangialosi et al. studied the physical aging of polycarbonate (PC) by monitoring the time evolution of permittivity and the free volume by positron annihilation lifetime spectroscopy (PALS)[35]. They considered free volume diffusion as the main mechanism and postulated that the films contained low-density regions which provided an internal surface area at which free volume holes could diffuse to and annihilate. The combination of internal annihilation mechanism into free volume diffusion model was shown to successfully model the observed aging behaviors. Thornton et al. [36] proposed a newly formulated diffusion model which covered time-dependent length scale and could predict the aging behavior in polymer thin films. In this new model, lattice contraction was attributed to a direct result of free volume 4


the motions of segments, confinements, interaction between molecules and substrates, and

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al. [37] has also employed a similar free volume holes diffusion model, with diffusion in films aluminum layers. Based on these conclusions [12, 34–36], we can speculate that for ultrathin polymer films (< 100nm), the lattice contraction and internal annihilation would not be the main factors while the free volume diffusion should dominate the physical aging behaviors. In this model, free volume holes diffuse from interior of films into the external surface where Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

holes annihilate, resulting in the volume contraction and physical aging in polymer ultrathin films. However, so far as we know, there is no simulation work to directly demonstrate the validity of FVDM from the molecular level for understanding the structural relaxation and physical aging of ultrathin polymer films. In the past two decades, some simulation progresses of structural relaxation and physical aging behaviors have been achieved in bulk polymer systems[38–43, 47–49]. For example, Andrejew et al. used the Monte Carlo simulation to investigate the physical aging of glassy polymers during the cooling process[38]. The bond-fluctuation model on a simple lattice with an energy associated with long bonds was applied to study the glassy state. This bond-fluctuation model has also been employed to describe the physical aging in polymeric materials recently[39]. Molecular dynamics (MD) simulation was used to study the effect of physical aging on the mechanical properties of a model polymer glass [40, 41], the impact of nanoparticles on physical aging of polymer nanocomposite [42], and the isothermal volume relaxation behavior of atactic polystyrene [43]. Some interesting MD simulations concerning the stringlike cooperative motions of supercooled liquid [44] have been employed to study the relaxation and physical aging behavior of coarse-grained polymer glass [45, 46]. Other simulations based on constitutive equations [47], commercial software such as Materials Studio [48], and generalized fractional Maxwell model [49] were also employed to simulate the physical aging of polymers. Most of these simulations have some interesting results for the physical aging behaviors in bulk polymers. By far, there is little simulation work in particular based on the FVDM to investigate the structural relaxation and physical aging behaviors in ultrathin polymer films. Here we use the dynamic Monte Carlo simulation by employing the vacancy diffusion mechanism (similar to FVDM) to study the structural relaxation in ultrathin polymer films. Recent experiments have demonstrated that the substrates will dramatically influence the 5


and annihilation at interfaces, to explain the Tg reductions in polystyrene thin films between


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View Article Online relaxation behaviors in polymer thin films [26]. In our simulation, we choose the freeDOI: 10.1039/C3CP53555J

standing polymer thin films, trying to eliminate the substrate effect. The paper is organized the inclusion of computational details based on the dynamic Monte Carlo simulation and we then add the vacancy diffusion process within the films and annihilation process at the surface layer. In section III, we discuss the relaxation behaviors of ultrathin polymer films and investigate the variation of relaxation rate as a function of start vacancy concentrations Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

and temperatures. In addition, we will compare our results with previous physical aging behaviors in experiments, and from the molecular level we explain the emergence of a peak of relaxation rate when decreasing temperatures. Finally, the conclusion with an outlook of presented procedure for further investigation is provided in section IV.

II.

MODEL AND SIMULATION DETAILS A.

Model of Polymer Free-Standing Ultrathin Films

Here we use the dynamic Monte Carlo simulation method to generate the polymer freestanding ultrathin films [50]. The lattice model is applied to simulate the coexisting situations of polymer chains and vacancies. We consider n polymer chains with each polymer consecutively occupying N lattice sites (N is the polymer chain length). The vacancies occupy Nv lattice sites, resulting in the volume of the film: V = nN + Nv . Our simulation is performed on a cuboid lattice with its size of 80 × 80 × (H + 20), where H is the pure thickness (the thickness where all the segments in polymer films are close-packed on the lattice) of the ultrathin films. In our simulation, the bond between nearest sites along the chain molecules could be along the lattice grid or along the diagonal lines, which could enhance the conformation sampling in Monte Carlo simulations [50]. The motion of polymer chains is generated through a microrelaxation model [50, 51] which allows each segment to change position with it’s neighboring vacancies or air sites, accompanied by the sliding motions between two “kink” segments along the counter of chain molecules if necessary. This procedure is similar to the theoretical consideration proposed by de Gennes by combining the jumping and sliding motions together to explain the variation of glass transition temperature in free-standing polymer films [52, 53]. Thus we believe the microrelaxation model with the

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as follows. In section II, we first set up a model of free-standing ultrathin polymer film with

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View Article Online inclusion of single-body jump and sliding motions would provide some resemblances to the DOI: 10.1039/C3CP53555J

molecular motions in real glasses. with the potential energy change


∆E aEss + bBsv + cEc = kB T kB T Ess Bsv Ec = (a +b + c) Ec Ec kB T

(2)

where Ess represents the non-bonded nearest-neighbor segment-segment interaction and Bsv is the mixing energy between segments − vacancies. The factors a and b are the total number of corresponding contact pairs. Ec is the bending energy for two adjacent bonds connected along the chain and c is the net number of non-collinear-connected bond pairs [54]. The reduced temperature T 0 = kB T /Ec is introduced in our simulation, where kB is the Boltzmann constant and T is the temperature. We use T instead of T 0 to represent this reduced temperature in the following. The unit of time evolution is defined as one Monte Carlo step (MCS), including the number of trial moves equal to the number of segments in the sample system. In our simulation, the vacancy sites represent the “excess” free volume generated by the polymer films quenched from high to lower temperatures (The physical meaning of the vacancy sites will be discussed in the following). Thus the vacancies can be occupied by segments without changing potential energy. Due to these reasons, we set Bsv /Ec = 0. To obtain the free-standing ultrathin films, we apply periodic conditions in the X and Y directions and set the non-bonded segment-segment interaction Ess /Ec = −1.0. This cohesive segment-segment interaction will force the system to form two free surfaces along Z direction, similar to previous Monte Carlo simulations of polymer films with free surfaces in vacuum [55, 56]. In our simulation box, apart from the polymer chains and vacancies, there are still a lot of lattice sites, which represent the air molecules in reality. To avoid the instability of ultrathin polymer films in vacuum, we introduce a mixing energy d(Bsa /Ec ) between polymer segments and air sites (no interaction between air sites and vacancies) in eqn. (2) with Bsa /Ec = 1.0 and d is the number of corresponding contact pairs. This repulsive interaction will separate the segments and air sites from each other and generate two phase separated free surfaces (shown in Fig. 3(a)) to stabilize the ultrathin polymer films, which is similar to previous reported free surfaces in polymer solutions [57]. 7


Conventional Metropolis sampling algorithm was employed in each microrelaxation step


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FIG. 2: (a) Illustration of the free-standing ultrathin polymer films. (b) Illustration of the vacancy diffusion process within the films and annihilation process at the surface layer. B.

Model of Vacancy Diffusion and Annihilation in Ultrathin Films

Here we apply the vacancy diffusion and annihilation mechanism into Monte Carlo simulation to investigate the structural relaxation (volume contraction) of ultrathin polymer films, trying to uncover the main factors that determine the relaxation rate (the key parameter that is concerned by experiments and industries due to it’s strong connection with the lifetime of polymer-based devices) of ultrathin polymer films. Figure 2(b) provides the illustration of vacancy diffusion and annihilation process in ultrathin polymer films. In the interior region of ultrathin films, the diffusion of vacancies will be obtained by the motions of chain segments, mainly the interchange of positions between vacancies and segments. For example, the trial move of segment C in Fig. 2(b) into vacancy B will be determined by the Metropolis sampling. Successful move of segment C will interchange the positions of segment C and vacancy B, resulting in the diffusion of vacancy B from it’s original position to new place. This process will diffuse the vacancies throughout the whole films. When the vacancies approach the surface region, the situation will be changed. In Fig. 2(b), vacancy A at the surface layer will change it’s position with segment located at the top position of film. This interchange process will diffuse the vacancy A into the air region. Once the vacancy approaches into the air region, this vacancy site will be occupied by the air molecules immediately due to the extremely high mobility of air molecules, resulting in the annihilation of vacancies in polymer ultrathin films[12, 30, 31, 34]. 8



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FIG. 3: (Color Online) Average density of segments and vacancies along Z axis (a) and the snapshot (b) of equilibrated ultrathin polymer film with chain length N = 400 at temperature T = 7.6. Green color in (b) shows the cross-section along Z axis in ultrathin film and yellow color shows the bonds locating at interfaces. III. A.

RESULTS AND DISCUSSION Preparation of Quenched Free-Standing Ultrathin Polymer Films

When polymer films are quenched from high to lower temperatures, they will generate some “excess” free volumes, which represent the deviations of quenched systems from equilibrium state (the lower the temperature, the more the “excess” free volumes), within the polymer films [1–3, 58]. To investigate the structural relaxation behaviors of quenched polymer films by simulations, the first step is to prepare a quenched polymer film. However, in our lattice model, it is impossible to directly generate the “excess” free volume sites by quenching the films from high to lower temperatures due to the fact that all segments are set 9



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View Article Online on lattices and the system is incompressible [59]. In this case, we choose an alternative way DOI: 10.1039/C3CP53555J

to prepare the quenched polymer films. We provide some vacancy sites within the polymer film. When we quench the system to lower temperatures (such as T = 5.0), this equilibrated free-standing polymer film with some vacancy sites could be considered as the quenched film containing some “excess” free volumes [60]. In this case, these vacancy sites represent the “excess” free volumes generated by the polymer thin films quenched from high to lower Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

temperatures in reality. Figure 3 shows the typical average density and morphology of equilibrated polymer ultrathin film including some vacancy sites at temperature T = 7.6. The system includes n = 160 polymer chains with each chain length of N = 400 and Nv = 3368 vacancy sites. As can be seen in Fig. 3(a), the average density of segments reaches a plateau between z = 13 and z = 19 and gradually drops down to 0 when z is smaller than 13 or larger than 19. The enhanced distribution of vacancies within these interfaces can also be observed, which is similar to the situation in reality that polymer chains near free surface take high “excess” free volumes. Figure 3(b) shows the corresponding morphology of polymer ultrathin film, in which we can find a clearly flat free-standing thin film located in the Z axis. In the following section, we will systematically investigate the structural relaxation process of this quenched free-standing ultrathin film by applying the vacancy diffusion mechanism into the Monte Carlo simulation and discuss in detail the temperature-dependence of the relaxation rate in ultrathin polymer films.

B.

Structural Relaxation in Ultrathin Polymer Films

In this section, we will study in detail the time-evolution of reduced volume of quenched free standing ultrathin polymer films. The system includes n = 160 polymer chains with each chain length of N = 400 (Other chain lengths of N = 100 and 800 were also considered and similar relaxation behaviors were observed). First we relax the film into equilibrium at T = 7.6 with vacancy concentration φv = Nv /(nN + Nv ) = 0.05. Then we quench the system into T = 5.0 and set the configuration of film at this time as the initial state (V (0) = nN + Nv (0)). As time proceeds, due to the motion of segments, the vacancies will gradually annihilate from the two free surfaces of the ultrathin film and result in the decrease 10


films at high temperature (such as T = 7.6) and perform the simulation to equilibrate the

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FIG. 4: (a) Variation of reduced volume V (t)/V (0) of ultrathin polymer film quenched to T = 5.0 as a function of time (in unit of MC steps). The red line shows the best fit of original data during the relaxation region with the slope of −0.035. Time-evolution of average density in the middle and surface regions within ultrathin polymer films are shown in (b) and (c) correspondingly. Inset shows the positions to obtain the average density.

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Article Online of volume for the film: V (t) = nN + Nv (t). Figure 4(a) provides the variation of View reduced DOI: 10.1039/C3CP53555J

volume V (t)/V (0) as a function of time. It is clearly shown that the volume of ultrathin films of film decreases little and V (t)/V (0) shows a stable plateau. Then the film volume drops quickly and decreases linearly with logarithm of time, indicating a linear structural relaxation process within this time region. The red line in Fig. 4(a) gives the best fit of original data with the slope of −0.035, reflecting the relaxation rate during this stage. Finally, vacancy Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

annihilation process becomes so slow that the V (t)/V (0) approaches a stable plateau, where the film could be considered as in the equilibrium state [61]. The three typical stages of time-evolution for the reduced volume of ultrathin film shown in Fig. 4(a) are similar to the physical aging process reported in experiments for glassy bulk polymers [64] and polymer thin films [6, 17, 25, 26]. Previous MD simulations could also generate the reduction of specific volume during the relaxation process [43]. In these off-lattice models, there indeed exists some vacancy diffusion processes implicitly. However, it is not easy to clarify delicately how the vacancies diffuse within films and to detail the relation between vacancy diffusion and chain architectures. In our lattice Monte Carlo simulation, the diffusion and annihilation of vacancies are carried out by the jump motions between segments and vacancies and the sliding motions of “kink” segments along the counter of chain molecules. In this case, the architecture of macromolecules will influence the sliding motions of segments and also the diffusion process of vacancies, and further affect the volume relaxation process shown in Fig. 4(a). From this viewpoint, the incorporation of vacancy diffusion model into lattice Monte Carlo simulations is still meaningful and could be extended to study the anomalous structural relaxations in ultrathin polymer films with complex architectures [19, 20]. During the relaxation process, as vacancies annihilate from the two free surfaces of ultrathin films, the films will gradually become densed. In our simulations, we calculate the average density of segments in the middle region (see the inset in Fig. 4(b)) of ultrathin films. The time-evolution of average segmental density during the relaxation stage is shown in Fig. 4(b), in which we can find that the average density increases linearly with the logarithm of time. This densification behavior of polymer thin films can be monitored in experiments by the fluorescence measurements of mobility-sensitive chromophores-labeled films [25, 26] and the intensity measurements of dielectric strength [6]. For example, Priestley et al. employed fluorescence methods to monitor the structural relaxation of supported PMMA 12


decreases with time (MCSs) and shows three different stages. At the initial stage, the volume

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FIG. 5: (a) Time-evolution of reduced volume V (t)/V (0) of polymer films for distinct φv0 at temperature T = 5.0. (b) V (t)/V (0) and the corresponding β(t) as a function of time at φv0 = 0.05 and T = 5.0. (c) The ratio between β and φv0 for distinct start vacancy concentrations.

films [25, 26]. The fluorescence intensity of TC1(4-tricyanovinyl-[N-(2-hydroxyethyl)-Nethyl]aniline)-labeled films, which was proportional to the average density of labeled areas, increased with logarithm of time during the relaxation process. Recent measurements performed by Napolitano et al. [6] with the dielectric spectroscopy show that the dielectric strength, related to the mean square dipole moment and also the density of films, shows

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simulation result shown in Fig. 4(b) is similar to the structural relaxation behaviors of 4(c) that the average density near the surface region decreases linearly with the logarithm of time. This can be attributed to the fact that as film becomes densed, the surface layers gradually shrink towards the bulk region of thin films, resulting in the reduction of density at fixed surface positions. Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

In above simulation, we start the vacancy diffusion and annihilation process with the vacancy concentration of φv0 = Nv (0)/(nN + Nv (0)) = 0.05. As the vacancy diffusion rate can be connected with the vacancy concentration [31], the φv0 might affect the structural relaxation rate of ultrathin polymer films. Recent experiment has also demonstrated that the physical aging in thin films is highly related to the amount of “extra free volume” created by packing frustration at the interface [58]. Figure 5(a) shows the time-evolution of reduced volume of polymer thin films with distinct φv0 . It is shown that as φv0 increases, the relative magnitude of relaxations during the linear relaxation region becomes larger, indicating a faster relaxation behavior. To clarify the effects of start vacancy concentrations on the structural relaxation process of ultrathin films, we perform simulations with a series of φv0 and calculate the structural relaxation rates of corresponding films. We will first introduce a method to determine the relaxation rate β of polymer ultrathin films based on the simulation data. There are many methods in experiments to calculate β according to the distinct experimental procedures. Detailed comparison of these methods could be found in the literature [17]. Here we will summarize and compare two main procedures and employ one to calculate the relaxation rate in our simulation. The conventional procedure of determining the physical aging rate in glassy polymers is provided by Struik [2]: β=−

1 dV (t) V∞ d log t

(3)

Here, V∞ is the equilibrium volume of polymers when the sample is aged to equilibrium. However, the value of V∞ is hard to achieve in practice and usually obtained by extrapolating the equilibrium liquid line of V (T ) to the aging temperature. The estimated value from this method might be larger than the practice value of the aging temperature [17]. More importantly, when considering the system with distinct aging rates, such as polymer 14


polymer thin films based on these experiments [6, 25, 26]. Interestingly, we can find in Fig.

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uncertainty of V∞ becomes so significant that it would make the direct comparison of the Another method to calculate the physical aging rate is by normalizing the data to the volume V0 instead of V∞ [17, 25, 26]:


β=−

1 dV (t) d(V (t)/V0 ) =− V0 d log t d log t

(4)

where V0 is the volume at the reference time t = 0. The particular choice of the normalization value (V∞ and V0 ) has been demonstrated to produce the β within experimental errors [17]. Here, trying to directly compare our results with recent experiments [17, 25, 26] and also for simplicity, we choose eqn 4 to calculate the structural relaxation rate of polymer ultrathin films and the V0 represents the volume of the film equilibrated at T = 7.6. It is also worth mentioning that the procedures described by eqns 3 and 4 are available only when the entire relaxation process could be clearly monitored, while not suitable for describing the measurements in laboratory time scales for the fast relaxations higher above Tg (fully aged states) or slow relaxations deeply lower than Tg (unrelaxed states). Figure 5(b) gives the relaxation rate at different time t based on the variation of reduced volume of ultrathin film with φv0 = 0.05 at T = 5.0. It is shown that the relaxation rate β is nearly equal to zero at initial stage and then increases towards a peak during the linear variation region of the reduced volume, and finally decreases to zero when the system approaches the equilibrium. During the linear relaxation region (the shaded area in Fig. 5(b)), the mean value of β(t) is nearly identical to the absolute value of the slope of best fit shown in Fig. 4(a). In the following, we employ this mean value of β(t) to represent the relaxation rate of ultrathin polymer films. In Fig. 5(a), it is qualitatively shown that as start vacancy concentration φv0 increases, the relative magnitude of relaxations during the linear relaxation region of reduced volume becomes larger, resulting in the enhanced relaxation rate β. This implies that the ratio between β and φv0 should be an invariant of φv0 . The normalized β/φv0 is plotted as a function of φv0 , which is shown in Fig. 5(c). We can find that β/φv0 takes nearly the same value when φv0 varies, indicating that the start vacancy concentration φv0 should have little influence on the response of relaxation rate to the external environments, such as temperatures, confinements, and so on.

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distinct aging rates meaningless.


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FIG. 6: (a) Structural relaxation rate β of ultrathin films as a function of temperature. The two symbols represent the β with different start vacancy concentrations φv0 = 0.05 and φv0 = 0.025. (b) Variation of β/φv0 as a function of temperature for distinct φv0 . C.

Temperature-Dependence Relaxation Rate of Ultrathin Films

In this section, we will investigate the response of relaxation rate to the temperatures and further study the influence of start vacancy concentration φv0 on the variation behavior of relaxation rates. Figure 6(a) shows the structural relaxation rate β of polymer ultrathin films when the films are quenched to different temperatures T from two distinct φv0 . We can find that with decreasing temperature, the relaxation rate increases first and then declines after passing through a maximum value. These results are consistent with the experimental reported physical aging rates in glassy bulk polymers[64] and thin films[17–20]. For example, for a 29 ± 1nm thick PS film, as temperature decreased from 368K to about 314K, the aging rate increased first and then decreased after passing through a peak at about 350K[18], 16



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View Article Online which were consistent with our simulation results. In experiments, the agingDOI: rates of bulk 10.1039/C3CP53555J

polymer and thin films are in the orders of 10−3 ∼ 10−4 while our simulation results yield model that accelerates the motions of segments compared with that in reality. Despite the presence of difference in the magnitude, the similar response behaviors of relaxation rates to temperatures both in experiments [17–20, 64] and simulations show the effectiveness of vacancy diffusion model in understanding the aging behaviors in glassy polymer films. In Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

addition, we have calculated the relaxation rates of ultrathin films from two start vacancy concentrations: φv0 = 0.05 and φv0 = 0.025. The main difference shown in Fig. 6(a) is that the absolute mean value of β for φv0 = 0.05 is larger than that for φv0 = 0.025. This is due to the fact that relaxation rate is proportional to the φv0 as mentioned above, resulting in the larger β for higher φv0 . However, it is clearly shown that the two initial conditions produce similar variation behaviors of relaxation rate β and generate the maximum relaxation rate at nearly the same temperature T = 5.0. More interestingly, when we normalize the relaxation rates at distinct temperatures with the corresponding initial φv0 , the variation curve of β/φv0 gradually approaches a master curve, which is clearly shown in Fig. 6(b). This demonstrates again that the start vacancy concentrations of ultrathin films have little effects on the response of relaxation rate to external variations of environments, such as temperatures. From above, we have obtained the responses of relaxation rates β of ultrathin films to temperatures from our simulation results, which are consistent with experiments [17–20, 64]. Similar behaviors are also observed when the chain length takes the value of N = 100 and 800. The emergence of a peak in the variation of β as a function of T indicates that there might exist two competing mechanisms that contribute distinctly to the relaxation rate of ultrathin films. In the next, we will give detailed evidences from the molecular levels to explain the emergence of a peak during the variation of relaxation rate as a function of temperatures. First we will provide a quantitative description of the motion of segments in ultrathin films. The relationship between segment mobility and structural relaxation of free-standing polymer films has been investigated recently by dielectric relaxation spectroscopy method [65–67]. The segment mobility in previous MD simulation has been defined as the maximum distance traveled by a segment from it’s initial position during ∆t [68, 69]. Here a similar 17


the relaxation rate with the magnitude of 10−2 . This can be attributed to our coarse-grain


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FIG. 7: (Color Online)(a) Time evolution of instantaneous segment mobility in ultrathin films at distinct temperatures. Inset shows the variation of segment mobility α(t) as a function of the logarithm of time at T = 5.0. (b) Variation of average α(t) as a function of temperature.

procedure can be employed in our Monte Carlo simulation to define the segment mobility. The instantaneous segment mobility (∆t = 1 MC step) in ultrathin films can be defined as: α(t) = 1 −

n X N X

δ[rij (t) − rij (t − ∆t)]/Ntotal

(5)

i=1 j=1

where rij (t) represents the position of segment j in the i-th chain at the t-th MC step within the ultrathin films and Ntotal is the total number of segments. With this definition, we can obtain the time-evolution of segment mobility in ultrathin films quenched to distinct temperatures, which is shown in Fig. 7(a). It is shown that as time proceeding, the segment mobility shows a similar behavior to that of the reduced volume of thin films during the relaxation process (see the inset of Fig. 7(a)). In the beginning of Fig. 7(a), the vacancy concentration at different temperatures takes nearly the same value of 0.05, leading to nearly 18



DOI: 10.1039/C3CP53555J

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the same segment mobilities in ultrathin films. However, when t is larger than 300, View 000Article MCOnline DOI: 10.1039/C3CP53555J steps, the segment mobility approaches a stable value, corresponding to the situation that equilibrium states. In this case, the differences between vacancy concentrations at distinct temperatures are so small that its contributions to segment mobility could be ignored. At this time, the obtained segment mobilities of ultrathin polymer films are mainly determined by the relaxation temperatures. We calculate the average segment mobility of ultrathin films Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

with the time ranging from 400, 000 to 500, 000 MC steps and show the results in Fig. 7(b). It is clearly shown that the average segment mobility α(t) in ultrathin films decreases with lowering the temperatures, which will result in the slowing down of vacancy diffusion and annihilation processes in ultrathin films and also the decrease of corresponding structural relaxation rate. Next we will give detailed description of another mechanism that determines the relaxation rate of ultrathin films quenching to low temperatures. In the above section, we have discussed the influence of start vacancy concentration φv0 on the relaxation rate of ultrathin films. It is shown in Fig. 5(a) that at fixed temperature T = 5.0, the relaxation rate β of ultrathin film increases with φv0 . Here, when we consider the variation of β of ultrathin film at fixed φv0 = 0.05 quenched to different temperatures T , the interesting behaviors of β shown in Fig. 6 draw us back to reconsider the accurate initial vacancy concentration φvi where the relaxation of ultrathin film begins. When the film is quenched from high temperature to T = 5.0 with φv0 = 0.05, the system will first experience an initial plateau, and then starts the relaxation behavior, just as shown in Fig. 4(a). The intersection point between the initial plateau and the relaxation curve gives the precise start point with the accurate initial vacancy concentration φvi , which is shown in Fig. 8(a). Here, the accurate φvi is different from the start vacancy concentration φv0 in the following aspect. Just as shown above, the φv0 is set at the beginning of the simulation and takes the same value of 0.05 (or 0.025, see Fig. 6) when quenching the films to distinct temperatures. This process is somewhat artificial and the φv0 can not provide meaningful physical pictures. However, as time proceeds within the initial plateau (see Fig. 5(b)), the volume reduction caused by the rearrangements of segments and chain conformations in ultrathin polymer film will greatly weaken the artificial factors of φv0 , thus the accurate φvi obtained from the intersection point (see Fig. 8(a)) could provide insightful physical understandings, which will be discussed in 19


the vacancy annihilation process is so slow that the polymer films could be considered as in


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FIG. 8: (a) Illustration of the position of accurate initial vacancy concentration φvi at the timeevolution curve of the reduced volume V (t)/V (0) for ultrathin polymer films. (b) The variation of φvi as a function of temperature.

the next section. Figure 8(b) depicts the accurate initial vacancy concentration φvi when the ultrathin film is quenched to different temperatures. It is clearly shown that the accurate φvi where the ultrathin films start the relaxation processes increases when decreasing the temperature. The main reason for this behavior can be attributed to the fact that as temperature decreases, the reduction of reduced volume within the initial plateau will become smaller, resulting in the increase of accurate φvi (see Fig. 8(a)). This phenomenon could be compared with the qualitative description of the deviations of quenched polymer films by the temperature difference in experiments [6, 17–20], which will be discussed in detail in the next section. Just as mentioned above, increasing the initial vacancy concentration φvi will enhance the relaxation rate of ultrathin films[70], indicating that quenched the ultrathin films to lower temperatures will increase its relaxation rates correspondingly. 20



DOI: 10.1039/C3CP53555J

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View Article Online Based on the two competing mechanisms mentioned above, we can provide aDOI: more detailed 10.1039/C3CP53555J

picture from the molecular level to explain the emergence of a peak of relaxation rate (see the average segment mobility α(t) in ultrathin polymer films will decrease correspondingly. This will slow down the diffusion and annihilation processes of vacancies within ultrathin films, resulting in the decline of relaxation rates at corresponding temperatures. At the same time, quenching the polymer films to lower temperatures will enlarge the accurate Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

initial vacancy concentration φvi where the relaxation process begins, leading to an increase of structural relaxation rate correspondingly. These two mechanisms compete when the films are quenched to lower temperatures T , resulting in the emergence of a peak during the variation of relaxation rate as a function of temperature.

D.

Discussion

In above, we have shown that due to the vacancy diffusion and annihilation processes caused by the motion of segments, the reduced volume of quenched ultrathin polymer films shows some typical structural relaxation behaviors, such as the linear increase of average density with the logarithm of time and the peak emergence of relaxation rate when decreasing temperatures. These behaviors are consistent with some experimental results of physical aging behaviors in glassy polymer thin films [6, 17–20, 25, 26]. However, when we put our simulation results to compare with that in experiments, we should be very careful because there exists some general differences between simulations and experiments. For example, the timescales in simulations are usually within microseconds while that in experiments often last to hours or days [3]. Here we will discuss in detail these differences, trying to rationalize our simulation results in understanding the real relaxation and aging behaviors in experiments. Nearly all the experimental measurements on the structural relaxation and physical aging behaviors of polymer films were performed at temperatures below Tg [17–20]. In this case, the relaxation and aging behaviors could be lasted to macroscopic time scales, such as hours, days, or even weeks [3]. This long-term relaxation process below Tg would not be directly investigated by molecular simulations due to the fact that the time-scales in molecular simulations can only approach microseconds (for coarse-grained models). However, as 21


Fig. 6) when the film is quenched to lower temperatures. When decreasing the temperatures,


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View Article Online is well known, when polymers are cooled down from melt-like states to glassy states, its DOI: 10.1039/C3CP53555J

responses to environmental stimuli show similar behaviors and obey the time-temperature long-time aging behaviors below Tg should probably be similar to the relaxation behaviors at temperatures higher above Tg , where the timescales is short enough to be captured by simulations. This assumption has not yet been validated by experiments due to the fact that there is little experiments concerning on the structural relaxations at temperatures higher Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

than Tg (Probably due to the fact that this short-time relaxations is hard to be monitored in experiments and might also have little practical applications). In our simulation, the reduced temperature, for example T 0 = kB T /Ec = 5.0, can be transferred into the real temperature, T = 5.0 ∗ Ec /kB , with Ec the energy difference between the gauche and trans states along polymer chains. Here we choose polyethylene as a real polymer for the rough estimation of real temperature T (other polymers will lead to similar results). For polyethylene, the energy difference Ec takes the value of about 0.5kJ/mol [72], resulting in the real temperature T = 5.0 ∗ Ec /kB ∼ = 5.0 ∗ 0.5 ∗ 103 Jmol−1 /(1.38 ∗ 10−23 JK −1 ) ∼ = 300K. The glass transition temperature (Tg ) for bulk polyethylene usually takes the value of about 193K [73]. Considering the deviations of Tg from bulk values for thin films [74], we can conclude that the temperature of 300K indicates the melt-like polymer film in reality. The striking consistency between our simulation results for polymer melts and experimental results of aging behaviors for polymer glasses is of great significance in the following aspects. First, we demonstrate that polymer films at temperatures above Tg could generate similar structural relaxation behaviors observed in glassy polymer films in experiments and these relaxation behaviors could be compared with that in glassy films through TTS principle. Second, the outlined procedure validates the effectiveness of the molecular simulations with vacancy diffusion mechanism in understanding the structural relaxation and physical aging behaviors in glassy polymer films. Third, our work could be easily extended to study the effect of molecular architectures, such as star-shaped macromolecules, on the structural relaxation behaviors of polymer films, which has already been investigated recently by experiments [19, 20]. Another point that we want to stress is the characterization of the deviations from equilibrium states for quenched polymer films. In experiments, the widely used quantity for characterizing the deviations is the temperature difference of 4T = Ta − Tg , where Ta is the 22


superposition (TTS) principle[71]. Based on the TTS principle, one could expect that the

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Article Online aging temperature and Tg is the glass transition temperature of polymer films [6,View 17–20]. DOI: 10.1039/C3CP53555J

When polymer films are quenched to lower aging temperatures Ta , the absolute value of 4T rium state. The stronger deviation will drive a faster aging process of the quenched polymer film towards equilibrium, indicating a larger aging rate. This qualitative description of the deviations with 4T provides a simple and intuitive understanding of deviations of quenched glassy polymer films. However, in our simulation, we could provide a more detailed descripPublished on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

tion of the deviations from the molecular levels. As mentioned above, we use the vacancy sites to characterize the “excess” free volumes generated by quenched polymer films. In Fig. 8(b), we found that the accurate φvi , where the linear relaxation behaviors of ultrathin films began, showed a strong temperature dependence. It is clearly shown that when decreasing the temperature, the accurate φvi increases correspondingly, similar to the qualitative description of deviations by 4T . Thus the accurate φvi could be employed in our simulations to represent the deviations of quenched polymer films from equilibrium states. Compared with the qualitative description of the deviations by 4T = Ta − Tg in experiments, the φvi obtained from our simulation results could provide more detailed information about the deviations of quenched polymer films. For example, in Fig. 8(b) we can find that at the extremely low temperatures, the increase of deviations becomes slower, indicating that the deviations of quenched polymer films would probably become saturated at the extremely low temperatures. Recently, Cangialosi et al. showed a double-step enthalpy recovery with DSC measurements in several glass-forming polymers [75]. This interesting thermodynamic study indicates the existence of two physical aging processes (fast and low processes) in glassy polymers, which can probably be connected to the kinetic viewpoint of dynamic heterogeneity in diverse glassy systems, such as molecular, colloidal glasses, and granular materials [76]. The phenomenon of dynamic heterogeneity emerges in the supercooled state of glass-formers, where the relaxation is spatially heterogeneous with regions that are faster and slower than the average. It is reasonable to speculate that dynamic heterogeneity also exists in glasses at temperatures lower than Tg . Thus the physical aging in glasses should naturally include two steps, just as shown in recent experiment [75]. However, it would be a great challenge to demonstrate this assumption in simulations based on the following reason. Currently, most computer simulations of glassy behaviors are performed at temperatures higher than the 23


becomes larger, indicating a stronger deviation of quenched polymer film from the equilib-


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View Article Online real Tg [77]. This procedure would be rationalized by TTS principle when considering the DOI: 10.1039/C3CP53555J

variation of general properties, such as viscosity, fast relaxation time, and so on [71]. Howsupercooled state, the simulations should be performed at very low temperatures, where the whole relaxation takes very long time and it would be very hard to wait for the emergence of second relaxation in simulations. Just as mentioned above, our simulations are performed at temperatures higher than Tg where no dynamic heterogeneity emerges. Thus our simulation Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

can not generate the second aging process observed in experiments, and the second plateau shown in Fig. 4(a) represents the equilibrium state within our simulation framework. Also it is worth noting that the experimental results compared with our simulations were performed in hours or days, which corresponded to the first aging process reported by Cangialosi et al [75]. Thus strictly speaking, our simulations are consistent with the fast aging process reported in experiments. It would be interesting to develop new simulation methods in the near future by considering the dynamic heterogeneity to account for the emergence of second aging process reported in experiments [75]. In the end, we would like to emphasize that although the great significance of structural relaxation and physical aging for glassy polymer films shown in practical applications, the current description of glassy state of polymers is still in controversial both in theories and simulations [76], which limits the direct descriptions from the molecular levels of the relaxation and aging behaviors for glassy films at present. Also, the long relaxation time in glassy polymer films (usually hours or days) [3] is hard to achieve within the simulation time scales. In these cases, the simulation of structural relaxations in polymer films at temperatures above Tg with the inclusion of vacancy-diffusion mechanism, as well as the investigation of the responses of relaxation rate β to environments (temperatures, confinements, and so on), would be helpful for understanding the diverse relaxation behaviors in glassy polymer films from the molecular levels.

IV.

CONCLUSION

In conclusion, we perform a Monte Carlo simulation with the incorporation of vacancy diffusion model to investigate the structural relaxation and physical aging behaviors in ultrathin polymer films. Results show that the local average density of segments increases 24


ever, to account for the specific behaviors (such as dynamic heterogeneity) emerged in the

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View Article Online linearly with the logarithm of time, which is consistent with recent fluorescenceDOI: intensity and 10.1039/C3CP53555J

dielectric strength measurements. By quenching films to lower temperatures, the relaxation also consistent with the corresponding experimental results. The reason for this behavior can be attributed to two competing mechanisms: the segment mobility and the accurate initial vacancy concentration φvi . When temperature decreases, the segment mobility decreases while φvi increases, leading to a peak of relaxation rate as a function of temperature. Our Published on 07 October 2013. Downloaded by Tulane University on 08/10/2013 21:07:01.

results demonstrate that the incorporation of vacancy diffusion model into MC simulations could indeed provide new insights from the molecular levels to understand the physical aging behaviors of glassy ultrathin polymer films. The outlined approach can be extended to study from the molecular levels the responses of relaxation rates of glassy ultrathin films to distinct conditions, such as molecular architectures, nano-confinements, various substrates, and so on. Acknowledgments We appreciated the helpful discussions with Prof. G¨ unter Reiter and Prof. Simone Napolitano. We also appreciated the helpful comments and suggestions from the referees. This work was supported by National Natural Science Foundation of China (Grant Nos. 20825415 and 21274061) and National Basic Research Program of China (Grant No. 2011CB606100), and China Postdoctoral Science Foundation (Grant No. 2013M531319).

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“excess” free volume spaces. However, this off-lattice model is not suitable for the delicate

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095701-1-5.

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DOI: 10.1039/C3CP53555J

[76] L. Berthier and G. Biroli, Rev. Mod. Phys., 2011, 83, 587-645.



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TOC caption: Illustration of vacancy diffusion and annihilation processes in ultrathin polymer films.



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Self-doping of ultrathin insulating films by transition metal atoms.