Slowing Down versus Acceleration in the Dynamics of Confined

Aug 31, 2012 - The rotational and translational segmental dynamics were analyzed for each film thickness in different film layers, for a wide range of...
0 downloads 11 Views 614KB Size
Download PDF
Article pubs.acs.org/Macromolecules

Slowing Down versus Acceleration in the Dynamics of Confined Polymer Films Chrysostomos Batistakis,*,†,‡ Alexey V. Lyulin,† and M. A. J. Michels† †

Theory of Polymers and Soft Matter (TPS), Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ‡ Dutch Polymer Institute, P.O. Box 902, 5600 AX Eindhoven, The Netherlands ABSTRACT: We have performed molecular dynamics (MD) simulations on coarse-grained polymer films which are confined between two attractive crystalline Lennard-Jones substrates with three different substrate−substrate separations. Two different polymer−substrate interactions strengths have been studied. Detailed analysis of the structural properties of each film showed a layering of the monomers close to the polymer−substrate interface and a preferential orientation of the bonds parallel to the substrate surface; both depend on substrate attraction strength and temperature, but not on film thickness. The rotational and translational segmental dynamics were analyzed for each film thickness in different film layers, for a wide range of temperatures and for both substrate attraction strengths. For all simulated films, the segmental dynamics was found to be faster than that in the bulk. For relatively thick films and energetically neutral polymer−substrate interaction, a dramatic slowing down of the polymer mobility was found close to the polymer−substrate interface, when compared with the middle of the film, thus providing stiffness enhancement due to the presence of the attractive substrates. With decreasing substrate− substrate separation these gradients in stiffness became overlapping. However, this did not lead to an overall stiffness enhancement in the film, as expected; instead, a large shift toward lower overall mobility was found for the thinner films, which was attributed to finite-size scaling effects. weakening filler−filler and polymer−filler bonds, but conclusive experimental evidence at the required length scales is lacking. Much of what is known on the reinforced filled elastomers is due to detailed experimental studies with carbon black as filler particles5−15 because these have been widely used in industry over the past decades. Special attention is therein given to the role of the fractal aggregate structure of carbon black. Studies with AFM9,10 and small-angle scattering13,15 have given insight in the shape and size of the fillers, confirming the organization of the particles into larger aggregates. These aggregates are formed with polymeric bridges between the filler particles. Therefore, the strength of adsorption of the polymers to the filler particles seems to be very important to the mechanical reinforcement. In a recent NMR and X-ray study,16 the effect of the local strain-induced crystallization of the polymer between the CB particles is investigated, and this effect is shown to be correlated with macroscopic mechanical behavior. Studies by SAXS and SANS17,18 of polymer composites filled with monodisperse silica particles showed a similar fractal network as for CB and similar behavior under loading. An important difference between silica and CB fillers is that silica has much less bonding to the polymers; the strength of adsorption

1. INTRODUCTION Elastomeric polymers, colloquially called “rubbers”, are very versatile materials and find their use both in simple household and in demanding high-tech conditions: examples include tires, many types of high performance plastics which are used e.g. in the automobile industry to replace steel, products for leisure and sport, and structural parts of advanced equipment.1−3 For many applications, e.g. seals and tires, more stiffness and wear resistance is needed, and this is realized by adding hard submicrometer-size particles like carbon black (CB) or silica. These form an additional and reinforcing network through the polymer, and the resulting composite exhibits much higher stiffness and extreme resistance to both fracture and abrasion but still proves highly elastic. Nevertheless, the mechanism for mechanical reinforcement is not sufficiently understood. Specifically for the tire applications, the decrease of the rolling resistance is getting very important, for both environmental and marketing reasons. The main cause of the high rolling resistance in the tires is the so-called Payne effect: the reversible loss of rigidity at larger strain amplitudes and the associated viscous energy losses. Even 10% strain amplitude is enough to decrease the storage modulus by 1 order of magnitude.4 The Mullins effect is also very important: the irreversible and strain-history-dependent softening of the filled elastomer. Both effects are believed to be associated with © 2012 American Chemical Society

Received: April 16, 2012 Revised: July 20, 2012 Published: August 31, 2012 7282

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

of glassy polymeric bridges between the filler particles, thus providing a realistic scenario of the Payne effect. The scope of the present simulation study is to focus on the influence of the polymer−filler interaction strength on the polymer mobility between two fillers and to study the possible gradient of segmental dynamics in polymer films as a function of both the distance from the filler surface and temperature. Since filler particles create a reinforcing percolating network inside elastomers, the assumption of a modified polymer mobility close to the filler surface implies that the filler−filler distances are extremely important for the mechanical behavior of the composite. To study the effect of the filler−filler separations, we simulated coarse-grained polymer melts that are confined between two crystalline substrates. The polymer− substrate interactions have been chosen to be of a simple Lennard-Jones (LJ) type, so that the interaction strength can be easily modified. The effect of the substrate−substrate distance has been studied by creating films of different thicknesses, comparable to or much larger than the single-chain radius of gyration. In section 2, we give a full representation of the simulated model and describe the exact procedure of the sample preparation. In section 3, we present structural properties of the systems we created. Simulated data for the glass-transition temperature far away from the substrates are also included. In section 4, results are presented of the segmental orientational and translational mobility at various distances from the substrates, for different polymer−substrate interaction strengths, and for different temperatures. A comparison between results for different film thicknesses follows. In section 5 a discussion is given of the above methods and results, and the paper ends with conclusions in section 6.

depends on the surface chemistry of the particles. Silica surfaces have been modified by the use of coupling agents19,20 and plasma treatment,21,22 and in this manner the polymer−surface interaction can be well controlled. It has been suggested4 that the polymers around the filler create a layer that exhibits much lower mobility in comparison to the bulk polymers at the same temperature. This suggestion has also been proved by recent NMR experiments for carbonblack-filled rubbers.23 In other studies of silica-filled rubbers,24,25 the existence of a gradient in the polymer glasstransition temperature close to the silica surface has been measured by NMR. This gradient was successfully related to the temperature- and frequency-dependent mechanical behavior of the composite.24,25 Strain-induced softening of the percolating glassy bridges between the filler particles is then held responsible for the nonlinear mechanical behavior. However, it is hard to have (experimentally) a full overview of the polymer dynamical properties close to fillers, mostly because of the filler structure and the small size of the polymer layer surrounding the filler. Molecular simulations can help here and give a detailed picture of the polymer−filler interface behavior. They offer the additional advantage that the range and the strength of the polymer−filler interaction can be easily tuned. A series of molecular dynamics (MD) simulations of polymers close to fillers or between fillers (the fillers are usually represented as walls) have shown inhomogeneous dynamics near the polymer−filler interface, when compared with that in the bulk. Torres et al.26 showed that the glass-transition temperature of supported polymer films on attractive walls can be higher or lower than that of the bulk when using different wall attraction strengths, the difference being more pronounced for thinner films. Similar results were found by Starr et al.,27 who simulated a filler particle surrounded by a coarse-grained polymer melt. They found that the polymer glass-transition temperature could either increase or decrease close to the filler by using attractive or excluded-volume polymer−filler interactions, respectively. Anisotropic dynamics was found in simulations of confined 1,4-polybutadiene between two graphite walls,28 where a slowing down of the film dynamics was observed when getting close to the walls. Varnik et al.29 simulated confined polymer melts between repulsive walls and studied the effect of the wall-to-wall distance on the average film dynamics. They found an acceleration of the film dynamics close to the polymer−wall interface. Simulating films of different thicknesses (i.e., wall-to-wall distances), they found that thinner films exhibit faster dynamics. All the previous experimental and simulation results indicate that the polymer−filler interaction can have a huge effect on the polymer dynamics in the vicinity of the filler surface, but the sign of the effect (lower or higher mobility, higher or lower Tg) is not always clear. Slowing down of the polymer mobility observed close to attractive fillers could be the dominant effect in the stiffness enhancement by the fractal filler aggregates. Higher concentrations of the filler particles in the polymer matrix may result in much smaller distances among them, so that the immobilized surrounded polymer layers may overlap and create extremely stiff glassy bridges. On the basis of this assumption and in parallel with the experimental work of Berriot et al.,24,25 Long, Merabia, and Sotta30,31 simulated composite samples with a few thousand particles by dissipative particle dynamics. With this approach they were able to study the dynamics under loading and unloading of yield and rebirth

2. MODEL AND METHODS 2.1. Model Description. In the present simulations we study a coarse-grained polymer model in order to avoid as much as possible chemical details and to investigate the generic phenomena of the polymer−filler reinforcement. We used bead−rod polymer chains, where each chain consists of 80% particles of type A and 20% particles of type B at random positions in the chain (i.e., a random copolymer). The reason for choosing two types of particles is to avoid crystallization. The bonded monomers along a chain are connected with rigid bonds. We assume no bending potential between neighboring bonds and neither torsion interactions. The systems we simulated consist of 25 chains (50 beads per chain), which are confined between two crystalline substrates. Each substrate is composed of three layers of LJ spheres on an FCC lattice. All nonbonded monomer pairs interact via a truncated LennardJones potential: ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σ σ ⎪ ⎪ 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ if r ≤ rc ⎝r⎠ ⎦ VLJ(r ) = ⎨ ⎣⎝ r ⎠ ⎪ ⎪0 if r > rc ⎩

(1)

with a cutoff at rc = 2.5σ. Coulomb interactions are not taken into account. The calculations are carried out in reduced LJ units. We take m = mA = 1, σ = σA = 1, and ε = εA = 1 as the units of mass, length, and energy, respectively. As a unit of time, we use τ = σ(m/ε)1/2. We measure the temperature in units of ε/kB, the pressure in units of ε/σ3, and the density in units of m/σ3. For 7283

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

the B particles we used εB = 1, σB = 1.2, and mB = (1.2)3. The bond length was chosen to be σ. The LJ parameters of the substrate particles were chosen to be σs = 0.85 and εs ≫ 1. Regarding the substrate−polymer interaction, the σsp was calculated according to Lorentz−Berthelot rule σij = 1/2(σii + σjj), although the εsp was chosen to be either εsp = 0.1 or εsp = 1, corresponding to weakly and energetically neutral adsorbing substrates (since the polymer−polymer interaction parameter is εpp = εA = εB = 1). At the very initial stage of the sample preparation, only the repulsive part of the LJ potential (eq 1) between the polymers and the wall particles has been used: ⎛ σ ⎞12 VLJ = 4ε⎜ ⎟ ⎝r⎠

needed to achieve the desired density in each system. As explained earlier, the initial polymer−substrate interaction was purely repulsive (see eq 2) in order to avoid immediate adsorption of the chain ends to the substrates. The systems were allowed to relax in an NVT ensemble at very high temperature (T = 10) in a high-mobility state and for sufficiently long time (12 000−30 000τ, depending on the system). This relaxation was followed by uniaxial compression (along the z-axis) in an NPT ensemble. For each system, the external pressure was adjusted in order to achieve the same final density for all the three films. A second constant-volume relaxation followed the compression. The quality of the equilibration was checked by measuring the individual-chain radius of gyration (Figure 2, black squares):

(2)

The ε was chosen here to be εsp = 1. In order to study the effect of the confinement width, three films of different thicknesses (i.e., substrate−substrate separations) have been created, in such a way that the total density ρ inside each polymer film is approximately the same, with the same amount of simulated particles in each case; the density in our films is ρ ≈ 1.18. Consequently, substrates with larger lateral dimensions must be used in order to create thinner films. Periodic boundary conditions were applied in all three directions. The leapfrog variant of the velocity Verlet algorithm has been used to integrate the Newton’s equations of motion with an integration time step Δt = 0.0005τ. The reason for choosing such a small time step is the very strong interaction between the substrate particles. For the simulations (equilibration and production), we used both NVT and NPT ensembles, with the Nose−Hoover thermostat and the Berendsen thermostat and barostat, respectively. The time constant for both thermostats was chosen to be τt = 0.3τ, and the time constant for the barostat was chosen to be τp = 1.5τ. The PLincs algorithm,32 which is a parallel version of the Lincs33 algorithm, has been used for constraining the bonds. All simulations have been performed with the versions 4.5.3 and 4.5.4 of the open source MD simulation package GROMACS,34 and all results have been produced after averaging over five independent samples on each film thickness. 2.2. Equilibration and Annealing. Every film was created in the following way: the polymer chains were initially completely stretched along the z-axis and put far away from the substrates (the distance between the chain ends and the substrates was initially 25σ) in a large simulation box (Figure 1). This results in an initial volume 4−8 times larger than the final one. Consequently, the polymer−polymer interactions are relatively weak. Because of the presence of the crystalline substrates inside the simulation box, compression in the xyplane of this box is not possible without breaking the crystallinity. For that reason, the box dimensions in the xyplane where chosen fixed, so only in the z-axis compression was

Rg2 =

∑ mi ri ⃗ 2 ∑ mi

(3)

Figure 2. Black (■): radius of gyration for individual chains for film thickness S = 22.5 after equilibration at very high temperature (T = 10) and for repulsive polymer−substrate interaction. Very small fluctuations (3−4%) are observed, which is an indication of wellequilibrated samples. Red (●), blue (▲), and green (▼): the Iαα (α = x, y, z) values of the inertia tensor divided by the total mass ∑mi.

where ri⃗ is the position of a monomer with respect to the chain center of mass; we verified that it fluctuates not more than 3− 4%. This is an indication of very well equilibrated samples. Another proof of equilibration is isotropy of the sample structure. For that purpose, the components

∑ mi(yi 2 + zi 2), Iyy = ∑ mi(xi 2 + zi 2), Izz = ∑ mi(xi 2 + yi 2 )

Ixx =

(4)

of the inertia tensor were calculated (Figure 2; red circles, blue triangles, and green inverse triangles for the Ixx, Iyy, and Izz, respectively). Small variations are visible between the three components, where the Izz is only slightly larger, which means that the chains are in random-coil configuration.35 In the next stage of the equilibration, the full LJ potential was used for the polymer−substrate interactions, according to eq 1. For each film thickness, we studied two cases of the interaction strength: εsp = 0.1 and εsp = 1. Each system was again allowed to relax in an NVT ensemble for a long time (6000τ) at temperature T = 10. The systems were cooled down by keeping the initial density constant (NVT ensemble) until T = 0.2 with a cooling velocity of 0.0003[(ε/kB)/τ]. During the cooling

Figure 1. Snapshot of the system configuration at the beginning of the equilibration procedure. Initially the polymer chains are completely stretched and well separated from the substrates. The planar dimensions are not changing during the simulation. Compression in the z-direction until the desired density has been reached leads to the final film thickness. 7284

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

procedure, the configurations at different temperatures were saved for further production runs to calculate the dynamical properties of the polymer chains.

3. STRUCTURAL PROPERTIES 3.1. Ordering of Monomers. The monomer-bond orientations were analyzed in each film as a function of the distance to the substrate. To do so, we calculated the bond orientation with respect to the z-axis, using the second-order Legendre polynomial: 3 1 P2(z) = ⟨cos2 θ ⟩ − (5) 2 2 where θ is the angle between a unit vector along the z-axis and the bond vector b⃗(t) between two bonded neighboring monomers at any time t. Within each film the order parameter was calculated in different film layers of thickness d = 0.8−1 (depending on the overall film thickness). The brackets denote an average over all bonds as well as a time average in each layer. The results for T = 1.3 are summarized in Figure 3, where a comparison of the bond ordering in the three different films is

Figure 3. Bond orientation of the monomers with respect to the z-axis for the three different film thicknesses at temperature T = 1.3 and for the substrate−polymer interaction strength εsp = 1.

made. In the middle of each film the bonds are oriented randomly and the order parameter P2 is very close to zero. The behavior observed close to the substrates is similar for all films. The bonds which are in the range of the LJ substrate potential tend to have a preference for orientation parallel to the substrate surface. For the thicker film (S = 22.5, black squares), the oriented area is only a small fraction of the total film volume and the bonds in the rest of the film are randomly oriented. Since the oriented area is not changing for thinner films (S = 14, red circles; S = 7.5, blue triangles), the volume fraction of the disordered area will decrease with decreasing film thickness. Similar behavior was observed when calculating the order parameter at other temperatures. In section 4.2.3 we study the possible influence of this effect on the film dynamics. 3.2. Density Profiles. The density profiles for the simulated systems were measured for different temperatures and for polymer−substrate interactions both εsp = 0.1 and εsp = 1, but also immediately after finishing the first part of the equilibration (at T = 10 with repulsive substrates). In Figure 4a, a comparison of the density profiles for the three different film thicknesses is made for the latter case.

Figure 4. (a) Density profiles for the three different film thicknesses, for a repulsive substrate potential and at very high temperature (T = 10). (b) Effect of the substrate potential on the density profile of the thicker film at T = 1. A stronger potential results in higher ordering close the substrates. (c) Effect of temperature on the monomer layering close to the substrates. Higher ordering is observed when decreasing the temperature.

By analyzing Figure 4a, we are able to distinguish two different density regimes. Near the polymer−substrate interfaces, substrate layers are formed. Because of the presence of the walls, the polymer beads exhibit strong ordering and sharp density variations are visible. The peak-to-peak distance is approximately σ in all three films. The small differences between the left and right peaks of the density profile are connected with small deviations in the interval width used for calculating the local densities. For that reason, we ignore these differences and we consider the samples completely symmetric. 7285

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

It can also be seen that stronger confinement results in slightly weaker density variations (less monomer ordering) close to the substrates. When the ordering has decayed, a middle layer of bulk behavior is observed and the density profile becomes smooth. The ordered area constitutes only a small volume fraction of the thick film (S = 22.5), but it starts to dominate when decreasing the substrate−substrate distance; for the thinnest film (S = 7.5) the middle layer has disappeared. The monomer layering close to the substrates highly depends on the polymer−substrate interaction strength (Figure 4b) but also on the temperature (Figure 4c). Repulsive substrates induce a weaker layering of monomers in the substrate layers. In this case the observed ordering is only an entropic effect. As soon as a full LJ potential is applied, the layering is enhanced due to the attractive part of the substrate potential (Figure 4b). In Figure 4c, we see the temperature dependence of the density profile for film thickness S = 22.5, in the case of εsp = 1. A very smooth density profile is observed for extremely high temperatures, with a relatively small ordering close to the walls. However, as the temperature decreases, the number of monomers that are ordered in layers due to the wall attraction increases. 3.3. Glass-Transition Temperature. An estimation of the glass-transition temperature (Tg) in the thicker (S = 22.5) film can be given from the temperature dependence of the density occur when it is cooled down under a constant external pressure. Depending on the value of this pressure, the Tg can dramatically change. We adjusted the external pressure in order to have ρ ≈ 1.18 (and so film thickness S ≈ 22.5) when the temperature is T ≈ 1. The polymer−substrate interaction parameter was chosen to be εsp = 0.1. The film was equilibrated at very high temperature T = 10 and cooled down with a cooling velocity 0.0003[(ε/kB)/τ] to T = 0.2 under a constant pressure p = 13. As explained in section 2, due to the crystalline nature of the substrates, the pressure is applied only along the z-axis. The bulk Tg has been also calculated under the same external pressure, although in this case, the pressure in the bulk polymer was applied in all three directions. In the temperature range T = 0.2−1.5, the configurations have been saved every temperature step T = 0.05, and after running further production runs in a NPT ensemble, the average density at each T was measured. The temperature dependence of the density for both the film and the bulk is plotted in Figure 5a. The glass-transition temperature is determined by fitting two straight lines in the low- and high-temperature regions and finding the intersection point. We find the glass-transition temperature of the film to be Tg = 0.8 ± 0.05, although for the bulk it was calculated equal to Tg = 0.88 ± 0.05. Changes of Tg compared to the bulk have been reported also for polystyrene and PMMA supported films.36,37

Figure 5. Density−temperature relation under constant-pressure cooling for the bulk and for the thicker film S = 22.5 with polymer−substrate interaction strength εsp = 0.1. The intersection point of the two straight lines in the low- and high-temperature regions gives the simulated glass-transition temperature.

averaging as well as an averaging over all the bonds. For the calculation of the P2 locally (in separate layers of a film) the problem arises how to assign the bonds to different layers. Our definition is that a bond belongs to a specific layer if the middle of the bond is inside the layer at time t = 0. The drawback here is that a bond can jump outside the layer or can fluctuate between two layers during the simulation and affect the final result. The probability of a jump to happen is strongly dependent on the layer width, which should be big enough to ensure that the full relaxation of the majority of the bonds takes place inside the layer. Some bonds will jump outside the layer before they have fully relaxed, but the produced error can be made small enough in the majority of the layers (see Appendix). The effect of the polymer−substrate interaction strength has been studied for all film thicknesses, where the orientational segmental dynamics has been analyzed in different layers of width l ≈ 1.25σ (for the appropriateness of the chosen width, see Appendix). Bulk sample was also created using similar procedure as described in section 2.2 with the density the same as that in the films. In Figure 6, a typical example is shown of

4. DYNAMICAL PROPERTIES 4.1. Orientational Mobility. In order to have insight into the effect of the walls on the polymer dynamical behavior, the local orientational mobility in different film layers has been studied with the help of the autocorrelation function (ACF): P2(t ) =

3 ⃗ 1 (b (0)b ⃗(t ))2 − 2 2

Figure 6. Decay of the bond-orientation ACF P2(t) in different layers parallel to the substrates, for temperature T = 1, substrate attraction εsp = 1, and film thicknesses S = 22.5 and S = 7.5; z is the distance from the nearest substrate in each case. Since the films are symmetric with respect to the their two substrates, the results have been averaged over the symmetric regions.The red curves correspond to the KWW fits according to eq 7.

(6)

where b⃗ is the bond vector between two neighboring beads as measured at time 0 and time t. The brackets denote a time 7286

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

Figure 7. (a) Temperature dependence of the orientational-relaxation times in different layers of the thicker film (S = 22.5). The black squares correspond to εsp = 0.1 and the red circles to εsp = 1. (b) Orientational-relaxation times at T = 1.1 in different film layers and for different film thicknesses. On the vertical axis, the distance z from the bottom substrate has been normalized by the film thickness S. The arrow corresponds to the bulk relaxation time at the same density and temperature with the polymer films.

the ACF behavior in a semilog plot for temperature T = 1, polymer−substrate interaction εsp = 1, and film thicknesses S = 22.5 and S = 7.5. For S = 22.5, in layers close to the substrates (i.e., for 0 < z ≤ 1.25) the relaxation is 2 orders of magnitude slower than in the middle of the film (7.25 ≤ z ≤ 11.25), whereas this behavior is strongly reduced for thickness S = 7.5, where much faster relaxation is observed. Similar curves were produced for εsp = 0.1 and for other temperatures. In order to extract the P2 relaxation times in each layer, we fitted the final parts (Figure 6, red curves) of the calculated P2 ACF’s with the Kohlrausch−Williams−Watts (KWW) stretched-exponential function:38 ⎛ ⎛ t ⎞β⎞ P2(t ) = α exp⎜ −⎜ ⎟ ⎟ ⎝ ⎝τ⎠ ⎠

mobility polymer bridge between the substrates. A comparison between the local orientational dynamics of all three simulated films is made in Figure 7b. For film thickness S = 14, 11 layers have been used and for S = 7.5, six layers. The relaxation times in all cases were extracted for polymer−substrate interaction strength εsp = 1 and for temperature T = 1.1. In Figure 7b, we see that the difference between the layers close to the substrate and the middle of the film is getting smaller in thinner films, as a consequence of the overlap of the former regions. Surprisingly, also a shift to lower relaxation times is observed when decreasing the film thickness. Near the polymer− substrate interface this speed-up of relaxation is even 3 orders of magnitude when changing the film thickness from S = 22.5 to S = 7.5. So, contrary to expectation, stronger confinement results in an acceleration of the average film dynamics. 4.2. Translational Mobility. 4.2.1. Film-Averaged Dynamics. For all film thicknesses the segmental translational mobility has been studied by calculating both the layer-resolved and the film-averaged mean-square displacements

(7)

where α ≤ 1, τ is the characteristic relaxation time, and β is the parameter taking into account the nonexponential nature of the relaxation process. In each layer the P2 relaxation times were calculated for each sample separately and then averaged (see Figure 7a). The parameter β was left free during the calculation but showed only small fluctuations around an average value of ∼0.5. At high temperatures (T = 6.7) there is hardly a measurable difference in the dynamical behavior between the different layers in the film. However, as the temperature decreases (T = 1.8), we are able to distinguish a modification region near the substrates (of typical width 6σ) in which the polymers exhibit slower orientational dynamics, highly dependent on the distance from the substrates. This dependence is getting much stronger for still lower temperatures (T = 1). When the temperatures are relatively low, the strength of the polymer−substrate interaction has a further slowing-down effect. In the layers closest to the substrate, relaxation times even up to 2 orders of magnitude higher are observed when taking the εsp = 1 instead of εsp = 0.1 for the substrate potential. The main conclusion from Figure 7a is that attractive substrates lead to a slowing down of the segmental orientational dynamics near the substrates. Consequently, one would expect that by decreasing the film thickness (i.e., the substrate−substrate distance), the average slowing-down effect would also increase, since the volume fraction of the modified regions should increase for thinner films, leading to the emergence of low-

d(t )2 = ⟨Δ r ̅ 2⟩

(8)

for polymer−substrate interaction strengths εsp = 0.1 and εsp = 1; the chosen temperature range is T = 0.6−1.5. In Figure 8, a comparison is made between the film-averaged displacements for each film and the bulk values at a relatively high temperature (T = 1.1). A similar behavior is observed for every film thickness. At very short times (0.01τ−0.1τ) we distinguish the ballistic regime with a slope close to 2. The ballistic regime is followed by a plateau, where the tagged monomer is trapped in a cage created by its neighboring monomers (“cage effect”). As soon as the monomer escapes from the cage after some characteristic time τα (the so-called α-relaxation time39,40) the subdiffusive Rouse regime follows, with a slope 0.54. This is slightly higher than the Rouse exponent 0.5 normally expected; the difference is due to the small length of the simulated chains. In Figure 8 we see that the thicker the film, the larger the time needed for the monomers to escape from the cage, while this time is still smaller than the time needed in the bulk. Similar to the acceleration of the orientational dynamics when decreasing the film thickness (Figure 7b), acceleration in the average 7287

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

The temperature dependence of the relaxation times for film thickness S = 22.5 and for both εsp = 0.1 and εsp = 1 but also for the bulk can been seen in Figure 9b. To extract Tc, we fitted the data with eq 10 for each sample. The results are summarized in Table 1. We see that for both εsp = 0.1 and εsp = 1, there is a clear decrease of the mode-coupling critical temperature Tc when decreasing the film thickness. The observed shift of Tc to lower values again indicates that at fixed T > Tc the translational relaxation is faster for thinner films. 4.2.2. Translational Segmental Dynamics in Layers. The translational motion has also been studied in separate layers of the thicker film (S = 22.5). Again, we define that a monomer belongs to a specific layer if the center of mass of this monomer belongs to this layer at time t = 0. We focus on the thicker film (S = 22.5) because it shows a pronounced difference in mobility between the different film regions. Figure 10 depicts the local translational mobilities for the middle layer (7.5 < z < 11.25) and the layer close to the substrates (0 < z < 1.25) and compare with the bulk values, both for εsp = 0.1 and εsp = 1 and for temperature T = 1. For weakly attractive substrates (εsp = 0.1) negligible mobility differences occur between the middle layer and the layers closest to the substrate, but a pronounced delay in the mean-square displacement occurs close to the polymer− substrate interface if the attraction strength is increased (εsp = 1). The bulk values are close to these of the closest to the substrate layer (for εsp = 1). 4.2.3. Anisotropic Translational Motion. The possible anisotropy (i.e., parallel as compared to perpendicular displacement to the substrate) of the monomer motion in different film layers was examined by calculating the projections of the meansquare displacements in the xy-plane (dx2) and along the vertical axis z (dz2):

Figure 8. Comparison of the film-averaged mean-square translational displacements for three simulated film thicknesses and the bulk data; substrate−polymer interaction is εsp = 1 and for relatively high temperature T = 1.1. The dashed (black), dash-dotted (blue), and dash-dot-dotted (green) lines correspond to film thicknesses S = 22.5, S = 14, and S = 7.5, respectively, and the short-dash-dotted to the bulk mean-square displacement. The short-dashed vertical lines correspond to the α-relaxation time for each film thickness.

translational mobility (decrease of τα) is thus observed in thinner films. An example of the temperature dependence of the dynamics can be seen in Figure 9a, were the temperature dependence of the mean-square translational displacement is plotted for film thickness S = 22.5 and εsp = 0.1. In order to extract the characteristic time of the τα-relaxation, we fitted the final parts (in the Rouse regime, Figure 9a, red lines) of all curves and for every film thickness, with the power law41,42 ⟨Δr 2(t )⟩ ≈ (Dα t )α

(9)

The exponent α was held constant and equal to the average value over all temperatures in each case. According to the mode-coupling theory,39,40 the characteristic time τα = Dα−1 of the α-relaxation diverges algebraically at some critical temperature Tc just above Tg: τα ∝ (T − Tc)−γ

dxy 2 =

⟨Δx 2⟩ + ⟨Δy 2 ⟩ 2

and

dz 2 = ⟨Δz 2⟩

(11)

dxy2 was divided by 2 to make it comparable to dz2. In Figure 11a we see the average monomer motion in different layers for film thickness S = 22.5. In the middle of the film, where bonds between the monomers are randomly oriented (see Figure 3),

(10)

Figure 9. (a) Example of the temperature dependence of the film-averaged mean-square translational displacement for film thickness S = 22.5 and εsp = 0.1. The subdiffusive Rouse regime shows a straight line with average (over all temperatures) slope equal to 0.58. (b) Semilog plot of the evolution of the characteristic times τα for film thickness S = 22.5 and for both εsp = 0.1 and εsp = 1. The solid red lines correspond to MCT-based fit, according to eq 10. 7288

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

Table 1. Mode-Coupling Critical Temperatures Tc and the Critical Exponents γ for Different Film Thicknesses and for Different Polymer−Substrate Interactions εsp = 0.1 εsp = 1.0

S = 22.5

S = 14

S = 7.5

bulk

Tc = 0.85 ± 0.002 γ = 2.02 ± 0.06 Tc = 0.79 ± 0.004 γ = 2.89 ± 0.08

Tc = 0.80 ± 0.009 γ = 1.42 ± 0.09 Tc = 0.76 ± 0.005 γ = 2.34 ± 0.07

Tc = 0.67 ± 0.001 γ = 1.75 ± 0.07 Tc = 0.67 ± 0.002 γ = 1.74 ± 0.07

Tc = 0.92 ± 0.002 γ = 3.02 ± 0.04

5. DISCUSSION In our study, we created sufficiently equilibrated thin films of thicknesses S = 22.5, S = 14, and S = 7.5, confined between atomistically detailed substrates. The radius of gyration of individual chains was fairly constant in all films and was equal to 3.4 (Figure 2), so the thickness S is in the range of 2.2−6.6Rg. All simulations took place in the temperature range T = 0.6− 10. Both repulsive and weakly attractive or energetically neutral substrates have been used. Analysis of the density profiles of all films showed a layering of the monomers, in a range of Rg from the substrates (Figure 4); this layering is familiar from liquids and associated with entropic effects (as in the case of repulsive substrates) but becomes more pronounced when increasing the substrate attraction strength (Figure 4b). In the same range, a preferential bond orientation parallel to the substrate was observed (Figure 3). A middle range defines the rest of the film; it shrinks for decreasing S and vanishes for very thin films (S ≈ 2Rg).When cooling down under a constant external pressure, by measuring the temperature dependence of the density of the thicker film, a decrease in Tg was found as compared to the bulk. Detailed analysis of the orientational dynamical behavior in the thick film (S = 22.5) showed higher relaxation times in the layers close to the substrates when comparing with the middle of the film; the relaxation times were extracted from the KWW38 (eq 7), and the β was close to 0.5. The increase of the relaxation times in approach of the substrates is small for weakly attractive substrates but is orders of magnitude larger when increasing substrate attraction (Figure 7a) and also larger than in the bulk; the mobility in the rest of the film was found to be much faster. The same picture was obtained by analyzing

Figure 10. Mean-square displacements of the bulk and in different regions of the thick film (S = 22.5), for polymer−substrate interaction εsp = 0.1 (blue) and εsp = 1 (red). The dash-dotted lines correspond to the mean-square displacement d2 of the layer closest to the substrate (0 < z < 1.25), the dashed lines to the middle layers (7.25 < z < 11.25), and the dash-dot-dotted line to the bulk values. The temperature here is T = 1.

the monomer motion is isotropic. However, anisotropic motion is observed near the polymer−substrate interface, where the monomer ordering in the xy-plane results in a preferential monomer motion in that plane. Independent of the small observed anisotropy of the monomer motion, there is a significant slowing down effect in each direction. Similar behavior is observed for film thickness S = 7.5 (Figure 11b), where slowing down effect is smaller but again in all directions.

Figure 11. (a) Mean-square displacements parallel and transversal to the substrate surface for film thickness S = 22.5, temperature T = 1.1, and substrate attraction εsp = 1. The black (dashed) and green (dash-dotted) lines correspond to the dxy2 and dz2 displacements, respectively, for the monomer motion in the middle of the film, and the red (dotted) and blue (dash-dot-dotted) to the dxy2 and dz2 displacements for the monomer motion close to the polymer−substrate interface. (b) Similar to (a) but for film thickness S = 7.5. 7289

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

the α-relaxation process in different film layers (Figure 10), indicating a coupling of the orientational and translational dynamical behavior. For thinner films, the middle layer shrinks (S = 14), and ultimately vanishes (S = 7.5), so an increase in the total stiffness would be expected when decreasing the film thickness. This is the assumption used in the literature31 in order to explain the Payne effect. Immobilized polymer layers created close to filler surfaces may overlap by increasing filler concentration (i.e., decreasing filler−filler distance). Under deformation, the polymer glassy bridges which were created will break and the reinforcement is rapidly decreasing. Indeed, the dramatic difference in the mobility between the middle and side layers observed for film thickness S = 22.5 at low temperatures and for energetically neutral polymer substrate interactions is getting smaller for thickness S = 14 and almost disappears when S = 7.5, where the bulk-like middle region has vanished. However, an overall shift to much lower relaxation times was observed for the smaller films, providing the opposite from what was expected (Figure 7b): our thinner films exhibit faster dynamics. So, in contrast to explanations suggested so far for the Payne effect, in terms of overlapping immobilized polymer layers at high filler concentration which break under deformation, our simulations suggest that the influence of substrate attraction strength and low temperature may be counteracted by the effect of decreasing film thickness. To substantiate the surprising findings, the α-relaxation times were extracted for different film thicknesses and temperatures and for both weakly attractive and energetically neutral substrates, and after fitting (eq 10), the mode-coupling critical temperatures were calculated (Table 1). In Figure 9b we found that above T = 0.9, the relaxation times are higher when using εsp = 1 instead of εsp = 0.1, although the effect inverts at lower temperatures. A possible explanation can be that close to and below T = 0.9, when εsp = 1, a glassy layer is created in the vicinity of the polymer−substrate interface, so effectively the film thickness is decreasing and additional finite-size effects are present, leading to some acceleration in the dynamics in the middle of the film (compared to the case of εsp = 0.1), which is absent at higher temperatures (Figure 10). Further investigation of this effect is needed. Adam and Gibbs43 introduced the idea of “cooperative motion” in order to explain the dynamical behavior close to Tg. When the temperature approaches Tg, monomer motion is a result of collective rearrangements. Assuming the cooperatively moving monomers create a cluster of size ξ, the cluster relaxation time should be τcl ∝ ξν, where ν is a critical exponent.44 The dramatic increase of the relaxation times near the Tg is then a result of the divergence of the cluster size when the temperatures decreases. Edwards and Vilgis45 showed that the concept of cooperative motion alone is enough to result in the Vogel−Fulcher−Tamman (VFT) law. Varnik et al.29 assumed that for confined polymers, the cluster size will have the film thickness as an upper plateau value. When the cluster size reaches the film size, i.e. S = ξ, the relaxation is becoming size dependent. Varnik et al. simulated confined polymer films between smooth and repulsive walls. Their analysis of the local dynamical behavior indicated an acceleration of the polymer dynamics close to the filler surface for high film thicknesses. By decreasing the film thickness, they also found a shift of the mode-coupling critical temperature to lower values. Their explanation was based on the effect of the repulsive walls on the local dynamics, in combination with finite-size effects.

Alternative explanations may be sought from details of the dynamics, e.g., anisotropy in the diffusion in the oriented side layers, in which parallel diffusion might be accelerated preferentially for thinner films. By analyzing the monomer motion in different directions and close to the substrates, where there is an ordering of the bonds, a slightly preferential motion parallel to the substrate surface (xy-plane) was found both for thick and thin films (Figure 11). The motion perpendicular to the substrate is becoming more difficult due to the presence of the substrates but also because of the substrate attraction. However, this anisotropy is minor in each case and small compared to the acceleration of the average diffusion in the side layer when the film thickness decreases from 6.6Rg to 2.2Rg. Also, the diffusion in the middle accelerates, although less so; the dynamics becomes more homogeneous. In summary, a slowing down of the polymer dynamics in regions close to an attractive surface has been confirmed. However, the synergistic effect on the polymer mobility when such low-mobile regions overlap, as supposed to explain the Payne effect, has not been seen. Decreasing film thickness results to more mobile films, as observed in the past for repulsive surfaces29 and explained by finite-size scaling near the glass-transition temperature. In the present study, only weakly attractive (εsp = 0.1εpp) and energetically neutral (εsp = εpp) substrates were used. A broader comparison of cases is needed to investigate the competition between the confinement effects and the effects of the substrate interaction. A range of stronger attractions should be explored and chemical binding via coupling agents such as used in silica-filled rubbers; some simulations with polymer grafted on silica were already performed elsewhere.46,47 Also, rough surfaces and surface curvature have been shown to be relevant aspects.48

6. CONCLUSIONS We compared coarse-grained polymer films confined between two crystalline attractive substrates for three different film thicknesses (substrate−substrate separations) and two different monomer−substrate interaction strengths. We found oscillations in the monomer-density profiles near the substrates, which depend on the polymer−substrate interaction strength and also on temperature. By examining the bond orientation, we saw that inside the range of the substrate potential there is a preferential orientation of the bonds parallel to the substrate surface. The substrate attraction strength also influences the film dynamics as was seen from the segmental orientational and translational mobility in different layers within each film. For relatively thick films (6.6Rg), a dramatic slowing down of the polymer dynamics was found in the vicinity of the substrate surface when compared with the middle of the film, thus proving the existence of a strong gradient in mobility near each surface. For thin films (2.2Rg) this gradient almost disappear due to the overlap of the two low-mobility regions. However, in contrast to what was expected this did not lead to low-mobility bridges in thin films. Thinner films rather showed a strong increase in overall mobility, consistent with finite-size effects near the mode-coupling singularity.

■

APPENDIX

In order to calculate the orientational and translational segmental diffusion locally, the bonds (or monomers) have to be assigned to specific film layers (see section 4). This labeling leads to statistical errors, since a bond (or a monomer) can 7290

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

Figure 12. (a) Mean-square translational displacements within different film layers for film thickness S = 22.5, substrate−polymer interaction εsp = 1, and temperature T = 1.5. The times shown are the α-relaxation times in each film layer. (b) Vertical monomer displacements over the same film areas as shown in (a). We see that the α-relaxation times measured in (a) correspond to a vertical displacement of 0.17σ.

in τα; in this case, the jumps to the neighboring layers on either side give compensatory effects. In the boundary layers, monomers leave only in one direction, so this leads to an error of approximately 13.5/2 = 6.75%.

jump outside the layer during the simulation. The smaller the layer, the higher the probability for a bond or monomer to jump outside the layer. One possible solution to this problem is not to take into account those bonds (or monomers) that do jump outside during the simulation run. This means that only the slower monomers will contribute to the mobility, and the result will inevitably be different from reality. On the other hand, calculation of the mobility of all monomers again leads to errors, if a high percentage of monomers (this depends on the layer width) jumps out of the layer before the average relaxation time. In the main text, we have chosen the second method, with a layer width l ≈ 1.25σ. A possible way to check the appropriateness of this width is by measuring the average vertical displacement of the monomers assigned to a layer at t = 0. An average vertical displacement dz = 1.25 at time t means that 100% of the monomers will leave the layer during this time. So the goal is to have segmental orientational and translational relaxation at much shorter times, in order to minimize the error. For example, with a calculated α-relaxation time and a known average vertical displacement during this time, one can semiquantitatively predict the fraction of monomers which jump outside. In Figure 12, we see an example of the error calculations for the layer-resolved mean-square translational displacement. We measured the total (vertical and parallel) mean-square displacements (Figure 12a) and the vertical displacements (Figure 12b) of the polymer beads in different film layers for film thickness S = 22.5, substrate−polymer interaction εsp = 1, and temperature T = 1.5. The α-relaxation times for each of the three film layers correspond to an average vertical displacement of approximately 0.17σ. Given the layer width l ≈ 1.25σ and assuming the monomers are equally distributed, only a fraction f = 2 × 0.17/1.25 ≈ 0.27 of the monomers are in position to leave the layer at time t = τα. Since the monomer motion is random, 50% of the monomers will move up and the other down which means that only f/2 monomers will leave the layer in time t = τα, leading to an error of 13.5%. When neighboring layers exhibit similar relaxation times (i.e., around the bulk regime), this error is not important since part of the relaxation of the jumping monomers takes place in another layer of the same mobility. The error is also small when there is a gradient

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is part of the research program of the Dutch Polymer Institute, project #701. It was also sponsored by the Stichting Nationale Computerfaciliteiten (National Computer Facilities Foundation, NCF) for the use of supercomputer facilities, with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization from Scientific Research, NWO). We thank Daniel Bonn (University of Amsterdam, experimental part of the project) as well as the industrial partners SKF and Michelin for helpful discussions.

■

REFERENCES

(1) Garces, J. M.; Moll, D. J.; Bicerano, J.; Fibiger, R.; McLeod, D. G. Adv. Mater. 2000, 12, 1835−1839. (2) Njuguna, J.; Pielichowski, K. Adv. Eng. Mater. 2003, 5, 769−778. (3) Eustache, R.-P. Handbook of Condensation Thermoplastic Elastomers; Wiley-VCH Verlag GmbH and Co. KGaA: Weinheim, 2006; pp 263−281. (4) Wang, M. J. Rubber Chem. Technol. 1998, 71, 520−589. (5) Heinrich, G.; Klüppel, M. Filled Elastomers Drug Delivery Systems; Advances in Polymer Science; Springer: Berlin, 2002; Vol. 160, pp 1− 44. (6) Heinrich, G.; Klüppel, M.; Vilgis, T. A. Curr. Opin. Solid State Mater. Sci. 2002, 6, 195−203. (7) Klüppel, M. Filler-Reinforced Elastomers/Sanning Force Microscopy; Advances in Polymer Science; Springer Berlin, 2003; Vol. 164, pp 1− 86, 10.1007/b11054. (8) Klüppel, M.; Heinrich, G. Kautsch. Gummi Kunstst. 2005, 58, 217−224. (9) Donnet, J.; Bansal, R.; Wang, M. Carbon Black: Science and Technology; Dekker: New York, 1993. 7291

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292

Macromolecules

Article

(45) Edwards, S. F.; Vilgis, T. Phys. Scr. 1986, 1986, 7. (46) Ndoro, T. V. M.; Böhm, M. C.; Müller-Plathe, F. Macromolecules 2012, 45, 171−179. (47) Ghanbari, A.; Ndoro, T. V. M.; Leroy, F.; Rahimi, M.; Böhm, M. C.; Müller-Plathe, F. Macromolecules 2012, 45, 572−584. (48) Scheidler, P.; Kob, W.; Binder, K. J. Phys. Chem. B 2004, 108, 6673−6686.

(10) Yurekli, K.; Krishnamoorti, R.; Tse, M. F.; McElrath, K. O.; Tsou, A. H.; Wang, H.-C. J. Polym. Sci., Part B: Polym. Phys. 2001, 39, 256−275. (11) Kohls, D. J.; Beaucage, G. Curr. Opin. Solid State Mater. Sci. 2002, 6, 183−194. (12) Sternstein, S. S.; Ramorino, G. K.; Jiang, B.; ZHU, A.-J. Rubber Chem. Technol. 2005, 78, 258−270. (13) Ehrburger-Dolle, F.; Hindermann-Bischoff, M.; Livet, F.; Bley, F.; Rochas, C.; Geissler, E. Langmuir 2001, 17, 329−334. (14) Koga, T.; Takenaka, M.; Aizawa, K.; Nakamura, M.; Hashimoto, T. Langmuir 2005, 21, 11409−11413. (15) Koga, T.; Hashimoto, T.; Takenaka, M.; Aizawa, K.; Amino, N.; Nakamura, M.; Yamaguchi, D.; Koizumi, S. Macromolecules 2008, 41, 453−464. (16) Dupres, S.; Long, D. R.; Albouy, P.-A.; Sotta, P. Macromolecules 2009, 42, 2634−2644. (17) Botti, A.; Pyckhout-Hintzen, W.; Richter, D.; Straube, E. Physica A 2002, 304, 230−234. (18) Ehrburger-Dolle, F.; Bley, F.; Geissler, E.; Livet, F.; Morfin, I.; Rochas, C. Macromol. Symp. 2003, 200, 157−168. (19) ten Brinke, J. W.; Litvinov, V. M.; Wijnhoven, J. E. G. J.; Noordermeer, J. W. M. Macromolecules 2002, 35, 10026−10037. (20) ten Brinke, J. W.; Debnath, S.; Reuvekamp, L. A. E. M.; Noordermeer, J. W. M. Compos. Sci. Technol. 2003, 63, 1165−1174. (21) Tiwari, M.; Noordermeer, J. W. M.; van Ooij, W. J.; Dierkes, W. K. Polym. Adv. Technol. 2008, 19, 1672−1683. (22) Tiwari, M.; Dierkes, W. K.; Noordermeer, J. W. M.; Datta, R. N.; Talma, A.; van Ooij, W. J. Kautsch. Gummi Kunstst. 2009, 62, 306− 312. (23) Litvinov, V. M.; Orza, R. A.; Klüppel, M.; van Duin, M.; Magusin, P. C. M. M. Macromolecules 2011, 44, 4887−4900. (24) Berriot, J.; Montes, H.; Lequeux, F.; Long, D.; Sotta, P. Macromolecules 2002, 35, 9756−9762. (25) Berriot, J.; Lequeux, F.; Monnerie, L.; Montes, H.; Long, D.; Sotta, P. J. Non-Cryst. Solids 2002, 307−310, 719−724. (26) Torres, J. A.; Nealey, P. F.; de Pablo, J. J. Phys. Rev. Lett. 2000, 85, 3221−3224. (27) Starr, F. W.; Schrøder, T. B.; Glotzer, S. C. Phys. Rev. E 2001, 64, 021802. (28) Yelash, L.; Virnau, P.; Binder, K.; Paul, W. Phys. Rev. E 2010, 82, 050801. (29) Varnik, F.; Baschnagel, J.; Binder, K. Phys. Rev. E 2002, 65, 021507. (30) Long, D.; Sotta, P. Macromolecules 2006, 39, 6282−6297. (31) Merabia, S.; Sotta, P.; Long, D. Macromolecules 2008, 41, 8252− 8266. (32) Hess, B. J. Chem. Theory Comput. 2008, 4, 116−122. (33) Hess, B.; Bekker, H.; Berendsen, H. J. C.; Fraaije, J. G. E. M. J. Comput. Chem. 1997, 18, 1463−1472. (34) van der Spoel, D.; Lindahl, E.; Hess, B.; van Buuren, A. R.; Apol, E.; Meulenhoff, P. J.; Tieleman, D. P.; Sijbers, A. L. T. M.; Feenstra, K. A.; van Drunen, R.; Berendsen, H. J. C. Gromacs User Manual version 4.5, www.gromacs.org, 2010. (35) Solc, K.; Stockmayer, W. H. J. Chem. Phys. 1971, 54, 2756− 2757. (36) Keddie, J. L.; Jones, R. A. L.; Cory, R. A. Europhys. Lett. 1994, 27, 59. (37) Keddie, J. L.; Jones, R. A. L.; Cory, R. A. Faraday Discuss. 1994, 98, 219−230. (38) Williams, G.; Watts, D. C. Trans. Faraday Soc. 1970, 66, 80−85. (39) Leutheusser, E. Phys. Rev. A 1984, 29, 2765−2773. (40) Hansen, J. P.; Levesque, D.; Zinn-Justin, J. Liquids, Freezing and the Glass Transition; Elsevier: Amsterdam, 1991. (41) van Zon, A.; de Leeuw, S. W. Phys. Rev. E 1999, 60, 6942−6950. (42) Lyulin, A. V.; Balabaev, N. K.; Michels, M. A. J. Macromolecules 2002, 35, 9595−9604. (43) Adam, G.; Gibbs, J. H. J. Chem. Phys. 1965, 43, 139−146. (44) Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Oxford University Press: New York, 1971. 7292

dx.doi.org/10.1021/ma300753e | Macromolecules 2012, 45, 7282−7292