Effect of Chain Flexibility and Interlayer Interactions on the Local

Jan 3, 2018 - Layered polymeric systems are widely used in membrane separation applications; chain mobility in these layered systems is a key consider...
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Effect of Chain Flexibility and Interlayer Interactions on the Local Dynamics of Layered Polymer Systems Sriramvignesh Mani and Rajesh Khare* Department of Chemical Engineering, Texas Tech University, Box 43121, Lubbock, Texas 79409-3121, United States S Supporting Information *

ABSTRACT: Layered polymeric systems are widely used in membrane separation applications; chain mobility in these layered systems is a key consideration in the design of the membranes. The transport properties of membrane polymers can be significantly altered by the perturbations in chain dynamics induced by the presence of an interface and also by the topological properties of the polymers constituting the layered systems. In this work, we use molecular dynamics (MD) simulations to determine the effects of polymer backbone flexibility and interlayer interactions on the glass transition and chain dynamics of polymer layers in the layered systems. We observed that the onset of glass transition of the entire layered system is governed by the stiffer polymer layer and is independent of the type of interactions between the layers. However, the interlayer interactions govern the strength of the glass transition of the entire layered system. Polymer mobility, on the other hand, exhibits a strong dependence on both the chain flexibility and the interlayer interactions. In systems with attractive interactions between the layers, the fully flexible polymer chains at the interface have a lower mobility than those in the bulk region of the layer; the behavior differs from that of rigid polymers, which have a higher mobility at the interface compared to that in the bulk. On the other hand, when the interactions between the layers are repulsive, each layer acts as a free-standing film with chains in both the layers exhibiting higher mobility at the interface. polymer−substrate interactions,31,32 and flexibility of polymer chains.33,34 For membrane separation applications, solute transport through the polymeric membrane is influenced by chain dynamics in the membrane. Recently, we and our collaborators studied use of bulk polyacrylate gels as candidate materials for separating dilute alcohol−water mixtures by a pervaporation process.35−37 Using molecular simulations, we studied the dynamics of water and ethanol molecules in these polyacrylate gels.36,37 A significant conclusion of the study was that water molecules exhibit dynamic coupling with the polymer chains due to their strong hydrogen-bonding interactions while ethanol molecules do not exhibit such coupling. This finding suggests the possibility that the observed dynamic coupling can be used to tune the separation efficiency of the polyacrylate gels. However, supporting the membrane on another polymer for gaining mechanical strength can significantly alter the chain dynamics in the membrane due to the complex interplay of interface induced chain confinement and interactions between the support and the membrane polymers. Any resulting deviations from the bulk behavior of polymers can, in turn, affect transport of solvent molecules if the two are dynamically coupled as was observed in our earlier study. Thus, the main

I. INTRODUCTION Polymeric membranes that are used in industry for applications such as fuel cells1,2 and separation of organic solvents3,4 are often supported on another polymer for mechanical strength,5,6 thus creating a layered system. Some of the commonly used layered polymer systems for separation applications consist of polydimethylsiloxane (PDMS), polyamide, or poly(N,Ndimethylaminoethyl methacrylate) membrane layer that is supported on polyimide (PI), poly(vinylidene fluoride) (PVDF), polysulfone (PSF), or cellulose acetate (CA).7−11 The properties of these layered assemblies depend on both the interlayer interactions (e.g., electrostatic interactions, hydrogenbonding interactions) and interfacial chain dynamics. A large number of previous studies12−20 of polymer films have focused on the glass transition in free-standing films or polymer films supported on substrates or confined between hard surfaces such as silica or gold.12,13,21 These have indicated varied behavior of the shift in the glass transition temperature (Tg) of the polymers. Specifically, a reduction in Tg for freestanding films, either an increase or a decrease in Tg for supported polymers, and no significant change in Tg for confined polymers were observed. In addition to the Tg values, differences in density,22,23 strength, and width of the glass transition,24,25 chain conformations,26−28 and chain diffusion29,30 were also observed between geometrically confined polymers and their bulk counterparts. These diverse behaviors have been attributed to the presence of free-surface layers, © XXXX American Chemical Society

Received: July 17, 2017 Revised: December 8, 2017

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DOI: 10.1021/acs.macromol.7b01519 Macromolecules XXXX, XXX, XXX−XXX

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polymer chains. The incompatible length scales of LJ potential and the harmonic bond potential prevent crystallization of the polymer chains. In what follows, we report all of the quantities in the reduced LJ units. A reduced quantity can be converted to its analogue in real units using the parameters of the bead− spring model, i.e., based on the mapping scheme used for coarse-graining the polymer system. For example, the Kremer− Grest model which is used extensively in polymer physics has been mapped to several realistic polymer systems.54 When mapped to polyethylene (PE) chains, each bead of the model represents three PE monomers, and the reduced time step of 0.002 and the reduced temperature of T* = 1 correspond to values of 1.32 × 10−13 s and 448 K in real units, respectively. We further note that the glass transition temperature (Tg) of fully flexible bead−spring chain systems is well characterized in the literature;48,55 the reported values are in the range of 0.4− 0.6. The MD simulations in this work were carried out using the LAMMPS simulation package.56 The temperature and pressure of the systems were maintained at the desired set point by the Nosé−Hoover thermostat and barostat, respectively.57,58 A time step of 0.002 was used to integrate the equations of motion. Periodic boundary conditions were applied in all directions. The coordinates of the atoms were stored at desired intervals to calculate the quantities of interest. The initial structures of the layered polymer systems were prepared by the simulated annealing polymerization technique.59,60 In this approach, beginning with a reaction mixture of monomer beads, the polymer chains are built by connecting the spatially closest reacting beads that are identified using the simulated annealing multivariable optimization technique. The layered systems studied in this work were prepared by a similar methodology in which the reaction mixtures of the individual layers were polymerized separately (i.e., during the polymerization process, bonds were not allowed to be formed between beads belonging to different layers). The chains thus formed in the different layers were assigned different kθ values leading to a difference in their flexibility. Specifically, the kθ value of one of the layers was fixed at 0, thus representing a layer of fully flexible chains. For the other layer, kθ value of either 1.5 or 3 was used to obtain chains with different levels of flexibilities. In addition to studying the effect of chain stiffness, another goal of this work is to study the effect of interlayer interactions on the chain dynamics in the layers. Thus, two types of interactions between the chains comprising the layers were considered: (i) LJ interactions truncated at a distance of 2.5, these systems being referred to as type I systems; (ii) LJ interactions truncated at a distance of 1.12. Truncation of potential at 1.12 leads to the purely repulsive Lennard-Jones or the Weeks− Chandler−Andersen (WCA) potential.61 These systems are referred to as type II systems in what follows. All of the layered systems were equilibrated for 10 million steps using MD simulations at a reduced temperature and reduced pressure of 1.5 and 0, respectively. The stability of the layered system was ensured by monitoring the variation in the width of the interfacial region as a function of time. For both type I and type II systems, the interfacial widths were nearly constant even at the highest temperature (T* = 0.9) at which we report results. These small variations in the interfacial width indicate that the polymer layers remain separate on the simulation time scale, thus confirming the stability of the layered structure. In experimental situations, stability of the layered systems could be a concern if the polymers constituting the layers have favorable interactions, in which case the chains in the adjoining layers

focus of this work is to elucidate the effects of a polymer support on the chain dynamics in the supported polymers, examples of which are mentioned earlier. The insights gained from the work will be valuable for selecting appropriate supports for the membranes. A few experimental and simulation studies describing the chain dynamics at the interface formed by two polymers can be found in the literature. These studies have primarily focused on the interdiffusion of the polymer chains at the interface.38−42 Rather than diffusion of the entire chains, our focus is on studying the effects of interlayer interactions and chain flexibility on the segmental dynamics and glass transition of each polymer constituting the layered system. Recently, Baglay and Roth43,44 and Lang et al.45 reported variations in the Tg of individual layers in layered polymeric systems as a function of the hardness of the confining layer and the interfacial interactions. In this work, we have studied the effects of both chain flexibility differences between the layers and interlayer interactions on the glass transition and chain dynamics in layered polymeric systems. Furthermore, in order to decipher the mechanism by which the layers affect each other, properties of the interfacial and bulk regions of each layer were determined separately. Clear differences were observed between the chain dynamics, local Tgs, and relaxation times in these regions. A coarse-grained, bead−spring model of polymer chains that allows for the investigation of the basic physics of the problem is used in this work. The coarse-grained models have previously been used to capture effects such as the dependence of polymer chain dynamics on the film thickness,46−49 effects of interfacial interactions on the Tg,18,45,49 structure of polymer chains near the interface,18,46,48 effects of polymer topology or the presence of side groups on the Tg of free-standing films,33,50−52 and the molecular weight dependence of Tg of free-standing films.53 In the coarse-grained model used in this study, the differences in the characteristics of the polymer membranes and the support are captured by varying (1) the backbone flexibility of the polymer chains and (2) the interfacial interactions between the layers. Further details of the model and the methodology used in this work are given in the next section.

II. SIMULATION METHODOLOGY The polymer chains were represented by a semiflexible bead− spring model that was recently used by Shavit and Riggleman.33 In this model, the nonbonded interactions between the beads are calculated via the standard 6−12 shifted Lennard-Jones (LJ) potential, while the bonded interactions are modeled using a harmonic bond potential and a cosine angular potential as follows: bond lengths:

bond angles:

U B(r ) = kB(r − σ )2

U θ(θ) = kθ(1 + cos θ)

(1) (2)

The parameters in the potential are assigned the same values as those employed by Shavit and Riggleman.33 Specifically, a force ϵ constant value of kB = 1000 σ 2 was used. Chains of different flexibilities were modeled by using kθ = 0, 1.5, and 3 ϵ. Here, σ and ϵ denote the diameter of the polymer beads and the depth of Lennard-Jones potential well, respectively. The model with kθ = 0 ϵ corresponds to fully flexible chains, while the model with kθ = 3 ϵ corresponds to the least flexible chains. LennardJones interactions were evaluated between all the beads in the B

DOI: 10.1021/acs.macromol.7b01519 Macromolecules XXXX, XXX, XXX−XXX

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identified by evaluating the coefficient of volume thermal expansion (CVTE) as a function of temperature. A.1. Individual Polymers: Stiffer Chains Have a Higher Tg and a Wider Glass Transition. The specific volume as a function of temperature for the individual polymers (i.e., not layered) is shown in Figure 2. The Tg values for the polymers

could interpenetrate the layers by the process of diffusion. Our model systems correspond to the experimental scenario where the stacked system is cooled to the glassy region on a time scale that is smaller than the time scale of chain diffusion. In the practical separation applications where the polymeric membranes are supported on a rigid polymeric substrate, the membrane polymers are typically lightly cross-linked and hence are not capable of interdiffusing thus maintaining the integrity of the layered system. Following the equilibration stage, the systems were cooled gradually to a temperature of T* = 0.9. A sample simulation box of a layered polymer system so obtained is shown in Figure 1.

Figure 2. Specific volume as a function of temperature for individual polymers (i.e., not layered) with kθ = 0 (black circle), kθ = 1.5 (red square), and kθ = 3 (green plus). Inset shows the coefficient of volume thermal expansion (CVTE) as a function of temperature. Error bars are smaller than the size of the symbols. Figure 1. Simulation box of a type I layered system with a layer of semi flexible chains (kθ = 1.5) and a layer of fully flexible chains (kθ = 0).

with kθ = 0, 1.5, and 3 are 0.43, 0.49, and 0.58, respectively. The uncertainty in Tg values is less than 0.002. These calculated Tg values are in excellent agreement with the values reported in the literature for the same systems.33 From Figure 2 inset (CVTE plot), two prominent effects of chain flexibility on the glass transition process can be observed: (i) width of the glass transition region decreases with increasing flexibility of the polymer chains, and (ii) the onset of the glass transition region shifts to a lower temperature with increasing flexibility of the polymer chains. In addition, chains with the lowest flexibility (kθ = 3) exhibit slightly higher CVTE in the rubbery state, while CVTE values in the glassy state are the same for all the systems. A.2. Layered Systems: Flexibility Differences between the Layers Affect the Width of the Glass Transition Region while Interactions between the Layers Affect the Strength of the Glass Transition. The glass transition process of the layered systems is governed by the interplay of flexibility differences between the different layers and also by the interactions between the layers. In what follows, we compare the glass transition behavior of type I (LJ interactions) and type II (WCA interactions) layered systems by first focusing on the volumetric properties of the entire layered assemblies. Figure 3 shows the variation in specific volume as a function of temperature for type I layered systems. These systems exhibit behavior similar to that of the individual polymers in that a single Tg is observed (see Table 1). The variation of specific volume with temperature for the layered systems with WCA interactions between the layers (type II systems) is shown in Figure 4. Analogous to the type I systems, type II systems also show a transition from the rubbery to the glassy state at a single temperature. However, a weak signature of a

For the purpose of comparison, individual polymer systems (i.e., not layered) with kθ values 0, 1.5, and 3 were also simulated. These individual polymer systems were exactly the same as the corresponding layers in the layered systems and thus consisted of 82 polymer chains in each case. Among these 82 polymer chains, 81 chains had 500 LJ beads per chain and the remaining chain had 324 beads. All of the quantities reported in this work were averaged over three replicas for each system.

III. RESULTS AND DISCUSSION A. Volumetric Properties and Tg of Individual and Layered Polymer Systems. Volumetric properties of the layered systems were determined and compared with those of the individual polymers, which we consider to be the base case. For this purpose, the volumetric properties of both the layered and the base case individual polymer systems were determined by following the protocol adopted in our previous work.62 Briefly, the systems were cooled in a stepwise manner from a high temperature of T* = 0.9 to a low temperature of T* = 0.1. Each cooling stage consisted of reducing the temperature by 0.05 followed by MD simulation for a duration of 10 million steps. The specific volume of the system at a particular temperature was determined by averaging over the last half of the simulation run at that temperature. The Tg was then obtained as the point of intersection of the linear fits to the rubbery and glassy regions on the volume−temperature (V−T) plot. For this purpose, the rubbery and glassy regions were C

DOI: 10.1021/acs.macromol.7b01519 Macromolecules XXXX, XXX, XXX−XXX

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distinct Tgs corresponding to the Tgs of respective individual polymers can be observed for type II layered systems when the interactions between the layers are weak compared to the interactions within the layers (i.e., ϵ11, ϵ22 ≫ ϵ12). From the CVTE plots (see Figure 3 and 4 insets), it can be seen that the onset temperature for glass transition of kθ = 0, 3 layered systems is higher than that of the kθ = 0, 1.5 systems. Also, the CVTE in the rubbery region is higher for type II layered systems than the corresponding type I layered systems. An important observation is that (see Table 1) the Tgs of the entire type I and type II layered systems with given flexibility values of the chains are independent of the interactions between the layers. This observation is consistent with the CVTE plots in the insets of Figures 3 and 4, which show similar width of the glass transition region for the corresponding systems in type I and type II cases. On the basis of the above observations, it can be concluded that the interactions between the layers affect only the strength of the glass transition (i.e., the difference between the CVTE values of rubbery and glassy regions) but not the values of the Tgs of the overall systems. Additionally, from these results, it is apparent that the CVTE of type I and type II layered systems with given chain flexibility values exhibit a qualitatively similar trend with temperature as that of the CVTE of the polymer comprising the layer with the lower flexibility. In particular, for both type I and type II systems, the onset temperature and the end temperature of the glass transition region for the layered systems with kθ = 0, 1.5 and kθ = 0, 3 coincide approximately with their values for the kθ = 1.5 and kθ = 3 individual systems, respectively. The above behavior suggests that the width of the glass transition in layered systems is governed by the least flexible layer. In summary, interactions between the layers affect the strength of the glass transition of the layered systems whereas the flexibility differences between the layers affect the width of the glass transition of the layered systems. These conclusions apply to the entire layered assembly. In what follows, we probe the mechanisms underlying these observations by focusing on the properties of the interfacial and bulk (i.e., far away from the interface) regions of each of the layers in the layered systems. A.3. Individual Layers in the Layered Systems: Volumetric Properties of Regions Far Away from the Interface Are Very Similar to Properties of the Individual Polymers. The first task in this analysis process was to determine the interfacial and bulk regions in the layered systems. For this purpose, the density profiles within each layer in the layered systems were determined by dividing the entire simulation box into a fixed number of bins of size between 6 and 7 (given the small variation in the dimensions of the simulation box between different structures and also at different temperatures, the bin size varied slightly from system to system). A typical density profile of a layered system is shown in Figure 5. Based on the density profile, the bulk and interfacial regions were defined as follows. For each polymer layer, its bulk region was defined as the region that is devoid of the other polymer, i.e., the region in which the density of the other polymer is zero (shown as bulk in Figure 5). The interfacial regions, on the other hand, were identified as those regions in which chains of both polymers exist and the density of each layer varies from zero to its bulk region value (shown as interface in Figure 5). The width of the total interfacial region in type I systems is around 13. Owing to the lack of attractive interactions between the layers, a sharp interfacial region is seen

Figure 3. Specific volume as a function of temperature for the entire type I layered systems. The systems are represented in the following colors: layered systems with kθ = 0, 1.5 (blue circle) and layered systems with kθ = 0, 3 (brown square). Inset shows the coefficient of volume thermal expansion (CVTE) as a function of temperature. Error bars are smaller than the size of the symbols.

Table 1. Tgs for Layered Systemsa type I type II

kθ = 0, 1.5 layered systems

kθ = 0, 3 layered systems

0.47 0.47

0.51 0.52

a

Values are reported for the entire layered system in each case. The uncertainty in the Tg values is less than 0.002.

Figure 4. Specific volume as a function of temperature for the entire type II layered systems. The systems are represented in the following colors: layered systems with kθ = 0, 1.5 (blue circle) and layered systems with kθ = 0, 3 (brown square). Inset shows the coefficient of volume thermal expansion (CVTE) as a function of temperature. Error bars are smaller than the size of the symbols.

second Tg can be observed for the kθ = 0, 3 type II layered system from Figure 4 inset. This observation suggests that two D

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in the study of volume-temperature behavior. The temperature variation of specific volume of the bulk regions so identified for the layers in kθ = 0, 1.5 and kθ = 0, 3 type I layered systems is shown in Figure 6. For the purposes of comparison, the specific volumes of respective individual polymers from Figure 2 are also included. From Figure 6, it can be seen that the bulk regions of each of the layers in type I layered systems have the same specific volumes as those of their respective individual polymer systems. Similar behavior is also observed in type II layered systems (see Figure S2). We thus conclude that the effects of the density variation in the interface do not propagate deep into the layers, and the regions far away from interface exhibit volume−temperature behavior that is very similar to that of the single polymer (nonlayered) systems. B. In Layered Systems with Attractive Interactions between Layers (Type I Systems), Interfacial Dynamics of Each Layer Is Affected by the Glass Transition Behavior of the Other Layer, While in Layered Systems with No Attractive Interactions between Layers (Type II Systems), Each Layer Acts as a Free-Standing Film. In order to investigate the effects of the interface on the polymer chain mobility, the chain dynamics in the interfacial and bulk regions were determined in the temperature range T* = 0.9 to T* = 0.3. For this purpose, the dynamics of the beads in these regions were quantified using the self-part of the van Hove distribution function. Mathematically, the self-part of the van Hove distribution function is expressed as follows:63

Figure 5. Density profile of kθ = 0 (dark brown circle) and kθ = 1.5 (dark cyan square) layers in kθ = 0, 1.5 type I layered system at temperature T* = 0.3.

in type II systems (see Figure S1 in the Supporting Information). The mean density (or specific volume) values of each of the layers in the regions far away from the interface (i.e., bulk regions) were taken as the average over all the bins in those respective bulk regions. These mean density values were used

Gs(r, t ) =

1 N

N

∑ δ⟨(r + ri(t ) − ri(0))⟩ i=1

(3)

Figure 6. Specific volume (red circles) as a function of temperature for bulk regions (i.e., regions far away from interface) of (A) kθ = 0 layer in kθ = 0, 1.5, (B) kθ = 1.5 layer in kθ = 0, 1.5, (C) kθ = 0 layer in kθ = 0, 3, and (D) kθ = 3 layer in kθ = 0, 3 type I layered systems. For comparison purposes, specific volume values for the corresponding individual polymer systems are shown as open blue squares. E

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Figure 7. van Hove distribution function of the chains in the interface (open blue square) and the bulk regions (filled red circle) of the kθ = 0 layer in kθ = 0, 1.5 type I layered systems. For comparison purposes, the van Hove distribution function of individual kθ = 0 (black dashed line) system is also shown.

Figure 8. van Hove distribution function of the chains in the interface (open blue square) and the bulk regions (filled red circle) of the kθ = 1.5 layer in kθ = 0, 1.5 type I layered systems. For comparison purposes, the van Hove distribution function of individual kθ = 1.5 (black dashed line) system is also shown.

Here, ri(t) represents the position vector of bead i at time t and N denotes the total number of beads. The van Hove functions are normalized such that ∫ 4πr2Gs(r, t) dr = 1. The van Hove distribution function denotes the probability of finding a bead at location r at time t provided the same bead was at origin at time t = 0. In order to evaluate the dynamics of the polymer

chains as a function of the distance from the interface, the simulation box was divided into bins of thickness 6−7 as described earlier. The polymer beads were assigned to specific bins based on their initial location (i.e., at measurement time t* = 0). The van Hove distribution functions were then calculated for each of the bins using eq 3 to characterize the chain F

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Figure 9. van Hove distribution function of the chains in the interface (open blue square) and the bulk regions (filled red circle) of the kθ = 0 layer in kθ = 0, 1.5 type II layered systems. For comparison purposes, the van Hove distribution function of individual kθ = 0 (black dashed line) system is also shown.

attractive interfacial interactions which tend to retard (accelerate) chain dynamics of the fully (semi)flexible chains. At very high temperature (T* = 0.9), both flexible (kθ = 0) and stiff (kθ = 1.5) chains have enough thermal kinetic energy to overcome the attractive interfacial interactions. The interface thus has a slightly higher chain mobility than the bulk at these conditions. As the temperature is lowered, the relative importance of attractive interaction increases and the layers start affecting each other’s dynamics. Thus, for the flexible chains (kθ = 0), at T* = 0.65, bulk and the interface have the same mobility, while at T* = 0.5 interfacial dynamics is significantly slower than the bulk dynamics due to the influence of the other layer that is composed of relatively inflexible chains (kθ = 1.5). Opposite behavior is exhibited by the stiffer chains (kθ = 1.5). In their case, for both T* = 0.65 and T* = 0.5, the interfacial region has a higher mobility than the bulk region due to the influence of the mobile layer of the flexible chains. Deep in the glassy state (T* = 0.3), chain dynamics is severely hindered in all cases, and the interfacial and bulk regions show the same behavior of chain dynamics. Additionally, it can be observed that the dynamics in the bulk regions of each of the layers are not affected by the interfacial interactions and are the same as the dynamics of their respective individual polymer systems. These trends in the mobility of fully flexible and semiflexible chains in the interfacial region can be compared with those predicted by the recent limited mobility lattice model.64,65 In particular, the lattice model for free volume dynamics was applied to a bilayer system in which the materials constituting the layers showed different free volume mobilities.64 Furthermore, the adjacent sites in the two layers across the interface interacted with each other and thus affected each other’s free volume state; i.e., the system had a strong interface, analogous to our system with attractive interactions between the layers. Similar to our simulation observations, the lattice model results also showed that in the interfacial region the

dynamics at a given distance from the interface. The calculations were performed up to a time t* = 2000; this time value was chosen so that in this time period, the majority of the beads remained in the bins to which they were assigned based on their initial location. We also note that over this time period, the dynamics of beads (as captured by the van Hove distribution function) are quantitatively similar in the directions parallel and normal to the interface. The van Hove functions at different temperatures and for interface and bulk regions of each of the layers in kθ = 0, 1.5 type I layered systems at t* = 2000 are shown in Figures 7 and 8. At a high temperature of T* = 0.9, which is well above the onset temperature of glass transition of both the kθ = 0 and 1.5 individual polymer systems (see Figure 2), the mobility of the chains in the interfacial region in kθ = 0 layer is higher than that of the chains in the bulk region as seen from the slightly wider van Hove distribution function of the interfacial layer. On reducing the temperature to T* = 0.65, the mobility of the fully flexible chains in the interfacial and the bulk regions effectively become equal. On further cooling to T* = 0.5, the interfacial chain mobility decreases compared to that in the bulk region. At temperatures below the Tg of both the layers (T* = 0.3), the chain mobility in both interfacial and bulk regions in the kθ = 0 layer significantly reduces and becomes equal. The kθ = 1.5 layer shows qualitatively similar behavior as the kθ = 0 layer at the highest (T* = 0.9) and the lowest (T* = 0.3) temperatures. However, at the two intermediate temperatures, the interfacial region of kθ = 1.5 layer has a higher chain mobility compared to the chain mobility in its bulk region. A similar trend in the dynamics of fully flexible and semiflexible chains is also observed for kθ = 0, 3 type I layered systems (see Figures S3 and S4). We interpret these results as follows. In general, the chains at the interface have a propensity for higher mobility compared to the bulk; this tendency is opposed (augmented) by the G

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Figure 10. van Hove distribution function of the chains in the interface (open blue square) and the bulk regions (filled red circle) of the kθ = 1.5 layer in kθ = 0, 1.5 type II layered systems. For comparison purposes, the van Hove distribution function of individual kθ = 1.5 (black dashed line) system is also shown.

Figure 11. Debye−Waller factors (DWF) of the interfacial (open blue square) and the bulk regions (filled red circles) in the kθ = 0, 1.5 layered systems. Figure shows DWF for the (A) kθ = 0 layer in type I systems, (B) kθ = 1.5 layer in type I systems, (C) kθ = 0 layer in type II systems, and (D) kθ = 1.5 layer in type II systems.

A similar analysis for kθ = 0, 1.5 type II layered systems is shown in Figures 9 and 10. In this case, for both polymers (i.e., polymers of higher and lower flexibility), the interfacial region has a higher chain mobility than the corresponding bulk region of a given layer at all temperatures except at the lowest

mobility of the layer that has a slower free volume dynamics is enhanced compared to the mobility in its bulk region (i.e., away from the interface), while mobility of the layer with a faster free volume dynamics is reduced compared to the mobility of its bulk region. H

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Table 3. Tgs of the Interfacial and the Bulk Regions in the kθ = 0, 3 Layered Systemsa

temperature studied where deep in the glassy state, the interfacial and bulk regions have effectively the same chain mobility. This observation suggests that in the absence of attractive interactions between the layers, each layer acts as a free-standing film with the interfacial region within the layer exhibiting higher chain mobility than the bulk region. The observation is consistent with the prior work of Lang et al.45 where it was observed that when the interactions between the layers are purely repulsive, the Tg of each layer corresponds to its free-standing film Tg value. Similar behavior is also exhibited by kθ = 0, 3 type II layered systems (see Figures S5 and S6). Furthermore, the dynamics in the bulk regions of the individual layers are not perturbed by the interface analogous to the earlier observation for the type I systems. C. Interfacial and Bulk Regions of the Individual Layers Have Different Tgs. The differences in the chain mobilities in the interfacial and bulk regions of each of the layers suggest that these regions will turn glassy at different temperatures. Following previous work, the Debye−Waller factor (DWF), which provides a measure of the mobility of the atoms on molecular time scales, was used to estimate the Tgs of the bulk and interfacial regions.66−68 For this purpose, the DWF of these regions was calculated at different temperatures, and then the local Tg was defined as the temperature at which the temperature dependence of DWF changes. In this work, the DWF was obtained as the mean-squared displacement at time t* = 1000. This value of time was chosen since the bulk regions of each of the layers in the layered systems exhibit a caging behavior on that time scale at the lowest temperature (T* = 0.3) studied. The variation of DWFs with temperature for each of the layers in the kθ = 0, 1.5 and kθ = 0, 3 type I and type II layered systems is shown in Figure 11 and Figure S7, respectively. As seen from the figures, in all cases (type I and type II systems and interfacial and bulk regions), the DWFs show linear dependence on temperature albeit with different slopes in the rubbery and the glassy states. In fact, the DWF is effectively constant in the glassy state. The Tg was obtained for each case as the point of intersection of linear fits to the DWF values in the glassy and the rubbery states. The local Tgs in the bulk and interfacial regions of individual layers so obtained are listed in Table 2 and Table 3. Consistent

kθ = 0, 3 layered systems kθ = 0 type I type II a

a

bulk

interface

bulk

interface

0.45 (0.021) 0.36 (0.005)

0.51 (0) 0.51 (0.005)

0.49 (0.014) 0.47 (0.009)

interface

0.42 (0) 0.42 (0)

0.49 (0.008) 0.39 (0.025)

0.67 (0.008) 0.65 (0.024)

0.57 (0.057) 0.62 (0.025)

Values in the parentheses denote the uncertainty in the Tgs.

hbulk h Tg,bulk + interface Tg,interface (4) H H Here, ⟨Tg⟩, hbulk, hinterface, H, Tg,bulk, and Tg,interface are the average Tg of a film, thickness of the bulk region, thickness of the interfacial region, total thickness of the film, Tg of the bulk region, and Tg of the interfacial region, respectively. Following the same approach, the average Tgs of each of the layers in both type I and type II systems were modeled by applying the twolayer model to the layers separately. For this purpose, each layer was considered to be divided into interfacial and the bulk regions. The average Tg of the individual layers (i.e., Tg on the left-hand side of eq 4) was obtained by calculating the DWF and its temperature dependence. For this calculation, all of the polymer beads of a layer were considered together without dividing them into bulk and interfacial regions. For the purposes of evaluating the right-hand side of eq 4, Tg values of the different regions that are given in Tables 2 and 3 were used. The thicknesses of the bulk and interfacial regions were obtained from the density profiles mentioned earlier. The Tg values obtained from the DWF calculations and from the layer model are compared in Tables 4 and 5. It is clear that the layer model predictions are in good agreement with the Tg values of the layer that were determined directly from simulations. Thus, the layer model captures the overall dynamics of each of the layers in the multilayered systems.

kθ = 1.5

0.42 (0.009) 0.42 (0)

bulk

⟨Tg⟩ =

kθ = 0, 1.5 layered systems

type I type II

interface

the enhanced dynamics in the interfacial regions, interfacial regions of layers of both fully flexible and semiflexible chains turn glassy at a lower temperature than the corresponding bulk regions. It is clear from these local Tg results that the interface and the bulk regions of each of the layers in both type I and type II layered systems exhibit different dynamics. For thin film systems that also exhibit similar spatial dynamic heterogeneity, the two-layer model25,69 and its variant, the three-layer model,16 have been extensively used in the literature to model the average properties (e.g., Tg and CVTE) of the entire film in terms of contributions from each of the regions (bulk and interface) that are weighted by their thickness. The resulting expression for the film Tg in the layer model is as follows:

Table 2. Tgs of the Interfacial and the Bulk Regions in the kθ = 0, 1.5 Layered Systemsa kθ = 0

kθ = 3

bulk

Values in the parentheses denote the uncertainty in the Tgs.

Table 4. Average Tg of the Individual Layers in the kθ = 0, 1.5 Layered Systemsa

with the prior discussion, differences in the mobility between the interfacial and bulk regions in the layered systems lead to different Tgs in these regions. In type I layered systems, due to the attractive interactions between the layers, the interfacial region in the layer of fully flexible chains undergoes a transition to a glassy state at a higher temperature than the corresponding bulk region. The interfacial region of the semiflexible layer, on the other hand, turns glassy at a lower temperature than the corresponding bulk region. For type II layered systems, due to

kθ = 0, 1.5 layered systems kθ = 0

type I type II a

I

kθ = 1.5

⟨Tg⟩

layer model prediction

⟨Tg⟩

layer model prediction

0.42 (0.005) 0.38 (0.005)

0.43 0.41

0.49 (0.012) 0.50 (0.005)

0.50 0.50

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as the time at which the scattering function decayed to a value of 0.3. For this purpose, the scattering function was fitted to the Kohlrausch−Williams−Watts (KWW) stretched exponential functional form expressed as follows:72

kθ = 0, 3 layered systems kθ = 0

type I type II a

kθ = 3

⟨Tg⟩

layer model prediction

⟨Tg⟩

layer model prediction

0.43 (0) 0.41 (0.005)

0.45 0.42

0.64 (0.005) 0.64 (0.017)

0.63 0.65

⎡ ⎛ t ⎞β⎤ Fs(q, t ) = exp⎢ −⎜ ⎟ ⎥ ⎣ ⎝τ⎠ ⎦

Here, τ and β are the relaxation time and stretching exponent, respectively. The KWW fit function was then used to determine the time at which the scattering function decays to a value of 0.3. These calculations were performed only in the temperature range of 0.9−0.5 for type I systems and 0.9−0.55 for type II systems, i.e., in the rubbery state, since the scattering function decay was extremely slow at low temperatures. The structural relaxation time values determined from the scattering functions are plotted in Figure 12 and Figure S8 as a function of temperature. Consistent with the previous results, in type I layered systems, due to dynamic coupling between layers, the dynamics of fully flexible chains in the interfacial region are slower than that of the chains in the corresponding bulk regions at temperatures in the vicinity of the Tg of the semiflexible layer. The semiflexible chains in the interfacial region, on the other hand, exhibit enhanced dynamics compared to these chains in the corresponding bulk regions. In type II layered systems in which both layers essentially act as free-standing films, the dynamics of both fully flexible and semiflexible chains are enhanced in the interfacial regions compared to the chains in the bulk regions, and this difference in mobility increases with decreasing temperature. The structural relaxation times exhibit a super-Arrhenius behavior as can be seen from the figures. For glass-forming

Values in the parentheses denote the uncertainty.

D. Local Glass Transition Dynamics of Each of the Layers Deviate from Vogel−Fulcher−Tammann (VFT) Behavior below Their Respective Tg. To gain further insight into the chain dynamics, the structural relaxation time (τα) of bulk and interfacial regions was quantified. For this purpose, the self-part of the intermediate scattering function was calculated as follows:70 Fs(q, t ) =

1 N

N

∑ ⟨exp[−iq·(rj(t ) − rj(0))]⟩ j=1

(6)

(5)

Here, q is the wave vector and rj(t) is the position vector of bead j at time t. Because of the anisotropic nature of the systems in the z direction (i.e., normal to the plane of the layers), the scattering function was evaluated only in the x−y plane. A wavenumber of 5.98 that corresponds to the location of the first peak in the structure factor was used for these calculations. Thus, wave vectors with magnitude 5.98 were generated in the x−y plane, and the scattering functions were then obtained by averaging over 12 such vectors. Following prior literature,33,51,71 the structural relaxation time was defined

Figure 12. Structural relaxation time (τα) of the interfacial (open blue square) and the bulk regions (filled red circles) in the kθ = 0, 1.5 layered systems. Figure shows temperature dependence of τα for the (A) kθ = 0 layer in type I systems, (B) kθ = 1.5 layer in type I systems, (C) kθ = 0 layer in type II systems, and (D) kθ = 1.5 layer in type II systems. Dotted lines are the VFT fits to the data. J

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while semiflexible chains in the interface exhibit higher mobility than their bulk regions. In type II layered systems where the interactions between the layers are repulsive, each layer acts as a free-standing film with the chain mobility being higher at the interface than in the bulk region in all cases. Our results indicate that in layered polymer systems, chain dynamics in a given layer is affected by both the interactions with the other layer and flexibility of the chains in the other layer. These findings have implications in membrane separation applications where it has previously been noted that if a penetrant has strong interactions with the polymer, its mobility is coupled with the membrane polymer chain mobility. Our finding that the interfacial and bulk regions in the layers have different chain mobilities also has implications for membrane design. For example, usage of a stiff polymer that has attractive interactions with the polymer membrane as a support will retard the membrane polymer dynamics at the interface at temperatures well above its Tg. This effect can, in turn, alter the penetrant dynamics in the interfacial region. Thus, the relative amounts of the interfacial and bulk regions could be a key design consideration for such layered systems, this aspect can be changed by tuning the pore size of the support in the case of porous membranes. The model used in this work considered only the physics of the problem, and the effects of specific interactions such as hydrogen bonding were not considered. Currently, work using atomistic simulations is underway to decipher the effects of these specific interactions on the chain dynamics in layered polymeric systems.

systems, the super-Arrhenius behavior of structural relaxation time with temperature is usually described by the Vogel− Fulcher−Tammann (VFT) functional form as follows:71,73 ⎛ DT0 ⎞ τα = A exp⎜ ⎟ ⎝ T − T0 ⎠

(7)

Here, A, D, and T0 are the fitting parameters. The VFT fits to the temperature-dependent structural relaxation times are also shown in the figures. The VFT equation quantitatively describes the temperature-dependent behavior of both the layers in type I and type II systems with few exceptions. The deviations from VFT behavior are seen only at the lower end of the temperature range investigated for the following systems: kθ = 1.5 chains in bulk regions of type I systems, kθ = 3 chains in bulk regions of both type I and type II systems, and kθ = 3 chains in interfacial regions of both type I and type II systems. We note that the temperatures at which these deviations from the VFT behavior are observed are all below the corresponding Tgs for these regions as determined from DWF (see Tables 2 and 3). In other words, consistent with observations for other glass-forming systems (e.g., naturally aged amber74), the local relaxation times of our systems exhibit VFT behavior above the Tg but show deviation from it below the Tg. Furthermore, these deviations suggest that the relaxation times of the chains in the interfacial and the bulk regions will not diverge at a finite temperature below the Tg. In closing, we note that at the temperatures where the deviation from the VFT behavior is observed, the relaxation time value is approximately 1000, i.e., the same as the time scale used for the evaluation of DWF (and hence the Tg), thus further confirming that the deviation from VFT behavior occurs below the Tg.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b01519. Density profile of type II layered systems, variation in the specific volume of bulk regions with temperature for type II layered systems, van Hove distribution functions of the interface and bulk regions of the layers in kθ = 0, 3 type I and type II layered systems, temperature dependence of the Debye−Waller factors of the interfacial and bulk regions of kθ = 0, 3 type I and type II layered systems, and structural relaxation time as a function of temperature for interfacial and bulk regions of kθ = 0, 3 type I and type II layered systems (PDF)

IV. SUMMARY AND CONCLUSIONS Molecular simulations were used to determine the effects of chain flexibility and interlayer interactions on the glass transition process and chain dynamics in layered polymer systems. For both type I layered systems (i.e., systems with attractive interactions between the layers) and type II layered systems (i.e., systems with no attractive interactions between the layers), the width of the glass transition process is governed by the stiffer polymer chains as these chains exhibit a wider transition region. However, due to the repulsive interactions between the layers, the strength of the glass transition of type II layered systems is higher than that of the corresponding type I layered systems. Nevertheless, in both type I and type II layered systems, volumetric properties and chain dynamics in the bulk regions (i.e., regions far away from the interface) of each of the layers are not perturbed by the presence of the other layer, and these bulk regions exhibit behavior similar to that of their respective individual polymers. There is a clear difference in chain mobilities in the interfacial and bulk regions of the layers in various systems which leads to differences in chain relaxation times and local Tgs of these regions. In general, the relaxation times show VFT behavior above Tg but deviate from it below the Tg. The relative difference in the mobility of the polymer chains in the interface and the bulk regions is determined by the effective interactions between the polymer layers and the chain flexibility. In systems with attractive interactions between the layers (type I), the dynamics of the interfacial regions of each of the layers is affected by the other layer. Fully flexible chains in the interfacial region have lower mobility compared to their bulk regions,

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (R.K.). ORCID

Rajesh Khare: 0000-0002-8859-766X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based on the work supported by the National Science Foundation (NSF) under Grant NSF CMMI-1335082. We also acknowledge the High Performance Computing Center (HPCC) at Texas Tech University at Lubbock for providing computational resources for running molecular simulations. K

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