Nonmonotonic Glass Transition Temperature of Polymer Films

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Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Nonmonotonic Glass Transition Temperature of Polymer Films Supported on Polymer Brushes Hoyeon Lee,† Vaidyanathan Sethuraman,‡ Yeongsik Kim,† Wooseop Lee,† Du Yeol Ryu,*,† and Venkat Ganesan*,‡ †

Department of Chemical and Biomolecular Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Korea Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712, United States

‡

S Supporting Information *

ABSTRACT: We modulated the grafting density (σ) and chain length of polystyrene (PS) brushes on substrates to probe their effect on the glass transition temperature (Tg) in overlaying PS thin films. The Tg of PS films was analyzed as a function of brush thickness (or σRg2), where Rg is the radius of gyration of PS brushes. Our results indicate that PS films below 90 nm exhibit a nonmonotonic dependence of Tg on grafting density with the maximum Tg occurring near the regime of wetting−dewetting transition of the overlaying melt. The maximum Tg was found to be higher than the value of bulk PS system. Computer simulation results suggest that such trends arise as a consequence of the environment of enhanced friction presented by the brush layer and the overlap between the polymer film and the grafted layer.

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INTRODUCTION Interfacial interactions modulated by polymer brushes1−3 have attracted considerable attention in the past few decades due to the potential impact on a wide range of physicochemical properties such as wettability,4−12 adhesion,13−16 slip,17,18 and lubrication.19−22 In this regard, significant interest has arisen in the interactions between a polymer melt and a substrate grafted with polymer brushes,23 especially for the situation of polymers supported on chemically identical polymer brushes. For such cases, the wetting−dewetting behavior is dependent on three parameters: N, P, and σ, where N and P are the degrees of polymerization of polymer brush and melt, respectively, and σ denotes the grafting density (i.e., grafted chains per unit surface area). Scaling ideas24−26 and self-consistent field theory23 results have shown that sparse brush grafting densities and/or short polymers lead to the situation where the brush is solvated (wetted) by the polymer melt. In contrast, for denser grafting densities and/or longer polymers, the polymer melt is expelled from the brush (dewets). Such behaviors are reflected both in the effective interfacial tension between the brush and the melt and in the effective interactions between two grafted surfaces in a polymer melt.23,24 Studies which have probed the role of bare, ungraf ted substrate−polymer interactions of thin polymer films have concluded that, in general, unfavorable polymer−substrate enthalpic interactions result in a depression of Tg (relative to the bulk conditions), whereas favorable polymer−substrate enthalpic interactions lead to an increase in the Tg of polymer films.27−41 However, studies on the polymer melt−brush interfacial behavior and their interactions have been a © XXXX American Chemical Society

complicated issue due to the influence of the restricted mobility of grafted chains.36,39,40,42−52 For instance, Nealey et al. studied Tg of hydroxyl-terminated polystyrene films that were thermally annealed on a Si substrate. Interestingly, an elevation of Tg for PSOH films below 100 nm was attributed to the interfacial interaction between ungrafted polymer melt and grafted chains.43 Meanwhile, it was suggested by Tsui et al. that a decrease in Tg of PS films on PS brushes is comparable with that on the Si substrate.39 In a recent work,45 we probed the glass transition behavior of PS films supported on chemically identical polymer brushes with varying molecular weight and observed trends consistent with such general considerations. Explicitly, a greater decrease in Tg of thin PS films was observed in the PS films supported on extremely short and dense brushes. Such results were explained with the excess positive interfacial energy of melts in contact with short and dense brushes.23 From the simulation perspectives, furthermore, there has been a wealth of studies which probed the cooperative motion of polymers under confinement 53−56 and the propagation length scale of such motion for different geometric constraints.57−59 Despite the results discussed above, the connection among wetting−dewetting phenomena, interfacial energies, and the glass transition of polymer melts supported on polymer brushes is far from resolved.52 For instance, in an earlier work, Tsui and co-workers reported the glass transition behavior of PS films Received: February 7, 2018 Revised: May 8, 2018

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DOI: 10.1021/acs.macromol.8b00290 Macromolecules XXXX, XXX, XXX−XXX

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dependence of Ψ at constant λ (560 nm) was chosen for better resolution in determining Tg by the intersection temperature of the two linear regressions between the glassy and rubbery regimes. The in situ measurements of the films were conducted to collect ellipsometric parameter of Ψ when the temperature increased from 30 to 170 °C at a heating rate of 2 °C/min.

supported on random copolymer brushes of polystyrene-rpoly(methyl methacrylate) (PS-r-PMMA).40 They varied the composition of the brushes to tune the interfacial energy with the PS40 and demonstrated that the Tg of overlaying polymer melt cannot be rationalized exclusively in terms of the interfacial energy between the melt and the brush. Such contrasting observations (and others52) raise the following questions: (i) What is the influence of grafted layers on the dynamics of overlaying polymer films? (ii) Is there indeed a correlation between the wetting−dewetting phenomena and the overall dynamics of the polymer films? In this paper, we present results from in situ ellipsometric experiments for a system of a polymer melt supported on the grafted layers of chemically identical polymers, which indicate a nontrivial influence of grafted polymers on the Tg of overlaying films. We used hydroxyl end-functionalized polystyrenes (PSOHs) to modulate grafting density (σ) and chain length of PS brushes attached to the substrates. The thicknessdependent Tg of the overlaying PS films was measured as a function of brush thickness (and σRg2, where Rg denotes the radius of gyration of PSOH melts). Surprisingly, we observe a nonmonotonic variation of the Tg of the PS films with the grafting density of the brushes manifesting a maximum value exceeding the bulk Tg of PS melt. We present computer simulation results for the local segmental dynamics in overlaying polymer melt and identify the competing factors of the enhanced friction arising from the polymer brush and wetting−dewetting transition as the physics underlying our experimental observations.

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SIMULATION DETAILS Computer Simulations. We employed coarse-grained molecular dynamics simulations to characterize the local segmental dynamics of the homopolymer film in the presence of grafted substrates.27,29,30,60 The simulation system consisted of both “free” homopolymer and grafted chains within the simulation box. We considered a symmetric system where both the top and bottom surfaces contained grafted homopolymers sandwiched by the free homopolymers. In this manner, the simulation was rendered symmetrical to isolate specific influence of grafted polymers, while eliminating the potential effects arising from the other (air) interface. Both the homopolymer and grafted chains were modeled using a bead−spring model wherein the nonbonded interactions were described using a Lennard-Jones (LJ) potential: ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ d d ⎪ ⎪ 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ + ϵ, r ≤ rc ⎝ r ⎠ ⎥⎦ Unb(r ) = ⎨ ⎢⎣⎝ r ⎠ ⎪ ⎪ 0, r > rc ⎩

(1)

where ϵ, d, and rc represent the strength of interaction energy, diameter of the particles, and the cutoff radius for the interactions (rc = 2 × 21/6d), respectively. The bonded interactions were described using finite extensible nonlinear elastic (FENE) potentials of the form

EXPERIMENTAL METHODS

Polystyrene (PS) was synthesized by the living anionic polymerization of styrene in cyclohexane solution using sec-butyllithium as an initiator at 45 °C under purified argon environment. Number-average molecular weight (Mn) and dispersity (Đ = Mw/Mn) of PS were characterized to be 98 600 g/mol and 1.03, respectively, which were measured by size-exclusion chromatography (SEC) with PS standards. Hydroxyl end-functionalized polystyrenes (PSOHs, purchased from Polymer Source), PSOH-06 (Mn = 6000 g/mol and Đ = 1.07) and PSOH-14 (Mn = 14 000 g/mol and Đ = 1.09), were used for a grafting-to method to prepare the PS brushes to the substrates. To obtain large-area uniformity of PS brushes, PSOH solutions (1 wt %) in toluene were spin-coated onto a 4 in. Si substrate at 800 rpm for 60 s, and the films were annealed at 130 °C for various times under vacuum. The film was thoroughly cleansed with toluene (as a good solvent) to remove unattached PSOHs until the film thickness became consistent; this process forms a layer thickness of PS brushes onto the native oxide surface (∼2 nm) of Si substrates. PS brushes were further reannealed at 130 °C for 12 h under vacuum to remove residual solvent. PS films were fabricated onto PS brushes by spin-coating typically at 2000−4500 rpm for 60 s. The film thickness was controlled by varying the concentration (1.0−4.0 wt %) in toluene to produce 30−120 nm. Subsequently, the thermal annealing of PS films was done at 120 °C for 12 h under vacuum to ensure thermal equilibrium at the interfaces between the PS melts and the brushes, which is above Tg ∼ 102.6 °C of PS melt (Figure S1). The film samples were quenched to room temperature for less than 1 min (at a cooling rate of −100 °C/min) immediately after the second thermal annealing, and then the samples were stored at room temperature under vacuum. To characterize Tg of PS films, a lab-made heating stage was installed at a vacuum chamber in the spectroscopic ellipsometer (SEMG-1000, Nanoview Co.), which was operated at an incidence angle of 70° using a halogen light source at wavelength (λ) ranging from 350 to 850 nm (or 1.5 to 3.5 eV). Although the two ellipsometric parameters of Ψ and δ are associated with film thickness using a Cauchy model, the temperature

⎡ ⎡⎛ d ⎞12 ⎛ d ⎞6 ⎤ ⎛ r ⎞2 ⎤ 1 Ub(r ) = − kR 0 2 log⎢1 − ⎜ ⎟ ⎥ + 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ + ϵ ⎝ r ⎠ ⎥⎦ ⎢⎣ ⎢⎣⎝ r ⎠ 2 ⎝ R 0 ⎠ ⎥⎦ (2)

where k and R0 represent the spring constant and the maximum extend of the bond, respectively. In all our simulations k and R0 were fixed at 30 and 1.5d, respectively. In accordance with the FENE model, the second term in the bonded interaction is cutoff at r = 21/6d. Finally, the interactions between the polymer chains and the surfaces were modeled using a purely repulsive potential of the form ⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ d d ⎪ ⎪ 4ϵ⎢⎜ ⎟ − ⎜ ⎟ ⎥ + ϵ, r ≤ rc ⎝ ⎠ ⎝ r ⎠ ⎥⎦ Uwall(r ) = ⎨ ⎢⎣ r ⎪ ⎪ 0, r > rc ⎩

(3)

1/6

where the cutoff radius is set at r = 2 d. Note that while the nonbonded interaction between the monomers (Unb) have both attractive and repulsive components for the energy, the interaction between the monomers and the walls (Uwall) is described using purely repulsive potentials. The initial configuration for the simulation is generated by placing the homopolymer at random inside the simulation box. The grafted polymers were placed (“grafted”) at random on the substrate. Care was taken to avoid overlap of the grafted monomers while placing them inside the simulation box. The degree of polymerization of the homopolymer chains (NHP) used in the simulations was fixed at 56. The initial overlap between the homopolymer monomers was then removed using B

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We note that Ipg is a measure that embodies the interpenetration of free polymers into the grafted chains rather than a length scale quantifying the width of interpenetration.23,45,63−66 We also note that a very similar measure of overlap between the brush and free polymers had been used in the earlier study of Pastorino et al.18,67 For quantifying the local segmental dynamics as a function of distance from the surface, the simulation box was divided into bins of size 1.0d. Subsequent to defining the regions of interest, the polymer segments in a given layer at t = 0 were identified, and the dynamic property of the particular layer was then obtained as an ensemble average of the property of the polymer segments that are present in the corresponding layer at t = 0.30,68−71 The local segmental dynamics in the thin film was then characterized as a function of the distance |zmid − z0| from the interface, where z0 represents the location of the bottom surface and zmid denotes the midpoint of any given bin in the plane normal to the interface. We computed the relaxation times from the mean-squared displacements in the directions parallel to the thin film (g∥(z,t)) for all the monomers in a bin at a distance zi at t0 using

a slow push-off method. To this end, the grafted monomers were placed at random in such a way that the distance between any two grafted points was at least 21/6d. The grafting density, σ is given by σ = ng/LxLy, where ng, Lx, and Ly are the number of chains and the lengths of the simulation box in the direction parallel to the substrate, respectively. Subsequently, the system was evolved in a NVT ensemble using a Nòse−Hoover thermostat with a relaxation time of 0.5τ, where τ represents the monomer relaxation time. The “grafted monomers” were immobilized during the course of simulation by setting their initial velocities to identically zero. The grafted monomers were also excluded while performing the Verlet integration, thus effectively immobilizing the graft for the entire course of the simulation. The z-position of the grafted monomer was kept at 0.01σ away from both the surfaces to avoid numerical difficulties while performing MD calculations. The total momentum of the system was zeroed out every 100τ to avoid numerical discrepancies. All simulations were performed using LAMMPS.61 The timesteps used in equilibration and production cycles were 0.005τ and 0.012τ, respectively. Simulations were performed at two separate temperatures, T = 0.55 and at T = 0.80 in Lennard-Jones (LJ) units, which correspond to reduced temperature, T/T* = kB/ϵ. The glass transition temperature (Tg) of a system for bulk conditions is found to be Tg = 0.45 in LJ units.27,28,62 Hence, our choice of temperatures corresponds to 1.2Tg (a temperature close to Tg) and 1.74Tg (a temperature far from Tg) of a system without graft. As we demonstrate below, the qualitative features of the results are independent of the temperature. Without loss of generality, ϵ and d were set at 1.0 kBT and 1, respectively. All simulations were carried out for at least 50 × 106 timesteps. The direction normal to the substrate denoted as the Z direction in the subsequent discussion. Periodic boundary conditions were applied only in the X and Y directions. The film thickness (h) was chosen large enough such that the center of the film did not exhibit substrate effects. In all our simulations, h was set at 80d for T = 0.80 and at 120d for T = 0.55. The height of the thin film was chosen such that the bulk dynamics shows a plateau near the center of the film for all grafting densities analyzed. The densities for the systems with different grafting densities were chosen such that the net pressure (P) of the system is maintained constant. The lowtemperature simulations were performed at P = 1.15 ± 0.1ϵd−3, and the high-temperature simulations were performed at P = 2.82 ± 0.1ϵd−3. To this end, the X and Y box lengths were adjusted iteratively in such a way that the density which corresponds to the required pressure is obtained. Such dimensions varied between 17d and 24d. The ratio of the MWs between the homopolymer and the grafted chains is denoted using f MW. The number of chains on each surface is parametrized by the grafting density, σ. The static quantities probed in this work include the number density profiles of the polymer melt (ρp(z)) and the grafted polymer (ρg(z)). Such quantities are calculated as the ratio of the average number of a given type of monomer in every bin to the total numbers of monomers in the system. Subsequently, the amount of interpenetration (Ipg) or amount of overlap is computed as the convolution of the polymer and grafting densities using Ipg =

∫0

Lz

ρp (z)ρg (z) dz

N

g (z , t − t0) =

∑i = 1 (δ(z − zi(t0))) × [ri(t ) − ri(t0)]2 N

∑i = 1 (δ(z − zi(t0))) (5)

where ri(t) denotes the position vector of the ith monomer in the direction parallel to the thin film. Subsequently, the relaxation times (τ(z)) are calculated as the time for which g∥(z,t) = gxx + gyy = d2, where gxx and gyy represent the meansquared displacements in the X and Y directions, respectively.27,29 Temperature Regime of Simulations. One of the concerns in the literatures pertaining to simulations on the glass transition behavior is to establish that the simulations are indeed performed near or below Tg. To this end, before presenting the results in the subsequent sections, we note two important results regarding the simulation parameters and the relaxation times obtained in this work. (i) As noted above, the lowest reduced temperature utilized in all the simulations is set to T = 0.55, which is approximately 1.2 times the critical temperature (Tc = 0.45). Such choice of simulation parameters set the system near Tg rather than inside the supercooled regime. (ii) Representative dimensional relaxation times for the polymer−(ungrafted) substrate systems are provided in Figure S2. Therein, we note that the relaxation times near the midfilm are 400τ for the bare surface system, which correlates to approximately 0.4 ns in real time. Such relaxation times in simulations are below the relaxation times in the supercooled regime in experiments that typically vary between microseconds and computational cost involved in simulations below Tg.

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RESULTS A grafting-to method with PSOH-06 and PSOH-14 was utilized to attach the PS brushes to the substrates, and the grafting density (σ) of the brushes was modulated by controlling annealing time at a constant temperature of 130 °C under vacuum. The σ, corresponding to the number of brushes per unit area (chains/nm2), is calculated by σ = ρhbNA/Mn, where ρ and hb are the mass density (1.05 g/cm3) of PS and brush thickness measured with ellipsometry, respectively, and NA is Avogadro’s number. The grafting density is normalized to a dimensionless σRg2 for comparison.

(4) C

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Macromolecules Figure 1 shows the brush thickness (hb) (or σRg2) of PS brushes prepared during thermal annealing the films at 130 °C.

Figure 1. Brush thickness (hb) and σRg2 of PS brushes prepared with PSOH-06 and PSOH-14 as a function of annealing time at a constant temperature of 130 °C.

As the annealing time increases, the hb increases rapidly until an early stage of ∼30 min and approaches slowly to the maxima. This second-stage retardation above 30 min reflects that the diffusion of hydroxyl end groups of PSOHs onto a Si substrate is remarkably hindered due to the PS brushes that were already present on the surface. Higher hb and σRg2 of PS brushes from PSOH-14 indicate that longer PS brushes are more stretched out of the substrate, although the overall increasing trends of PS brushes from PSOH-06 and PSOH-14 are similar in hb and σRg2 with increase in the annealing time. The maximum hb and σRg2 are obtained for the two PSOHs: 5.24 nm and 2.72 (σ = 0.606 chains/nm2) for PSOH-06 and 11.05 nm and 5.24 (σ = 0.499 chains/nm2) for PSOH-14, respectively. Figure 2 presents the ellipsometric angle (Ψ) of PS films supported on a PSOH-06 brush with σRg2 = 2.72 as a function of temperature, where the thickness of PS films is hereafter referred to as the net film thickness subtracted brush thickness (hb) from total film thickness. For a 120 nm PS film, a Tg is determined to be 102 °C by the two different slopes of Ψ between the glassy and rubbery regimes as well as the two different thermal expansions of the film, as indicated with the red arrows. As the film thickness decreases to 90, 60, and 30 nm, the overall Ψ increases with temperature, and the Tg evaluated from the different slopes decreases to 97.8, 94.3, and 82.9 °C, respectively. Compared with a measured bulk Tg (102.6 °C of PS melt) (Figure S1), a noticeable decrease in Tg of PS films with decreasing film thickness is attributed to the autophobic behavior between the PS melts and brushes. Under a highly stretched (or higher σRg2) PS brushes, there is a stronger influence arising from the entropically unfavorable interface between the PS melts and brushes. This thickness dependence of Tg behavior is a good agreement with our prior result.45 Figure 3 displays the temperature dependent Ψ for 30 nm PS films supported on PSOH-06 and PSOH-14 brushes with various σRg2, in which the Ψ values are normalized and shifted

Figure 2. Ellipsometric angle (Ψ) of PS films supported on a PSOH06 brush with σRg2 = 2.72, which was measured at a selected wavelength of 560 nm as a function of temperature. The Tg of PS films was determined by the two different slopes of Ψ between the glassy and rubbery regimes, as indicated with the red arrows. The dotted line indicates a consistent bulk Tg (102.6 °C) of PS melt.

by a factor of 0.01. The Tg is determined by the intersection temperature of the two linear regressions between the glassy and rubbery regimes because the different slopes of Ψ reflect two different thermal expansions of the films, as marked with the red arrows. When the σRg2 of PSOH-06 brush decreases from 2.72 to 1.05, as shown in Figure 3a, it is evident that the Tg increases from 82.9 to 107.4 °C and then decreases to 93.9 °C at a lower σRg2 = 0.593 having a few PS bushes on the native oxide surface. This tendency of Tg for PSOH-06 brush is analogous to that for PSOH-14 brush, as shown in Figure 3b, explicitly demonstrating a possible existence of the maximum D

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Figure 3. Temperature-dependent Ψ for 30 nm PS films supported on (a) PSOH-06 and (b) PSOH-14 brushes with various σRg2, in which the Ψ values are normalized and shifted by a factor of 0.01 to avoid overlap. The Tg of PS films is determined by the two different slopes of Ψ between the glassy and rubbery regimes, as indicated with the red arrows. The dotted line indicates a consistent bulk Tg (102.6 °C) of PS melt.

Figure 4. Tg of PS films supported on (a) PSOH-06 brushes and (b) PSOH-14 brushes as a function of brush thickness (hb), where σRg2 of PS brushes is indicated on the top abscissa. The dotted line indicates a consistent bulk Tg (102.6 °C) of PS melt.

Tg in PS films as a function of the grafting density for PS brushes. Figure 4a displays Tg of PS films supported on PSOH-06 brush as a function of hb, where σRg2 of PS brushes is indicated on the top abscissa. The Tg of the PS films was measured more than seven times with new pieces of a large sample, and the value was averaged as a mean value. The values of Tg of 120 nm PS films are seen to be relatively close to the bulk Tg (102.6 °C) of PS melt (Figure S1), irrespective of hb (i.e. grafting density, σRg2) of PS brushes. However, when the film thickness decreases to 90 nm, a weak maximum Tg appears at hb ≃ 2.22. With further decrease in film thickness, both the magnitude and the occurrence of maxima in Tg become more pronounced.

More interestingly, the maximum value in Tg is seen to be larger than the value (102.6 °C) of the bulk PS melt (indicated as dashed lines). To elucidate the chain length effect of PS brushes upon the above behavior, the Tg of PS films supported on PSOH-14 brush were also evaluated, and the results are shown in Figure 4b. Consistent with the results for PSOH-06, the trends for PSOH-14 also display a maximum Tg (larger than that of the bulk PS melt) at an intermediate grafting density. The location of the maximum Tg occurs at a comparable value (but slightly lower value) of σRg2 compared to the PSOH-06 brush. To obtain insights into the physics underlying the above results, we utilized computer simulations on a coarse-grained E

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Figure 5. Density profiles for melt (solid lines) and graft (dotted lines) for f MW = 0.50 at (a) T = 0.80 and (b) T = 0.55, where T (in LJ units) correspond to reduced temperature, T/T* = kB/ϵ.

model of the system. As a first step toward understanding the dynamical behavior of polymer melts deposited on grafted polymer layers, we present equilibrium results quantifying the density profiles for the polymer melt and the brush for different grafting densities. In Figure 5a, we display the number density profiles (due to symmetry of this situation, we display only results for half the film thickness) of both the homopolymers (ρp(z)) and the grafted polymers (ρg(z)) (normalized to the total number of monomers in the system) as a function of grafting densities for a molecular weight ratio (MW of grafted polymer to MW of homopolymer), f MW = 0.50. The results displayed correspond to density profiles smoothed at T = 0.80 to eliminate the oscillations arising from the repulsive bead− bead interactions, and the original unsmoothed density profiles are displayed in Figure S3. From Figure 5a, it is seen that at low grafting densities the melt chains are able to penetrate the entire brush up to the substrate. With increase in the grafting density of the brush, the distance to which the homopolymers penetrate the brush is seen to monotonically decrease. Qualitatively, similar results were obtained for the case of f MW = 0.16 at T = 0.80, as shown in Figure S4. The results obtained at T = 0.55 are shown in Figure 5b and are seen to be qualitatively similar to the results displayed for T = 0.80. The density profiles can be utilized to explicitly quantify the interpenetration of the homopolymer with grafted polymers (Ipg, eq 4), as displayed in Figure 6 as a function of σRg2. At low grafting densities, the overlap between the homopolymer and the brush is small due to the smaller number of grafted chains present in such situations. With increase in the grafting density, we observe that the overlap increases as a consequence of both the increased number of grafted chains and the penetration of homopolymers into the grafted layer. However, beyond a critical σRg2, the overlap begins to decrease with further increase in the grafting density. The latter reflects the physics arising from dewetting which is accompanied by the reduced penetration and the subsequent expulsion of the homopolymers from the grafted layers. Our results also suggest that the wetting−dewetting transition is prone to occur at a lower threshold σRg2 for smaller MW brush. In comparing the results for f MW = 0.50 with f MW = 0.16, we identify that the maximum overlap occurs at a lower value of grafting density for latter

Figure 6. Interpenetration of graft and melt polymers (normalized by the maximum value in each case) as a function of grafting density at T = 0.80.

systems. Such results are consistent with the stronger propensity for dewetting in the systems with shorter grafted polymers.23,25 We note that Pastorino and co-workers have reported similar results in the context of analyzing the behavior of brush−polymer interfaces under shear flow.18 To understand further the influence of the grafted layers on the dynamics of overlaying polymer melt, we probed the local segmental dynamics in the overlaying polymer film (eq 5). Figures 7a and 7b display the local segmental relaxation times of the homopolymers (normalized by the corresponding midfilm values) as a function of distance from the substrate with various grafting densities for f MW = 0.50 and f MW = 0.16. At zero grafting density (bare substrate), it can be seen that there exists an acceleration in the segmental dynamics near the substrate (z = 0). Such an acceleration results from the repulsive interactions between the melt and the substrate and is consistent with the results of previous studies modeling the behavior of polymer melts near repulsive surfaces.27,29 For sufficient distance from the substrate, z, the relaxation times are seen to approach the midfilm value, reflecting the finite range of F

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Figure 7. Local segmental relaxation times as a function of distance from the grafted surface with various grafting densities for (a, b) T = 0.80 and h = 80d and (c, d) T = 0.55 and h = 120d at (a, c) f MW = 0.50 and (b, d) f MW = 0.16. Rg2 = 6.85σ2 for N = 28 ( f MW = 0.50). Rg2 for N = 8 was calculated by assuming grafted Gaussian chains. The midfilm values used for normalization are computed by averaging over the middle part of the film (0.25 < z < 0.5).

probed the thickness dependent Tg and dynamics of polymer melts have attributed such positive interfacial energies (or equivalently, unfavorable enthalpic interactions) with accelerated dynamics.33,34,72−76 A surprising outcome of our above results is the demonstration that interfacial energy alone may not always serve as a good identifier for the dynamical effects arising from grafted substrates. In this regard, our conclusions broadly agree with the results of the study of Tsui et al.,40 which probed the Tg of polymer films on random copolymer brushes. For the system probed by Tsui et al.,40 due to the enthalpic interactions between unlike monomers, we expect an even more complex influence of the brushes on the local segmental dynamics of the polymers.

the interfacial interactions. Qualitatively similar results were obtained at low temperatures as well and are displayed in Figures 7c and 7d. With further increase in the grafting density of the brush, the segmental dynamics of overlaying polymer melt in the overlap region with the brush is substantially hindered, to an extent where the local relaxation times become even higher than that of the bulk (free) homopolymers near the middle of the film. Such a behavior can be understood by noting that as a consequence of being grafted, the dynamics of brushes is expected to be substantially slowed. As a result, the homopolymer melt chains located inside the brush find themselves in an environment that has inherently slower segmental dynamics relative to the midfilm conditions, resulting in a significant increase in the local relaxation times within the grafted layers. An interesting observation arising from a comparison of the results of Figures 3 and 4 is that the local segmental dynamics of the polymer film becomes slower than midfilm relaxations even for grafting densities that may be classified as dewetting regimes. As discussed in the Introduction, such regimes are often characterized by a positive interfacial tension between the melt and the brush, and numerous earlier studies which have

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DISCUSSION In summary, our simulation results above point to two main observations regarding the influence of grafted layers on overlaying polymer film. First, the overlap between the free polymers and brush segments exhibits a nonmonotonic dependence on the grafting density of the brushes. Second, the segmental dynamics of polymer of the free polymers in the overlap region is seen to be substantially hindered due to the slow dynamics of the grafts themselves. Prior to commenting G

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Macromolecules on the connection between such observations and the experimental results, we point out that there is still a lack of complete clarity on the physical mechanisms underlying the thickness dependence of Tg of polymer films. In general, unfavorable polymer−substrate interactions promote a thickness-dependent decrease of Tg, whereas, in contrast, favorable polymer−substrate interactions result in an increase in Tg relative to the bulk Tg.32,40 Such results have in turn been rationalized by invoking the influence of the polymer−substrate interactions upon the polymer segmental dynamics.30 Explicitly, unfavorable (favorable) polymer−substrate interactions usually lead to an acceleration (slowing) of the polymer segmental dynamics near the substrate. Such perturbations have been speculated to propagate through the film to influence the overall Tg of polymer thin films. While the preceding ideas have found wide acceptance, there is still a lack of clarity on the mechanism by which such local perturbations to dynamics propagate over the length scales observed in experiments (in some cases, ranging to several Rgs of the polymer36,37,77). While a number of phenomenological models have been proposed,41,78−81 to our knowledge, there has been no computer simulations invoking micromechanical models for polymer chains which have reproduced such long length scale effects. To establish the connection between our simulation results and experimental observations, we note that in an earlier study45 we examined the Tg behavior of polymer thin films supported on extremely short and dense brushes. Therein, the thickness dependence of Tg was observed to be more pronounced for the films on the shorter brushes with the high grafting density. To rationalize the observations, in such a context, we invoked the combined effects of the interpenetration width between the brush and the polymer (computed using a scaling theory) and a therein unproven hypothesis that the free polymer segments in the overlap region exhibit slower dynamics relative to the polymer away far from the bare surface (midfilm region). We combined such features within a phenomenological percolation model proposed by Long and Lequex to explain the thickness dependence of Tg of thin polymer films.78 For such a purpose, we incorporated the surface interaction effects in such percolation models by including a “skin of influence” around the surface in consideration. We proposed that such a skin represents the zone over which the polymer (matrix) dynamics is affected by the surface. We considered the influence of increasing the zone of influence of the substrate while keeping its strength a constant, and demonstrated that a larger zone of influence leads to a more pronounced thickness dependence of Tg (see Figure 8). Overall, the simulation results presented in Figures 5−7, when considered in conjunction with our earlier results from the percolation model,45 suggest a possible explanation of the physics underlying the experimental observations of Figure 4. Specifically, the results of Figure 6 demonstrate that the overlap in between the overlaying polymer and the grafts, akin to the “skin of influence” in our earlier terminology, exhibits a nonmonotonic dependence with the grafting density of the brush. Further, our results of Figure 7 corroborate the earlierproposed (albeit, unproven) hypothesis that the dynamics of polymer segments in the overlap region are indeed slowed relative to the polymer near the middle region of the film. We propose that together such effects are responsible for the increase in Tg above that of the midfilm relaxation values, as observed in the results displayed in Figure 7. With further

Figure 8. Percolation model results for ΔTg (defined as Tg(h) − Tg(h = ∞)) as a function of film thickness (expressed in lattice units) for different skin thicknesses denoted by Δ. Reproduced with permission from ref 45.

increase in the grafting density, the local segmental dynamics of the polymer film does become more retarded in the overlap region (Figure 7). The overlap of the melt with the brush also decreases (Ipg, Figure 6), and hence the extent of the region of hindered polymer dynamics becomes smaller. As a result, the influence of the brush on average polymer melt dynamics, such as Tg, is expected to become mitigated. For such regimes, the polymer film can be expected to reflect only the polymer interface on the other end (which is capped by air in experiments) and manifest a Tg which is lower than the average value of PS melt in the middle of the film.

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CONCLUSION

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ASSOCIATED CONTENT

The results presented in this paper identify two surprising conclusions on the dynamics and Tg of polymer films supported on polymer brushes. First, we observed a nonmonotonic dependence of the Tg on the grafting density of the brush. To our knowledge, such a nonmonotonic trend Tg has not been reported in earlier studies. Such behavior can be understood as a consequence of the combined influences of the enhanced segmental relaxations in the polymer−brush overlap regions and the reduced overlap between brush and melt. Second, we observed that the local dynamics of polymer films supported on polymer brushes exhibits a complex behavior arising from the influence of the hindered dynamics of the brush on the segmental dynamics of the overlaying polymer melt. Such results suggest that interfacial energies, while serving as a useful quantity for understanding polymer dynamics on bare substrates, possess only a limited utility in more complex situations such as polymers supported on polymer brushes.

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00290. Figures S1−S4 and Table S1 (PDF) H

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (D.Y.R.). *E-mail: [email protected] (V.G.). ORCID

Du Yeol Ryu: 0000-0002-0929-7934 Venkat Ganesan: 0000-0003-3899-5843 Author Contributions

H.L. and V.S. contributed equally to the article. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS V.G. acknowledges funding in part by grants from the Robert A. Welch Foundation (Grant F1599), the National Science Foundation (DMR-1306844), to King Abdullah University of Science and Technology (OSR-2016-CRG5-2993-1). Acknowledgment is also made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research (56715-ND9). D.Y.R. acknowledges funding by the NRF grants (2017R1A2A2A05001048, 2017R1A4A1014569) funded by the Ministry of Science, ICT & Future Planning (MSIP), Korea. H.L. acknowledges funding by the Yonsei University Research Fund (2016-12-0011).

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