Glass Formation near Covalently Grafted Interfaces - American

Mar 24, 2015 - Specifically, as in nanocomposites and nanoconfined glass-formers, we find that near-interface mobility suppression is mediated by...
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Glass Formation near Covalently Grafted Interfaces: Ionomers as a Model Case Dihui Ruan and David S. Simmons* Department of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States ABSTRACT: In materials ranging from ionomers to semicrystalline polymers, the glass transitionthe central phenomenon determining the dynamic and mechanical behavior of amorphous materialscan reflect profound alterations by covalent grafting to surfaces or inclusions. The understanding of these alterations in ionomers has long been underpinned by the view that these grafts serve as “tethers”, suppressing dynamics of attached chains out to a distance related to their molecular stiffness. Here we describe simulations of model ionomers suggesting the need for a fundamental reconsideration of this understanding. Specifically, as in nanocomposites and nanoconfined glass-formers, we find that near-interface mobility suppression is mediated by cooperative rearrangements intrinsic to glass-forming liquids rather than by a unique covalent tethering effect. While remaining consistent with many earlier predictions, these results bring dynamics near ionomeric aggregates and other grafted interfaces into agreement with a body of literature suggesting the universal presence of a cooperative dynamic length scale in glass-forming liquids that dominates their relaxation behavior and establishes their fundamental length scale of local dynamic gradients.

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INTRODUCTION The precipitous suppression in linear-regime segmental dynamics upon approach to the glass transition is the most importantand perhaps most poorly understoodfeature of the dynamic and mechanical behavior of soft materials. In such technologically relevant materials as semicrystalline polymers, ionomers (charge-containing polymers), and grafted nanocomposites, covalent grafting to a particle or surface appreciably modifies this transition, with corresponding effects on the material’s properties and performance. For example, ionomersion-containing polymersexhibit strong spatial gradients in segmental dynamics near interfaces between charged ionic aggregates that are characteristic of their morphology and the surrounding neutral monomer domains, with the covalent bonds between the aggregated ionic moieties and attached neutral chain segments serving as trans-interfacial grafts (for example, see Figure 1 for a snapshot1 of a simulated such interface from the present study). This effect also generally yields an enhancement in the ionomer’s overall segmental relaxation time and glass transition temperature, Tg,2−9 and can introduce an additional high-temperature relaxation process that is sometimes termed a “second Tg”.5−7,10−13 Similarly, semicrystalline polymers frequently exhibit a “rigid amorphous fraction” of material with suppressed segmental dynamics,14 with this observation commonly being associated with the presence of dangling or bridging amorphous chains that are effectively grafted to the surface of crystalline domains. In both cases, these phenomena can profoundly alter transport and mechanical properties. The predominant understanding of these observations is underpinned by the intuition that covalent surface-grafting serves as a kind of tether that suppresses a © 2015 American Chemical Society

grafted chain’s segmental mobility out to a distance related to its molecular stiffness.14,15 Recent advances in the understanding of polymer glass formation suggest that this “bond-centric” view may not capture the dominant mechanism driving alterations in segmental dynamics near grafted surfaces. Specifically, a recent body of literature16−19 suggests that dynamic alterations near interfaces more generally may instead propagate via cooperative segmental rearrangements that are not directly related to chain connectivity.20−22 These rearrangements more generally underpin dynamics in fragile (non-Arrhenius) glass-forming liquids.23−27 This alternative mechanism has been shown to dominate near ungrafted interfaces in simulated polymer films20−22 and nanocomposites,28−30 which can exhibit large shifts in glass transition temperature and segmental relaxation time.16−19,31−33 Indeed, favorable surface interactions in these systems can yield a local suppression in dynamics similar to that observed in ionomers and semicrystalline polymers.29,34−42 A parallel series of experimental studies have reported similar trends in grafted and ungrafted polymer nanocomposites,43−46 with these observations often rationalized in this context based on a model for interfacial alterations of the percolation of dynamically slow, spatially heterogeneous domains that appear to be near-universal features of glass forming liquids near Tg.47−49 Several of these studies have directly suggested an equivalence between covalently grafted and strongly attractive interfaces, calling into question the “bond-centric” view of grafted interface effects in the context of nanocomposites. This Received: January 7, 2015 Revised: March 12, 2015 Published: March 24, 2015 2313

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factors other than the chain persistence length and therefore generally cannot be altered without chemical alteration of the polymer itself. In contrast, interfacial gradients in dynamics in systems described by the “cooperativity model” are sensitive to factors including the introduction of small-molecule diluents,52−54 such as antiplasticizers, and possibly the “fragility” of glass formation, a measure of the abruptness and “nonArrheniusness” of segmental relaxation upon approach to Tg.55 Establishing the role of these hypothesized mechanisms is thus critical to both the fundamental understanding and rational design of these materials. This problem is especially important in ionomers, given their applications in settings ranging from fuel cells to golf balls and the profound impacts on their segmental dynamics of the presence of ionic moieties. In ionomers, the established understanding of these effects is provided by the longstanding and highly successful model of Eisenberg, Hird, and Moore (EHM), which posits that local suppression in the mobility of ionomer segmental dynamics near nanoscale ionic aggregates emerges from the cross-linking effects of these aggregates and extends to a distance associated with the chain’s persistence length.15 Intriguingly, Eisenberg later reported on a strong parallel between dynamics in ionomers and those in ungrafted polymer nanocomposites.6 In addition to the work in nanocomposites described above, other studies have recently reported that near-particle gradients in such systems are controlled by cooperative motion rather than persistencelength considerations.28 These observations, taken together, suggest that the “tether” view of local dynamics in ionomers may require modification. Such a reimagining of the role of covalent grafting would be expected to have a substantial impact on the understanding and design of these materials. Accordingly, here we employ molecular simulations to test the validity, in several model ionomers, of the “bond-centric” and “cooperativity-centric” models described above.

Figure 1. Snapshots of single ionic aggregates with grafted chains at T = 1.0 in the pendant (a) and semitelechelic (b) ionomers (produced in VMD1). Blue beads are neutral monomers, light gray beads are covalently grafted ions, and dark gray beads are counterions. The uncharged monomers in each image are cut away in the front half of the aggregate to allow the ions to be seen.

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METHODOLOGY Simulation Model. Simulations employ a model ionomer, composed of neutral bead−spring linear polymer chains56 with a single charged bead attached to each chain.57 Monomer interactions are based on an attractive extension of the Kremer−Grest model,56 which has been widely applied in the study of polymer glass formation.18,58−62 Within this model, bonded beads interact via a standard finitely extensible nonlinear elastic (FENE) potential56,63 and nonbonded beads interact via a 12−6 LJ potential63

overall body of literature suggests a fundamentally different view of near-interface gradients in dynamics; namely, interfacial dynamic gradients are governed by some intrinsic dynamic correlation length in supercooled liquids rather than by bondconnectivity considerations. These observations raise a fundamental question: do covalent attachments f undamentally change the physics by which an interface alters nearby segmental dynamics and glass formation, or are they essentially indistinguishable f rom strong physical attractions for these purposes? This question has practical implications for the behavior of systems containing grafted interfaces. First, these two hypotheses predict distinct temperature dependences of the size of reduced-mobility regions: if near-interface dynamics propagate via intrachain bond correlations, their size should mirror the typically weak temperature dependence of the chain persistence length; if the “cooperativity” model dominates, their size should exhibit a strong, roughly Arrhenius, growth upon cooling.38,50,51 As a consequence, in the latter case, suppressed-mobility domains can exhibit a percolation transition on cooling; in the former, percolation is controlled by the typical distance between interfaces or particles, but not temperature. Furthermore, within the bond-centric model, the above range is insensitive to

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ E LJ = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠

(1)

with a cutoff distance of 2.5σ. Charged beads additionally interact via a pairwise Coulombic potential qiqj ECoul = 4πεr ε0r (2) where q is an ion charge, ε0 is vacuum permittivity, and εr is the dielectric constant. Direct calculation of charge−charge interactions is cut off at 8σ with longer range Coulombics computed via a particle−particle particle−mesh (pppm) solver.64 In accordance with prior work,57 Coulombic interactions are parametrized to mimic the strength of electrostatic interactions 2314

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second half of equilibration runs at that temperature, after which the system is equilibrated for another 103τLJ in the NVT ensemble prior to data collection in the NVT ensemble. Simulation Analysis. Static structure is quantified by the structure factor S(q):

in fully neutralized poly(ethylene-co-acrylic acid) (PEAA)/ (Na+) ionomers. Specifically, we use εr = 4 and a fundamental electron charge e of 11.80 in LJ units to obtain a ratio of Coulombic to LJ interaction strengths consistent with these experimental ionomers. Based on this mapping, each polymer bead represents three CH2 units, yielding σ ≅ 0.4 nm.57 The molecular weights of three CH2 units, a COO− group, and Na+ are 42, 44, and 23, respectively, so that monomer and anion mass is chosen as 1.0 and counterion mass as 0.5 in LJ units. If T = 1.0 in LJ units corresponds roughly to room temperature, then the LJ unit of time τLJ ≅ 1 ps. Each simulated ionomer chain consists of a linear 20-bead chain of neutral monomers with a single attached anion of charge −1e, where all of these beads employ ε = σ = 1. In order to control for the role of grafting functionality, we simulate two types of ionomers, shown in Figure 2: a pendant ionomer with

S(q) =

1 N

N

N

∑ ∑ exp(−iqri) exp(iqrj) (3)

i=1 j=1

where q is the wave vector, ri is the position of the ith bead, and N is the number of beads.67 Free volume and vibrational dynamics are quantified by the Debye−Waller factor ⟨u2⟩, defined as the mean-square displacement at 0.99τ LJ.68 Segmental relaxation dynamics are quantified via the self-part of the intermediate scattering function (Fself(q,t)): Fself (q, t ) =

1 N

N

∑ ⟨exp[−iq(rj(t ) − rj(0))]⟩ (4)

j

where rj(t) and rj(0) are the positions of particle j at times t and 0, respectively, and where we employ wavenumber q = 7.07, comparable to the first local peak in the structure factor. Fself is comparable to time-resolved incoherent neutron scattering results. We define the α segmental relaxation time τα as the time at which Fself(q,t) = 0.2, employing a Kohlrausch− Williams−Watts (KWW) stretched exponential fit69,70 for data smoothing and interpolation. Throughout most of this article, a computational dynamic glass transition temperature Tg is defined as the temperature at which τα = 103τLJ, the time scale at which these simulations begin to fall out of equilibrium. Since a typical set of runs will not include a simulation at exactly this relaxation time, we first perform a fit of relaxation-time data for in-equilibrium temperatures less than 1.0 to a Vogel−Fulcher−Tamman form71,72

Figure 2. Schematic of the pendant ionomer (a) and semitelechelic ionomer (b) considered in this study. Blue beads, gray beads, and black beads represent neutral chain monomers, covalently bound ions, and counterions, respectively.

a charge attached to the 10th out of 20 neutral beads and a semitelechelic ionomer with a charge attached to the 20th out of 20 neutral beads. The mole fraction of charged groups incorporated into chains in both systems is approximately 5%, similar to many experimental systems. In the case of the pendant ionomer, simulations also include a 50/50 ionomer/ homopolymer blend in order to control for the effect of ionomer concentration. Pure ionomer simulations each contain 100 ionomer chains, while the 50/50 ionomer/homopolymer blend contains 50 ionomer chains and 50 20-bead homopolymer chains. Each system is 100% neutralized with free counterions of charge 1e, ε = 1, and σ = 0.5. Results of ionomer simulations are compared to prior simulations of glass formation in the corresponding pure homopolymer. Simulated are performed within the within the LAMMPS63 (Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics environment and integrated via a rRESPA integrator65 with a time step of 0.005τLJ for nonbonded and 0.001τLJ for bonded interactions. Simulations employ the Nosé−Hoover thermostat and barostat, as implemented in LAMMPS, for temperature and pressure control with damping parameter 2.0.63 Four initial configurations for each system are generated in Packmol66 (Packing Optimization for Molecular Dynamics Simulations) and initially equilibrated for 5 × 104τLJ at pressure P = 0 and T = 1.5. Each configuration is then quenched at P = 0 to low T at a rate of 10−5T/τLJ, with configurations saved regularly through the quench. Configurations at each temperature are then subject to an additional equilibration time τeq at P = 0, where τeq equals 104τLJ for 1.0 ≤ T ≤ 1.5 and 105τLJ for T < 1.0. Data from a given temperature is only employed if it satisfies the equilibration criterion τα ≤ 0.01τeq, where τα is the mean neutral monomer relaxation time at that temperature. After equilibration, the simulation box is resized to the equilibrium density as determined from the

ln τα = A +

DT0 T − T0

(5)

where A, D, and T0 are fitting parameters. Tg is then determined based on a negligible extrapolation of this fit to τα = 103τLJ. The kinetic fragility index m is then defined as the slope on an Arrhenius plot of τα at this Tg:73 ⎡ d log τ ⎤ α ⎥ m=⎢ ⎢⎣ d(Tg /T ) ⎥⎦ T=T

g

(6)

Relaxation and glass formation data are additionally obtained locally by performing this analysis on subsets of monomers after binning them as a function of their distance to the nearest ion at an initial time. For the purposes of more direct comparison with experimental time scales, we also define an extrapolated glass transition temperature Tgext, which we obtain by performing a very large extrapolation of the VFT form to a LJ time scale of 1014τLJ, equivalent to roughly 100 s in real units. However, we emphasize that this convention is not applied throughout most of the paper because large shifts in the high-temperature relaxation time of the segments near ionic domains result in aphysical results when Tg is defined by this large extrapolation. Specifically, these shifts produce large changes in apparent local fragility that lead to unphysical extrapolations such as a prediction of a suppressed Tg near ionic aggregates at time 2315

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Macromolecules scales 1011 times slower than those probed here. In contrast, as described below, over all temperatures measured, the relaxation time near the clusters is enhanced. Given the extraordinarily long range of this extrapolation and the aphysicality of this result, we primarily rely upon the more directly probed computational Tg and refer the extrapolated value only for the purpose of estimating the approximate size scale of slow domains expected near the experimental Tg.

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RESULTS Ionomer Morphology and Glass Transition. We begin by confirming that these simulations yield morphology consistent with previously studied experimental and simulated ionomers, which generally possess nanoscale ionic aggregates and exhibit an “ionomer peak” in scattering data reflecting the interaggregate distance.74 As shown by Figure 3, ions in these Figure 4. Structure factor (Soverall(q)) versus wavenumber (q) for pendant ionomer (□), pendant ionomer/homopolymer blend (△), semitelechelic ionomer (○), and pure polymer (◇).

Figure 3. Ionic aggregates at T = 1.3 and T = 1.0 for the pendant ionomer (a, b) and the same temperatures for the semitelechelic ionomer (c, d) after an equilibration period of 105τLJ.

simulations indeed organize into aggregates, with the structure factor exhibiting an ionomer peak (Figure 4) at wavenumbers 0.95/σ and 0.71/σ for the pendant and semitelechelic systems, respectively. These wavenumbers suggest mean interaggregate spacings of 6.6σ and 8.8σ. Also consistent with prior simulations,75 ionic aggregates at high temperature take the form of relatively disordered extended structures (Figure 3a,c) but collapse into more ordered, bilayer platelets upon cooling (Figure 3b,d). As shown in Figure 5, this transition is marked by a splitting of the peak in the ion−ion structure factor corresponding to intra-aggregate correlations. The ratio of wavenumber q for these two peaks, approximately 21/2, is consistent with scattering from a KCl crystal, which forms an FCC lattice but yields a structure factor indistinguishable from that of a simple cubic lattice because the two scatterers comprising the basis have the same scattering cross section. This characterization is also consistent with the images in Figure 3. We note that this “split peak” feature is not visible in the overall structure factor (Figure 4), likely due to the small aggregate size, again consistent with the absence of such a

Figure 5. Plots of ion−ion structure factor (Sion−ion(q)) versus wavenumber (q) from q = 5 to q = 10 for ions within the pendant ionomer (a) and the semitelechelic ionomer (b) at T = 1.3 (●), T = 1.2 (◆), T = 1.1 (▲), and T = 1.0 (■).

feature in the overall structure factor of experimental ionomers. The aggregate size also becomes temperature-independent below this temperature (see Figure 7), consistent with an aggregate solidification transition. As shown in Figure 6, this 2316

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tration of ions in this system. These results suggest a rate of ∼1.4% Tg enhancement per mole percent ion. Based on a Tg of 237 K for amorphous polyethylene,76 this corresponds to an enhancement of approximately 3.3 K per mole percent ion, consistent with the typical range of Tg enhancement in experimental ionomers.74 The insensitivity of this result to ionomer type (pendant vs semitelechelic) indicates that the exact steric details of the polymer’s covalent linkage to these ionic aggregates do not substantially impact its segmental dynamics. In contrast, if the shift in segmental dynamics Tg was dominated by chain connectivity effects, one might expect to see a greater enhancement for pendant ionomers, which should experience greater steric stiffening near the aggregate because each ion is covalently bonded to two rather than one chains (visible in Figure 1). In addition to Tg, we also quantify changes in the kinetic fragility index m of glass formation upon addition of ions. m quantifies the rate, ∂ (log τα)/∂(Tg/T), at which the structural relaxation time τα grows with cooling upon approach to Tg.55 Together with Tg, m provides information on the strength of the temperature dependence of segmental relaxation dynamics, rather than simply providing the temperature at which equilibrium is lost. Fragility is also closely connected to the cooperative segmental motions that appear to underpin nearinterface dynamics in nanoconfined polymers and polymer nanocomposites. Results indicate that the overall fragilities of glass formation m of these three systems are suppressed by about 3.0% relative to the neat polymer, again insensitive to details of chain connectivity. Given that reductions in fragility are commonly observed in nanoconfined polymers,29,39,52,77−85 this observation is consistent with the supposition that ionomer segmental dynamics behave in an essentially equivalent manner to those in nanocomposites and nanoconfined polymers. Near-Aggregate Gradients in Dynamics. To further study the impact of ionic aggregates on monomer glass formation in ionomers, we sort polymer segments into bins of thickness 0.5σ based on their distance to the nearest ion and compute τα vs T and the corresponding Tg and m separately for segments in each bin. As shown by Figure 8b−d, the segmental relaxation time is generally enhanced over all temperatures studied for segments nearer to the ionic aggregates. As shown by Figure 9a, Tg is, correspondingly, strongly enhanced for segments nearest to an ionic aggregate and decays with increasing distance. This result is roughly consistent with the experimental results of Miwa et al.,86 which similarly indicated a near-aggregate enhancement in Tg that decayed with increasing distance. Miwa et al. interpreted this result as support for the EHM model’s “cross-linking” view, since their extrapolated range of local mobility alterations was comparable to the persistence length of the polymer in its neat state. However, our results are not consistent with this interpretation: for our simulation, the range of Tg effects is considerably greater than the chain persistence length, which is approximately 0.7σ. Indeed, in this study, bulk Tg is never recovered even at a distance of 5σ from the nearest aggregate (corresponding to ∼2.0 nm in real units), likely because of a lack of sufficient distance between adjacent aggregates. Instead, we note that the trends in Tg and m shown in Figure 9a are in most respects quantitatively consistent with the same properties, shown in Figure 9b, near nongrafted attractive crystalline nanoparticles in a similar model polymer.28 The two systems exhibit excellent agreement in the extent of nearsurface enhancement in Tg, near-surface suppression in m, and

Figure 6. Pair-interaction energy vs temperature for pendant ionomer (■) and semitelechelic polymer (●). The solid and dashed circles indicate the location of the aggregate structural transition and neutralmonomer calorimetric glass transition, respectively. Inset shows the ion−ion pair-interaction energy, exhibiting a signature of the aggregate structural transition.

Figure 7. Mean number of ions per aggregate as a function of temperature for the semitelechelic (●), pure pendant (■), and pendant/homopolymer blend (▲) ionomers. Error bars are the standard deviation of the mean aggregate size across four runs.

transition is also accompanied by a feature in the system’s energy vs T curve. A previous simulation study argued that this aggregate-ordering transition is the origin of the observation of two apparent glass transitions in the mechanical response of ionomers.75 The Tg associated with the dynamics of uncharged monomers within these ionomers is enhanced with respect to the neat polymer, consistent with experimental results. As shown by Figure 8a, the segmental relaxation time is enhanced for the ionomeric systems, with a commensurate enhancement in Tg. This enhancement is insensitive to placement of the charged groups within the chain, with both pendant and semitelechelic ionomers exhibiting a 7.0% enhancement in Tg; indeed, the relaxation time data shown for these two systems in Figure 8a are nearly indistinguishable. The 50/50 ionomer/homopolymer blend exhibits a 3.3% Tg enhancement, roughly consistent with the reduced concen2317

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Figure 8. (a) Overall relaxation time vs temperature for semitelechelic (●), pure pendant (■), and pendant/homopolymer blend (▲) ionomers and the pure homopolymer (◆). (b−d) Relaxation time as a function of temperature for monomers at varying distances from the nearest ion (provided in the legend of each figure), for the pendant (b), pendant/homopolymer blend (c), and semitelechelic (d) ionomers. Solid lines are VFT fits to the relaxation time data. Vertical dashed lines denote overall (a) or local (b−d) Tg’s determined via the computational convention described in the Methodology section. The arrow in (b−d) points in the direction of shorter distance to the nearest ion.

Figure 9. (a) Tg (solid points) and m (empty points), normalized by their values in the neat homopolymer, as a function of distance from the nearest aggregate for the pendant ionomer (■), the pendant ionomer/homopolymers (▲), and the semitelechelic ionomer (●). (b) Data reproduced from ref 28 for the same properties plotted vs distance from a nanoparticle surface, with different symbols representing different nanoparticle concentrations.

long-range enhancement of m as well as the location of a crossover from m-suppression to m-enhancement. Further-

more, the dramatic suppression in near-aggregate fragility is consistent with measurements of reduced activation energy at 2318

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Macromolecules the Tg of near-filler material in polymer nanocomposites.6 We emphasize that no effort was made to design the present model to achieve this correspondence. Instead, the interface effect involved is apparently sufficiently universal that nearly the same results are obtained when these distinct interfaces are introduced to comparable model polymers. The result is obtained despite the large difference in size between the ionomeric aggregates and nanoparticles: the former contain on average ∼25 or ∼70 ions depending on the ionomer type; the latter were modeled as hollow icosohedra with the outer shell alone composed of 356 particles. This correspondence strongly suggests that covalent cross-linking between the chains and the ionic aggregate is equivalent to a strong noncovalent surface attraction and does not otherwise play a qualitative role in the physics of near-aggregate segmental mobility and Tg. More generally, a single temperature measurement (Tg) is of limited capacity in drawing conclusions regarding the physical origins of this gradient in dynamics. As suggested in the Introduction, a more rigorous test should compare the temperature dependence of the range over which structural relaxation time varies near aggregates to the temperature dependence of the persistence length. If the gradient in segmental dynamics is indeed dominated by the chain persistence length, then the two should exhibit the same temperature dependence to within a scaling factor. In order to perform this test, we employ the same binning procedure described above to determine the segmental relaxation time τα of the polymer segments as a function of distance from the nearest ion. As shown by Figure 10a, at all temperatures studied, τα is enhanced near the aggregate and decays toward a pure-state value with increasing distance. Consistent with prior simulation work examining dynamics near interfaces,38,50 we define for each temperature an interfacial length scale ξτ over which relaxation dynamics vary based on the criterion |τα(z)/τα pure − 1| = 0.2

Figure 10. τα (a) and ⟨u2⟩ (b) normalized by their pure-polymer values vs distance to the nearest ion in the pendant ionomer, at T = 1.2 (●), T = 0.8 (▲), and T = 0.6 (■). Solid lines are fits to eq 8, and the dashed line indicates the criterion for ξτ and ξ⟨u2⟩.

(7)

after first smoothing τα(z) by fitting it to the functional form

⎛ τ (z ) ⎞ ln⎜ αpure ⎟ = Az B + C ⎝ τα ⎠

As shown in Figure 11, ξτ is approximately 1.5σ at high temperature and grows in a roughly Arrhenius manner by a factor of ∼3 on cooling, while the persistence length lp remains essentially constant over the same temperature range. This lack of a correspondence between ξτ and lp firmly excludes the persistence length as a central determining factor of the size of the reduced-mobility regions in these simulations. Nor do local relaxation dynamics track with local free volume alterations. Specifically, we also calculate the range ξ⟨u2⟩ over which the Debye−Waller factor ⟨u2⟩a measure of local free volume89 varies near the clusters via the same procedure employed above for τα. Results indicate that, like the persistence length, ξ⟨u2⟩ is nearly temperature invariant. This observation also firmly rules out a free volume layer model for the reduction in mobility near clusters, consistent with results near free surfaces.38,50 Instead, the Arrhenius temperature dependence of ξτ observed here is consistent with results for alterations in dynamics near the interfaces of polymer thin films and nanocomposites.38,50,51,90−92 ξτ is found to extrapolate to approximately 9σ−14σ at the extrapolated experimental Tg (characterized by a relaxation time of τα = 1014 in LennardJones units, roughly equivalent to 100 s), as shown by Figure

(8)

where A, B, and C are fitting parameters. Examples of this process are illustrated in Figure 10a,b; previous work in polymer films has demonstrated that its details do not qualitatively affect results.38,50 However, we emphasize that ξτ measures the length scale over which segmental dynamics recover their bulk behavior; it explicitly does not measure the size of a glassy domain surrounding the aggregates. Recent work has demonstrated that the latter definition, which employs a “fixed time scale” cutoff for the size of immobile domains rather than measuring the scale over which dynamics are altered, will always exhibit a strong temperature dependence due to convolution with the overall dependence of the segmental α relaxation time on temperature.50,87,88 When comparing results based on the present construction to experiment, it is therefore essential to carefully distinguish between metrologies that probe the size of a domain of “reduced mobility” relative to bulk (equivalent to the present method) and those that measure the size of an “immobile” domain as defined by some time scale convention (not equivalent to the present method). 2319

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time scale equal to the peak time in the non-Gaussian parameter. Stringlike rearrangements are then identified by finding those mobile particles that replace one another within a threshold of 0.6σ on the same time scale. As shown by the Figure 11b inset, ξτ is linearly related to L over the entire temperature range of glass formation. This correspondence indicates that the size of suppressed-mobility regions in ionomers is determined by the scale of cooperative rearrangements inherent to supercooled liquid dynamics rather than by chain connectivity effects. Test of the Adam−Gibbs Theory of Glass Formation in Ionomers. Finally, we test the proposition that segmental relaxation dynamics during glass formation in these ionomers are determined by the scale of cooperative rearrangements, as they appear to be more generally in glass-forming liquids. The seminal theory positing this relationship is that of Adam and Gibbs.23 Recent work has indicated that, if this theory holds for a given system, then τα should depend on L via the equation22,27,29 ⎡ L(T ) Δμ ⎤ τα(T ) = τ∞ exp⎢ ⎥ ⎣ LA kBT ⎦

(9)

where LA is the CRR size at the high-temperature onset of glass formation TA, kB is the Boltzmann constant (equal to unity in LJ units), and τ∞ is the high-temperature relaxation time and is treated as a fitting parameter. Δμ = ΔH − TΔS, where ΔH and ΔS are the high temperature activation enthalpy and entropy, respectively. ΔH is specifically equal to the high-temperature slope of the system’s relaxation time when plotted on an Arrhenius plot, and ΔS is evaluated via the equation ΔS = −kB ln(τA/τ0), where τA is also a fitting parameter from the Arrhenius fit above and τ0 is the vibrational relaxation time, taken to be 0.1 LJ time units, consistent with prior work.22,27 TA is defined as the temperature where the relaxation time most nearly deviates from the high-temperature Arrhenius fit by 10% and is found to range from 0.82 to 0.86 in LJ temperature units for these systems. Based on eq 9, if ln(τα/τ∞) is plotted against L/LA·Δμ/kBT, all glass-forming materials obeying these physics should collapse to a line of slope 1. Moreover, Hanakata et al. have pointed out that ξ and L should be related as22

Figure 11. (a) Length scales ξτ and ξ⟨u2⟩, mean string length L, and chain persistence length lp versus T for pendant ionomer (■), pendant ionomer/homopolymer blend (▲), semitelechelic ionomer (●), and pure polymer (◆). (b) Arrhenius plot of ξτ and L versus T; solid lines are an Arrhenius extrapolation, and dashed lines indicate extrapolated ξτ at extrapolated experimental monomer Tgext where τα extrapolates to 1014τLJ; inset shows linear correlation between ξτ and L; symbols have the same meaning as in (a).

⎛ξ ⎞ L − 1 = A⎜ τ − 1⎟ LA ⎝ ξA ⎠

11b, consistent with results in simulated thin films.21 Within those nanoconfined polymers, ξτ is associated with the length scale of cooperatively rearranging regions (CRRs) in the glassforming liquid,20−22 the existence of which was postulated by Adam and Gibbs to underlie dynamics in supercooled liquids.23 Although the Adam−Gibbs theory did not specify the molecular form of these CRR’s, a number of studies in simulated molecular glass-formers and experimental colloids have suggested that, at least in these systems, they take the form of one-dimensional, cooperative, “stringlike” rearrangements.22,25,26,93−96 Accordingly, in order to test whether the same relationship between interfacial dynamics and cooperative motion holds here, we quantify the size of these cooperative rearrangements by calculating the mean number of neutral segments L participating in “stringlike” cooperative rearrangements via a protocol that is well-established in the literature.25,27,93,97 Within this protocol, segments are identified as mobile if they exhibit a displacement greater than that expected within a Gaussian distribution of displacements on a

(10)

where ξA is the value of ξτ at the onset temperature TA and A is a fitting parameter. Inserting this equality into the Adam− Gibbs relation suggests that, if the size of the restricted mobility region is indeed determined by the CRR size, then the relaxation time in the system should be related to ξτ as22 ⎡⎛ ξ (T ) ⎞ Δμ ⎤ ⎥ τα(T ) = τ∞ exp⎢⎜A τ + 1 − A⎟ ⎢⎣⎝ ξA ⎠ kBT ⎥⎦

(11)

As shown in Figure 12, the collapses corresponding to eqs 9 and 11 are both successful in all systems considered in this study. This success provides additional evidence that glass formation in these ionomers is driven by cooperative motion associated with universal glass physics and not with persistence length or cross-linking considerations particular to these systems. Furthermore, these results indicate that the Adam− Gibbs entropy theory of glass formation is an excellent candidate for describing the glass transition of ionomers. 2320

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systems. In fact, these features of near-interface dynamics are not specific to attractive surfaces; comparable effects, albeit leading to a mobility enhancement, are found near free surfaces and poorly wetting substrates.21,22 This correspondence emphasizes the generality of this type of near-interface modification in mobility; the essential physics are not limited to surfaces exhibiting covalent grafting or otherwise long-lived adhesion. From a practical standpoint, these results indicate that it is possible for suppressed-mobility domains to be isolated at high temperature but percolate at low temperature, consistent with recent experimental observations of a growth in reduced-mobility domains in nanocomposites upon cooling.46 Moreover, they suggest that these interfacial gradients should be susceptible to the introduction of small-molecule diluents, such as plasticizers and antiplasticizers, that have been shown to suppress such effects in nanoconfined polymers.52,53,98 While remaining consistent with many of the predictions of the original EHM model, these conclusions bring ionomer glass formation into line with a decade of results suggesting the universal presence of a cooperative length scale of dynamics in glass-forming liquids that both dominates their growing relaxation time upon cooling and establishes the fundamental length scale for local dynamic gradients with these materials. We expect that these conclusions should be applicable to a range of other systems incorporating covalently grafted interfaces, including grafted substrates, polymer brushes, grafted nanocomposites, and semicrystalline polymers. This possibility should prompt a reexamination of the understanding of segmental dynamics and glass formation in these systems, with implications for their rational design.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (D.S.S). Notes

The authors declare no competing financial interest.

Figure 12. Test of eq 9 (a) and eq 11 (b) from the main text for the pendant ionomer (■), the pendant ionomer/homopolymer blend (▲), the semitelechelic ionomer (●), and the pure polymer (◆, part a only).

■

ACKNOWLEDGMENTS This material is based on work supported by the National Science Foundation under Grant DMR1310433. The authors thank Dr. Robert Weiss, Dr. Kevin Cavicchi, and Dr. Jack Douglas for helpful discussions and editorial advice.

■

CONCLUSIONS Results from these molecular dynamics simulations of several model ionomers are inconsistent with the intuition that covalent grafting leads to a qualitatively distinct “tether” effect on local segmental dynamics, in which mobility restrictions propagate through the chain over a range scaling with the chain persistence length. Instead, they indicate that the range of mobility suppression tracks with the scale of cooperative rearrangements in the glass-forming liquid, consistent with results in nanocomposites,28−30 nanoconfined materials,22,50 and glass-forming liquids more broadly. Several key observations support this result: near-interface gradients in glass transition temperature and fragility are in quantitative agreement with results from simulated ungrafted nanocomposites;39 the range of alterations in segmental dynamics is found to grow in an essentially Arrhenius manner on cooling, inconsistent with the temperature insensitivity of the chain persistence length but consistent with observation in nanoconfined systems;38,50,51,90−92 and the Adam−Gibbs theory,23 relating segmental relation dynamics to the extent of cooperative motion, is found to quantitatively describe dynamics in these

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