© 2015 Dihui Ruan All Rights Reserved

Home , Glass ionomer cement

DIHUI RUAN

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Dihui Ruan

May, 2015 GLASS FORMATION BEHAVIOR OF MODEL IONOMERS

Dihui Ruan

Thesis

Approved: Accepted:

______Advisor Dean of the College Dr. David S. Simmons Dr. Eric J. Amis

______Faculty Reader Interim Dean of the Graduate School Dr. Robert A. Weiss Dr. Rex D. Ramsier

______Faculty Reader Date Dr. Kevin A. Cavicchi

______Department Chair Dr. Sadhan C. Jana ii

ABSTRACT

Ionomers – polymers with bonded ionic groups – are important because of their wide applications in various fields. The glass formation behavior of ionomers has direct impact on their properties and performance, and therefore demands a deeper understanding. The ionic groups collapse into distributed ionic aggregates in the ionomers. The region surrounding the aggregates shows dynamic suppression among uncharged polymer chains, and the size of this region is correlated to chains’ persistence length. This phenomenon is a consequence of the fact that there is bond connectivity between aggregates and polymers, which differs from materials with only non-bonded interfaces, like composites. Here, we perform molecular dynamics simulations of model ionomers to test this conventional view and propose a fundamental reconsideration of grafted micro-phase materials like ionomers. Based on our results, the covalent grafts are not the central factor determining linear segmental dynamics and glass formation.

Instead, we find that they are equivalent to strong physical attractions, as in ungrafted nanocomposites and nanoconfined glass-formers, where near-interface mobility suppression is mediated by cooperative rearrangements intrinsic to glass-forming liquids, rather than by a unique covalent ‘tethering effect’. This conclusion indicates the need for a revised understanding of glass formation and segmental dynamics in materials incorporating covalent grafting.

iii

DEDICATION

My Family

AKNOWLEDGEMENTS

I would like to thank my advisor, Dr. David S. Simmons, for his patience and friendliness in leading me not only on the path of my research but also on other parts of my life and career. Also, I’m thankful to my other committee members: Dr. Kevin A.

Cavicchi for discussion on the structure of ionic aggregates, and Dr. Robert A. Weiss for suggestions on the manuscript. Finally, I would like to thank all my colleagues, friends, and family for supporting my research.

TABLE OF CONTENTS

Page

LIST OF TABLES ...... viii

LIST OF FIGURES ...... ix

CHAPTER

I. INTRODUCTION ...... 1

II. METHODOLOGY...... 8

Simulation Model...... 8

Simulation Analysis ...... 11

III. RESULTS AND DISCUSSIONS ...... 15

Ionomer Morphology and Glass Transition ...... 15

Near-aggregate Gradient in Dynamics ...... 25

Test of the Adam-Gibbs Theory of Glass Formation in Ionomers ...... 37

IV. CONCLUSIONS ...... 41

V. FUTURE WORK...... 43

REFERENCES ...... 44

APPENDICES ...... 51

APPENDIX A. CALCULATION OF DEBYE-WALLER FACTOR FO FLEXIBLE IONOMERS ...... 52

APPENDIX B. CALCULATION OF DEBYE-WALLER FACTOR FOR PENDANT IONOMERS ...... 53

APPENDIX C. CALCULATION OF DEBYE-WALLER FACTOR FOR PURE POLYMERS ...... 54 vi

APPENDIX D. CALCULATION OF SEGMENTAL RELAXATION FOR FLEXIBLE IONOMERS ...... 55

APPENDIX E. CALCULATION OF SEGMENTAL RELAXATION FOR PENDANT IONOMERS ...... 56

APPENDIX F. CALCULATION OF SEGMENTAL RELAXATION FOR PURE POLYMERS ...... 57

APPENDIX G. OVERALL EXTRAPOLATED GLASS TRANSITION TEMPERATURES ...... 58

APPENDIX H. VFT EXTRAPOLATION IN BINNED ANALYSES FOR FLEXIBLE IONOMERS ...... 59

APPENDIX I. VFT EXTRAPOLATION IN BINNED ANALYSES FOR PENDANT IONOMERS ...... 60

APPENDIX J. BINNED EXTRAPOLATED GLASS TRANSITION TEMPERATURE ...... 61

vii

LIST OF TABLES

Table Page

3.1 List of overall monomer-part Tg and m for the semi-telechelic ionomer (Kθ = 0), the 50/50 pendant ionomer/homopolymer blend (Kθ = 0), the pendant ionomer (Kθ = 0, 1.5, and 3.0) compared with the corresponding neat polymer...... 22

viii

LIST OF FIGURES

Figure Page

2.1 Schematics of the pendant (a) and the semi-telechelic (b) ionomer models including neutral monomer beads (blue), anion beads (light grey), and free cation beads (black)...... 8

3.1 (a): Plot of structure factor (Soverall) versus wave number (q) for all the particles in the systems at T = 1.0. (b): Plot of ion-ion structure factor (Sion-ion(q)) versus wave number (q) from q = 5 to q = 15 for ions at T = 1.3 (dashed lines) and T = 1.0 (solid lines). The included systems are the pendant/homopolymer Kθ = 0 (green), the semi-telechelic ionomer Kθ = 0 (red), the pendant ionomer Kθ = 0 (blue), the pendant ionomer Kθ = 1.5 (purple), and the pendant ionomer Kθ = 3.0 (grey)………………………………………………………………………...... 15

3.2 VMD snapshots for ionic aggregates at T = 1.3 (top) and T = 1.0 (bottom) in pure ionomer systems. Dark grey beads and silver beads represent for the bonded anions and the free cations respectively...... 16

3.3 Snapshots of single ionic aggregates with grafted chains at T = 1.0 in the pendant ionomer (a) and the semi-telechelic ionomer (b) with polymer chains (colorful), bonded anions (silver), free cations (dark grey)...... 17

3.4 Plot of average number of ions in an ionic aggregate in the semi-telechelic ionomer Kθ = 0 (●), the 50/50 pendant ionomer/homopolymer blend Kθ = 0 (green ■),the pendant ionomer Kθ = 0 (blue ■), the pendant ionomer Kθ = 1.5 (▲), and the pendant ionomer Kθ = 3.0 (♦)...... 17

3.5 Plots of the peak value of ion-ion structure factor within wavenumber q = 5-9 (a) and q = 9-15 (b) at T = 1.0 for the semi-telechelic with Kθ =0 (●), the 50/50 pendant ionomer/homopolymer blend with Kθ =0 (green ■), the pendant ionomer with Kθ =0 (blue ■), 1.5 (▲), 3 (♦)...... 19

3.6 Ion-ion pair-interaction energy (a) and overall pair-interaction energy (b) vs. temperature for the semi-telechelic ionomer Kθ = 0 (●), the pendant ionomer Kθ = 0 (■), the pendant ionomer Kθ = 1.5 (▲), and the pendant ionomer Kθ = 3.0 (♦)...... 20

3.7 Monomer-part τα versus T (a) and monomer-part τα normalized by the neat-state τα versus T normalized by the neat-state Tg (b) for simulated systems: the semi- ix

telechelic ionomer Kθ = 0 (●), the 50/50 pendant ionomer/homopolymer blend Kθ = 0 (green ■), the pendant ionomers with Kθ = 0 (blue ■), 1.5 (▲), and 3 (♦), and the pure systems with Kθ = 0 (×), 1.5 (-), and 3 (+)...... 21

3.8 Monomer-part τα versus T as a function of distance r from the nearest ion for the pendant/homopolymer blend with Kθ = 0 (b), the semi-telechelic ionomer with Kθ = 0 (c), and the pendant ionomer with Kθ = 0 (a), Kθ = 1.5 (d), and Kθ = 0 (e)...... 25

3.9 (a): Tg (solid markers) and m (hollow markers) normalized by their values in the neat homopolymer with the same Kθ, as a function of distance from the nearest ion for the semi-telechelic ionomer Kθ = 0 (●), the 50/50 pendant ionomer/homopolymer blend Kθ = 0 (green ■), the pendant ionomer Kθ = 0 (blue ■), the pendant ionomer Kθ = 1.5 (▲), and the pendant ionomer Kθ = 3 (♦). (b) Data for the same properties plotted versus distance from an attractive nanoparticle surface, with different markers representing different nanoparticle concentrations.59 ...... 26

3.10 Plots of τα normalized by the corresponding pure-polymer value versus distance from the nearest ion at T = 1.2 (●), T = 0.8 (▲), and T = 0.6 (■). Solid and dashed lines are fits to equ (3.3) and the criterion for ξτ in equ (3.1)...... 29

3.11 Plots of normalized by pure-state value versus distance from the nearest ion at T = 1.2 (●), T = 0.8 (▲), and T = 0.6 (■). Solid and dashed lines are fits to equ 2 (3.4) and the criterion for ξ in equ (3.2)...... 30

2 3.12 Length scales ξτ , ξ , and mean string length L versus T normalized by Tg of the neat polymer with the same Kθ for (a): pure polymer with Kθ = 0 (orange), semi- telechelic ionomer with Kθ = 0 (red), pendant ionomer/homopolymer blend with Kθ = 0 (green), pendant ionomer with Kθ = 0 (blue), and (b): pendant ionomer with Kθ =1.0 (blue), 1.5 (purple), Kθ = 3 (grey)...... 34

3.13 (a): Plot of Kuhn length b verus T for the pure systems with Kθ = 0 (×), 1.5 (җ), and 3 (+). (b): Arrhenius plot of ξτ versus T, where solid lines are an Arrhenius extrapolation, and dashed lines indicate extrapolated ξτ at an extrapolated EX experimental monomer Tg ; inset shows linear correlation between ξτ and L for the flexible systems: the pendant ionomer (blue ■), the 50/50 pendant ionomer/homopolymer blend (green ■), and the semi-telechelic ionomer (●)...... 35

3.14 Plots of non-Gaussian parameter versus time t at T = 1.5 (dashed line) and T = 0.7 (solid line) for (a): the pendant ionomer (blue), the 50/50 pendant ionomer/homopolymer blend (green), and the semi-telechelic ionomer (red) with Kθ = 0; (b): the pendant ionomer with Kθ = 0 (blue), 1.5 (purple), 3 (grey); (c): the pure polymer with Kθ = 0 (orange), 1.5 (pink), 3 (aqua)...... 36

3.15 (a): Illustration for Arrhenius correlation between τα and T, where τα has an

Arrhenius dependence on T when T ln(τα / τA) / Ea on the y axis equals 1. (b): Plot of enthalpy-entropy compensation, the slope of which is the compensation temperature. The solid line indicates a linear relation. Including systems are the pendant ionomer (blue ■), the 50/50 pendant ionomer/homopolymer blend (green ■), and the semi-telechelic ionomer (●) with Kθ = 0; (b): the pendant ionomer with Kθ = 0 (■), 1.5 (▲), 3 (♦); (c): the pure polymer with Kθ = 0 (×), 1.5 (җ), 3 (+). .. 37

3.16 Test of Adam Gibbs theory based on equ (3.5) and (3.8) for the pendant ionomer (blue ■), the 50/50 pendant ionomer/homopolymer blend (green ■), the semi- telechelic ionomer (●) and the pure polymer (×) with Kθ = 0...... 40

CHAPTER I

INTRODUCTION

Ionomers – ion-containing polymers – have received attentions from academics and industry for decades because they are widely applied in the fields of elastomers1,2, dental composites3–7, electrolytes8,9, fuel cells10,11, and golf balls12,13. Throughout the ionomer, ionic groups mixed with neutralizers undergo aggregation via a micro-phase separation driven by the polar ions and nonpolar polymers, and also balanced by the electrostatic force of ions and elastic force from polymer chains.14–17 The ionomer’s glass formation, reflecting its linear-regime segmental dynamics, has a huge impact on the properties and performance of the material, such as transport18–20 and mechanical21–23 properties that are central to most applications. Since polymer chains are connected to aggregates via covalent bonds, the glass formation behavior of ionomers also raises profound questions regarding the physical origin of glass-forming dynamics in soft materials with micro- phase grafts, such as grafted nanoparticle composites, substrate-bonded films, and semicrystalline polymers or copolymers.

The existence of ionic aggregates, formed by the collapse of the bonded ionic groups and neutralizers, are directly shown by small-angle x-ray scattering (SAXS), where there is a peak located at a small scattering angle.24 This peak is therefore called the ‘ionomer peak’ as a common feature of ionomers, illustrating the long-distance inter-

1 aggregate spacing correlation14–16. As is postulated by Eisenberg25, the ionic groups can collapse into ionic aggregates either including or excluding polymer chains, investigated under dielectric26, infrared27,28 and Raman29 measurements. Both the interpretation of the

‘ionomer peak’ and the existence of polymer-included aggregation are validated by a famous hard-sphere model modified by Yarusso and Cooper30, based on the core-shell model established by Macknight et al.31, that any two spherical ion-rich domains are separated by surrounding shells composed of uncharged polymer chains. However, the intra-aggregate structure is less clear, partly as a result of the small size of aggregates

(1~150nm)32,33, which are constrained by the balance among ion interaction, steric hindrance from polymer chains, interfacial tension or interaction, and possible crystallinity.34 It is worth noting that a lamellar model24, deviating from the commonly- believed spherical shape of ionic aggregates and including a short-range intra-aggregate correlation, has been originally proposed by Macknight’s group to track the features in

SAXS. Both the structure and the size of these ionic aggregates are sensitive to the polymer backbond, the placement and randomness of the ionic groups in the polymer chain, the type of the ionic group and neutralizer, and the degree of neutralization.4,22,35–37

With respect to computational studies on ionomers, ionic aggregates, including only ions, have been investigated as an extended, unordered stringy structure at relatively high temperature within both coarse-grained and atomistic molecular dynamics (MD) simulations38–42; additionally, Goswami and his coworkers43 have reported that ionic aggregates self-assemble into a condensed, well-ordered plane with two layers at low temperature. This structure can be understood as a planar model with a small lateral

2 extent14 with uncharged polymer chains located above and below, modified from the lamellar model of Macknight et al.24

The ionic aggregates collapse into a condensed, well-ordered structure from a relatively extended, disordered one via a self-assembly transition, observed by a high- temperature relaxation process within the simulation by Goswami et al.43 and termed as a

‘second Tg’. However, in physical experiments, the high temperature of an additional relaxation, investigated by a broad peak within mechanical response44–46, is interpreted as the Tg for those polymer chains whose dynamics are reduced by ionic aggregates. The introduction of ionic aggregates into the polymer matrix yields an enhancement in

18,22,44–47 ionomers’ glass transition temperature Tg via a strong spatial gradient in

45,48,49 suppressed dynamics approaching aggregates . The Tg of ionomers is more greatly enhanced with increasing ion concentration44,45,50, with an increase in the range of

1.2~5.9°C per mole of ionic groups48. This suppression in segmental dynamics has often been attributed to the crosslinking role played by ionic aggregates.14,51–53 This crosslinking effect is emphasized by Eisenberg, Hird, and Moore17, who suggest that the range of the restricted-mobility region surrounding the ionic aggregates is correlated with the stiffness of the polymer chains, characterized by the persistence length. This EHM model has since become a predominant view of the structure and dynamics of ionomers.

Since the persistence length is nearly temperature independent, we note that this view also implies that the size of the reduced-dynamics domain around aggregates should be insensitive to temperature.

The above view represents a significant difference from the behavior of materials near non-bonded interfaces. In these systems, similar alterations in glass transition temperature and the spatial gradient in dynamics have also been also found in nanoconfined materials, such as thin films54–56 and nanoparticle composites57–59.

59 According to simulations done by Betancourt et al. , Tg is enhanced in nanoparticle polymer composites with attractive interfaces, induced by a gradient in dynamics around the nanoparticle with respect to the bulk polymer. The comparability of the gradient in dynamics between ionomers and nanoconfined materials is explained by their similar morphology as heterogeneous systems, according to Tsagaropoulos and Eisenberg44,46. In contrast, for these nanoconfined materials, the effective range of the dynamics gradient has been found in simulation to exhibit a roughly Arrhenius dependence on temperature, and is correlated to the size of cooperatively rearranging regions (CRRs) underpinning dynamics in fragile (non-Arrhenius) glass-forming liquids.57,59–61 These CRRs are defined as subsets of correlated moving particles whose arrangement is independent of the surroundings. It has been predicted by Adam and Gibbs62 that relaxation depends on temperature via the size of CRRs determined by the configurational entropy, as shown in the following equation:

z     0 exp  , (1.1) kB T  where τα is segmental relaxation time, τ0 is vibrational relaxation time, Δμ is the limiting activation free energy, and z is the size of CRRs evaluated as the average number of particles included in CRRs. Adam and Gibbs have not given a specific definition for the shape of the cooperatively rearranging regions (CRRs), leading to the difficulty in direct 4 characterization of the CRRs’ size. While in simulations46,61,63–65, a string-like cooperative motion has been observed in simulations of glass formers, such that mobile particles in a string-shaped group move by replacing their former one’s position. The size of the ‘strings’ in this model, which captures most of the features of CRRs relevant to the dynamics, is computed by the mean string length, defined as the average number of particles included in the same string-like motion.63

In nanoconfined materials, the chains’ stiffness, as characterized by the persistence length, is not central to the glass-forming dynamics, but it is a factor of the kinetic fragility, defined as the slope of an Arrhenius plot of relaxation time versus temperature at Tg in an Angell plot:

d log  m    , (1.2) d Tg / T    TT g where τα is the segmental relaxation time. It indicates how rapidly the relaxation is slowed down around Tg. For example, the fragility as well as the glass transition temperature Tg of polymers generally grows with increasing stiffness of the backbond, as predicted in the generalized entropy theory66 of glass formation, which is an extension of the Adam-Gibbs theory62.

With respect to the similar trends in dynamics within ionomers and nanoconfined materials, do covalent attachments fundamentally transform the physics by which an interface alters nearby segmental dynamics and glass formation, or are they essentially indistinguishable from strong physical adsorption for these purposes? A recent study by

Papon and coworkers67 have reported that the existence of covalent bonds connecting the

5 polymers with the fillers is not essential to the dynamic gradient, with similar results in filler-grafted elastomers by Berriot68. These results imply that the covalent bonds are, instead of central to the physical origin, equal to an adhesion to the surface of ionic aggregates in ionomers for the purpose of glass-forming dynamics. Is this really the case?

If near-interface dynamics propagate via intra-chain bond correlations, their size should mirror the typically weak temperature dependence of the chain persistence length, accompanied with a huge leap as a consequence of increasing stiffness; if the

‘cooperativity’ model dominates, their size should exhibit a strong, roughly Arrhenius, growth upon cooling instead.

To understand the nature of the impact of covalent grafting on the glass transition in ionomers, we perform molecular dynamics (MD) simulations on model ionomers with the ionic groups placed either on the end or the side of the uncharged backbond with varying the chain stiffness. Our discussions begin with a consistency check of the morphology of ionic aggregates compared with the results in previous experiments and

17 simulations. We test the bond-centric view of the EHM model by comparing overall Tg and spatial Tg gradient among systems, accompanied with more general interfacial length scales defined from structural relaxation and vibration dynamics. Additionally, the size of

CRRs is characterized by the mean string length via a well-established protocol developed in the string model69,65,70 and is then compared with the interfacial length scales. Finally, a test on Adam-Gibbs theory is performed in terms of the mean string length and the interfacial length scale. These results prove that the chain connectivity between the polymer chains and the ionic aggregates is not critical for the purpose of

6 glass formation in ionomers; rather, the glass formation of ionomers is consistent with that of nanoconfined materials.

CHAPTER II

METHODOLOGY

Simulation Model

We perform molecular dynamics (MD) simulations of coarse-grained bead-spring model ionomers within the LAMMPS17,27,31 (Large-scale Atomic/Molecular Massively

Parallel Simulator) environment, in order to study (pendant/semi-telechelic) ionomers’ glass formation.

Figure 2.1 Schematics of the pendant (a) and the semi-telechelic (b) ionomer models including neutral monomer beads (blue), anion beads (light grey), and free cation beads

(black).

As shown in Figure 2.1, model ionomers are composed of 20 neutral beads on each backbond, designed to compare with 20-bead linear homopolymers. To make the pendant and semi-telechelic ionomers, an anion bead (-1e) is bonded to either the 10th or the 20th uncharged bead on each chain. Within (either pendant or semi-telechelic) ionomer simulations, 100 ionomer chains are 100% neutralized by cation beads (+1e), while a

50/50 pendant ionomer/homopolymer blend is additionally simulated for the purpose of a dilute ionomer system.

Interactions among beads are computed in Lenard-Jones (LJ) non-dimension units71 based on an attractive extension of the Kremer-Grest model72 commonly employed to study polymers’ glass formation. Within these models, bonded beads interact via a standard finitely extensible nonlinear elastic (FENE) potential71,72:

2  2 r  EKRbond 0.5 bond 0 ln 1    , (2.1) R 0  

2 where spring constant Kbond is 30 /  , maximum extent of the bond R0 is 1.5 , and r is distance. Bonds’ angles are computed via a cosine angular potential71:

EK  1  cos   , (2.2)

73 where bending coefficient Kθ is set as 0, 1.5 , and 3 for different bond stiffnesses .

Non-bonded beads interact via a 12/6 Lenard-Jones (LJ) potential71:

12 6  LJ    LJ  ELJ4 LJ      for r  R c , (2.3) r   r  

where, LJ sets the diameter with a cutoff distance Rc as 2.5 . Charged beads additionally interact via a Coulombic interaction potential71, directly calculated within cutoff Rc = 8 as:

qi q j Ecoul for r  R c , (2.4) 40  rr where qi and qj are charges of ion pairs, ε0 is vacuum permittivity, and εr (=4.0) is the dielectric constant. Longer-range Coulombic interactions beyond the cutoff Rc are

9 computed via a particle-particle particle-mesh (pppm) solver74, with further optimization applied for the reason that most particles are uncharged in systems.

Based on the model based on poly(ethylene-co-acrylic acid) (PEAA) fully neutralized with sodium (Na+) established by Hall et al.38, each neutral polymer bead is set as 1.0 (  0.4nm ) in diameter, mapped to three CH2 units (~ 0.4nm) along a polyethylene backbond; the diameter of each monovalent anion and cation bead is scaled as 1.0 and 0.5 representing a COO– group (~ 0.4nm) and a Na+ (~ 0.2nm). The masses of uncharged monomers, anions, and counterions are chosen to be 1.0, 1.0, 0.5 in

LJ units respectively, based on their molar masses of 42, 44, and 23 g/mol. The time in

LJ units   1ps , and room temperature (300K) corresponds to T 1.0 in LJ units. By choosing the dielectric constant εr as 4.0, the fundamental electron charge e is converted into 11.80 in LJ units in order to obtain proper Coulombic and LJ interactions consistent with the properties of poly(ethylene-co-acrylic acid) (PEAA) fully neutralized with sodium (Na+) for the purpose of glass formation.

We perform MD simulations on the pendant ionomer with K  0 , a 50/50 blend of

pendant ionomer with homopolymer with K  0 , the semi-telechelic ionomer with

K  0 , the pendant ionomer with K 1.5, and the pendant ionomer with K  3.0 , and in each case we compare results to simulations of the corresponding pure (charge-free) linear homopolymer with the same Kθ. For each system, four initial configurations are generated by Packmol75 (Packing Optimization for Molecular Dynamics Simulations) in a cubic simulation box. Simulations employ the Nose-Hoover thermostat and barostat71

10 within damping parameter 2.0 and a rRESPA76 integrator with a timestep of 0.005 for non-bonded and 0.001 for bonded interactions.

We begin each simulation with a 5 104 run in the NPT ensemble at T 1.5 and

P  0 for each configuration. The system is then quenched to a low temperature at a rate of 105T / and P  0 with configurations saved at regular temperature intervals. For each saved configuration, an additional equilibration run is employed at P  0 with the length of time varying from 104 for 1.0T  1.5 with selected T interval 0.025 to 105 for T 1.0 with selected T interval 0.02. After that, the simulations boxes of four configurations at each temperature are deformed to an identical equilibrium density determined by the second half average of equilibration runs within 103 , and then equilibrated for 103 . Finally, data is collected in the NVT ensemble. This data is then analyzed via methods implemented in AMDAT77 (Amorphous Molecular Dynamics

Analysis Toolkit), a MD analysis code written by Simmons’ Research Group.

Simulation Analysis

Average number of ions in an aggregate within simulated ionomer systems is calculated at each temperature as the total number of ions divided by the number of ion clusters, where a cluster is defined as a subset of ions within which the bonded anions (

1.0 ) and the free cations ( 0.5 ) are continuously connected using a cutoff distance of

1.0 from one another, evaluated by a cluster computation implemented in LAMMPS71.

Pair interaction energy among either ions or all the beads within each system is calculated based on thermodynamic data computed by LAMMPS71, in order to track the calorimetry and investigate the transition upon cooling. 11

Static structure is quantified by the structure factor S(q):78

1 NN S(q ) exp(  i q  ri )exp( i q  r j ) , (2.5) N i1 j  1

th th where q is the wave vector and ri and rj are the positions of the i and j bead within the calculation among N particles.

Free volume and vibrational dynamics are quantified by the Debye-Waller Factor

, defined as the length scale of a ‘cage’ within which particles rattle around. is calculated as the mean-squared displacement at a crossover time (~1τ) from ballistic to

‘cage’ limited motion, which is set as 0.99τ in our systems79. This important parameter has been proved as a measurement of free volume80 defined as the unoccupied volume in a system. A free volume theory relates this local packing into properties of glass formers, such as segmental relaxation given by:80

2/3  exp  /   , (2.6)   c f   where υc is a constant, < υf> is the free volume, and τα is the segmental relaxation time.

Non-Gaussian behavior is quantified by the non-Gaussian parameter, calculated from the mean-squared displacement ():

3 r t4 non-Gaussian parameter =2  1, (2.7) 5 r t2 the peak time of which is considered as the time when maximum cooperative motion takes place.70

Segmental relaxation dynamics are quantified via the self-part of the intermediate scattering function Fself(q,t), which is comparable to characterization by time-resolved incoherent neutron scattering in experiments:

1 N F (q ,t ) exp  i q  r t  r 0  , (2.8) self   j  j    N j where rj(t) and rj(0) are the positions of particle j at time t and an initial time. Also, we employ wave number q  7.07 , comparable to the first peak in the structure factor of a standard FENE 20-bead linear homopolymer. The alpha segmental relaxation time τα is

defined as the time when Fself (q ,t ) 0.2, after employing a Kohlrausch-Williams-Watts

(KWW) stretched exponential fit81,82 to smooth and interpolate the data after the fast beta relaxation (within 0.99τ):

 t   Fselft |q 7.07  A exp     , (2.9)    where A, τ and β are fitting parameters.

The dynamic glass transition temperature Tg is defined as the temperature threshold at which these simulations fall out of equilibrium within the timescale (103τ). In another word, the definition of Tg in our study is the temperature at which the segmental

3 3 relaxation time  10  . The   data, which is less than 10 τ, for temperatures less than

1.0 is fit to a Vogel-Fulcher-Tamman (VFT) form:83,84

 DT  ln  A 0 , (2.10)   T  T  0

13 where A, D, and T0 are fitting parameters. The computational Tg is then determined by a negligible extrapolation of this fit, since the data within the super-cooled region are considered as in-equilibrium from the simulations whose equilibration time is no less

2 3 than 10  ( 10 for TT  g ) within these systems at each temperature.

The kinetic fragility index m is then defined as the slope on an Angell plot

85 (Arrhenius) of τα at Tg:

 d log  m     . (2.11)  d Tg / T     T T g

In order to compare some results in our simulation with physical experiments, we

EX defined an extrapolated glass transition temperature (Tg ) by an extrapolation from the

14 2 VFT functional form to a temperature at which  10  (  10 s ), which is comparable

EX to the criterion in experiments to define the glass transition temperature. Tg is not usually used in our study, because the huge extrapolation involved can lead to aphysical trends. To keep the precision of our work, most of discussions are based on the computational Tg defined above.

Additional local dynamics analyses on vibration and relaxation are performed on each subset of uncharged polymer beads by binning them as a function of the distance to their nearest ion at an initial time. For the purpose of discussion, we denote analyses only on bonded anion and counterion beads as ‘ion’ properties, while analyses on only uncharged polymer beads and all the particles within these systems are denoted as

‘monomer’ and ‘overall’ properties, respectively.

CHAPTER III

RESULTS AND DISCUSSIONS

2 1.8 a b 1.8 1.6 1.6 1.4 1.4 1.2 ) 1.2 ) q q ( (

1 ion 1 - overall ion S 0.8 S 0.8 0.6 0.6 0.4 0.4 0.2

0 0.2 0 5 10 15 20 25 5 7.5 10 12.5 15 q q

Figure 3.1 (a): Plot of structure factor (Soverall) versus wave number (q) for all the particles in the systems at T = 1.0. (b): Plot of ion-ion structure factor (Sion-ion(q)) versus wave number (q) from q = 5 to q = 15 for ions at T = 1.3 (dashed lines) and T = 1.0 (solid lines). The included systems are the pendant/homopolymer Kθ = 0 (green), the semi- telechelic ionomer Kθ = 0 (red), the pendant ionomer Kθ = 0 (blue), the pendant ionomer

Kθ = 1.5 (purple), and the pendant ionomer Kθ = 3.0 (grey).

Ionomer Morphology and Glass Transition

We perform MD simulations on ionomer systems with about 5mol% ionic groups bonded to the neutral backbond and an additional diluted system of the 50/50 pendant

15 ionomer/homopolymer blend with about 2.4mol% ionic groups for longer distance range among ionic aggregates, all fully neutralized with free counterions. The pendant ionomer and the semi-telechelic ionomer are employed to ascertain how the placement of ionic groups affects dynamics, while varying the bending coefficient (Kθ) in the pendant model is aimed to clarify the correlation between chains’ stiffness and ionomers’ dynamics.

Figure 3.2 VMD snapshots for ionic aggregates at T = 1.3 (top) and T = 1.0 (bottom) in pure ionomer systems. Dark grey beads and silver beads represent for the bonded anions and the free cations respectively.

Figure 3.3 Snapshots of single ionic aggregates with grafted chains at T = 1.0 in the pendant ionomer (a) and the semi-telechelic ionomer (b) with polymer chains (colorful), bonded anions (silver), free cations (dark grey).

Avg.of#Ions in Aggregatean 20

10 0.3 0.5 0.7 0.9 1.1 1.3 1.5 T Figure 3.4 Plot of average number of ions in an ionic aggregate in the semi-telechelic ionomer Kθ = 0 (●), the 50/50 pendant ionomer/homopolymer blend Kθ = 0 (green ■),the pendant ionomer Kθ = 0 (blue ■), the pendant ionomer Kθ = 1.5 (▲), and the pendant ionomer K = 3.0 (♦). θ 17

The ‘ionomer peak’ around q  0.70 / 0.95 /  appears in all the systems with ionic groups in Figure 3.1(a), consistent with previous work on ionomers’ morphology14–

16. This peak at a low q reflects the inter-aggregate correlations investigated by scattering data, which is a common feature of ionomers. Within these simulated systems, the mean distance among ionic aggregates is 6.6 ~ 8.8  2.6 3.5nm . The images made by the trajectories of ions in Figure 3.2 and the average number of ions in the aggregate from

Figure 3.4 demonstrate that the size of ionic aggregates in the pendant ionomer with Kθ =

0 is smaller compared with the other pure ionomer systems both at a higher T and lower

T. The formation of larger aggregates is attributed to the weakened entropy from polymer chains, caused by the reduction of chains bonded to each anion bead between the pendant ionomer (Figure 3.3(a)) and the semi-telechelic ionomer (Figure 3.3 (b)), as well as the increasing stiffness tuned by bending coefficient Kθ. Also, these images and plots indicate an intriguing self-assembly transition at T 1.3 1.0 in which relatively disordered extended ionic aggregates with changing sizes collapse into more ordered bilayer platelets with a more stable size upon cooling, found in the prior simulation by Goswami et al.43 This high-temperature transition corresponds to a ‘peak-splitting’ behavior in the ion-ion structure factor at q  5/ ~15/  (0.17 0.50nm) , in a wave number range reflecting the intra-aggregate correlation, as shown in Figure 3.1(b). As a sanity check, the heights of the two peaks corresponding to intra-aggregate correlation are evaluated in

Figure 3.5 as an indirect characterization of the aggregate size. These results are consistent with the average number of ions in the aggregate as shown in Figure 3.4, in that the size of ionic aggregates is enhanced by reducing the number of chains bonded to

18 each anion and increasing the stiffness of polymer chains. The bilayer aggregates are ordered into two layers of a body-centered cubic (BCC) lattice, as shown by the two split peaks located at wave numbers with a ratio approximately 21/2, even though the structure seems like an FCC lattice such as NaCl. This is because the structure factor S(q) of the bonded anion beads and free cations is computed at the same time based on the two scatterers with the same scattering cross section. Thus, two FCC lattice of the two ions are cancelled into one BCC lattice such as KCl. The intra-aggregate correlation indicated by peaks’ splitting is difficult to capture both in the overall structure factor (Figure 3.1

(a)) within our simulations and in the ionomers’ scattering experiments, because of the small aggregate, whose mean length is 3 4.5 (1.2~1.8nm) in our simulations

2 2 a b 1.9 1.9

1.8 1.8

1.7 1.7

1.6 1.6 Peak Peak Value Peak Peak Value 1.5 1.5 = 5~9 = 9~15 q q

)| 1.4 1.4 )| q q ( ( ion

1.3 ion 1.3 - - ion ion S 1.2 S 1.2

1.1 1.1

1 1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0.3 0.5 0.7 0.9 1.1 1.3 1.5 T T Figure 3.5 Plots of the peak value of ion-ion structure factor within wavenumber q = 5-9

(a) and q = 9-15 (b) at T = 1.0 for the semi-telechelic with Kθ =0 (●), the 50/50 pendant ionomer/homopolymer blend with Kθ =0 (green ■), the pendant ionomer with Kθ =0 (blue

■), 1.5 (▲), 3 (♦).

19 according to the average number of ions in the aggregates (Figure 3.4), based on a bilayer square platelet model after the ordering transition as an estimation. Figure 3.6 shows that the existence of this transition is also supported by the feature observed in the ion-ion energy versus T (a), corresponding to the high-T feature in the overall energy (b), where the aggregate-ordering assembly shows increasing discontinuity in calorimetry from the flexible pendant ionomer to the stiff pendant ionomers to the flexible semi-telechelic ionomer, consistent with the sharpness of those split peaks in the ion-ion structure factor

(Figure 3.1 (b)) and the two peak height values (Figure 3.5) at low T. The ordering transition within ionic aggregates, combined with glass transition among neutral monomers, is argued as the origin of two apparent glass transitions observed in the mechanical response of ionomers by a previous simulation.43

-2.9 -6.5 a b -3 -7

-3.1 -7.5 -3.2 ion - -8 overall ion E E -3.3 -8.5 -3.4

-9 -3.5

-3.6 -9.5 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 T T

Figure 3.6 Ion-ion pair-interaction energy (a) and overall pair-interaction energy (b) vs. temperature for the semi-telechelic ionomer Kθ = 0 (●), the pendant ionomer Kθ = 0 (■), the pendant ionomer Kθ = 1.5 (▲), and the pendant ionomer Kθ = 3.0 (♦).

Glass transition temperature Tg of overall simulated systems is calculated based on the segmental relaxation time τα by fit into a VFT function form as shown in Figure

3.7(a). There is no significant difference among the segmental relaxation time of ionomer systems normalized by the corresponding pure polymer with the same bending coefficient Kθ (Figure 3.7(b)), indicating that neither the chain connectivity with the aggregates nor the stiffness of chains is central to the suppression of the segment mobility as a result of the introduction of ionic aggregates into the polymer bulk. The enhancement of the glass transition temperature Tg of uncharged monomers within these

103 2.5 a b

2 10 ) 1.5 pure α τ α τ / α τ 1

101 ln(

0.5

Tg 100 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 pure T T / Tg

Figure 3.7 Monomer-part τα versus T (a) and monomer-part τα normalized by the neat- state τα versus T normalized by the neat-state Tg (b) for simulated systems: the semi- telechelic ionomer Kθ = 0 (●), the 50/50 pendant ionomer/homopolymer blend Kθ = 0

(green ■), the pendant ionomers with Kθ = 0 (blue ■), 1.5 (▲), and 3 (♦), and the pure systems with Kθ = 0 (×), 1.5 (-), and 3 (+).

systems relative to the corresponding pure polymer with the same bending coefficient Kθ

is consistent with experimental results. To note, the comparison in Tg between the

ionomers and the corresponding pure polymer with the same stiffness is supported by Tg

of all simulated systems increasing with Kθ, in agreement with predictions of the

66 generalized entropy theory of glass formation. As shown in Table 3.1, Tg is enhanced

by 7.0% with respect to the neat polymer for the pure pendant ionomer and the semi-

telechelic ionomer both with Kθ = 0, indicating Tg enhancement is insensitive to the

placement of the ionic group on the neutral polymer chain. Otherwise, based on the EHM

86 model , there would be weaker enhancement in Tg in the semi-telechelic ionomer

compared with the pendant ionomer, because of the reduction in steric hindrance from

two to only one chain covalently bonded to each anion in ionic aggregates (Figure 3.3).

Table 3.1 List of overall monomer-part Tg and m for the semi-telechelic ionomer (Kθ = 0), the 50/50 pendant ionomer/homopolymer blend (Kθ = 0), the pendant ionomer (Kθ = 0,

1.5, and 3.0) compared with the corresponding neat polymer.

semi-telechelic pendant/homo pendant pendant pendant

Kθ 0 0 0 1.5 3.0

Tg 0.476 0.459 0.476 0.545 0.650

pure Tg 0.445 0.445 0.445 0.514 0.615

pure Tg/Tg - 1 7.0% 3.3% 7.0% 6.1% 5.8%

m 15.40 15.38 15.41 19.71 23.87

pure m 15.87 15.87 15.87 21.54 27.59

pure m/m - 1 -2.9% -3.0% -2.9% -8.5% -13.5%

The 50/50 pendant ionomer/homopolymer blend exhibits a 3.3% enhancement in

Tg, roughly consistent with the reduced concentration of ions in this system. If the stiffness of polymer chains increases mobility restriction by ionic aggregates as pointed

17 out in the EHM model , the enhancement in Tg would be much stronger relative to the pure polymer with the same Kθ, with comparison to the flexible pendant ionomer. In contrast, the enhancement in Tg of these stiff pendant ionomers is similar or even weaker with respect to the flexible one, indicating the contribution of ionic aggregates to Tg enhancement is weakened by increasing chains’ stiffness. These results are consistent with a typical range of Tg enhancement in experimental ionomers; a 1.2~1.4% Tg enhancement per mole-percent ion observed in our simulations corresponds to an enhancement of approximately 2.8~3.3 K per mole-percent ion48, as based on amorphous

87 polyethylene with a Tg of 237K . As a result of the introduction of ionic groups into uncharged polymers, the enhancement in Tg is irrelevant to the placement of the ionic group to the polymer chain, slightly weakened by the stiffness of chains, but sensitive to the ion concentration for a 100% neutralized ionomer system.

The kinetic fragility m associated with those uncharged monomers in each system is suppressed with respect to the neat polymer with the same bending coefficient Kθ, showing the similarity to nanoconfined polymers57,59 where the suppression of m is commonly observed with respect to the pure state polymer. Also, this comparison in fragility m between the ionomers and the pure polymers is supported by the growth of fragility m with the increase in stiffness tuned by the bending coefficient Kθ, consistent with the prediction by the generalized entropy theory66. Intriguingly, the suppression of m

23 is insensitive to both the ion content and the chain connectivity to ionic aggregates on the uncharged polymer chain, shown (in Table 3.1) by a ~ 3.0% m suppression with respect to the neat polymer in the semi-telechelic, the pendant ionomer/homopolymer blend, and the pendant ionomer system with Kθ = 0. In contrast, the suppression in m rises dramatically with increasing Kθ, raised by ≥ 5% for every 1.5 increase in Kθ. The increasing contribution of ionic aggregates to fragility suppression of uncharged monomers is associated with a larger adjustment in packing as a result of more favorable formation of ionic aggregates in ionomers, and more loose packing in the neat polymer caused by increasing stiffness. This indicates that introducing ionic aggregates can tune the kinetic fragility m (which characterizes the abruptness of the increase in segmental relaxation upon Tg), and this is meaningful for processing of polymers and is relevant to the glassy modulus.

103 r = 0.75 a r = 1.25 r = 1.75 r = 2.25 r = 2.75 102 r = 3.25 = 0 ) =

θ r = 3.75 K

pendant( pendant( 1

α 10 τ

Tg 100 0.4 0.5 0.6 0.7 0.8 0.9 1 T 103 103 r = 0.75 r = 0.75 b d r = 1.25 r = 1.25 r = 1.75 r = 1.75 r = 2.25 r = 2.25 102 r = 2.75 102 = 0 ) = r = 2.75 θ

r = 3.25 ) = 1..5 K

θ r = 3.25 r = 3.75 K r = 3.75 r = 4.25 r = 4.75

1 pendant( 1 10 r = 5.25 α 10 τ pendant/homo( α τ

Tg Tg 100 100 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 T T 103 103 r = 0.75 r = 0.75 c r = 1.25 e r = 1.25 r = 1.75 r = 1.75 r = 2.25 r = 2.25 102 r = 2.75 102 r = 2.75 = 0 ) = = 3 )

θ r = 3.25

θ r = 3.25 K r = 3.75 K r = 3.75 r = 4.25 r = 4.25 r = 4.75 pendant( pendant( telechelic( telechelic( - 1 α 1 10 τ 10 semi α τ

T g Tg 100 100 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0.7 0.8 0.9 1 T T

Figure 3.8 Monomer-part τα versus T as a function of distance r from the nearest ion for the pendant/homopolymer blend with Kθ = 0 (b), the semi-telechelic ionomer with Kθ = 0

(c), and the pendant ionomer with Kθ = 0 (a),25 K θ = 1.5 (d), and Kθ = 0 (e).

1.2 1.2 1.2 1.2

1.18 a 1.18 b 1.1 1.1 1.16 1.16 1 1 1.14 1.14 0.9 0.9 1.12 1.12 pure pure pure pure g g 1.1 0.8 1.1 0.8 m m T T / / / / g g

1.08 m 1.08 m T T 0.7 0.7 1.06 1.06 0.6 0.6 1.04 1.04 0.5 0.5 1.02 1.02

1 0.4 1 0.4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Distance From the Nearest Ion Distance From NP Surface

Figure 3.9 (a): Tg (solid markers) and m (hollow markers) normalized by their values in the neat homopolymer with the same Kθ, as a function of distance from the nearest ion for the semi-telechelic ionomer Kθ = 0 (●), the 50/50 pendant ionomer/homopolymer blend

Kθ = 0 (green ■), the pendant ionomer Kθ = 0 (blue ■), the pendant ionomer Kθ = 1.5 (▲), and the pendant ionomer Kθ = 3 (♦). (b) Data for the same properties plotted versus distance from an attractive nanoparticle surface, with different markers representing different nanoparticle concentrations.59

Near-aggregate Gradient in Dynamics

In order to quantify the effect of the ionic aggregates on local segmental dynamics of the polymer, we compute dynamics on monomers in bins with thickness 0.5 measured by the distance from the nearest ion. As shown in Figure 3.8, the Tg of the monomers is calculated after fitting the τα data as a function of temperature in each bin to a VFT functional form; the figure shows that τα of monomers is enhanced with decreasing distance from the nearest ion in each ionomer system. In Figure 3.9(a), the Tg of uncharged monomers increases as the aggregates are approached, but remains above the

26 value in the neat homopolymer at all distances within our computed range 3.75 ~ 5.25

(1.5~ 2.1nm), as a result of lacking sufficient distance between aggregates. This result is consistent with the trend of Tg decaying with increasing distance from the aggregates as

49 reported in the experimental study by Miwa et al. They interpret the gradient in Tg in their study as supporting the EHM model’s ‘crosslinking’ view17 because the extrapolated distance at which Tg enhancement vanishes is comparable to the persistence length of the neat polymer. In contrast, the effective length range of Tg enhancement is considerably greater than the chains’ persistence length in our simulations, for example, which is approximately 0.7 in those systems with bending coefficient Kθ = 0.

Additionally, the gradient of Tg enhancement by distance is similar in both shape and magnitude for the pendant ionomers with different Kθ, and hence, different persistence lengths. If the effective region size of the suppressed dynamics around the ionic aggregates is correlated to the chains’ stiffness, the Tg enhancement and m suppression should be greater with increasing bending coefficient Kθ at longer distance. Thus, the results in Figure 3.9(a) is inconsistent with the ‘restricted-mobility’ region around ionic aggregates which grows proportionally with the neat polymer’s persistence length, as stated in the EHM model17.

The trends in both Tg and m in Figure 3.9(a)&(b) resemble each other significantly, mostly with a quantitative consistence in the near-nanoparticle behavior of Tg and m, which are observed in simulations of the same model polymer employed in our study, but in the presence of attractive (non-grafted) crystalline nanoparticles59. Specifically, the two studies exhibit excellent agreement in the extent of near-surface enhancement in Tg,

27 near-surface suppression in m, and long-distance enhancement of m, as well as the location of a crossover from m-suppression to m-enhancement. We emphasize that no effort has been 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 the same model polymer even with the difference between the size of aggregates (3 ~ 4.5 ) and nanoparticles ( ~ 10.7 ). This observation strongly suggests that covalent crosslinking between the chains and the ionic aggregate is no different than a strong non-covalent surface attraction and does not otherwise play a significant role in the physics of near- aggregate segmental mobility and Tg, which is also observed in the grafted-nanoparticle polymer58,67 in experiments.

2 a

= 0 1.5 θ K ) | | ) pure 1 τα / τα = 1.2 pure α τ / 0.5 pendant α τ 0 ln(

-0.5 0.5 1 1.5 2 2.5 3 3.5 4 Distance From the Nearest Ion 2 2.5 b d = 0

θ 2

K 1.5 = 1.5 θ K ) | | ) 1.5 ) | | )

pure pure α 1 τα / τα = 1.2 τ pure pure τ / τ = 1.2 / α α α

τ 1 / / 0.5

pen/homo 0.5 α pendant τ α τ

ln( 0

-0.5 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.5 1 1.5 2 2.5 3 3.5 4 Distance From the Nearest Ion Distance From the Nearest Ion 2 4.5 c e = 3

1.5 θ 3.5 = 0 = K θ K ) | | )

) | | ) pure 1 τα / τα = 1.2 pure 2.5 α pure τ α / τ pure

/ / τα / τα = 1.2 0.5 1.5 semi pendant α α τ τ ln(

0 ln( 0.5

-0.5 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Distance From the Nearest Ion Distance From the Nearest Ion

Figure 3.10 Plots of τα normalized by the corresponding pure-polymer value versus distance from the nearest ion at T = 1.2 (●), T = 0.8 (▲), and T = 0.6 (■). Solid and dashed lines are fits to equ (3.3) and the criterion for ξ in equ (3.1). τ 29

0.1 a

= 0 0 θ K

) | | ) -0.1 pure > 2 -0.2

pendant -0.4 > 2 / pure = 0.8 u < -0.5 ln( -0.6 0.5 1 1.5 2 2.5 3 3.5 4 Distance From the Nearest Ion 0.1 0.1 b d = 0 = θ

0 1.5 = K 0 θ K ) | | )

-0.1 | ) -0.1 pure > pure 2 >

-0.2 2 -0.2

-0.4 pendant -0.4 > > 2 2 2 pure

2 2 2 pure

u / = 0.8 / = 0.8 u < -0.5 < -0.5 ln( ln( -0.6 -0.6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0.5 1 1.5 2 2.5 3 3.5 4 Distance From the Nearest Ion Distance From the Nearest Ion 0.1 0.1 c e = 3 =

0 θ 0 = 0 K θ K ) | | ) -0.1

) | ) | -0.1 pure > pure 2

> -0.2 -0.2 2

/ / -0.3 -0.3 semi pendant > 2 -0.4 > -0.4 2 u 2 2 pure 2 2 pure

u / = 0.8

< / = 0.8 -0.5 < -0.5 ln( ln(

-0.6 -0.6 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Distance From the Nearest Ion Distance From the Nearest Ion Figure 3.11 Plots of normalized by pure-state value versus distance from the nearest ion at T = 1.2 (●), T = 0.8 (▲), and T = 0.6 (■). Solid and dashed lines are fits to equ

(3.4) and the criterion for ξ 2 in equ (3.2). 30

In addition to the important characteristic temperature in dynamics (Tg), more general analyses are demanded to further study the physical origins of the dynamics gradient near ionic aggregates in ionomers. Thus, a length scale is defined below in order to test the persistence length dependence of the range over which the mobility of polymer chains is restricted by ionic aggregates as proposed in the EHM model17. This defined length scale and the persistence length should exhibit the same temperature dependence and scale with each other by a factor, if the gradient in dynamics is indeed dominated by the chain’s persistence length.

We employ the data characterizing the structural relaxation (τα) and the vibrational dynamics () as a function of distance from the nearest ion (r) from the binned analysis at each T. In Figure 3.10, τα is enhanced approaching the aggregate, but decays towards a pure-state value with increasing distance from the aggregate; while in Figure

3.11, is suppressed closer to the aggregate and resumes the pure-state value at a sufficient distance from the aggregate. According to prior simulation work examining dynamics near interfaces88, we define an interfacial length scale ξ as a distance at which dynamics have sufficiently recovered pure-state dynamics, in order to characterize the

2 effectivity of the gradient in dynamics in terms of τα and , based on the criteria:

 pure 1.2, (3.1) 

u2 pure  0.8 , (3.2) u2

2 after smoothing τα and as a function of distance r by using the functional form:

 ()r  B ln pure  Ar  C , (3.3)  

u2 () r  ln  ArB  C , (3.4) 2 pure  u  where A, B, C are fitting parameters, as illustrated in Figure 3.10 for τα and in Figure 3.11 for . According to the previous work in polymer films, the specific criteria and the fitting function do not have a qualitative impact on the results.54,88

pure ξτ grows by a factor of 2.5~3 cooling from T ≈ 2.2Tg , the corresponding neat

pure polymer Tg, down close to Tg in these systems with Kθ = 0 shown in Figure 3.12(a), with an Arrhenius behavior on temperature dependence shown in Figure 3.13(b). In

Figure 3.13(a), the persistence length of polymer chains, as investigated by the Kuhn length b, grows by a factor of 1~1.5 upon cooling within the same temperature range, which is much less dependent on T as compared to the growth of ξτ with T. This mismatch between ξτ and persistence length from our results is inconsistent with a correlation between the restriction range on mobility and the persistence length

17 postulated in the EHM model . To associate ξτ with some results in physical experiments, we employ an Arrhenius extrapolation of ξτ to an extreme value at the

EX 14 extrapolated Tg defined as the temperature at which  10  (~ 100s) , comparable with physical experiments. This size of the dynamics suppression described by ξτ ranges from 9 ~ 14 , scaling as 3.6~5.6nm in real units, comparable to the range of the dynamics alterations near the interfaces of polymer thin films and nanocomposites in simulations60,88–92. With respect to these pendant ionomers without any simple

32 dependence of ξτ on T, we do not observe a significant enhancement of either ξτ at the same T ratio relative to the pure-state Tg in Figure 3.12(b) or of the altered-mobility range

EX extrapolated at Tg , even though their persistence length is raised, as shown by the Kuhn length b increasing by a factor of ~3 tuned by Kθ from 0 to 3 in Figure 3.13. If the range over which the dynamics of polymer chains are suppressed by ionic aggregates is proportional to the chains’ persistence length, there will be an observable leap in both ξτ

EX at each normalized T and the extrapolated ξτ at Tg with increasing Kθ varying chains’ stiffness. Clearly, the persistence length of polymer chains is not central to determining the size of the restricted mobility region in our simulations.

Debye-Waller Factor is a length scale of particles rattling within a ‘cage’ at a short time79, and is a practical measure of free volume. can also be used to calculate kT/, which is proportional to the local modulus at high frequency66. Accordingly, the interfacial length scale ξ characterizes the gradient in dynamics, caused by the introduction of ionic aggregates into the homopolymer, in terms of free volume. The weak temperature dependence of ξ, ranging from 1~1.5σ in Figure 3.12, as compared to the Tg and m gradient in Figure 3.9, rules out the possibility of free volume dominating the origin of glass formation of ionomers, consistent with results near non-grafted interfaces59,88.

As is studied in those nanoconfined systems, ξτ is associated with the length scale of cooperatively rearranging regions (CRRs) as developed in the Adam-Gibbs theory85 in the glass-forming liquid. Due to the similarities between the simulated ionomer systems and the nanoconfined systems demonstrated above, it is interesting whether this

5.5 5 5 a b 5 4.5 4.5

4.5 4 4 ξτ 4 3.5 3.5 3.5 L 3 3 > 2 τ u ξ 3 < 2.5 2.5 ξ 2.5 2 2 2 1.5 1.5 1.5 ξ 1 1 1

0.5 0.5 0.5 1 1.3 1.6 1.9 2.2 1 1.3 1.6 1.9 2.2 pure pure T/Tg T/Tg

Figure 3.12 Length scales ξτ , ξ , and mean string length L versus T normalized by Tg of the neat polymer with the same Kθ for (a): pure polymer with Kθ = 0 (orange), semi- telechelic ionomer with Kθ = 0 (red), pendant ionomer/homopolymer blend with Kθ = 0

(green), pendant ionomer with Kθ = 0 (blue), and (b): pendant ionomer with Kθ =1.0

(blue), 1.5 (purple), Kθ = 3 (grey). correlation holds here, tested by calculating a mean number of segments L participating in the cooperative rearrangements via a protocol well-established and widely employed in the literature.69,65,70 The string analysis is computed on the mobile particles, defined as the particles moving farther than an Gaussian approximation at a time. The correlated moving particles consisting a ‘string’ are determined based on a criterion that one particle replace other particle within a radius of 0.6σc, where σc is the kissing distance between the centers of two beads. Then the mean string length L is calculated as the average number of segments participating in the same ‘string-like’ collective motion. Here, the mean string length L is only computed at the peak time (less than the segmental relaxation time τα) of the non-Gaussian parameter (Figure 3.14), corresponding to the 34 time of peak participation of particles in cooperative motion. As the bending coefficient

Kθ goes up (Figure 3.14 (b)&(c)), the peak time within τα gets harder to capture both in the pendant ionomer systems and the pure polymer systems due to the extra non-

Gaussian behavior invoked by chains’ stiffness. Thus, the string analyses are only performed on the flexible systems to deliver further discussion on cooperative motion.

5.5 2.8 b 5 a 2.3 4.5

4 1.8 3.5 ) τ ξ

b 3 1.3 ln( 2.5 5 0.8 2 τ 1.5 ξ 0.3 1 1.5 1.5 L 2.7 0.5 -0.2 1 1.3 1.6 1.9 2.2 0.2 0.4 0.6 0.8 1 pure T EX/T T / Tg g

Figure 3.13 (a): Plot of Kuhn length b verus T for the pure systems with Kθ = 0 (×), 1.5

(җ), and 3 (+). (b): Arrhenius plot of ξτ versus T, where solid lines are an Arrhenius extrapolation, and dashed lines indicate extrapolated ξτ at an extrapolated experimental

EX monomer Tg ; inset shows linear correlation between ξτ and L for the flexible systems: the pendant ionomer (blue ■), the 50/50 pendant ionomer/homopolymer blend (green ■), and the semi-telechelic ionomer (●).

Figure 3.12(a) demonstrates the similar growth of ξτ and L with T; additionally, the inset in Figure 3.13(b) shows that ξτ is linearly correlated to L over the entire range of glass formation in the flexible systems. This correspondence indicates that the size of

0.6 a

0.5

0.4

0.3

Naussian Parameter 0.2 - Non 0.1

0 10-3 10-1 101 103 105 t

1.2 0.4 b c

1 0.3 0.8

0.6 0.2

Naussian Parameter 0.4 Naussian Parameter - - 0.1 Non Non 0.2

0 0 10-3 10-1 101 103 105 10-3 10-1 101 103 105 t t

Figure 3.14 Plots of non-Gaussian parameter versus time t at T = 1.5 (dashed line) and T

= 0.7 (solid line) for (a): the pendant ionomer (blue), the 50/50 pendant ionomer/homopolymer blend (green), and the semi-telechelic ionomer (red) with Kθ = 0;

(b): the pendant ionomer with Kθ = 0 (blue), 1.5 (purple), 3 (grey); (c): the pure polymer with Kθ = 0 (orange), 1.5 (pink), 3 (aqua).

36 suppressed-mobility regions in ionomers is determined by the scale of cooperative rearrangements inherent to supercooled liquid dynamics rather than by chain connectivity effects.

2.4 3 a b 2.2 2.8 2 a E 1.8

) /) 2.6 A τ H / 1.6 Δ α τ 2.4

ln( 1.4 T T 1.2 2.2 1 ΔH = 1.74ΔS + 1.96 R2 = 0.9943 0.8 2 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 0 0.1 0.2 0.3 0.4 0.5 0.6 1 / T ΔS

Figure 3.15 (a): Illustration for Arrhenius correlation between τα and T, where τα has an

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, characterized by mean string length L in this thesis. A dependence of segmental 37 relaxation time τα on the size of cooperatively rearranging regions (CRRs) is established by Adams and Gibbs 62. Recent work has indicated that, if Adam-Gibbs theory holds for

59,60,70 a given system, τα should be correlated with L via the equation

LT     T     exp   , (3.5) LAB k T  where kB is the Boltzmann constant (equal to 1 in LJ units), τ∞ is the high temperature relaxation time that is treated as a fitting parameter, Δμ is the high temperature activation energy, and LA is the CRR size at the high-temperature onset of glass formation TA. After fitting τα and T from T = 1.5~1.0 to an Arrhenius functional form:

Ea   T    A exp  , (3.6) kB T  where τA and Ea are fitting parameters, TA is determined as the T at which τα deviates

~10% from this fit. It is illustrated in Figure 3.15(a) that τα has an Arrhenius dependence

on T when TEln /  A / a is equal to 1, and this quantity gets larger than 1 as τα deviates from this Arrhenius correlation with T. In equation (3.5), the high-temperature activation energy   HTS   , where ΔH and ΔS are the high temperature activation enthalpy and entropy, respectively. ΔH is equal to the fitting parameter Ea in the above

Arrhenius function form, and ΔS is evaluated via S   kBln( A  0 ) , where τA is also a fitting parameter from the Arrhenius fit above and τ0 is the vibrational relaxation time, taken to be 0.1τ in LJ units, consistent with prior work60,70. It is reported that there is a linear compensation between ΔH and ΔS60, which is also true in our simulation for all the systems as shown in Figure 3.15(b). Accordingly, a compensation T is defined as the

38 slope of the linear relation, which is Tcomp = 1.74, roughly 3~3.5Tg of all the systems. This extremely high Tcomp, based on the ionomers and pure polymers with bending coefficient

60 Kθ = 0~3, is different than the low Tcomp = 0.18 characterized in the nanofilms ; this discrepancy shows that the mechanism behind this Tcomp is still an open question.

  L  Based on the relation in equation (3.5), if ln   is plotted against , all    LAB k T glass-forming materials obeying these physics should collapse to a line of slope 1.

60 Moreover, Hanakata et al. have pointed out that ξτ and L should be related as

L   1 A  1  , (3.7) LAA  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

60 in the system should be related to ξτ as

 T     (TAA )  exp  1    , (3.8)     k T A  B  where the high-T relaxation time τ∞ is determined by the fit in equation (3.5) without refitting, Δμ is also the same as in equation (3.5), and ξA is the interfacial length scale ξτ at TA. If the Adam-Gibbs theory holds in our systems, it should collapse into a linear line

     T   with a slope of 1 when plotting ln   versus AA1   .    k T A  A  B

As shown in Figure 3.16, the collapses corresponding to equations (3.5) and (3.8) are both successful in all systems considered in this study. This success provides 39 additional evidence that glass formation in these ionomers is driven by cooperative motion associated with universal glass physics and not with persistence length or crosslinking 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.

10 8 a b 9 7 8 6 7 ) ) ∞ ∞ τ τ / /

α 6 α 5 τ τ

ln( 5 ln( 4 4 3 3

2 2 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8

L/LA·Δμ/T (A·ξτ/ξA+1-A)Δμ/T Figure 3.16 Test of Adam Gibbs theory based on equ (3.5) and (3.8) for the pendant ionomer (blue ■), the 50/50 pendant ionomer/homopolymer blend (green ■), the semi- telechelic ionomer (●) and the pure polymer (×) with Kθ = 0.

CHAPTER IV

CONCLUSIONS

We study glass formation behavior of ionomers by means of coarse-grained molecular dynamics simulations on model semi-telechelic and pendant ionomers.

Simulations exhibit the presence of reduced-mobility regions surrounding ionic aggregates, the sizes of which grow in a nearly Arrhenius manner on cooling. Our results are inconsistent with the prediction in the EHM model17 that the range over which the dynamics of polymer chains are suppressed around ionic aggregates is correlated with the persistence length as a parameter characterizing chains’ stiffness. These conclusions are supported by the fact that the temperature dependences of the suppressed-dynamics range and the persistence length are not equal, accompanied with the invariance in the size of these regions when quantified in ionomers over a broad range of persistence lengths.

Like previously studied filler-grafted composites67,68, bond connectivity between polymers and interfaces does not qualitatively transform the physical origin of glass formation in ionomers. Specifically, ionomers behave like nanoconfined materials in that their dynamics are determined by the size of CRRs, which grow upon cooling. These findings have practical implications for grafted micro-phase materials and their glass forming dynamics, for example, semi-crystalline polymers and brush copolymers.

More broadly, the consistency of glass formation among ionomers, filler-grafted materials and those materials without a grafted micro-phase, such as nanoparticle composites57–59 and thin films54–56, tends to merge the superficially different dynamical behaviors in various materials into a unified intuition. In this thesis, the success of the test of the Adam-Gibbs theory62 via a string-like cooperative motion model60,70 takes us one step closer to such an idealized goal.

CHAPTER V

FUTURE WORK

Further work will involve studies on the self-assembly of ionic aggregates as a tool to generate super molecules. Due to electrostatic interaction among ions, the morphology of ionic aggregates can be controlled by employing either an electric or magnetic field.

The structural control into a percolated or networked morphology will bring about improvement in ionomers’ transportation properties.

Theoretical work on glass forming dynamics can help to uncover the factors affecting the dependence of fragility on polymer chains’ stiffness in ionomers. Also, the ionic aggregates do not participate in cooperative motion with polymer chains as a consequence of huge difference on timescale for segmental relaxation. In addition to ionomers, it would be interesting to study other systems, like polymers with additives, within which different components compose cooperatively rearranging regions.

REFERENCES

1. Weiss, R. A., Izzo, E. & Mandelbaum, S. New Design of Shape Memory Polymers: Mixtures of an Elastomeric Ionomer and Low Molar Mass Fatty Acids and Their Salts. Macromolecules 41, 2978–2980 (2008).

2. Samuel, J., Xavier, T. & Kurian, T. High styrene–rubber ionomers, an alternative to thermoplastic elastomers. J. Appl. Polym. Sci. 85, 2294–2300 (2002).

3. Nicholson, J. W. Chemistry of glass-ionomer cements: a review. Biomaterials 19, 485–494 (1998).

4. Smith, D. C. Development of glass-ionomer cement systems. Biomaterials 19, 467– 478 (1998).

5. Wasson, E. A. Reinforced Glass-ionomer cements — a review of properties and clinical use. Clin. Mater. 12, 181–190 (1993).

6. McCabe, J. F. Resin-modified glass-ionomers. Biomaterials 19, 521–527 (1998).

7. Sarkar, N. K., El-Mallakh, B. & Graves, R. Silver release from metal-reinforced glass ionomers. Dent. Mater. 4, 103–104 (1988).

8. Smith, D. W. et al. A Microblock Ionomer in Proton Exchange Membrane Electrolysis for the Production of High Purity Hydrogen. Macromolecules 46, 1504–1511 (2013).

9. Zhang, H. & Shen, P. K. Recent Development of Polymer Electrolyte Membranes for Fuel Cells. Chem. Rev. 112, 2780–2832 (2012).

10. Kreuer, K.-D. Ion Conducting Membranes for Fuel Cells and other Electrochemical Devices. Chem. Mater. 26, 361–380 (2014).

11. Holdcroft, S. Fuel Cell Catalyst Layers: A Polymer Science Perspective. Chem. Mater. 26, 381–393 (2014).

12. Molitor, R. Golf ball cover compositions comprising a mixture of ionomer resins. (1974). at

13. Sullivan, M. J., Melvin, T. & Nesbitt, R. D. Golf ball cover of neutralized poly(ethylene-acrylic acid) copolymer. (1990). at

14. Brain P., G. Review and critical analysis of the morphology of random ionomers across many length scales. Polym. Eng. Sci. 48, 1029–1051 (2008).

15. Weiss, R. A. et al. in Coulombic Interactions in Macromolecular Systems 302, 2–19

16. Utracki, L., Walsh, D. & Weiss, R. Polymer Alloys, Blends, and Ionomers - an Overview. Acs Symp. Ser. 395, 1–35 (1989).

17. Eisenberg, A., Hird, B. & Moore, R. B. A new multiplet-cluster model for the morphology of random ionomers. Macromolecules 23, 4098–4107 (1990).

18. Wong, C. & Clarke, J. H. R. Molecular dynamics simulation of microstructure and counterion transport in dry ionic heteropolymers. J. Chem. Phys. 116, 6795–6802 (2002).

19. Dimitrova, P., Friedrich, K. A., Vogt, B. & Stimming, U. Transport properties of ionomer composite membranes for direct methanol fuel cells. J. Electroanal. Chem. 532, 75–83 (2002).

20. Fan, Y. et al. Viscoelastic and gas transport properties of a series of multiblock copolymer ionomers. Polymer 52, 3963–3969 (2011).

21. Bellinger, M., Sauer, J. A. & Hara, M. Tensile Fracture Properties of Sulfonated Polystyrene Ionomers. 1. Effect of Ion Content. Macromolecules 27, 1407–1412 (1994).

22. Weiss, R. A. & Yu, W.-C. Viscoelastic Behavior of Very Lightly Sulfonated Polystyrene Ionomers. Macromolecules 40, 3640–3643 (2007).

23. Yoshikawa, K., Molnár, A. & Eisenberg, A. Rheological properties of blends of lithium- or sodium-sulfonated polystyrene ionomers with polyamide 6. Polym. Eng. Sci. 34, 1056–1064 (1994).

24. Roche, E. J., Stein, R. S., Russell, T. P. & Macknight, W. J. Small-angle x-ray scattering study of ionomer deformation. J. Polym. Sci. Polym. Phys. Ed. 18, 1497– 1512 (1980).

25. Eisenberg, A. Clustering of Ions in Organic Polymers. A Theoretical Approach. Macromolecules 3, 147–154 (1970).

26. Hodge, I. M. & Eisenberg, A. Dielectric and Mechanical Relaxations in a Nafion Precursor. Macromolecules 11, 289–293 (1978). 45

27. Andreeva, E. D., Nikitin, V. N. & Boyartchuk, Y. M. Infrared Studies of Ionic Aggregation of Ethylene- and Butadiene-Based Ionomers. Macromolecules 9, 238– 243 (1976).

28. Rouse, G. B., Risen, W. M., Tsatsas, A. T. & Eisenberg, A. Far-infrared studies of ion clustering in ionomers. J. Polym. Sci. Polym. Phys. Ed. 17, 81–85 (1979).

29. Neppel, A., Butler, I. S. & Eisenberg, A. Raman spectra of styrene-sodium methacrylate copolymers—evidence for ion clustering. J. Polym. Sci. Polym. Phys. Ed. 17, 2145–2150 (1979).

30. Yarusso, D. J. & Cooper, S. L. Microstructure of ionomers: interpretation of small- angle x-ray scattering data. Macromolecules 16, 1871–1880 (1983).

31. Macknight, W. J., Taggart, W. P. & Stein, R. S. A model for the structure of ionomers. J. Polym. Sci. Polym. Symp. 45, 113–128 (1974).

32. Winey, K. I., Laurer, J. H. & Kirkmeyer, B. P. Ionic Aggregates in Partially Zn- Neutralized Poly(ethylene-ran-methacrylic acid) Ionomers: Shape, Size, and Size Distribution. Macromolecules 33, 507–513 (2000).

33. Kirkmeyer, B. P., Weiss, R. A. & Winey, K. I. Spherical and vesicular ionic aggregates in Zn-neutralized sulfonated polystyrene ionomers. J. Polym. Sci. Part B Polym. Phys. 39, 477–483 (2001).

34. Sauer, B. B. & McLean, R. S. AFM and X-ray Studies of Crystal and Ionic Domain Morphology in Poly(ethylene-co-methacrylic acid) Ionomers. Macromolecules 33, 7939–7949 (2000).

35. Grady, B. P. & Moore, R. B. EXAFS Studies of Various Sulfonated and Carboxylated Cadmium Ionomers. Macromolecules 29, 1685–1690 (1996).

36. Vanhoorne, P. & Register, R. A. Low-Shear Melt Rheology of Partially-Neutralized Ethylene−Methacrylic Acid Ionomers. Macromolecules 29, 598–604 (1996).

37. Wouters, M. E. L., Goossens, J. G. P. & Binsbergen, F. L. Morphology of Neutralized Low Molecular Weight Maleated Ethylene−Propylene Copolymers (MAn- g-EPM) As Investigated by Small-Angle X-ray Scattering. Macromolecules 35, 208– 216 (2002).

38. Hall, L. M. et al. Ionic Aggregate Structure in Ionomer Melts: Effect of Molecular Architecture on Aggregates and the Ionomer Peak. J. Am. Chem. Soc. 134, 574–587 (2012).

39. Hall, L. M., Stevens, M. J. & Frischknecht, A. L. Effect of Polymer Architecture and Ionic Aggregation on the Scattering Peak in Model Ionomers. Phys. Rev. Lett. 106, 127801 (2011).

40. Hall, L. M., Stevens, M. J. & Frischknecht, A. L. Dynamics of Model Ionomer Melts of Various Architectures. Macromolecules 45, 8097–8108 (2012).

41. Bolintineanu, D. S., Stevens, M. J. & Frischknecht, A. L. Influence of Cation Type on Ionic Aggregates in Precise Ionomers. Macromolecules 46, 5381–5392 (2013).

42. Bolintineanu, D. S., Stevens, M. J. & Frischknecht, A. L. Atomistic Simulations Predict a Surprising Variety of Morphologies in Precise Ionomers. ACS Macro Lett. 2, 206–210 (2013).

43. Goswami, M., Kumar, S. K., Bhattacharya, A. & Douglas, J. F. Computer simulations of ionomer self-assembly and dynamics. Macromolecules 40, 4113–4118 (2007).

44. Tsagaropoulos, G. & Eisenberg, A. Dynamic Mechanical Study of the Factors Affecting the Two Glass Transition Behavior of Filled Polymers. Similarities and Differences with Random Ionomers. Macromolecules 28, 6067–6077 (1995).

45. Osborn, S. J. et al. Glass transition temperature of perfluorosulfonic acid ionomers. Macromolecules 40, 3886–3890 (2007).

46. Tsagaropoulos, G. & Eisenburg, A. Direct observation of two glass transitions in silica-filled polymers. Implications to the morphology of random ionomers. Macromolecules 28, 396–398 (1995).

47. Moore, T. T., Mahajan, R., Vu, D. Q. & Koros, W. J. Hybrid membrane materials comprising organic polymers with rigid dispersed phases. AIChE J. 50, 311–321 (2004).

48. Weiss, R. A. in Assignment of the Glass Transition 214–225 (ASTM, 1994).

49. Miwa, Y., Kondo, T. & Kutsumizu, S. Subnanoscopic Mapping of Glass Transition Temperature around Ionic Multiplets in Sodium-Neutralized Poly(ethylene-random- methacrylic acid) Ionomer. Macromolecules 46, 5232–5237 (2013).

50. Ma, X., Sauer, J. A. & Hara, M. Poly(methyl methacrylate) Based Ionomers. 1. Dynamic Mechanical Properties and Morphology. Macromolecules 28, 3953–3962 (1995).

51. Li, B., Pan, F., Fang, Z., Liu, L. & Jiang, Z. Molecular Dynamics Simulation of Diffusion Behavior of Benzene/Water in PDMS-Calix[4]arene Hybrid Pervaporation Membranes. Ind. Eng. Chem. Res. 47, 4440–4447 (2008). 47

52. Jiang, L. Y., Chung, T. S. & Kulprathipanja, S. Fabrication of mixed matrix hollow fibers with intimate polymer–zeolite interface for gas separation. AIChE J. 52, 2898– 2908 (2006).

53. Wakabayashi, K. & Register, R. A. Morphological Origin of the Multistep Relaxation Behavior in Semicrystalline Ethylene/Methacrylic Acid Ionomers. Macromolecules 39, 1079–1086 (2006).

54. Hanakata, P. Z., Douglas, J. F. & Starr, F. W. Local variation of fragility and glass transition temperature of ultra-thin supported polymer films. J. Chem. Phys. 137, 244901–244901–7 (2012).

55. Long, D. & Lequeux, F. Heterogeneous dynamics at the glass transition in van der Waals liquids, in the bulk and in thin films. Eur. Phys. J. E 4, 371–387 (2001).

56. Dequidt, A., Long, D. R., Sotta, P. & Sanséau, O. Mechanical properties of thin confined polymer films close to the glass transition in the linear regime of deformation: Theory and simulations. Eur. Phys. J. E 35, (2012).

57. Starr, F. W. & Douglas, J. F. Modifying Fragility and Collective Motion in Polymer Melts with Nanoparticles. Phys. Rev. Lett. 106, 115702 (2011).

58. Berriot, J., Montes, H., Lequeux, F., Long, D. & Sotta, P. Gradient of glass transition temperature in filled elastomers. Europhys. Lett. EPL 64, 50–56 (2003).

59. Betancourt, B. A. P., Douglas, J. F. & Starr, F. W. Fragility and cooperative motion in a glass-forming polymer–nanoparticle composite. Soft Matter 9, 241–254 (2012).

60. Hanakata, P. Z., Douglas, J. F. & Starr, F. W. Interfacial mobility scale determines the scale of collective motion and relaxation rate in polymer films. Nat. Commun. 5, (2014).

61. Aichele, M., Gebremichael, Y., Starr, F. w., Baschnagel, J. & Glotzer, S. c. Polymer- specific effects of bulk relaxation and stringlike correlated motion in the dynamics of a supercooled polymer melt. J. Chem. Phys. 119, 5290 (2003).

62. Adam, G. & Gibbs, J. H. On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming Liquids. J Chem Phys 43, 139–146 (1965).

63. Donati, C. et al. Stringlike Cooperative Motion in a Supercooled Liquid. Phys Rev Lett 80, 2338–2341 (1998).

64. Gebremichael, Y., Vogel, M. & Glotzer, S. C. Particle dynamics and the development of string-like motion in a simulated monoatomic supercooled liquid. J. Chem. Phys. 120, 4415–4427 (2004).

65. Zhang, H., Kalvapalle, P. & Douglas, J. F. String-like collective atomic motion in the interfacial dynamics of nanoparticles. Soft Matter 6, 5944 (2010).

66. Dudowicz, J., Freed, K. F. & Douglas, J. F. in Advances in Chemical Physics 137, 125–222 (Wiley, 2008).

67. Papon, A. et al. Low-Field NMR Investigations of Nanocomposites: Polymer Dynamics and Network Effects. Macromolecules 44, 913–922 (2011).

68. Berriot, J. et al. Filler–elastomer interaction in model filled rubbers, a 1H NMR study. J. Non-Cryst. Solids 307–310, 719–724 (2002).

69. Starr, F. W., Douglas, J. F. & Sastry, S. The relationship of dynamical heterogeneity to the Adam-Gibbs and random first-order transition theories of glass formation. J. Chem. Phys. 138, 12A541–12A541–18 (2013).

70. Betancourt, B. A. P., Douglas, J. F. & Starr, F. W. String model for the dynamics of glass-forming liquids. J. Chem. Phys. 140, 204509 (2014).

71. LAMMPS: Large-scale Atomic/Molecular Massively Parallel Simulator, http://lammps.sandia.gov.

72. Kremer, K. & Grest, G. S. Dynamics of entangled linear polymer melts: A molecular‐dynamics simulation. J. Chem. Phys. 92, 5057–5086 (1990).

73. Shavit, A. & Riggleman, R. A. Influence of Backbond Rigidity on Nanoscale Confinement Effects in Model Glass-Forming Polymers. Macromolecules (2013). doi:10.1021/ma400210w

74. Hockney, R. W. & Eastwood, J. W. Computer Simulation Using Particles. (1989).

75. Packmol: Packing Optimization for Molecular Dynamics Simulations, http://www.ime.unicamp.br/~martinez/packmol/.

76. Tuckerman, M., Berne, B. J. & Martyna, G. J. Reversible multiple time scale molecular dynamics. J. Chem. Phys. 97, 1990–2001 (1992).

77. AMDAT: Amorphous Molecular Dynamics Analysis Toolkit.

78. Hansen, J.-P. & McDonald, I. R. Theory of Simple Liquids. (Academic Press, 2006).

79. Simmons, D. S. & Douglas, J. F. Nature and interrelations of fast dynamic properties in a coarse-grained glass-forming polymer melt. Soft Matter 7, 11010–11020 (2011).

80. Starr, F. W., Sastry, S., Douglas, J. F. & Glotzer, S. C. What Do We Learn from the Local Geometry of Glass-Forming Liquids? Phys. Rev. Lett. 89, 125501 (2002).

81. Williams, G. & Watts, D. C. Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans Faraday Soc 66, 80–85 (1970).

82. Kohlrausch, F. Kohlrausch. Pogg Ann Phys. 119, 352 (1863).

83. Vogel, H. Das temperatur-abhängigkeitsgesetz der viskosität von flüssigkeiten. Phys Zeit 22, 645–646 (1921).

84. Fulcher, G. S. Analysis of recent measurements of the viscosity of glasses. J Am Ceram Soc 8, 339 (1925).

85. Cavagna, A. Supercooled liquids for pedestrians. Phys. Rep.-Rev. Sect. Phys. Lett. 476, 51–124 (2009).

86. Eisenberg, A., Hird, B. & Moore, R. B. A new multiplet-cluster model for the morphology of random ionomers. Macromolecules 23, 4098–4107 (1990).

87. Huang, D. & McKenna, G. B. New insights into the fragility dilemma in liquids. J. Chem. Phys. 114, 5621–5630 (2001).

88. Lang, R. J. & Simmons, D. S. Interfacial Dynamic Length Scales in the Glass Transition of a Model Freestanding Polymer Film and Their Connection to Cooperative Motion. Macromolecules 46, 9818–9825 (2013).

89. Scheidler, P., Kob, W. & Binder, K. Cooperative motion and growing length scales in supercooled confined liquids. Europhys. Lett. EPL 59, 701–707 (2002).

90. Peter, S., Meyer, H. & Baschnagel, J. Thickness-dependent reduction of the glass- transition temperature in thin polymer films with a free surface. J. Polym. Sci. Part B- Polym. Phys. 44, 2951–2967 (2006).

91. Scheidler, P., Kob, W. & Binder, K. The relaxation dynamics of a confined glassy simple liquid. Eur. Phys. J. E 12, 5–9 (2003).

92. Scheidler, P., Kob, W. & Binder, K. The relaxation dynamics of a supercooled liquid confined by rough walls. J. Phys. Chem. B 108, 6673–6686 (2004).

APPENDICES

APPENDIX A

CALCULATION OF DEBYE-WALLER FACTOR FOR FLEXIBLE IONOMERS

103 a

101 = 0) θ K

10-1 pendant ( pendant MSD| 10-3 T=0.4 T=0.5

T=0.8 T=1.5 10-5 10-3 10-1 101 103 105 t

103 103 b c

101 101 = 0) θ K = 0) = θ K 10-1 10-1 semi ( semi pen/homo ( pen/homo MSD| MSD| 10-3 10-3 T=0.4 T=0.5 T=0.4 T=0.5

T=0.8 T=1.5 T=0.8 T=1.5 10-5 10-5 10-3 10-1 101 103 105 10-3 10-1 101 103 105 t t

APPENDIX B

CALCULATION OF DEBYE-WALLER FACTOR FOR PENDANT IONOMERS

103 a

101 = 0) θ K

10-1 pendant ( pendant MSD| 10-3 T=0.4 T=0.5

T=0.8 T=1.5 10-5 10-3 10-1 101 103 105 t

103 103 b c

101 101

2 = 3) 2 θ = 1.5) θ K K

10-1 10-1 pendant ( pendant ( pendant MSD| MSD| 10-3 10-3 T=0.4 T=0.5 T=0.4 T=0.5

T=0.8 T=1.5 T=0.8 T=1.5 10-5 10-5 10-3 10-1 101 103 105 10-3 10-1 101 103 105 t t

APPENDIX C

CALCULATION OF DEBYE-WALLER FACTOR FOR PURE POLYMERS

a 103

1 10 = 0) θ K

pure ( pure 10-1 MSD|

10-3 T=0.4 T=0.5

T=0.8 T=1.5 10-5 10-3 10-1 101 103 105 t

103 103 b c

101 101 = 3) = 1.5) θ θ K K 10-1 10-1 pure ( pure pure ( MSD| MSD| 10-3 10-3 T=0.4 T=0.5 T=0.4 T=0.5

T=0.8 T=1.5 T=0.8 T=1.5 10-5 10-5 10-3 10-1 101 103 105 10-3 10-1 101 103 105 t t

APPENDIX D

CALCULATION OF SEGMENTAL RELAXATION FOR FLEXIBLE IONOMERS

1 a T=1.5 = = 0 θ

K 0.8 T=1.1

T=0.8 0.6 T=0.5

inPendant with 0.4 = 7.07 q )| )| t

( 0.2 self

F τα 0 10-3 10-1 101 103 105 t

1 1 b T=1.5 c T=1.5 = 0 θ T=1.1 = 0 K 0.8 T=1.1

θ 0.8 K T=0.8 T=0.8 0.6 0.6 T=0.5 T=0.5 in Semi in Semi with 0.4 0.4 in Pen/homo with = 7.07 q )| = 7.07 t q 0.2 ( 0.2 )| )| t self ( τ F self α τα F 0 0 10-3 10-1 101 103 105 10-3 10-1 101 103 105 t t

APPENDIX E

CALCULATION OF SEGMENTAL RELAXATION FOR PENDANT IONOMERS

1 a T=1.5 = = 0 θ

K 0.8 T=1.1

T=0.8 0.6 T=0.5

in in Pendant with 0.4 = 7.07 q )| t

( 0.2 self

F τα 0 10-3 10-1 101 103 105 t

1 1 b T=1.5 c =3 θ =1.5 T=1.1

0.8 K 0.8 θ K T=0.8

0.6 T=0.5 0.6

T=1.5

0.4 inPendant with 0.4 inPendant with T=1.1 = 7.07 q = 7.07 q )| )| 0.2 t 0.2 ( T=0.8 )| )| t ( self

τ F τ self α α T=0.5 F 0 0 10-3 10-1 101 103 105 10-3 10-1 101 103 105 t t

APPENDIX F

CALCULATION OF SEGMENTAL RELAXATION FOR PURE POLYMERS

1 a T=1.5

=0 0.8 T=1.1 θ K T=0.8

0.6 T=0.5 in in Pure with 0.4 = 7.07 q )| t ( 0.2 self F τα 0 10-3 10-1 101 103 105 t

1 1 b T=1.5 c

T=1.1 =3 = = 1.5

0.8 θ 0.8 θ K

K T=0.8

0.6 T=0.5 0.6 in in Pure with in in Pure with 0.4 0.4 T=1.5 = 7.07 q = 7.07 q

)| T=1.1 t )| )| ( t

( 0.2 0.2 self

self T=0.8 F

F τα τα T=0.5 0 0 10-3 10-1 101 103 105 10-3 10-1 101 103 105 t t

APPENDIX G

OVERALL EXTRAPOLATED GLASS TRANSITION TEMPERATURES

1014 pendant (0) pen/homo (0) 1011 semi (0) pure (0) pendant (1.5) 108 pendant (3) α

τ pure (1.5) 105 pure (3.0)

102

EX Tg 10-1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T

semi-telechelic pendant/homo pendant pendant pendant

Kθ 0.0 0.0 0.0 1.5 3.0 EX Tg 0.389 0.376 0.390 0.468 0.572 EX Tg | pure 0.367 0.367 0.367 0.449 0.553 EX EX Tg / Tg | pure - 1 6.1% 2.4% 6.1% 4.4% 3.5% mEX 196 197 196 263 312 EX m | pure 208 208 208 302 391 EX EX m / m | pure - 1 -2.9% -3.0% -2.9% -8.5% -13.5%

APPENDIX H

VFT EXTRAPOLATION IN BINNED ANALYSES FOR FLEXIBLE IONOMERS

1014 a r = 0.75 r = 1.25 r = 1.75 1011 r = 2.25 r = 2.75

8 r = 3.25 = 0 0 ) = 10 θ

K r = 3.75

105 pendant( pendant( α τ

102

EX Tg 10-1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T

1014 1014 r = 0.75 r = 0.75 b c r = 1.25 r = 1.25 11 r = 1.75 r = 1.75 10 1011 r = 2.25 r = 2.25 r = 2.75 r = 2.75 = 0) = = 0) = θ

8 θ K 10 r = 3.25 8 r = 3.25 K 10 r = 3.75 r = 3.75 r = 4.25 r = 4.25

5 telechelic( 5 10 r = 4.75 - 10 pendant/homo( pendant/homo( semi α r = 5.25 α τ τ 102 102

EX EX Tg Tg 10-1 10-1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T T

APPENDIX I

VFT EXTRAPOLATION IN BINNED ANALYSES FOR PENDANT IONOMERS

1014 r = 0.75 a r = 1.25 r = 1.75 1011 r = 2.25 r = 2.75

8 r = 3.25 = 0) = 10 θ

K r = 3.75

105 pendant( α τ

102

EX Tg 10-1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T

1014 1014 b r = 0.75 c r = 0.75 r = 1.25 r = 1.25 1011 r = 1.75 1011 r = 1.75 r = 2.25 r = 2.25 r = 2.75 r = 2.75

8 = 3 ) 8 = 1.5 1.5 ) = 10 θ 10 r = 3.25 θ

r = 3.25 K K r = 3.75 r = 3.75 r = 4.25 5 5 10 pendant( 10 pendant( α

α r = 4.75 τ τ

102 102

EX EX Tg Tg 10-1 10-1 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 T T

APPENDIX J

BINNED EXTRAPOLATED GLASS TRANSITION TEMPERATURE

0.6 a 0.55

0.5

0.45 EX

g 0.4 T

0.35

0.3 pen/homo (0) semi (0)

0.25 pendant (0) pendant (1.5) pendant (3) 0.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Distance from the Nearest Ion

450 b 400

350

300

250 EX

m 200

150

100 pen/homo (0) semi (0) pendant (0) pendant (1.5) 50 pendant (3) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Distance from the Nearest Ion