Effect of Copolymer Sequence on Mechanical Properties of Polymer

Eﬀect of Copolymer Sequence on Mechanical Properties of Polymer Nanocomposites from Molecular Dynamics Simulations

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Alex Trazkovich, B.S., M.S.

Graduate Program in Chemical Engineering

The Ohio State University

2019

Dissertation Committee:

Dr. Lisa M. Hall, Advisor Dr. Kurt Koelling Dr. Isamu Kusaka Dr. Yiying Wu c Copyright by

Alex Trazkovich

2019 Abstract

When incorporated into polymers, nanoparticles are known to modify the structure and dynamics of nearby polymer chains. Because nanoparticles have a high surface area to volume ratio, the properties of the polymer–nanoparticle interphase region can have a signiﬁcant eﬀect on the overall composite properties even at relatively low nanoparticle loading. In this work, we study the polymer–nanoparticle interphase region using molecular dynamics simulations, and we analyze the impact of a nanoparticle on local structure, dynamics, and viscoelastic properties.

Of particular interest here is a class of systems which consists of nanoparticles incorporated into two-component copolymers where one component of the copolymer interacts more favorably with the nanoparticle than the other. In these systems, modifying the particular copolymer sequence may modify the interphase properties, and composite properties may therefore be adjusted even while maintaining the same overall monomer ratio. These systems have been the subject of several simulation studies focused on nanoparticle dispersion and assembly; however, relatively little simulation work has focused speciﬁcally on the impact of copolymer sequence on properties of the copolymer–nanoparticle interphase.

ii We simulate a simple nanocomposite consisting of a single spherical nanoparticle surrounded by coarse-grained polymer chains. The polymers are composed of two different monomer types that differ only in their interaction strengths with the nanoparticle. By studying a series of regular multiblock copolymers with adjustable block length as well as a random copolymer, we examine the effect of copolymer sequence blockiness on the structure as well as the end-to-end vector autocorrelation, bond vector autocorrelation, and self-intermediate scattering function relaxation times as a function of distance from the nanoparticle surface. We find that, depending on block length, blocky copolymers can have faster or slower interphase dynamics than a random copolymer. Certain blocky copolymer sequences also lead to relaxation times near the nanoparticle surface that are slower than those of homopolymer systems composed of either component monomer.

To analyze viscoelastic mechanical properties in the interphase, we measure local atomic stress ﬂuctuations and use them to estimate the local stress autocorrelation as a function of distance from the nanoparticle. This local stress autocorrelation is then used to estimate the local dynamic modulus. This allows us to examine the eﬀect of adjusting copolymer sequence on the dynamic modulus as a function of both frequency of excitation and distance from the nanoparticle. Notably, we

ﬁnd that certain copolymer sequences can lead to a higher viscoelastic hysteresis in the interphase than either homopolymer system, suggesting that tuning copolymer sequence could allow for signiﬁcant control over nanocomposite dynamics.

iii To demonstrate a possible application of adjusting material properties using copolymer sequence, we brieﬂy consider a design challenge motivated by tire tread compounds, in which improving traction without sacriﬁcing fuel economy requires increasing high-frequency hysteresis while maintaining low-frequency hysteresis. By considering an additional set of sequences motivated by our results from studying regular multiblock copolymers, we show how further adjusting copolymer sequence can be used to make progress toward this goal.

iv To Pat, the reason I am here.

To Remi, for keeping my smile bright.

v Acknowledgments

My road to this degree has been a bit unusual, and as a result, there are several individuals to whom I owe more than the usual debt of gratitude owed by a typical

Doctoral student.

Thank you most to Dr. Lisa Hall, who took a chance on this odd robotics engineer and who tolerated the distraction and sporadic attendance as I split time between campus and my job. She has been a role model, an inspiration, and an excellent board game partner.

Of course, I never even would have met Dr. Hall without Dr. Pat Majors, without whom, quite literally, none of this would have happened. As my supervisor at Cooper

Tire, Dr. Majors repeatedly advocated for my degree, introduced me to Dr. Hall, and then supported me for years, giving me the ﬂexibility I needed to pursue my studies.

He has also been a valuable resource, a wealth of knowledge, and, I’m proud to say, a dear friend.

I also owe a more general thanks to Cooper Tire, which funded my first two semesters at Ohio State, and I would like to specifically thank Curt Selhorst, Jeff

Endicott, and Chuck Yurkovich, who, along with Dr. Majors, advocated to the administration on my behalf.

Now that I have moved on to SEA Ltd., I should also thank my supervisors, Dr.

Gary Heydinger and Jared Henthorn, for their kind support and ﬂexibility while I

vi wrote my dissertation, and I also thank Dr. Anmol Sidhu and An Nguyen, teammates who covered for me during occasional jaunts to campus.

Thank you to the Hall Research group, especially Kevin Shen, Jeﬀ Ethier, Dr.

Youngmi Seo, Dr. Janani Sampath, and Dr. Jon Brown, who always made me feel like a full member of the team despite my often ephemeral presence.

I also thank my Qualiﬁer, Candidacy, and Dissertation Defense committee mem- bers, Dr. Kurt Koelling, Dr. Isamu Kusaka, and Dr. Stuart Cooper, who challenged me in a way that signiﬁcantly improved this work.

I was fortunate enough to be aided during this study by several talented interns, most notably Tarik Akyuz and Mitchell Wendt, who each made major contributions to the analysis code and coauthored publications with me.

Finally, from the bottom of my soul, I thank my wife, Remi, for always taking care of me as the stress of pursuing a degree while working a full-time job took its toll.

Remi, you are the light of my life, and my heart ached every time I had to choose research instead of spending time with you. But guess what?

I’m done.

vii Vita

2010 ...... B.S. Robotics Engineering, Olin College of Engineering 2013 ...... M.S. Mechanical Engineering, Northwestern University 2014-present ...... Graduate Student, The Ohio State University

Publications

1. A.J. Trazkovich, T.H. Akyuz, L.M. Hall, “Eﬀects of Copolymer Sequence on Ad- sorption and Dynamics Near Nanoparticle Surfaces in Simulated Polymer Nanocom- posites”, Tire Science and Technology, 2019, In Press.

2. A.J. Trazkovich, M.F. Wendt, L.M. Hall, “Eﬀect of Copolymer Sequence on Structure and Relaxation Times Near a Nanoparticle Surface”, Soft Matter, 2018, 14, 5913–5921.

3. A.J. Trazkovich, M.F. Wendt, L.M. Hall, “Eﬀect of Copolymer Sequence on Local Viscoelastic Properties Near a Nanoparticle”, Macromolecules, 2019. doi: 10.1021/acs.macromol.8b02136.

Fields of Study

Major Field: Chemical Engineering

Areas of Interest: Molecular Simulations, Polymer Physics, Tire Mechanics

viii Table of Contents

Page

Abstract ...... ii

Dedication ...... v

Acknowledgments ...... vi

Vita...... viii

List of Abbreviations ...... xi

List of Symbols ...... xii

List of Figures ...... xv

1. Introduction ...... 1

1.1 Motivation ...... 1 1.1.1 Design Challenge in Tire Treads ...... 5 1.1.2 Coarse-Grained Modeling ...... 8 1.2 Structure of Dissertation ...... 10

2. Model and Simulation Methods ...... 12

2.1 Kremer-Grest Bead-Spring Model ...... 12 2.2 Polymer Nanocomposite System Details ...... 14

3. Structure and Relaxation Times of Homopolymer Systems ...... 18

3.1 Background ...... 18 3.2 Simulation Details ...... 19

ix 3.3 Eﬀect of Polymer–Nanoparticle Interaction Strength on Interphase Structure ...... 21 3.4 Eﬀect of Polymer–Nanoparticle Interaction Strength on Interphase Relaxation Times ...... 26

4. Structure and Relaxation Times of Copolymer Systems ...... 36

4.1 Background ...... 36 4.2 Simulation Details ...... 38 4.3 Eﬀect of Copolymer Sequence on Interphase Structure ...... 41 4.4 Eﬀect of Copolymer Sequence on Interphase Relaxation Times . . . 50 4.5 Discussion ...... 57

5. Eﬀect of Copolymer Sequence on Interphase Dynamic Modulus . . . . . 58

5.1 Background ...... 58 5.2 Definition of Dynamic Moduli and Relationship to Hysteresis . . . 62 5.3 Measurement Methods ...... 67 5.3.1 Influence of Bond Vibrations on Modulus Results ...... 72 5.3.2 Effect of Pre-Processing Data ...... 74 5.4 Results ...... 76 5.5 Discussion ...... 92

6. Adjusting Copolymer Sequence to Modify Hysteresis ...... 95

6.1 Motivation ...... 95 6.2 Comparison of Hysteresis in Original Systems ...... 97 6.3 Investigating Additional Sequences ...... 99 6.3.1 Triblock and New Regular Multiblock Copolymers . . . . . 99 6.3.2 Modiﬁcations to Random Copolymers ...... 106 6.4 Discussion ...... 108

7. Concluding Remarks ...... 111

7.1 Summary of Results ...... 111 7.2 Future Work ...... 114

Bibliography ...... 119

x List of Abbreviations

BVACF Bond Vector Autocorrelation Function EEACF End-to-End Vector Autocorrelation Function COM Centers of Mass FENE Finitely Extensible Nonlinear Elastic LAMMPS Large-scale Atomic/Molecular Massively Parallel Simulator LJ Lennard-Jones LSACF Approximation of the Local Stress Autocorrelation Function MD Molecular Dynamics MSD Mean-Squared Displacement NPT Isobaric, isothermal, and ﬁxed particle number ensemble NRI Nanoparticle Radial Vector NVE Microcannonical ensemble (ﬁxed volume, energy, particle number) PEO Polyethylene oxide PTHF Polytetrahydrofuran PRISM Polymer Reference Interaction Site Model SACF Stress Autocorrelation Function SBR Styrene-Butadiene Rubber

xi List of Symbols

A Monomer type with slightly unfavorable nanoparticle interaction A Constant of the soft-potential used for initial push-oﬀ ACFbv Bond vector autcorrelation function ACFee End-to-end vector autcorrelation function a, b Subscripts indicating axes α Monomer–nanoparticle interaction length scale in PRISM theory B Monomer type with favorable nanoparticle interaction BE Length of B blocks in BAB triblock copolymer sequence BL Repeating block length in regular multiblock copolymer sequence βee Stretching parameter for end-to-end vector relaxations βR Stretching parameter for relaxation process R ∆N Horizontal shift factor from monomer–nanoparticle LJ potential ∆t Autocorrelation time window length δ Viscoelastic phase shift by which stress lags behind strain δs Vertical shift factor set so that the LJ potential is equal to 0 at rc δt Timestep for numerical integration ∆Ed Energy dissipated in one stress-strain cycle ∆Es Maximum energy stored in one stress-strain cycle E0 Storage modulus (Young’s) E00 Loss modulus (Young’s) E∗ Complex modulus (Young’s) Viscoelastic strain 0 Peak viscoelastic strain during cyclic excitation ε Reduced LJ unit of energy εij LJ potential interaction strength between particle types i and j FR Relaxation function of process R FS Self-intermediate scattering function

Fija Force that particle j exerts on particle i in direction a G0 Storage modulus (shear) G00 Loss modulus (shear) G∗ Complex modulus (shear) gMN Pair correlation function between nanoparticle and monomer M θbv Angle between a bond vector and the NRI

xii θee Angle between a polymer end-to-end vector and the NRI k Self-intermediate scattering function wave-vector k Spring constant of FENE bond potential kB The Boltzmann constant M Subscript referring to interactions involving monomer type M m Reduced LJ unit of mass mi Mass of particle i N Subscript referring to interactions involving the nanoparticle N Polymer chain length; number of monomers per chain Nς Number of monomers in a shell NPT Isothermal-isobaric ensemble n Monomer number P Subscript referring to all monomers in the system P2 Second-order Legendre polynomial PB Local fraction of monomers that type B pi Bulk fraction of monomers that type i RAB Sequence with long random block terminated with short AB blocks RB Sequence with long random block terminated with short B blocks R, θ, φ Axes in spherical coordinates centered on the nanoparticle R0 Maximum length of FENE bond potential Rg Polymer radius of gyration Rbv Vector from a monomer to an adjacent, bonded monomer Ree Polymer chain end-to-end vector r Distance to the nanoparticle center rb Distance between two bonded monomers rc Cutoﬀ distance of LJ potentials rm Distance between two monomers ria Position of particle i along axis a rcom Position of a polymer center of mass rn Position of monomer n ς Subscript referring to a shell s Viscoelastic stress s0 Peak viscoelastic stress during cyclic excitation sab Viscoelastic stress in direction ab si Viscoelastic stress on particle i σ Reduced LJ unit of distance σij LJ potential interaction length scale for monomer types i and j σNP Eﬀective diameter of the nanoparticle T Temperature Tg Glass transition temperature t Time t0 First timestep of an autocorrelation window τ Reduced LJ unit of time

xiii τbv Bond vector autocorrelation function relaxation time τee End-to-end vector autocorrelation function relaxation time

τFS Self-intermediate scattering function relaxation time τR Relaxation time of process R UFENE FENE potential ULJ,ij LJ potential between monomer types i and j V System volume via Velocity of particle i in direction a x, y, z Axes in Cartesian coordinates ω Frequency of stress/strain cycle

xiv List of Figures

Figure Page

1.1 Representative snapshots of selected polymers from two simulated systems. Pink beads adsorb more strongly to the nanoparticle (purple) than cyan beads; the diﬀerent amounts of blockiness in the monomer sequences lead to diﬀerent preferred polymer conformations near the nanoparticle surface...... 4

1.2 Scanning electron microscope image of a tire tread compound with silica (light) dispersed in a styrene-butadiene rubber matrix (dark). .5

1.3 Contact forces between a tire tread and a road surface asperity (black arrows), resulting in a net force opposite the direction of travel (white arrow)...... 6

3.1 Pair correlation function of monomers with respect to the nanoparticle in the various homopolymer systems, as labeled. Radial ordering

increases with εNP...... 22

3.2 Bond vector orientation parameter, as deﬁned in the text, for the various homopolymer systems, as labeled. Bonds are assigned to shells around the nanoparticle based on the location of each bond’s center. The midpoint of each shell lies at the indicated distance, r, from the center of the nanoparticle, and each shell has a width of 0.25σ..... 23

xv 3.3 Root-mean-squared radius of gyration (a), and chain orientation parameter (b) for the various homopolymer systems, as labeled. Polymers chains are assigned to concentric spherical shells around the nanoparticle based on the location of each chain’s COM. The midpoint of each shell lies at the indicated distance, r, from the center of the nanoparticle, and each shell has a width of 1σ (x-axis labels are located on the shell boundaries). Although chains may wrap partially around the nanoparticle, adequate statistics were not obtained to report on the rare cases where a polymer’s COM resides at r < 3.5σ from the nanoparticle center...... 25

3.4 End-to-end vector autocorrelation function data of polymer chains in

the εNP = 1 system (a) and εNP = 8 systems (b) binned based on r, the distance of the chain’s COM from the surface of the nanoparticle. 27

3.5 End-to-end relaxation times of various homopolymer systems, as labeled, as a function of distance from the nanoparticle. Polymers are assigned to shells as described in the text, and shells are centered on the indicated radius and have a width of 1σ (x-axis labels are located on the shell boundaries). Measurement error increases with proximity to the nanoparticle because fewer polymer COMs reside in closer shells. A 90% conﬁdence interval of the measurement is approximately ±2.5 · 103τ at 5σ, ±1.5 · 103τ at 6σ, ±1 · 103τ at 8σ, and ±0.5 · 103τ at 15σ. Adequate statistics were not obtained to report on the rare cases where a polymer’s COM resides at r < 4.5σ...... 30

3.6 Bond vector relaxation times as a function of distance from the nanoparticle for the various homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are located on the shell boundaries), and bonds are assigned to shells as described in the text. Although measurement error increases with proximity to the nanoparticle because fewer bonds reside in closer shells, a 90% conﬁdence interval of the measurement is less than ±1τ in all shells considered...... 32

xvi 3.7 Relaxation times of the self-intermediate scattering function at |k| = 2πσ−1, reported as a function of distance from the nanoparticle for the various homopolymer systems, as labeled. Monomers are assigned to shells as described in the text, and shells are centered at the indicated distance and have width of 0.5σ (x-axis labels are located on the shell boundaries). Although measurement error increases with proximity to the nanoparticle because fewer monomers reside in closer shells, a 90% conﬁdence interval of the measurement is less than ±0.02 at all radii. 34

4.1 Schematic of the copolymer block sequences used in this work. Se- quences in the random copolymer system may vary between chains, and only one example is shown. Within each of the other systems, chains are all identical to the sequence shown...... 40

4.2 Clockwise from top left: structure of BL = 25, BL = 5, BL = 1, and Random systems. A monomers (εNA = 1) are shown in cyan, and B monomers (εNA = 5) are shown in pink...... 42

4.3 A monomer–nanoparticle (left) and B monomer–nanoparticle (right) pair correlation functions for the various copolymer systems, as labeled. Adapted from Ref. 1 with permission from the Royal Society of Chemistry...... 43

4.4 (a) gPN(r), the monomer–nanoparticle pair correlation function for the copolymer systems as well as the two homopolymer systems, as la-

beled, for comparison. (b) gBN(r)/gPN(r), the fraction of monomers at distance r that are type B, for the copolymer systems, as labeled. Adapted from Ref. 1 with permission from the Royal Society of Chem- istry...... 44

4.5 Bond vector orientation parameter, as deﬁned in the text above, for the various copolymer and homopolymer systems considered, as labeled. Bonds are assigned to concentric spherical shells around the nanoparticle based on the bond center. The midpoint of each shell lies at the indicated distance, r, from the center of the nanoparticle, and each shell has a width of 0.25σ. Adapted from Ref. 1 with permission from the Royal Society of Chemistry...... 46

xvii 4.6 Representative snapshots of selected polymers from three diﬀerent copolymer systems, showing, from left to right, example conformations of

adsorbed chains in the BL = 1, BL = 5, and BL = 25 systems. B monomers (pink) beads adsorb more strongly to the nanoparticle (purple) than A monomers (cyan)...... 47

4.7 Average radius of gyration of polymer chains as a function of distance from the center of the nanoparticle for the various copolymer and homopolymer systems, as labeled. Polymers are assigned to shells as described in the text, and shells are centered at the indicated radius and have a width of 1σ (x-axis labels are located on the shell boundaries). Although chains may wrap partially around the nanoparticle, adequate statistics were not obtained to report on the rare cases where a polymer’s center of mass resides at r < 3.5σ from the nanoparticle center. Adapted from Ref. 1 with permission from the Royal Society of Chemistry...... 48

4.8 Chain end-to-end orientation parameter, as deﬁned in the text above, for the various copolymer and homopolymer systems, as labeled. Chains are assigned to concentric spherical shells around the nanoparticle based on their center of mass, and shells have width of 1σ (x-axis labels are located on the shell boundaries). Adequate statistics were not obtained to report on the rare cases where a polymer’s center of mass resides at r < 3.5σ. Adapted from Ref. 1 with permission from the Royal Society of Chemistry...... 50

4.9 End-to-end relaxation times of the various copolymer and homopolymer systems, as labeled, as a function of distance from the nanoparticle. Polymers are assigned to shells as described in the text, and shells are centered on the indicated radius and have a width of 1σ (x-axis labels are located on the shell boundaries). Measurement error increases with proximity to the nanoparticle because fewer polymers reside in closer shells. A 90% conﬁdence interval of the measurement is approximately ±2.5 · 103τ at 5σ, ±1.5 · 103τ at 6σ, ±1 · 103τ at 8σ, and ±0.5 · 103τ at 15σ. Adequate statistics were not obtained to report on the rare cases where a polymer’s center of mass resides at r < 4.5σ. Adapted from Ref. 1 with permission from the Royal Society of Chemistry...... 51

xviii 4.10 Bond vector relaxation times as a function of distance from the nanoparticle for the various copolymer and homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are located on the shell boundaries), and bonds are assigned to shells as described in the text. Measurement error increases with proximity to the nanoparticle because fewer bonds reside in closer shells. A 90% conﬁdence interval of the measurement is approximately ±1τ at 5.5σ, ±0.5τ at 7σ, and ±0.1τ at 10σ and above. Adapted from Ref. 1 with permission from the Royal Society of Chemistry...... 53

4.11 Relaxation times of the self-intermediate scattering function at |k| = 2πσ−1, reported as a function of distance from the nanoparticle for the various copolymer and homopolymer systems, as labeled. Monomers are assigned to shells as described in the text, and shells are centered at the indicated distance and have width of 0.5σ (x-axis labels are located on the shell boundaries). Although measurement error increases with proximity to the nanoparticle because fewer monomers reside in closer shells, a 90% conﬁdence interval of the measurement is less than ±0.02 at all radii. Adapted from Ref. 1 with permission from the Royal Society of Chemistry...... 56

5.1 Complex modulus (a) and hysteresis (b) of the B homopolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. 73

5.2 Complex modulus (a) and hysteresis (b) of the B monomer system with bonds removed as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the main text...... 73

5.3 Complex modulus (a, c, e) and hysteresis (b, d, f) of the B homopolymer system with bonds removed as a function of distance from the nanoparticle, r. Monomers are assigned to shells as described in the text. Dashed lines report moduli calculated from LSACF data that is extended and filtered according to the procedure in Section 5.3. Solid lines report moduli calculated from raw LSACF data (a, b), LSACF data is filtered but not extended (c, d), and LSACF data that is extended but not filtered (e, f)...... 75

xix 5.4 Magnitude of complex modulus (a) hysteresis (b), storage modulus (c), loss modulus (d), and time-domain LSACF data (e) of the B homopolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data...... 77

5.5 Magnitude of complex modulus (a) hysteresis (b), storage modulus (c), loss modulus (d), and time-domain LSACF data (e) of the A homopolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data...... 81

5.6 Magnitude of complex modulus (a) hysteresis (b), storage modulus

(c), loss modulus (d), and time-domain LSACF data (e) of the BL = 1 copolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data...... 82

5.7 Magnitude of complex modulus (a) hysteresis (b), storage modulus

(c), loss modulus (d), and time-domain LSACF data (e) of the BL = 2 copolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data...... 83

5.8 Magnitude of complex modulus (a) hysteresis (b), storage modulus (c),

loss modulus (d), and time-domain LSACF data (e) of the BL = 25 copolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data...... 85

5.9 Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for the various copolymer and homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries)...... 86

xx 5.10 Dynamic mechanical reinforcement under normal stress at 9 rad/τ (a) and peak hysteresis under normal stress (b) for the various copolymer and homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries)...... 90

5.11 Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for each of the homopolymer systems studied, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries). Data points that lie outside the reported bounds are cases where δ is very close to π/2 (the system behaves nearly purely viscous), and tan(δ) is greater than 1,000...... 91

6.1 Hysteresis in the two closest shells to the nanoparticle, 5.5σ < r < 6.0σ (a) and 6.0σ < r < 6.5σ (b), for the various copolymer and homopolymer systems considered in Chapters 4 and 5, as labeled. . . 98

6.2 Schematic of the additional copolymer block sequences considered in ∗ this chapter. Chain length is 104 in the BL = 4 system, 102 in the ∗ BL = 3 system, and 100 in the triblock copolymer (BE) systems. . . 100

6.3 PB, the fraction of monomers at distance r that are type B, for the additional copolymer systems and reference copolymer systems from Chapters 4 and 5, as labeled...... 101

6.4 Bond vector relaxation times as a function of distance from the nanoparticle surface for the additional copolymer systems and selected copolymer systems from Chapters 4 and 5, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are located on the shell boundaries), and bonds are assigned to shells as described in the text. Measurement error increases with proximity to the nanoparticle because fewer bonds reside in closer shells...... 102

6.5 Hysteresis in the two closest shells to the nanoparticle, 5.5σ < r < 6.0σ (a) and 6.0σ < r < 6.5σ (b), for the additional copolymer systems and selected copolymer systems from Chapters 4 and 5, as labeled. Data that fall outside the bounds of the chart indicate systems that are nearly purely viscous (δ ≈ π/2) on a local scale...... 104

xxi 6.6 Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for the additional copolymer systems and selected copolymer systems from Chapters 4 and 5, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries)...... 104

6.7 Schematic of the additional primarily-random copolymer sequences introduced in this section. Grey segments indicated randomized se-

quences composed 50 A and 44 B monomers (in the case of the RB system) or 44 A and 44 B monomers (in the case of the RAB system). The randomized components may vary between chains within the system.106

6.8 PB, the fraction of monomers at distance r that are type B, for the various predominantly random systems, as labeled...... 107

6.9 Hysteresis in the two closest shells to the nanoparticle, 5.5σ < r < 6.0σ (a) and 6.0σ < r < 6.5σ (b), for the various predominantly random systems, as labeled...... 108

6.10 Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for the various predominantly random systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries)...... 109

7.1 Magnitude of complex modulus at 9 rad/τ (a) and peak hysteresis (b)

for the BL = 25 system under various stress modes (as labeled). Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries)...... 117

xxii Chapter 1: Introduction

1.1 Motivation

Because many properties of polymeric materials can be improved by the addition

of nanoparticles, polymer nanocomposites are in widespread use in commercial ap-

plications and have received signiﬁcant attention in the scientiﬁc community. This

has included a large body of simulation work focused on understanding and control-

ling the molecular scale features of nanocomposite materials.2–13 A clear picture has

developed that near nanoparticle surfaces, there is an interphase in which adsorbed

polymer chains have diﬀerent dynamics, rheology, and typical conformations than in

the bulk, and this is one reason why adding nanoparticles has a signiﬁcant impact on

overall composite properties.14–24

Nanoparticles have a high surface area to volume ratio, so the addition of even a

small volume fraction of particles causes a signiﬁcant percentage of polymer chains

to be incorporated in the interphase region. As the nanoparticle loading is increased,

the overall composite properties usually begin to more closely resemble those of the

interphase (up to a certain threshold, above which composite properties become in-

creasingly dependent on filler–filler network effects25,26). The relationship between interphase properties, the nanoparticle volume fraction, and the overall composite

1 properties has been characterized using analytical theories.27–32 Therefore, understanding and controlling the interactions between polymer chains and nanoparticle surfaces is key to predicting and tuning bulk material properties.

A critical factor that helps determine properties of the interphase is the polymer–

filler interaction strength. Favorable polymer–filler interactions are key to ensuring good nanoparticle dispersion. When interactions are unfavorable, nanoparticles have the tendency to aggregate due to entropic depletion effects, which usually has an adverse effect on the mechanical properties of the resulting composite. Therefore, in most commercial applications, polymer–filler interactions are favorable. This allows polymers to adsorb on the surface of the nanoparticle, and the dynamics of the polymers adsorbed on the nanoparticle are slowed.33–38 Adsorbed chains may also dynamically couple with nearby non-adsorbed chains, slowing the dynamics of the non-adsorbed chains and extending the effective range of the interphase.39 When polymer–nanoparticle interactiosn are favorable the local glass transition temperature in the interphase is elevated, and in more highly-loaded systems, the glass transition temperature of the overall composite can be noticeably increased.40–43 It is also possible to obtain an interphase region with faster dynamics than the bulk by selecting components such that the interaction is unfavorable,20 which has the effect of lowering glass transition temperatures.43–45 Polymer–filler interaction strength is dependent on the polymer type and nanoparticle surface chemistry, so interphase properties can be adjusted by functionalizing the nanoparticle surface or changing the polymer chemistry.

2 One class of systems that has been the subject of increased attention in recent

years consists of nanoparticles incorporated into blocky copolymer systems.46 Un- like in nanoparticle-homopolymer composites, some monomers in copolymer chains may interact more favorably with the nanoparticle than others.46 These systems have been the subject of several simulation studies focused on nanoparticle dispersion in microphase-separated block copolymers47–51 as well as the eﬀect of grafted copolymer chain sequence on nanoparticle interactions and self-assembly.52–55 However, relatively little simulation work has focused speciﬁcally on properties of the copolymer– nanoparticle interphase.

In systems where one component of the copolymer interacts more favorably with the nanoparticle than the other, the specific copolymer sequence would be expected to affect the adsorption dynamics of polymer chains interacting with the filler surface.

A small number of simulation studies have speciﬁcally examined properties of the copolymer–nanoparticle interphase. Chen and colleagues determined that structure and chain conformations around a nanoparticle depend on the relative interaction strength of the two monomer types56 and Martin et al. examined how structure and chain conformation were impacted by adjusting the copolymer sequence.57 Meanwhile, a recent experimental study by Helal and coworkers showed that copolymer sequence impacts interfacial glass transition temperatures.58 While most simulation studies include unfavorable interactions between unlike monomers (as is typically the case in experimental systems), a body of theoretical work, primarily from the Schweizer group, has shown that even when all monomer–monomer interactions are equal and monomers diﬀer only in their interactions with nanoparticles (as is the case in this

3 work), copolymer structure near nanoparticles still depends signiﬁcantly on copolymer sequence.59–62

In this work, we use molecular dynamics (MD) simulations to implement a similarly simple model where the only diﬀerence between monomers is the strength of their interaction with a nanoparticle. We show how copolymer sequence aﬀects interphase structure, chain conformations (Figure 1.1), and, perhaps most critically, interphase dynamics. This demonstrates that adjusting copolymer sequence can be a powerful tool for tuning material properties in polymer nanocomposites, and it suggests that modifying sequence may provide a method of addressing challenging problems in the

ﬁeld of polymer nanocomposite design. One example of such a design challenge is discussed in the following section.

Figure 1.1: Representative snapshots of selected polymers from two simulated systems. Pink beads adsorb more strongly to the nanoparticle (purple) than cyan beads; the diﬀer- ent amounts of blockiness in the monomer sequences lead to diﬀerent preferred polymer conformations near the nanoparticle surface.

4 1.1.1 Design Challenge in Tire Treads

Tire tread compounds are composed primarily of styrene-butadiene rubber (SBR) copolymer reinforced with nanoparticle filler (Figure 1.2), and the interaction between polymer chains and the filler particles is known to play a major role in tire traction, rolling resistance, and wear performance.4,16,63–68 These performance criteria are in- terdependent, and there are significant tradeoffs between them that are affected by tread material design.

Figure 1.2: Scanning electron microscope image of a tire tread compound with silica (light) dispersed in a styrene-butadiene rubber matrix (dark).

In particular, tire manufacturers have long recognized a strong tradeoﬀ between reducing rolling resistance and increasing traction. The challenge is that both rolling resistance and traction are highly dependent on mechanical hysteresis of the tread material. Low rolling resistance can be achieved by using tread materials that have minimal hysteresis in order to return as much stored energy as possible.65,69–72 On the other hand, good traction requires high tread material hysteresis, since energy

5 losses through hysteresis are a major mechanism of traction, particularly in wet,

winter, or otherwise lubricated conditions when adhesion eﬀects are signiﬁcantly re-

duced.64–66,68,73–77

Reducing rolling resistance while maintaining excellent traction is not an intractable challenge because the operating conditions associated with traction and rolling resistance are diﬀerent. Traction occurs as the result of very high-frequency interactions between the tire tread and road surface asperities (Figure 1.3). Account- ing for typical vehicle speeds, tread geometry, and the length scale of typical road surface asperities shows that tread material will be subject to excitations on the order of 102 through 108 Hz during braking.75,76,78 Meanwhile rolling resistance occurs because the normal forces in the leading half of the contact patch (where the tread material is being loaded) are greater than those in the trailing half (where the tread material is being unloaded). During a single tire rotation, each portion of the tread undergoes a single loading–unloading cycle, and so the frequencies relevant to rolling resistance are very close to those of typical tire rotation rates, which are between 100

and 102 Hz.65,70,71).

Figure 1.3: Contact forces between a tire tread and a road surface asperity (black arrows), resulting in a net force opposite the direction of travel (white arrow).

6 Therefore, one goal of this work is to understand how to tune hysteresis in poly-

mer nanocomposites, facilitating the design of materials that have high hysteresis at

the high-frequency operating conditions of traction while maintaining low hysteresis

at the low-frequency operating conditions of rolling resistance. A crucial mechanism

of hysteresis is the process of molecular desorption from and re-adsorption to ﬁller

surfaces.21,79 Many factors are involved in setting how the polymer adsorbs, including the polymer length, stiﬀness, and especially the local polymer–particle chemical interactions, which can be experimentally controlled by modifying the particle surface chemistry (depending on particle type), using surface functional groups or coupling agents, or using chemically diﬀerent polymers.26,27,80–83 In tire treads, perhaps the

most common and versatile coupling systems are silica nanoparticles used in com-

bination with various bifunctional organosilane coupling agents. By selecting the

correct organosilane, it is possible control the nanoparticle–SBR aﬃnity over a wide

range.25,84–88

SBRs used in tire treads are relevant to this work because, although they are usually synthesized with random sequence via emulsion free-radical or anionic solution polymerization, they can also be produced in controlled blocky configurations using various specialty techniques (albeit at a significantly higher cost).89,90 Moreover, nanoparticles can be functionalized such that one monomer (styrene or butadiene) adsorbs more strongly to the nanoparticle than the other. Even in the simplest case of bare silica, styrene monomers interact more strongly with the nanoparticles than butadiene monomers.91 Blocky SBRs have previously been studied and patented for use in tire treads,92–94 although this work has generally focused on the effect of microphase separation (caused by blockier sequences) and its effect on bulk properties.

7 Currently, the eﬀects of copolymer sequence are not well enough understood to allow detailed design of materials based on the impact of copolymer sequence on interfacial properties. Fully experimental investigation of potential copolymer conﬁgurations would be prohibitively expensive due to the large design space and the high cost of developing and synthesizing a given tailored SBR. Therefore, there is an opportunity to use the results of the current work to inform the design of tailored copolymer sequences for use in tire treads.

1.1.2 Coarse-Grained Modeling

MD simulations including nanoparticles and polymer are inherently diﬃcult to equilibrate; reaching the time and length scales required for particles to rearrange in such simulations would be very expensive or completely intractable for realistic particle sizes. Nanoparticle ﬁllers can have structure on length scales exceeding

100 nanometers, and relaxations can occur on a length scale of micrometers and a time scale of milliseconds. On the other hand, atomistic motions occur on a length scale of angstroms and a timescale of femtoseconds.17 Moreover, to collect reasonable ensemble statistics about relaxations, the simulation time window must be several times longer than the relaxation times of interest. In this work, we will use the coarse-grained, Kremer-Grest bead-spring model, a highly-eﬃcient simulation model in which polymers are represented as a chain of freely-jointed, coarse-grained “beads” connected by ﬁnitely extensible nonlinear elastic (FENE) bonds (“springs”).95,96 The

Kremer-Grest bead-spring model has been shown to be an eﬃcient and simple way to properly address molecular connectivity and structure, and it has been used with various nanoparticle types to allow simulation out to meaningful timescales using

8 reasonable computing resources.9,12,20,97 More details on this model can be found in

Chapter 2.

Using this coarse-grained model, we have performed MD simulations aimed at understanding important dynamic mechanisms relevant to nanocomposites, including typical tire tread materials, which are composed of carbon black or silica nanoparticles in crosslinked SBR along with various additives; however, here we develop this understanding by studying greatly simpliﬁed systems. Additives beyond the nanoparticles are not modeled, the polymers are purely linear, and our systems are not crosslinked.

Additionally, as noted above, the monomer–monomer interactions are all equal regardless of type, giving the systems no tendency to microphase separate in the bulk far from the nanoparticle.

These simpliﬁcations mean that the dynamics of our model systems are diﬀerent than the dynamics of tire tread materials and other real nanocomposite systems.

However, this simple model still allows us to better understand the underlying physical mechanisms governing the relationship between sequence and composite properties.

A final significant simplification is that, to decrease computational requirements, the nanoparticle loading fraction modeled is considerably lower than in many practical applications, including tire treads. To develop an understanding of trends that are applicable to materials with a much higher loading fraction, our simulations focus on the structure and dynamics of the interfacial region surrounding a single nanoparticle.

We expect the understanding that we develop regarding the properties of this interfacial layer can be used to qualitatively predict properties of systems with higher ﬁller

9 loading, as we can predict the fraction of material that is aﬀected by the particle depending on the volume fraction and the interfacial layer thickness that it determined form simulation.

1.2 Structure of Dissertation

The ultimate goal of the work herein is to understand how to modify material properties of the polymer–nanoparticle interphase by adjusting copolymer sequence.

To do this, we study a simple model, described in Chapter 2, consisting of a single nanoparticle surrounded by coarse-grained polymer chains. In Chapter 3, we use this model to determine the effect of polymer–nanoparticle interaction strength on interphase properties by studying a series of homopolymers systems, each with a different affinity for the nanoparticle. We measure a number of structural and dynamic properties as a function of distance from the nanoparticle surface, including chain end-to-end vector autocorrelation, bond vector autocorrelation, and self-intermediate scattering function relaxation times, and we study how polymer–nanoparticle interaction strength affects the progression of these properties from the nanoparticle surface to the bulk. In Chapter 4, we study the effect of copolymer sequence by performing the same analysis on a series of copolymer systems, each of which is composed of chains containing two monomer types that differ only in their affinities for the nanoparticle and are selected from the types studied in Chapter 3. These two monomer types are arranged in sequences that vary from system to system. Although the interphase properties studied in Chapters 3 and 4 reveal many relevant differences between the systems studied, they generally do not directly correspond to macroscopic properties that are easily measured experimentally and physically meaningful on a macro

10 scale. We are most interested in measuring dynamic modulus in the interphase region, but no well-established method exists to measure dynamic modulus locally from molecular dynamics simulation. Therefore, in Chapter 5, we develop a new method to estimate dynamic modulus on a local scale, and we use this method to examine the effect of copolymer sequence on interphase reinforcement and hysteresis as function of frequency of applied strain. In Chapter 6, we examine the height and shape of the frequency-hysteresis curves to determine how well each system addresses the tire tread material design problem introduced in Section 1.1.2. We then iterate on the most promising systems in an attempt to find copolymer sequences that increase hysteresis at higher frequencies while maintaining hysteresis at lower frequencies. Fi- nally, Chapter 7 provides some concluding remarks and a discussion of possible future research efforts.

11 Chapter 2: Model and Simulation Methods

2.1 Kremer-Grest Bead-Spring Model

Throughout this work, we use a model based on the standard Kremer-Grest bead-spring model95,96,98 in which polymers are freely-jointed chains of coarse-grained monomer beads. This model has previously been used for a variety of polymer systems, including systems incorporating copolymers,96,99–101 nanoparticles,9,12,20,97,102 or both.55,56

Here, we will use m, σ, and ε as the reduced units of mass, length, and energy, which are deﬁned based on the mass of a monomer and the length scale and strength of the interaction between nonbonded monomers. This further deﬁnes the reduced unit of time as τ = σ(m/ε)1/2. The mapping of ε, σ, and m to real experimental units depends on the system being modeled, the temperature considered, and the degree of coarse-graining, but the process is relatively straightforward. Mapping τ to real time units in coarse-grained systems is much more complex, since coarse-graining removes degrees of freedom, shortening timescales relative to those of atomistic simulations or experimental systems.103,104 If the beads in our system were considered to represent approximately the mass and length scales of Kuhn segments of polybutadiene, and if the temperature modeled was considered to be close to room temperature, then ε,

12 σ, m would take on values of approximately 2.5 kJ/mol, 0.99 nm, and 113 g/mol,

respectively.105 Using the most straightforward mapping (simply applying the equa-

tion τ = σ(m/ε)1/2) τ would then be approximately 2 × 10−10 seconds. Since this

simplistic mapping does not consider the eﬀect of lost degrees of freedom, the result

is eﬀectively a lower bound on τ. These numbers are provided only as a reference;

since our goal in this work is to understand basic physical mechanisms, we have in-

tentionally not selected a speciﬁc experimental system to model, so we will proceed

using reduced units only.

Bonded monomers are coupled by Finite Extensible Nonlinear Elastic (FENE)

potentials:

 2 1 2 rb − kR0 ln 1 − rb ≤ R0 2 R0 UFENE = , (2.1) 0 r > R0

where rb is the distance between the bonded monomers, R0 is the bond cutoﬀ distance,

set to 1.5σ, and k is a constant that sets the energy of the bond, for which we use the

standard value of k = 30ε/σ2, which is suﬃcient to prevent the chains from crossing

and the bonds from breaking.96

Monomer–monomer pairwise interactions are subject to a standard cut-oﬀ and

shifted Lennard-Jones (LJ) potential:

 σ 12 σ 6 4ε ij − ij + δ r ≤ r ij rm rm s m c ULJ,ij = . (2.2) 0 rm > rc

Here, rm is the distance between the monomers, εij is the interaction strength, σij is the interaction length scale, and the subscripts i and j refer to the types of the

1 two monomers involved in the interaction. The cutoﬀ distance, rc, is set to 2 6 σij for

13 bonded monomers and 2.5σij for non-bonded monomers. δs is a vertical shift factor

selected so that ULJ(rc) = 0. Note that while the original Kremer-Grest model used

1 only the repulsive portion of the LJ potential (cut oﬀ at 2 6 σij for both bonded and

non-bonded monomers), here we additionally include the attractive portion of the LJ

potential for nonbonded monomers. Although this increases the required computation

time, we believe that monomer–monomer attractive force plays an important role in

determining interphase properties, especially the dynamic modulus, which is studied

in Chapter 5.

2.2 Polymer Nanocomposite System Details

Our systems consist of 400 linear chains of length N = 100 (standard Kremer-

Grest chains of this length have fewer than two entanglements per chain on average106)

placed in a cubic simulation box. All monomers have mass 1m, and all monomer–

monomer interaction strengths and monomer–monomer interaction length scales are

equal, so σij = 1.0σ and εij = 1.0ε.

A single nanoparticle with eﬀective diameter 10σ and mass 1000σ is place in

the center of the simulation box, and periodic boundary conditions are applied.

Monomer–nanoparticle interactions follow a radially-shifted LJ potential, which has

been used to govern polymer–nanoparticle interactions in several prior simulation

studies of coarse-grained polymer nanocomposites.9,14,18,107 This potential is deﬁned

∞ r − ∆ ≤ 0  N  12 6  σ σ UNM = 4εNM r−∆ − r−∆ + δs 0 < r − ∆N ≤ rc , (2.3)  N N  0 r − ∆N > rc

14 where, the subscript M refers to monomer type, A or B, and we also use P to refer

to all monomers in the system. εNM is the strength of the interaction, and the shift

factor ∆N = (σNP − σ)/2 where σNP is the eﬀective diameter of the nanoparticle,

which we set to 10σ, yielding ∆N = 4.5σ. Here, r is the distance from the monomer

to the nanoparticle center.

The quantity r will also be used extensively in this work to characterize interphase properties as a function of distance from the nanoparticle. The precise meaning of r will depend on the property being measured and will be deﬁned in each context.

It will most often mean the distance of a monomer to the nanoparticle center (as in the equation for the monomer–nanoparticle interaction potential). However, when the property being considered is a property of polymer bonds, it will mean the distance of a bond midpoint from the nanoparticle center, and when the property being considered is a property of entire polymer chains, it will mean the distance of a polymer center of mass (COM) from the nanoparticle center. Note also that throughout this document, the phrase “distance from the nanoparticle” will, unless otherwise speciﬁed, mean r, the distance to the nanoparticle center. We will also occasionally

refer to the “surface” of the nanoparticle, which we deﬁne as σNP/2 = 5.0σ from the

nanoparticle center.

The monomer–nanoparticle interaction strength from Equation 2.3 is a key pa-

rameter that is adjusted in various parts of this study. In Chapter 3, we study

homopolymer systems, in which the interaction strength, εNP does not vary between

monomers within a given system, but does vary between systems, taking on integer

values between 1 and 8. In Chapter 4, we introduce copolymer systems, where chains

15 are composed of two monomer types (A and B) and the interaction strength εNM varies with monomer type M such that εNA = 1 and εNB = 5.

MD simulations were conducted using the open-source Large-scale Atomic/Molecular

Massively Parallel Simulation (LAMMPS) software package and applying the default equations of motion.108 A timestep of δt = 0.01τ was used throughout this study. Ini- tial chain conformations were generated as random walks with 0.96σ between bonded monomers, and monomer locations were rejected and regenerated if a location would fall inside the nanoparticle. Initial overlap between monomers was eliminated using a short soft “push-oﬀ” phase preceding equilibration in which the FENE bond interactions and the monomer–nanoparticle interactions were applied while the potential for all non-bonded monomer interactions was set to A[1 + cos(πrm/rc)] instead of

Equation 2.2. The strength of the soft potential, A, was increased linearly from 0 to

250 over the course of 50τ.

After push-oﬀ equilibration was performed in an isobaric-isothermal (NPT) ensemble. A Nosé-Hoover thermostat (damping parameter 1.0) held the reduced temperature at 1.0, and a Nosé-Hoover barostat (damping parameter 10.0) held the reduced pressure at 0. Equlibration ran for 200,000τ. This was 10 times as long as the bulk end-to-end vector autocorrelation function relaxation time (the time at which the function, which is normalized to 1 at time 0, drops below 1/e, which was approximately 1.9 × 104τ), Additioanlly, during this time, the monomers’ root mean squared displacement reached 16.8σ, more than three times the root mean-squared radius of gyration (approximately 5.1σ), suggesting that the polymers had suﬃcient time to

explore the simulation box and ﬁnd their preferred conformations. We also manually

conﬁrmed that all polymer chains which were initially adsorbed to the nanoparticle

16 desorbed at some point during the equilibration phase. Refer to Chapters 3 and 4 for more information regarding the end-to-end vector autocorrelation function and the mean squared displacement.

After equilibration in NPT, the systems were switched to a microcannonical

(NVE) ensemble by removing the thermostat and barostat and fixing the volume at the average of the previous one million timesteps. After this switch, the length of each side of the cubic simulation box was approximately 36σ. An additional equilibration of 50,000τ was performed after changing ensembles and before saving data for analysis in order to ensure that the system had time to adjust to the new fixed volume. The size of the simulation box was sufficient to ensure that, in all systems studied, with the nanoparticle defined as the center of the simulation box, the root mean-squared radius of gyration was within 2% of its bulk value for polymer whose centers were 6σ from the box edge, and the A and B monomer–nanoparticle pair distribution functions were within 2% of 1.0 by 2σ from the box edge. More detail on these measurements can be found in Chapters 3 and 4.

Data was saved for analysis at certain timesteps according to schemes that depended on the property being measured. The relevant data collection procedures are explained in the chapters in which each property is introduced.

17 Chapter 3: Structure and Relaxation Times of Homopolymer Systems

3.1 Background

The properties of the polymer–nanoparticle interphase are heavily inﬂuenced by

the polymer–ﬁller interaction strength. In most commercial applications, polymer–

ﬁller interactions are favorable, which helps prevent nanoparticle aggregation that

would otherwise occur due to entropic depletion. When interactions are favorable,

polymers adsorb on the nanoparticle, which results in slower dynamics and an in-

creased glass transition temperature in the interphase.33–38 When components are

selected such that interactions are unfavorable, interphase dynamics may be faster

than the bulk and the glass transition temperature lower.20 Polymer–ﬁller interaction strength is dependent on the polymer type and nanoparticle surface chemistry, so interphase properties can be adjusted by functionalizing the nanoparticle surface or changing the polymer chemistry. The eﬀect of homopolymer–nanoparticle interaction strength on composite properties has also been studied experimentally,25,26,109,110 using density functional theory,109 and in simulation20,111–113

A signiﬁcant body of prior work has suggested that the dynamics of the adsorbed polymer layer can be well described by a single parameter, εNP, which represents the

18 enthalpic gain of a monomer coming into contact with the nanoparticle versus its

reference state in the particle-free polymer solution.114–119 Speciﬁcally, the Polymer

Reference Interaction Site Model (PRISM) liquid state theory was used to study the

structure of polymer nanocomposites as a function of this parameter. Polymers were

modeled as freely jointed chains of hard core monomers, and the particle–particle

interactions were also purely hard core. The monomer–particle interactions were

modeled with a hard core and an exponential attraction, −εNP exp (−(r − rc)/α),

where α, the length scale of the attraction, is commonly set to 0.5, leaving εNP

as the sole adjustable parameter. At εNP = 0.55, the model successfully predicts the experimental scattering behavior of silica-polyethylene oxide (PEO),114 and at a lower

115 εNP, it predicts the scattering behavior of silica-polytetrahydrofuran (PTHF). The relative values of εNP corresponded to chemical diﬀerences in PEO and PTHF that aﬀect how strongly each polymer adsorbs to silica.

3.2 Simulation Details

Although the primary body of work described above used PRISM liquid state theory, our MD simulations implement a similar model in that the monomer–nanoparticle interactions are governed by a single adjustable interaction strength parameter. In this chapter, to explore the effects of adjusting this parameter, we report results for eight different homopolymer systems. The systems differ only in that the polymer– nanoparticle interaction strength εNM used in equation 2.3 varies between systems, taking on integer values from 1 to 8. Since the systems are homopolymers, we will refer to the attraction strength as εNP, with the subscript P indicating that the parameter applies equally to all monomers in the system. Note that since the nanoparticle

19 is larger than the monomers, εNP = 1 is eﬀectively a slightly repulsive interaction; as a monomer moves to the surface of the nanoparticle from the polymer bulk, it loses multiple interactions with other monomers but gains only a single interaction with the nanoparticle.

Here, we determine the eﬀect of polymer–nanoparticle interaction strength on several interphase structural properties, including the monomer–nanoparticle pair correlation function as well as various orientational parameters and the radius of gyration. We also devote considerable eﬀort to characterizing interphase dynamics, and we measure the end-to-end autocorrelation function, bond-vector autocorrelation function, and self-scattering function relaxation times. All properties the we consider are measured locally in the interphase as a function of distance from the nanoparticle, thereby providing an extensive characterization of the interphase properties.

That results that follow were calculated from data that was saved for analysis according to three diﬀerent schemes depending on the quantities of interest. For calculating static quantities including the radial distribution function, bond vector orientation, chain end-to-end vector orientation, and radius of gyration, data was saved every 100τ for 200,000τ. For calculating self-intermediate scattering and bond √ vector autocorrelation functions, data was saved at every power of 2 (rounded to

the nearest integer) for 1,000τ in 100 separate trajectories whose initial conﬁgurations

were separated by 1,000τ, well past the relevant relaxation times. For calculating the √ end-to-end autocorrelation function, data was saved at powers of 2 for 100,000τ in

200 separate trajectories whose initial conﬁgurations were separated by 50,000τ, also

past the relevant relaxation time.

20 Since the only chemical diﬀerence between the systems is the strength of absorp- tion to the nanoparticle, the structural and dynamic properties of the polymer phase suﬃciently far from the nanoparticle are independent of monomer–nanoparticle interaction strength. In fact, the bulk properties of the complete systems have only a negligible dependence on εNP since the nanoparticle loading fraction is only 2%.

Therefore, to examine the eﬀects of interaction strength, we have focused on analyz- ing local properties in the interphase.∗

3.3 Eﬀect of Polymer–Nanoparticle Interaction Strength on Interphase Structure

The structure of the interphase of monomers around the nanoparticle is quantiﬁed by the monomer–nanoparticle pair correlation function, gPN(r), the relative probability of ﬁnding a monomer at distance r from the nanoparticle compared to the probability in the bulk. Figure 3.1 reports gPN(r) for our homopolymer systems.

All the homopolymer systems yield gPN(r) data with a similar form, showing radial ordering around the nanoparticle with a comparatively large first peak and oscilla- tions with a wavelength of approximately 1σ that are characteristic of monomer-scale packing effects near a larger particle surface. The peak heights (and trough depths) increase with εNP, demonstrating that the degree of radial ordering near the nanoparticle increases with εNP. Within the range of interaction strength studied, the increase in the heights of the first few peaks and depths of the first few troughs is roughly

∗While the changes in local properties, especially dynamic properties, near the surface of nanoparticles are diﬃcult to fully characterize experimentally, some information about local structure and dynamics is accessible through techniques such as dielectric spectroscopy, 35,120 small angle X-ray and neutron scattering, 35 and bound rubber testing. 121

21 Figure 3.1: Pair correlation function of monomers with respect to the nanoparticle in the various homopolymer systems, as labeled. Radial ordering increases with εNP.

linear with increasing εNP. Farther than about 3σ from the surface nanoparticle, density is very close to the bulk value regardless of this interaction strength.

To further characterize the structure of the interphase on the monomer length scale, we measure bond orientations with respect to the nanoparticle using an orientation parameter based on the second-order Legendre polynomial.107 Speciﬁcally,

2 hP2(cos θbv)i = (3hcos θbvi − 1)/2), (3.1)

where θbv is the angle between the bond vector and the nanoparticle radial vector

(NRI, the vector from the nanoparticle center to the center of the feature of interest, in this case the midpoint of the bond). When the bond vectors tend to align parallel to the NRI, hP2(cos θbv)i > 0, while hP2(cos θbv)i < 0 indicates a tendency to align parallel to the nanoparticle surface, and hP2(cos θbv)i = 0 is consistent with random

orientation. To analyze bond orientation as a function of distance from the nanopar-

ticle, we assign bonds to concentric spherical shells around the nanoparticle based on

22 the bond center. Figure 3.2 reports hP2(cos θbv)i as a function of distance of the bond center from the nanoparticle center.

Figure 3.2: Bond vector orientation parameter, as deﬁned in the text, for the various homopolymer systems, as labeled. Bonds are assigned to shells around the nanoparticle based on the location of each bond’s center. The midpoint of each shell lies at the indicated distance, r, from the center of the nanoparticle, and each shell has a width of 0.25σ.

Bond ordering increases with εNP, roughly mirroring the local monomer density reported in Figure 3.1 data. This is a simple consequence of monomers being primarily arranged in structural shells (of high monomer density) separated by approximately the same distance as the length of a single bond. Bonds with their centers in areas of higher relative monomer density are more likely to be orientated parallel to the nanoparticle surface (with both bonded monomers residing in the same structural shell), and bonds with their centers in areas of lower relative monomer density are more likely to be oriented parallel to the NRI (with one bonded monomer each in two adjacent structural shells). As in the gPN(r) data, bond ordering increases with increasing εNP; however, whereas gPN(r) peaks increase roughly linearly with εNP,

23 orientation parameter peaks appear to asymptotically approach a value near that of

the εNP = 8 system. Far from the nanoparticle, bonds are randomly orientated.

To examine structure on a larger scale, we consider polymer chain conformations,

which we characterize using two measurements: the root-mean-squared radius of

gyration and a chain orientation parameter. The radius of gyration is deﬁned in the

standard fashion,

N 2 1/2 1 X 2 1/2 hRgi = h (rn − rcom) i , (3.2) N n=1

where rn is the position on the nth monomer in the chain and rcom is the position

of the chain COM. The chain orientation parameter is deﬁned analogously to the

bond vector orientation parameter, except that, in Equation 3.1, θbv is substituted

with θee, the angle between the polymer end-to-end vector and the vector from the nanoparticle to the polymer’s COM. Figure 3.3 reports these two measures of chain conformation as a function of distance of a chain’s COM from the nanoparticle center.

Near the nanoparticle, the large surface deforms polymer chains, making them more likely to align parallel to the surface as well as increasing Rg. Far from the

nanoparticle, chains are randomly oriented, and the root-mean-squared radius of

gyration stabilizes at approximately 5.1σ. There is also an intermediate distance,

between approximately 6.5 and 9.5σ from the nanoparticle, where Rg is slightly lower

than in the bulk. In this region, the nanoparticle surface slightly restricts polymer

conformations, compressing them somewhat in the nanoparticle radial direction but

not to enough of a degree to spread the chains out signiﬁcantly in the tangential

direction, as happens near the nanoparticle.

24 (a) (b)

Figure 3.3: Root-mean-squared radius of gyration (a), and chain orientation parameter (b) for the various homopolymer systems, as labeled. Polymers chains are assigned to concentric spherical shells around the nanoparticle based on the location of each chain’s COM. The midpoint of each shell lies at the indicated distance, r, from the center of the nanoparticle, and each shell has a width of 1σ (x-axis labels are located on the shell boundaries). Although chains may wrap partially around the nanoparticle, adequate statistics were not obtained to report on the rare cases where a polymer’s COM resides at r < 3.5σ from the nanoparticle center.

Overall, the monomer–nanoparticle interaction strength seems to have very lit-

tle impact on chain conformation according to either parameter considered. This is

likely because all monomers in each system have the same aﬃnity for the nanoparti-

cle, so the energy cost of swapping a monomer near the nanoparticle with an adjacent

monomer does not actually change with εNP. However, the lack of variation in chain

conformation may be somewhat counterintuitive in light of the large structural dif-

ferences noted at the smaller length scales considered in Figures 3.1 and 3.2.

25 3.4 Eﬀect of Polymer–Nanoparticle Interaction Strength on Interphase Relaxation Times

To estimate how relaxation times vary as a function of distance from the nanoparticle, we divide the polymer phase into concentric shells around the nanoparticle and consider relaxations in each shell separately, which is similar to our approach to characterizing interphase structure. Prior work has used the end-to-end vector autocorrelation function (EEACF) to characterize interphase polymer chain dynamics as a function of distance from both ﬂat surfaces17 and nanoparticles.37 The EEACF is deﬁned as

hR (t) · R (0)i : ee ee ACFee = 2 , (3.3) hRee(0) i where Ree(t) is the vector from the monomer on one end of a polymer chain to the monomer on the opposite end at time t. The EEACF measures the statistical correlation between a polymer’s conformation at time 0 and at time t. When t is small, the conformations are highly correlated and the EEACF has value near 1.

When t is suﬃciently large, the conformations become statistically decorrelated, and the value of the EEACF approaches 0. The rate at which the EEACF drops from 1 to 0 is a measure of how quickly polymer chain conformations decorrelate over time, and it is therefore a measure of chain mobility.

Using a similar approach to Refs. 17 and 37, we divide the system into a series of concentric shells centered on the nanoparticle and assign each polymer to the shell that contains the chain’s COM at the middle frame of the autocorrelation window.

Then, we calculate the EEACF separately for each shell. Polymers may move between shells during the course of the autocorrelation window, but this eﬀect is reduced

26 √ 44 by truncating the trajectories used for analysis at 2 timesteps (approximately

42,000τ), which is approximately twice the end-to-end vector autocorrelation function

relaxation time, as will be seen below.

By examining the relaxation behavior for polymers close to the nanoparticle sur-

face and comparing it to the bulk relaxation behavior, we can develop an understand-

ing of how dramatically the polymers are slowed near the nanoparticle. By measuring

the EEACF of shells of polymer at diﬀerent distances from the nanoparticle, we can

estimate how far out the interphase of slowed dynamics extends from the nanoparticle,

as we have done for the εNP = 1 and εNP = 8 systems in Figure 3.4.

For both systems, the dynamics very near the surface of the nanoparticle are slowed, and the dynamics trend toward those of the bulk at increasing distances from the nanoparticle. However, both the magnitude of this eﬀect and the length scale over which it occurs are signiﬁcantly greater in the εNP = 8 system.

(a) (b)

Figure 3.4: End-to-end vector autocorrelation function data of polymer chains in the εNP = 1 system (a) and εNP = 8 systems (b) binned based on r, the distance of the chain’s COM from the surface of the nanoparticle.

27 As mentioned above, it is important to note that while polymers are binned based on their locations during the middle time step of the autocorrelation windows, the polymers are mobile, so a chain’s COM may not stay within a particular shell throughout the entire window. However, we expect that most of the chains will stay near their initial shell over timescales not dramatically larger than the characteristic relaxation time. Of the chains that experience significant movement, some portion will spend more time closer to the nanoparticle, while others will spend more time farther away, so using the location at a single timestep is a reasonable first approximation. Using the middle timestep minimizes the maximum time between the time step that is used for indexing and the timesteps at either end of the autocorrelation window. Over very long timescales, we would expect all chains to converge to a behavior similar to the bulk regardless of initial location (the beginning of this convergence can be possibly be seen at times above approximately 104τ in Figure 3.4). However, for intermediate timescales, the shells are differentiated.

Although plots like Figure 3.4 give a good qualitative impression of the progression of a system’s interphase dynamics, it is difficult to compare such plots directly to compare multiple systems quantitatively. To quantify and more easily compare systems, we extract chain-scale relaxation times by fitting the end-to-end vector autocorrelation function with a stretched exponential model.9 The stretched exponential model is a phenomenological model of disordered systems that represents a relaxation process as the aggregate of a range of relaxation subprocesses, each with their own characteristic relaxation time. It is defined as

βR − t τR FR ≈ e , (3.4)

28 where FR is an overall relaxation function such as the EEACF, τR characterizes the

mean relaxation time of the distribution of subprocesses of FR, and βR is a stretching parameter related to the width of the relaxation time distribution. For the EEACF, we will refer to these parameters as τee and βee.

The data from each shell is ﬁtted with the stretched exponential model to extract

τee and βee. βee was found to be close to 0.55 and relatively independent of polymer– nanoparticle interaction strength and distance from the nanoparticle. On the other hand, τee was found to depend signiﬁcantly on both interaction strength and proximity to the nanoparticle. Plotting these relaxation times allows us to represent the data from each shell depicted in Figure 3.4 as a single point. This lets us directly examine the chain-scale relaxation times as a function of both εNP and distance from the ﬁller

surface. Figure 3.5 reports τee as a function of distance from the nanoparticle each of

the eight homopolymer systems.

In general, chain relaxation timescales increase with εNP and with proximity to

the nanoparticle, and chain dynamics decrease toward a common bulk value far from

the nanoparticle. In the εNP = 1 system, there appears to be an intermediate region

near the nanoparticle in which dynamics are actually slightly faster than in the bulk.

This is consistent with other work that has shown faster dynamics in nanocomposite

systems with slightly repulsive polymer–nanoparticle interactions. The exception is

at the closest reported shell, where the conﬁning eﬀect of the nanoparticle surface

appears to slightly outweigh the eﬀect of the repulsive interaction. A similar eﬀect

may also be present to a lesser degree in the εNP = 2 system, which could also be

eﬀectively a very slightly repulsive interaction due the number of monomer–monomer

interactions lost when a monomer approaches the nanoparticle from the bulk.

29 Figure 3.5: End-to-end relaxation times of various homopolymer systems, as labeled, as a function of distance from the nanoparticle. Polymers are assigned to shells as described in the text, and shells are centered on the indicated radius and have a width of 1σ (x-axis labels are located on the shell boundaries). Measurement error increases with proximity to the nanoparticle because fewer polymer COMs reside in closer shells. A 90% conﬁdence interval of the measurement is approximately ±2.5 · 103τ at 5σ, ±1.5 · 103τ at 6σ, ±1 · 103τ at 8σ, and ±0.5 · 103τ at 15σ. Adequate statistics were not obtained to report on the rare cases where a polymer’s COM resides at r < 4.5σ.

Chain relaxation times provide one method to characterize the width of the interphase. Here, we define the surface of the nanoparticle as 5σ, and we define the interphase as extending from the nanoparticle out to the last point where τee differs from the bulk value of 17.3 × 103τ by more than 10%. This yields interphases with approximate widths 0σ for εNP = 1 and εNP = 2, 1σ for εNP = 3, 2σ for εNP = 4, 3σ for εNP = 5 and εNP = 6, and 4σ for εNP = 7 and εNP = 8.

This analysis of interphase widths (and, in general, all the analysis of local end- to-end vector relaxation times) is confounded by the fact that τee is a property of the entire polymer chain, so it represents a physical process that is distributed over an area signiﬁcantly wider that the width of a single shell. Therefore, the analysis above should be treated only as a rough view of the progression of interphase dynamics rather than a precise measurement of local properties.

30 To provide a more localized description of relaxation, we also measure the bond

vector autocorrelation function (BVACF). Orientational relaxation of bonds has been

similarly studied by others,122 including as a function of distance from a nanoparti-

37,123 cle. The BVACF is deﬁned the same as the EEACF, except that Ree in Equation

3.3 is replaced by Rbv, the vector between adjacent bonded monomers. Equation 3.4 now yields τbv, the characteristic timescale of bond orientational relaxation. Impor- tantly, this characterizes dynamics on a length scale that is close to the width of a single shell.

Figure 4.10 reports τbv as a function of distance from the nanoparticle for each of the homopolymer systems studied. In this case, bonds are assigned to shells based on the location of the bond centers at the middle frame of the time window used for the calculation. Fits to determine τbv from Equation 3.4 are performed using data for t ≤ 10τ, during which time the MSD of the monomers is less than 1, meaning that the majority of monomers should remain in or near the shell to which they are assigned during the analysis time.

As with the end-to-end relaxations, bond-scale relaxation times generally increase with εNP. Here, it is clear that this increase is nonlinear, so that the eﬀect of raising

εNP by 1 on relaxation times increases with εNP. Also as with the EEACF data, bond relaxations generally slow with proximity to the nanoparticle (although the length- scale of the eﬀect is signiﬁcantly shorter than for the EEACF relaxation times).

Again, the exceptions are the εNP = 1 and εNP = 2 systems, in which, from the

bulk all the way to the second closest shell to the nanoparticle, relaxations speed up

with proximity to the nanoparticle. As before, the conﬁning eﬀect of the nanoparticle

surface causes both systems to exhibit slower relaxations in the ﬁrst shell than in the

31 Figure 3.6: Bond vector relaxation times as a function of distance from the nanoparticle for the various homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are located on the shell boundaries), and bonds are assigned to shells as described in the text. Although measurement error increases with proximity to the nanoparticle because fewer bonds reside in closer shells, a 90% conﬁdence interval of the measurement is less than ±1τ in all shells considered.

second, although in the εNP = 1 system, relaxations in the closest shell are still faster than in the bulk.

These measurements of bond relaxation times yield interphase width estimates of approximately 2σ for εNP = 1, 1σ for εNP = 2, 0.5σ for εNP = 3, 2σ for εNP

from 4 through 6, and 2.5σ for εNP = 7 and εNP = 8. Note that this includes

interphases of faster dynamics (for the εNP = 1 and εNP = 2 systems), which is

why interphase widths do not increase monotonically with εNP. Interfacial width is

known to depend on which dynamic (or structural) property is considered,34,37 and

the BVACF measures relaxations on a smaller length-scale than the EEACF, so it is

not surprising that the interphase widths calculated from BVACF relaxation times

diﬀer than those calculated from the EEACF.

To characterize local monomer mobility, we use the self-intermediate scattering

function FS(k, t), deﬁned as

32 Nς 1 X FS(k, t) = h exp[ik · (rn(t) − rn(0))]i, (3.5) Nς n=1

where rn(t) is the position of monomer n at time t, Nς is the number of monomers in the shell, and k is a wave vector.

In contrast to the bond vector autocorrelation, which measures local bond reorientation, the self-intermediate scattering function measures monomer translation and has previously been used by Ghanbari et al. to characterize the polymer-nanoparticle interphase.33 It also eﬀectively encodes the same information as the mean-squared displacement, which was used by Ndoro and colleagues in a similar context.37 Note that while this formulation of the self-intermediate scattering function is omni-directional, treating movement in all directions as equivalent, it would also be possible in future work to consider k vectors normal to and parallel to the nanoparticle surface in order to characterize monomer motion toward, away from, or parallel to the nanoparticle.

To characterize the interphase using FS(k, t) results, we apply a similar treat- ment as for the vector autocorrelations: we assign monomers to shells based on their locations at t/2 and then ﬁt the ensemble average of each shell with a stretched ex-

ponential model. We focus on changes in τFS(k,t), the characteristic relaxation time of

FS(k, t), across systems. As with the analysis of bond vector relaxation, ﬁts to deter-

mine τFS(k,t) are performed using data for t ≤ 10τ. Figure 3.7 plots these relaxation times as a function of distance from the nanoparticle for |k| = 2πσ−1, corresponding

to the monomer diameter length scale. The relaxation time constant is then related

to (though not precisely equivalent to) the timescale required for a monomer to move

1σ on average.

33 Figure 3.7: Relaxation times of the self-intermediate scattering function at |k| = 2πσ−1, reported as a function of distance from the nanoparticle for the various homopolymer systems, as labeled. Monomers are assigned to shells as described in the text, and shells are centered at the indicated distance and have width of 0.5σ (x-axis labels are located on the shell boundaries). Although measurement error increases with proximity to the nanoparticle because fewer monomers reside in closer shells, a 90% conﬁdence interval of the measurement is less than ±0.02 at all radii.

The results for all systems exhibit an oscillatory behavior that coincides with the ﬂuctuations in the monomer–nanoparticle pair distribution function seen in Fig- ure 3.1a; monomers in locations of high relative density due to structural ordering around the nanoparticle are seen to be less mobile over short timescales than the fewer monomers found in areas of low density at the middle of the time window

(these monomers may have been in the process of moving between higher density locations during the time window). Once again the εNP = 1 and εNP = 2 systems exhibit dynamics that are faster near the surface of the nanoparticle (this is also true for one low monomer density shell in the εNP = 3 system).

Deﬁning the interphase width as above, we see that the εNP = 1 and εNP = 2 systems have interphases that are approximately 1σ wide, and all the systems with

34 εNP ≥ 3 have an interphase width of 2.5σ. This is likely related to the monomer– nanoparticle interaction cutoﬀ distance, and this eﬀect is discussed in more detail in the next chapter, where we will consider copolymer systems.

35 Chapter 4: Structure and Relaxation Times of Copolymer Systems

*Text and ﬁgures in this section are adapted from Ref. 1 (Trazkovich, A. J.;

Wendt, M. F.; Hall, L. M. Soft Matter 2018, 14, 5913–5921) with permission from

the Royal Society of Chemistry.

4.1 Background

Many studies have addressed the role of block copolymers in nanoparticle disper-

sion47–51 and self-assembly.52–55 Sequence eﬀects have also been extensively studied

in the context of biological systems, where amino acid sequence can play a critical

role in the interaction between proteins and nanoparticles.124–126 However, compared

to the number of studies that have examined the eﬀect of polymer–ﬁller interac-

tion strength on interphase properties (as discussed in Chapter 3) few studies have

speciﬁcally explored the role of copolymer sequence in determining properties of the

polymer–nanoparticle interphase.

In a recent study,56 Chen et al. simulated regular multiblock copolymer chains of hydrophobic and hydrophilic monomers adsorbed to a hydrophobic nanoparticle. The authors focused on the formation of micelle-like structures of hydrophobic monomers, which could either form adsorbed on part of a nanoparticle or surrounding

36 the nanoparticle. The authors found that with all pairwise interactions equal to ε except for the hydrophobic–hydrophobic interactions, micelles preferentially formed adjacent to the nanoparticle when the interaction strength between the hydrophobic monomers was above a critical threshold close to ε. In another study,57 Martin and colleagues modeled a nanoparticle grafted with copolymer chains composed of alternating blocks of equal length. When one of the monomer types had a higher affinity for the nanoparticle, chain conformations around the nanoparticle were shown to depend on the length of the blocks used in the copolymer sequence. Generally, as block length increased, the average distance between the nanoparticle and the monomer type with higher affinity for the nanoparticle decreased while the average distance of the less adsorbing monomer type increased. However, some systems did not follow this expected behavior, depending on the strengths of the various monomer–monomer interactions as well as which block (high or low affinity) was grafted to the nanoparticle.

Although local interphase dynamics are much more diﬃcult to study experimentally than in simulation, a recent experimental study by Helal et al. presented some evidence that copolymer sequence impacts interfacial glass transition temperature. In a nanocomposite consisting of clay nanoparticles in polystyrene-b-poly(ethylene-co- butylene)-b-polystyrene, the glass transition temperature in the polymer–nanoparticle interphase (characterized by dielectric spectroscopy) was found to depend on the length of the polystyrene blocks.58

Prior simulation studies of copolymer nanocomposites have typically considered unfavorable interactions between unlike monomers (which are usually present in experimental systems); however, a body of theoretical work on copolymer-nanoparticle

37 systems from the Schweizer group considered monomers that differ only in their nanoparticle interactions, as we do here. Specifically, they showed that even when the only chemical difference between the monomers is the strength of their interaction with the nanoparticles (so the bulk behaves as a homopolymer with no tendency to microphase separate), copolymer structure in the vicinity of nanoparticles depends dramatically on the copolymer sequence.59–61 An adsorbed monomer layer was found to form around each nanoparticle (in which there was an increased concentration of adsorbing monomers), and the width of this phase increased with copolymer block length. Additionally, in systems with multiple nanoparticles, nanoparticles that were significantly smaller than the block length were found to have a much higher tendency to aggregate.

4.2 Simulation Details

Here, we now use MD simulations to show how copolymer sequence impacts structure as well as interphase dynamics, the latter of which was not addressed in the theoretical and simulation work described above. We have performed a series of simulations based on the same basic model as was used in Chapter 3, and we have measured all of the same interphase properties as a function of distance from the nanoparticle that were reported in that chapter.

Unlike in Chapter 3, where the monomer–nanoparticle interaction strength, εNP, varied between systems but did not vary between monomers within a given system, we now consider copolymer chains that each contain two types of monomers, A and

B. The properties of the two monomer types do not vary between systems, although the chain sequences into which they are arranged do. A and B monomers diﬀer

38 only in their interactions with the nanoparticle, εNA versus εNB. For A monomers, we selected εNA = 1, the lowest interaction strength that was studied in Chapter 2, which, as discussed earlier, is eﬀectively a slightly repulsive monomer–nanoparticle interaction. For B monomers, we selected εNB = 5, which we determined to be a good compromise between the required computational power (which would increase with εNB as the relevant relaxation times increased) and creating good diﬀerentiation between copolymer sequences.

Although our copolymer systems do not include the effect of unfavorable interactions between unlike monomers (which are present in most experimental systems), they can be considered to be a closer model of systems with low binary interaction parameters such as copolymers of poly(methyl methacrylate) (PMMA) and poly(ethylene oxide) (PEO) or poly(vinyl acetate) (PVA) and PEO, where the polymer components are very miscible with one another but may exhibit significant differences in their interactions with a nanoparticle. However, unlike in those experimental systems, our two model components do not have different glass transition temperatures.

In the systems explored in this chapter, each copolymer chain consists of an equal number of A and B monomers, which are arranged in diﬀerent conﬁgurations depending on the system. The majority of our copolymer systems are regular multiblock copolymer sequences with form [AxBx]y, where y = 100/(2x) and x is the length of each block. Depending on the system, x is set to 1, 2, 5, 10, 25, or 50. Hereafter, we refer to these systems by their block length as “BL = x”. These systems are in- spired by experimental multiblock copolymer sequences with adjustable block length, which have been successfully synthesized using a variety of methods in recent years;

39 accessible multiblock copolymer chemistries include poly(styrene-b-butadiene),89,127 poly(lactide-b-butadiene)128 and poly(styrene-b-methyl methacrylate).129 We additionally study a random copolymer system in which chain sequences are chosen by individually randomizing the order of a set of 50 A and 50 B monomers. Therefore, in the random system, sequences may vary between chains, which is not the case in the other systems; however, each individual chain still contains exactly 50% of each monomer type. In the particular set of random sequences generated by this procedure and used in this work, the average length of a block selected by choosing a random

B monomer was 2.99, the average length of the longest B block on each chain was

5.86, and the longest B block in the entire system was 13 monomers long. Figure 4.1 shows a schematic of each of the copolymer sequences used in our initial study.

Figure 4.1: Schematic of the copolymer block sequences used in this work. Sequences in the random copolymer system may vary between chains, and only one example is shown. Within each of the other systems, chains are all identical to the sequence shown.

40 Other than these adjustments, the simulations proceed exactly as did the homopolymer systems. Data is saved for analysis according to the same schemes described in Section 3.2. Except where noted for the monomer–nanoparticle pair correlation function, all methods used to calculate the results in Section 4.3 and 4.4, below, are identical to the methods used in Sections 3.3 and 3.4, respectively. For comparison, most ﬁgures in this chapter also reproduce results from A homopolymer and B homopolymer systems (equivalent to the εNP = 1 and εNP = 5 systems from

Chapter 2).

4.3 Eﬀect of Copolymer Sequence on Interphase Structure

Before we proceed to report quantitative results for the structure of the copolymer systems, we ﬁrst consider, qualitatively, four snapshots of our systems, which are shown in Figure 4.2. For all systems, there is an interphase around the nanoparticle in which the density of B monomers is signiﬁcantly higher than the density of A monomers. Increasing the block size widens this interphase, and the random copolymer system shows an interphase similar to that of the BL = 5 system. This is not surprising, since although the average block length of the random copolymer as determined by selecting a random B monomer is very close to 3, some blocks are longer, and longer B blocks likely aggregate preferentially around the nanoparticle.

The structure of the interphase of monomers around the nanoparticle is quantiﬁed by the monomer–nanoparticle pair correlation function, gMN(r), the relative probability of ﬁnding a monomer of type M at distance r from the nanoparticle compared to the probability in the bulk. Recall that we use the index P to refer to all monomers, while A refers to nonadsorbing monomers and B refers to adsorbing monomers. A

41 Figure 4.2: Clockwise from top left: structure of BL = 25, BL = 5, BL = 1, and Random systems. A monomers (εNA = 1) are shown in cyan, and B monomers (εNA = 5) are shown in pink.

small diﬀerence from our analysis of the homopolymer systems is that we can now separately measure gAN(r) and gBN(r) to quantify how A and B monomers order diﬀerently around the nanoparticle. gAN(r) and gBN(r) are reported in Figure 4.3.

The depletion of A and increased concentration of B near the surface that was

seen in the snapshots from Figure 4.2 is apparent in the ﬁrst couple peaks of gAN(r)

and gBN(r). This eﬀect increases with block length and is present to some degree for

all copolymer systems considered. Farther from the nanoparticle, the A and B peaks

become increasingly similar in size as a function of r. The amount of increase in B

42 (a) (b)

Figure 4.3: A monomer–nanoparticle (left) and B monomer–nanoparticle (right) pair correlation functions for the various copolymer systems, as labeled. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

concentration and the length scale over which the eﬀect persists increases with block length. As seen in the snapshots, the structure of the random copolymer system near the nanoparticle most closely resembles that of the BL = 5 system.

In Figure 4.4a, we also report gPN(r), the overall monomer–nanoparticle pair correlation function (results for homopolymer A and homopolymer B systems are included for comparison). Additionally, for the copolymer systems, we calculate gBN(r)/gPN(r), the ratio of the B–nanoparticle pair correlation function to the overall monomer– nanoparticle pair correlation function, which is equivalent to the probability that a randomly chosen monomer at distance r is type B.

Together, these two data sets encode the same information as the data that was reported in Figure 4.3, which is the more standard method of reporting the pair correlation function. However, Figures 4.4a and 4.4b provide a more direct way to understand the overall structure of monomers around the nanoparticle as well as the

43 (a) (b)

Figure 4.4: (a) gPN(r), the monomer–nanoparticle pair correlation function for the copolymer systems as well as the two homopolymer systems, as labeled, for comparison. (b) gBN(r)/gPN(r), the fraction of monomers at distance r that are type B, for the copolymer systems, as labeled. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

percentage of these monomers that are type B, thereby yielding a better understanding of the interphase of B monomers that forms around the nanoparticle.

From Figure 4.4a, we see that radial ordering for the copolymer systems generally falls between that of A homopolymer and B homopolymer systems. All copolymer systems (which include 50% B) have noticeably higher ﬁrst, second, and third peaks in gPN(r) versus the A homopolymer due to monomer adsorption. These peaks intensify with block length, and the systems with BL ≥ 10 are nearly indistinguishable from

the pure B system.

From Figure 4.4b, we see that all systems exhibit a phase around the nanoparticle

where the fraction of B monomers is above 0.5, and for BL ≥ 10, this phase is

nearly pure B. The width of this phase increases with the block length. Surrounding

this ﬁrst phase, most of the systems exhibit a secondary region where the fraction

44 of A monomers is above 0.5, although this eﬀect is minimal in the random and

BL = 50 systems. In the BL = 25 system, this secondary A-dominant phase extends to about 14σ from the center of the nanoparticle (the plot has been cropped for visual clarity). After this secondary A-dominant phase, all systems’ concentrations eventually converge to the bulk value (equal concentrations of A and B monomers).

Near the nanoparticle, the fraction of B monomers in the random copolymer system most closely resembles that of the BL = 5 system. This is consistent with the average

length of the longest B block in each chain in the random system and fact that the

longer B blocks will tend to preferentially aggregate around the nanoparticle. Out

of all the systems, the random copolymer system converges to equal concentrations

of A and B monomers over the shortest length scale. This is likely because in the

random system, the composition varies within and between chains, and a long B block

adsorbed to the nanoparticle may not be adjacent to a long A block, so the secondary

A dominant phase is not present.

The fraction of B monomers yields one way to measure the width of the static

interphase. For this purpose, we deﬁne the interphase as extending from the surface of

the nanoparticle (at r = 5σ) out to the last point where the fraction of B monomers

diﬀers from the bulk value of 0.5 by more than 5%. This yields interphases with

widths of 1.5σ for the random copolymer system, 2σ for BL = 1, 3σ for BL = 2, 3.5σ for BL = 5, 4σ for BL = 10, 7σ for BL = 25, and 4σ for BL = 50. Note that these values are somewhat sensitive to the precise cutoﬀ threshold used.

We characterize bond orientations in the copolymer systems using the same method as in Section 3.3. Figure 4.5 reports the bond orientation parameter as a function of distance of the bond center from the nanoparticle center. Recall that hP2(cos θbv)i > 0

45 indicates bonds that have the tendency to align normal to the nanoparticle surface, while hP2(cos θbv)i < 0 indicates a tendency to align parallel to the surface.

Figure 4.5: Bond vector orientation parameter, as deﬁned in the text above, for the various copolymer and homopolymer systems considered, as labeled. Bonds are assigned to concentric spherical shells around the nanoparticle based on the bond center. The midpoint of each shell lies at the indicated distance, r, from the center of the nanoparticle, and each shell has a width of 0.25σ. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

As with bond orientation in the homopolymer systems (see Figure 3.2), the trend in bond orientations in the copolymer systems roughly mirrors that of the local monomer density (Figure 4.4a). Compared to the homopolymer systems, there is relatively little variation between copolymer systems. Generally, the behavior of the copolymer systems falls between that of the A homopolymer and B homopolymer systems, and most of the variation within that range is explained by diﬀerences in gPN(r). However, the peak that the BL = 2 system exhibits at r = 6.0σ and that the BL = 5 system exhibits at r = 6.0σ and r = 6.25σ (both of which exceed even the peak exhibited by the B homopolymer system) are not as easily explained by difference in monomer structure. Likely, bonds in each of those systems have a greater

46 tendency to align perpendicular to the nanoparticle surface because of the sharp A–B phase boundary that occurs at that distance from the nanoparticle. This is seen in both the magnitude and the slope of the B to A transition that was reported in Figure

4.4b.

Polymer chain conformations are also impacted by copolymer sequence, which can be seen qualitatively in Figure 4.6, which depicts snapshots of typical chain conformations in three systems with various levels of blockiness. Chains with fewer, longer B blocks tend to exhibit more extended conformations as the B blocks adsorb to the nanoparticle and the A blocks reside in the secondary A-rich phase farther from the nanoparticle.

Figure 4.6: Representative snapshots of selected polymers from three diﬀerent copolymer systems, showing, from left to right, example conformations of adsorbed chains in the BL = 1, BL = 5, and BL = 25 systems. B monomers (pink) beads adsorb more strongly to the nanoparticle (purple) than A monomers (cyan).

To characterize polymer conformations quantitatively, we ﬁrst examine the radius of gyration (Rg) as a function of distance from the nanoparticle center. As with the

47 homopolymer systems in Section 3.3, polymers were assigned to concentric spherical

shells around the nanoparticle based on each chain’s COM. These results are reported

in Figure 4.7.

Figure 4.7: Average radius of gyration of polymer chains as a function of distance from the center of the nanoparticle for the various copolymer and homopolymer systems, as labeled. Polymers are assigned to shells as described in the text, and shells are centered at the indicated radius and have a width of 1σ (x-axis labels are located on the shell boundaries). Although chains may wrap partially around the nanoparticle, adequate statistics were not obtained to report on the rare cases where a polymer’s center of mass resides at r < 3.5σ from the nanoparticle center. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

Far from the nanoparticle, all systems have very similar Rg since there is no chemical diﬀerence between the monomers in the bulk. Between about r = 11σ and r = 8σ, Rg decreases slightly with proximity to the nanoparticle, then for r < 8σ,

Rg increases sharply as r decreases. In this latter region, Rg tends to increase with

BL in the multiblock copolymer systems while the results of the two homopolymer systems are very similar. Notably, for BL = 1, Rg is signiﬁcantly lower than either homopolymer system throughout this entire region, while for BL ≥ 25, Rg is

signiﬁcantly higher. In terms of radius of gyration, the random copolymer system

48 most closely resembles the BL = 25 system. The BL = 50 system has a signiﬁcantly higher Rg than any of the other systems out to a radius of 10σ; since the BL = 50 chains are diblock copolymers, the chain conformation is more extended when the B block is absorbed and the A block is outside the B-dominant phase surrounding the nanoparticle. All the other sequences allow for multiple B blocks on the same chain to adsorb on the nanoparticle, making a more compact conformation. Overall, there is signiﬁcantly more variation in radius of gyration between the copolymer systems than was observed between the homopolymer systems (see Figure 3.3a).

We also measure copolymer chain orientations. Figure 4.8 reports hP2(cos θee)i as a function of distance of the polymer’s center of mass from the nanoparticle center.

Compared to the radius of gyration, there is not as clear a trend in terms of copolymer block length, although all the copolymer systems tend to align less parallel to the nanoparticle surface than the homopolymer systems. Close to the nanoparticle, in all systems, chains tend to orient parallel to the nanoparticle surface, although this eﬀect is noticeably lower in the BL = 50 system. At intermediate distances, between approximately 10σ and 15σ, the copolymer chains show a slight tendency to align more perpendicular to the surface before settling back to purely random orientations far from the nanoparticle; all the copolymer chains are asymmetrical to some degree, and in the block copolymer systems one end of each chain is always adsorbing while the other end is not. The adsorbing end of the chain will tend to be closer to the nanoparticle while the non-adsorbing end remains farther away. Only the BL = 1 system (in which there is minimal assymetry) and the homopolymer systems do not exhibit an orientation parameter above 0 at any distance.

49 Figure 4.8: Chain end-to-end orientation parameter, as deﬁned in the text above, for the various copolymer and homopolymer systems, as labeled. Chains are assigned to concentric spherical shells around the nanoparticle based on their center of mass, and shells have width of 1σ (x-axis labels are located on the shell boundaries). Adequate statistics were not obtained to report on the rare cases where a polymer’s center of mass resides at r < 3.5σ. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

4.4 Eﬀect of Copolymer Sequence on Interphase Relaxation Times

To characterize interphase dynamics in the copolymer system, we follow the same procedures that were used for the homopolymer systems in Section 3.4. Recall that to measure chain-scale relaxation times, we divide the system into a series of concentric shells centered on the nanoparticle and assign each polymer to the shell that contains the chain’s COM at the middle frame of the autocorrelation window. We measure the EEACF of chains assigned to each shell, and the data from each shell is ﬁtted with the stretched exponential model (Equation 3.4) to extract the mean end-to-end relaxation time, τee, which was found to depend signiﬁcantly on copolymer sequence and proximity to the nanoparticle. These results are reported in Figure 4.9.

50 Figure 4.9: End-to-end relaxation times of the various copolymer and homopolymer systems, as labeled, as a function of distance from the nanoparticle. Polymers are assigned to shells as described in the text, and shells are centered on the indicated radius and have a width of 1σ (x-axis labels are located on the shell boundaries). Measurement error increases with proximity to the nanoparticle because fewer polymers reside in closer shells. A 90% conﬁdence interval of the measurement is approximately ±2.5 · 103τ at 5σ, ±1.5 · 103τ at 6σ, ±1 · 103τ at 8σ, and ±0.5 · 103τ at 15σ. Adequate statistics were not obtained to report on the rare cases where a polymer’s center of mass resides at r < 4.5σ. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

Generally, chain relaxation timescales increase with proximity to the nanoparticle.

In the blocky copolymer systems, increasing block length tends to increase interphase relaxation times (bulk relaxation times are equal in all systems since polymer properties are identical in the bulk). The length scale over which relaxations are slowed also appears to increase with block length. The interphase chain relaxation times of the random copolymer system are similar to those of the BL = 5 and BL = 10 systems.

In contrast to the overall trend, in the closest shell reported, the BL = 50 system exhibits statistically signiﬁcantly shorter relaxation times than several other systems with shorter block lengths. In this system, the chain COM is rarely in the shell closest to the nanoparticle surface because of the tendency of chains to orient with their A blocks far from the surface. Chains that happen to be in an unfavorable

51 conformation with COM near the surface will relatively quickly reorient to a more favorable orientation.

Interestingly, above a threshold of approximately BL = 5, the interphase chain relaxation times are slower than for the B homopolymer system despite the fact that the B homopolymer system contains twice as many adsorbing B monomers as the copolymer systems. This may be because, in the copolymer systems, there is an energy barrier associated with swapping an adsorbed B block with another B block on the same chain, and this energy barrier increases with the length of the non-adsorbing

A blocks that separate the B blocks. In the B homopolymer system, adsorbing chain segments may be swapped without experiencing this barrier.

As before, we characterize the interphase width by defining the interphase as extending from the nanoparticle surface (at r = 5σ) out to the last point where τee differs from the bulk value of 17.3 × 103τ by more than 10%. This yields interphases with approximate widths of 9σ for the random copolymer system, 1σ for BL = 1, 3σ for BL = 2, 5σ for BL = 5, 7σ for BL = 10, 7σ for BL = 25, 5σ for BL = 50, 3σ for homopolymer B, and 0σ for homopolymer A. Note that the interphases according to chain relaxations are significantly wider for some of the blockier systems than for the

B homopolymer system (or even compared to more strongly-adsorbing homopolymer systems studied in Chapter 2). This may be partly a consequence of the fact that the interphase radius of gyration tends to be larger in the copolymer systems (Figure

4.7) than in the homopolymer systems (Figure 3.3a), allowing chains to inﬂuence a wider area around the nanoparticle.

We also measure bond vector relaxations, recalling that this is done by assign- ing bonds to the shell that contains the bond center at the middle frame of the

52 autocorrelation window and then measuring τbv, the mean relaxation time of bond reorientation, in each shell.

Figure 4.10 reports τbv as a function of distance from the nanoparticle for each of the systems studied. As with the end-to-end relaxations, bond-scale relaxation times in the B homopolymer and all copolymer systems increase with proximity to the nanoparticle, although the length scale over which this eﬀect occurs is shorter than for the chain relaxations. Compared to the end-to-end relaxations, there is relatively less variation between the copolymer systems, with the behavior of all the copolymer systems strongly resembling that of the B homopolymer for r ≥ 7. There is also signiﬁcantly less variation between copolymer systems than between the various homopolymer systems considered.

Figure 4.10: Bond vector relaxation times as a function of distance from the nanoparticle for the various copolymer and homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are located on the shell boundaries), and bonds are assigned to shells as described in the text. Measurement error increases with proximity to the nanoparticle because fewer bonds reside in closer shells. A 90% conﬁdence interval of the measurement is approximately ±1τ at 5.5σ, ±0.5τ at 7σ, and ±0.1τ at 10σ and above. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

53 Bond relaxation times very close to the nanoparticle appear to depend primarily

on the local percentage of adsorbing monomers (Figure 4.4b). For BL ≤ 10, bond relaxations in the closest shell (r = 5.5σ) increase with BL, as does the fraction of B

monomers. For BL ≥ 10, the relaxation times plateau at nearly the same value as the

B homopolymer system. This corresponds to the copolymer block length threshold

above which the fraction of B monomers in the phase adjacent to the nanoparticle’s

surface is nearly indistinguishable from 1. However, the random copolymer system

exhibits bond relaxations in the ﬁrst shell that are slower than the BL = 5 system

(and equivalent to those of the longest block length copolymer systems) despite having

a locally adsorbed B-dominant phase that is less pure than that of the BL = 5

system. In the r = 6σ shell, several of the shorter block length systems as well as the random system exhibit signiﬁcantly slower bond relaxations than the B homopolymer system. This may be attributable to the very sharp transitions in the percentage of

B monomers observed at approximately that distance from the nanoparticle in the

BL = 2 and BL = 5 systems, and it may be related to the bond orientation peaks

observed in Figure 4.5; A–B bonds that are aligned across that boundary may be

disproportionately slow to relax as a result of the sharp phase transition.

In contrast to what was seen with the copolymer chain relaxation data (and with

the homopolymer bond relaxation data reported in Figure 3.6), the apparent inter-

phase width according to bond relaxation times is relatively independent of copolymer

sequence, with all systems converging to the bulk value approximately the same dis-

tance from the nanoparticle surface. For several systems, this is a shorter length

scale than the convergence of either the composition or the chain relaxation times.

Note that this is, in part, a consequence of our particular model, which deﬁnes the

54 two monomer types as identical except for their interactions with the nanoparticle.

If inter-monomer interactions were adjusted to vary with monomer type, then it is

likely that the interphase width according to local dynamic properties such as bond

relaxations would be more similar to the interphase width according to composition.

In contrast, in our system, the bond relaxation times converge to the bulk value after

about 2.5σ from the nanoparticle surface, which is the cutoﬀ distance of the monomer–

nanoparticle interaction potential, the distance out to which monomer-scale dynamics

are directly impacted by their interaction with the nanoparticle. On the other hand,

in the case of chain-scale relaxation times, apparent interphase eﬀects can persist

out to distances signiﬁcantly beyond the 2.5σ cutoﬀ because polymers whose chain

centers are beyond the cutoﬀ distance may still contain some monomers which are

much closer to the nanoparticle. In fact, the widest interphases according to chain

relaxation times were seen to be about 7σ, which is approximately the sum of the

cutoﬀ distance and the radius of gyration.

To characterize local monomer mobility, we again use the self-intermediate scatter-

ing function, FS(k, t), assign monomers to shells based on their locations at t/2, and

then extract τFS (k, t), the characteristic relaxation time of FS(k,t). Figure 4.11 plots these relaxation times as a function of distance from the nanoparticle for |k| = 2πσ−1.

As in the homopolymer systems, the results for all copolymer systems exhibit an oscillatory behavior that coincides with the ﬂuctuations in the monomer–nanoparticle pair correlation function seen in Figure 4.4a. Except in the A homopolymer system, monomer-scale motions slow with proximity to the nanoparticle. Generally, FS(k, t) relaxation times for the copolymer systems fall between those of the A homopolymer and B homopolymer systems. Relaxation times in the blocky systems are seen to

55 Figure 4.11: Relaxation times of the self-intermediate scattering function at |k| = 2πσ−1, reported as a function of distance from the nanoparticle for the various copolymer and homopolymer systems, as labeled. Monomers are assigned to shells as described in the text, and shells are centered at the indicated distance and have width of 0.5σ (x-axis labels are located on the shell boundaries). Although measurement error increases with proximity to the nanoparticle because fewer monomers reside in closer shells, a 90% conﬁdence interval of the measurement is less than ±0.02 at all radii. Adapted from Ref. 1 with permission from the Royal Society of Chemistry.

increase with block length, with the relaxation times of the BL = 25 and BL = 50 systems being nearly indistinguishable from those of the homopolymer B system.

The behavior of the random copolymer system is most similar to that of the BL = 5 system.

The apparent interphase width according to the self-intermediate scattering function is approximately 2.5σ and is relatively independent of copolymer sequence. As with the interphase width according to bond-vector relaxation times (see discussion above), this is likely linked to the cutoﬀ distance of the monomer–nanoparticle interaction potential.

56 4.5 Discussion

Using a series of AB copolymer sequences, we have discovered that, for many static and dynamic properties considered, adjusting sequence can tune the interphase to take on a range of properties between extremes bounded by the homopolymer

A and homopolymer B systems. Moreover, depending on the interphase property of interest, in some cases it is possible to produce interphase behavior that is outside the region bounded by the homopolymer A and homopolymer B systems. This is most clear in the case of radius of gyration and in the case of end-to-end relaxation times, where certain copolymer sequences with longer block lengths produce less compact chains with slower relaxations than even the pure B system.

Although it was previous understood that adjusting copolymer sequence could have a significant effect on material properties through the mechanism of microphase separation, our results demonstrate that even when microphase separation is not a factor, copolymer sequence is potentially a powerful tool for tuning properties of nanocomposite systems. Although nearly all real copolymer systems have monomer– monomer interactions that depend on monomer type, the effect of sequence on copolymer– nanoparticle interactions that we studied here will also be present. Thus, we expect that a better understanding of this mechanism will be useful when designing new materials. Toward this goal, in the next chapter, we extend our analysis to analyze the effect of copolymer sequence and nanoparticle interactions on dynamic modulus and hysteresis, which are mechanical properties that have a clear macroscopic meaning and are highly relevant to material design.

57 Chapter 5: Eﬀect of Copolymer Sequence on Interphase Dynamic Modulus

Text and ﬁgures adapted and reprinted with permission from A.J. Trazkovich.;

M.F. Wendt.; L.M. Hall. Macromolecules, 2019. doi:10.1021/acs.macromol.8b02136.130

5.1 Background

One material property of signiﬁcant interest to material designers is the elastic

modulus, which measures a material’s stiﬀness and resistance to deformation under

static load. Generally, introducing nanoﬁller with favorable interactions with the

polymer matrix increases modulus, and adjusting surface chemistry to increase the

polymer–ﬁller interaction strength tends to increase modulus further.27,80,111 Several

analytical theories exist to predict composite modulus, often using three-phase mod-

els that account for the moduli of the polymer matrix, the ﬁller, and the interphase

region as well as the relative volume fractions of each.131–134 Meanwhile, in blocky

copolymer systems, it is well known that strong enough unfavorable interactions be-

tween unlike monomers may lead to microphase separation, which may in turn aﬀect

the composite modulus.135,136 However, the eﬀect of copolymer sequence on com-

posite modulus via the mechanism of modifying polymer–nanoparticle interactions

58 is much less well-understood. A signiﬁcant barrier to this understanding is that the

local modulus in the polymer–nanoparticle interphase region has been much less ex-

tensively characterized than the bulk modulus.

Elastic modulus may be measured locally using atomic force microscopy, and ex-

perimental studies have used this technique to demonstrate that modulus increases

in the interphase between polymer and hard surfaces, provided that the polymer

has good aﬃnity for the surface.137–139 Static modulus may also be measured locally in simulation, where a local sum of interatomic forces can be used to calculate the

“Born term,” the theoretical instantaneous elastic modulus that would be experienced in response to a uniform, aﬃne, inﬁnitesimal strain.140,141 Alternatively, the

second derivative of free energy with respect to strain can by used to derive a more

complex formulation of the local modulus which is the sum of a Born term, a kinetic

energy term, and a stress ﬂuctuation term that accounts for nonaﬃne local particle

movement. A substantial body of work has developed and used this technique to

study molecular scale mechanical heterogeneities in polymer glasses and crystalline

materials.142–145

Although these simulation studies have provided a clear method to measure local

static modulus, and meanwhile experimental studies have shown that static modulus

increases near nanoparticles, none of this particular body of work considered local

dynamic modulus. Dynamic modulus, which varies as a function of the frequency of

applied strain, can be described as the sum of a storage modulus and a loss modulus,

which respectively are the components of the response in phase and out of phase with

the applied oscillatory strain. The ratio of the loss modulus to the storage modulus

gives the ratio of the amount of energy lost to the amount of energy returned in a

59 single cyclic excitation, and it is therefore a measure of the material’s viscoelastic hysteresis.

Dynamic moduli and, in particular, hysteresis, are important in any application where a polymer nanocomposite is subject to cyclic or changing loads. In ﬁlled rubbers, low hysteresis is desirable when the goal is to reduce energy loss and heat build-up, while high hysteresis is desirable in applications where the goal is to dis- sipate energy under cyclic load. In some applications, the design challenge may be even more complex, requiring low hysteresis under certain operating conditions while maintaining high hysteresis under a diﬀerent set of operating conditions. One example is the design challenge discussed in Section 1.1.1, where maintaining low rolling resistance (and therefore good fuel economy) in tire treads requires low hysteresis at the relatively low loading frequencies associated with tire rotation,65,70,71 while achieving good traction requires high hysteresis at the relatively high loading frequencies associated with tread surface deformation during sliding across road surface asperities.75,76

These types of design challenges have led material designers to employ a variety of methods to control hysteretic properties. Introducing nanofiller is known to impact energy dissipation, and material hysteresis can be either increased or decreased depending on the filler size, loading, and polymer–filler chemistry. One strategy that has been used to adjust hysteresis has been the development of tailored blocky copolymers.93,94,146,147

Unlike in nanoparticle-homopolymer composites, some monomers in copolymer chains may interact more favorably with the nanoparticle than the others.46When multiple polymer components interact near the nanoparticle surface, the resulting

60 dynamics include interesting and sometimes surprising eﬀects. For example, a significant body of work from the Akcora group has studied nanoparticles coated in highly adsorbing chains of a homopolymer with relatively high glass transition temperature

148–150 (Tg) that are dispersed in a matrix of a diﬀerent homopolymer with a lower Tg.

When the temperature is above the Tg of the bulk chains but below to Tg of the adsorbed chains, the adsorbed chains are in a glassy state and so are dynamically decoupled from the surrounding matrix. Meanwhile, at temperatures above Tg of both polymers, the adsorbed chains are mobile and so dynamically couple with the surrounding chains, dramatically increasing reinforcement. The result is a system that increases in modulus and transitions from a more liquidlike to an elastic state as temperature is increased, demonstrating one striking result that can be achieved when two polymer components with diﬀerent aﬃnities for a nanoparticle interact near a surface.

In copolymer-nanoparticle composites, as in nanoparticle-reinforced blends of homopolymers, some monomers in the system may interact more favorably with the nanoparticle than others,46 although unlike in homopolymer blends, monomers with different affinities for the nanoparticle may be present on the same chain. In these conditions, the specific copolymer sequence would be expected to affect the adsorption of the polymer chain to the filler surface. Since a crucial mechanism underlying hysteresis is the adsorption-desorption process,21,79 adjusting sequence in copolymer- nanoparticle composites potentially provides a way to significantly alter hysteresis and other viscoelastic properties. Thus, we seek to develop a better understanding of viscoelastic poperties in the polymer–filler interphase, especially in terms of their dependence on copolymer sequence.

61 Since a clear method to characterize local dynamic modulus and hysteresis is currently lacking, we now develop a new method to analyze atomic stress fluctuations and estimate the dynamic mechanical properties in the interphase, and we use this method to show that copolymer sequence has a significant effect on interphase material properties. By binning groups of atoms based on their distance from the nanoparticle, we calculate a local stress autocorrelation function. We then apply a method analogous to a method commonly used to relate the bulk stress autocorrelation to the bulk modulus. We treat the result as an estimate of the local dynamic modulus, which allows us to develop and analyze the resulting estimations of local storage modulus, loss modulus, and hysteresis as a function of copolymer sequence and distance from the nanoparticle. Although a true measurement of the local dynamic modulus is frustrated by the fact that particles move during the course of the simulation window associated with a given frequency of excitation, our results nevertheless provide perspective about the effect of copolymer sequence on the local properties in the polymer–nanoparticle interphase.

5.2 Deﬁnition of Dynamic Moduli and Relationship to Hys- teresis

In this section, we review the deﬁnitions of the various dynamic shear moduli as well as how they can be used to calculate hysteretic energy loss under a given stress or strain cycle. An ideal viscoelastic compound subject to an imposed sinusoidal shear stress s(t) will respond with a sinusoidal strain (t) oﬀset by a phase angle such that

(t) = 0 sin ωt, (5.1)

62 and

s(t) = s0 sin(ωt + δ), (5.2) where ω is the frequency of the applied stress, δ is the phase angle (by convention deﬁned as the amount by which strain lags behind stress), s0 is the maximum applied stress, and 0 is the amplitude of the resultant strain, which, in any viscoelastic material, is a function of ω. This relationship can be reformulated as follows:

s(t) = s0 sin(ωt + δ) (5.3)

= s0 cos δ sin ωt + s0 sin δ cos ωt (5.4)

0 00 = 0(G sin ωt + G cos ωt), (5.5) with

s G0 = 0 cos δ, (5.6) 0 and

s G00 = 0 sin δ. (5.7) 0 G0, known as the storage modulus, is the magnitude of the elastic (in phase) component of the response, and G00, known as the loss modulus, is the magnitude of the viscous (out of phase) component of the response. Both are functions of ω.

By deﬁnition, s0/0 is the ratio between the maximum amplitude of the input stress and the maximum amplitude of the resultant strain (or, equivalently, the relative maximum amplitude of stress that would result for a given amplitude of applied

63 strain). This is somewhat analogous to the elastic modulus, which gives the ratio of

stress to strain experienced by a material under static conditions. Solving for s0/0

from Equations 5.6 and 5.7 yields

s q 0 = G02 + G002 = |G0 + iG00| := |G∗|, (5.8) 0 where the quantity G∗ := G0 + iG00 is known as the complex modulus. |G∗|, the

magnitude of the complex modulus, is therefore an important parameter describing

the ratio of input to output responses in viscoelastic materials, and it can be calculated

directly from the storage and loss moduli.

For any viscoelastic material, the hysteretic energy loss per unit volume in one

cycle, ∆Ed, can be measured from the work expended to generate the cycle. This

can be calculated from the integral

I ∆Ed = s d, (5.9)

where the integral is performed over the period of one complete cycle. Meanwhile,

the maximum energy stored elastically in one sinusoidal cycle, ∆Es, can be calculated

according to

Z 0 1 I ∆Es = s d − s d. (5.10) 0 4

Note that the ﬁrst term of Equation 5.10 calculates the work required to deform the system over one quarter of a cycle, from zero strain to maximum strain, and the second term accounts for the portion of the work that goes into nonrecoverable hysteretic losses (each quarter cycle is fundamentally symmetric and thus generates an

64 equal portion of the overall energy loss). The remainder is therefore the recoverable,

elastic energy stored in the system at maximum deﬂection.

To determine ∆Ed in terms of moduli, we use Equations 5.1, 5.4, and 5.7 to

evaluate the integral from Equation 5.9 in the time domain, as follows:

Z 2π/ω d ∆Ed = s dt (5.11) 0 dt Z 2π/ω = (s0 cos δ sin ωt + s0 sin δ cos ωt)(0ω cos ωt) (5.12) 0

= π0s0 sin δ (5.13)

2 00 = π0G . (5.14)

while using an analogous method to evaluate Equation 5.10 yields

Z π/2ω d 1 Z 2π/ω d ∆Es = s dt − s dt (5.15) 0 dt 4 0 dt Z π/2ω 1 = (s0 cos δ sin ωt + s0 sin δ cos ωt)(0ω cos ωt) − π0s0 sin δ (5.16) 0 4 1 1 1 = s cos δ + π s sin δ − π s sin δ (5.17) 2 0 4 0 0 4 0 0 1 = s cos δ (5.18) 2 0 0 1 = 2G0. (5.19) 2 0

Often, material designers are interested in the ratio of energy lost to energy returned under cyclic deformation,

∆E 2π s sin δ d = 0 0 = 2π tan δ. (5.20) ∆Es 0s0 cos δ

65 Thus, tan(δ) is proportional† to the ratio of energy lost (viscously) to energy returned (elastically), and it is an important and commonly-used measure of hysteresis.

As a material property, tan(δ) measures how viscous versus how elastic the material is, and it can vary from 0 (in purely elastic materials) to inﬁnity (in purely viscous materials). This is further elucidated by noting that dividing Equation 5.7 by Equation

5.7 shows

G00 tan δ = , (5.21) G0 so the ratio of energy lost to energy returned is proportional to tan(δ), which is in turn proportional to the ratio between the magnitudes of the in-phase and out-of- phase components of the material response. The measurement tan(δ) is commonly used to quantify hysteresis in polymer systems, and in the analysis that follows, when we refer to “hysteresis”, we will speciﬁcally mean the value of tan(δ) unless otherwise noted.

G0, G00, |G∗|, and tan(δ) are all forms of dynamic moduli. Any two of these parameters fully describe a system’s viscoelastic response and are suﬃcient to calculate the remaining two. Each modulus gives a diﬀerent perspective regarding a material’s behavior, so our analysis in the remainder of this chapter will include all four. How- ever, we give somewhat more attention to |G∗| and tan(δ), which give, respectively, direct measurements of the magnitude and phase shift of the output response relative to a given input. In our view, these two parameters often yield the most intuitive understanding of the system dynamics.

†The constant of proportionality may change slightly if a diﬀerent deﬁnition of elastic energy is used, e.g. some works consider the average elastic energy stored instead of the maximum elastic energy stored.

66 Finally, note that when a system is subject to loading in the normal direction,

its mechanical response can be characterized by the normal moduli, which are also

referred to as the Young’s moduli. The equations for these moduli can be derived using

exactly the same procedure as was used for shear moduli in this section, although by

convention, the normal shear, normal loss, and normal complex moduli are notated

as E0, E00, and E∗, respectively.

5.3 Measurement Methods

Data was saved for analysis at every timestep in ﬁve trajectories that each ran for

100τ (10,000 timesteps) and whose initial conﬁgurations were separated by 50,000τ, well past twice the end-to-end vector autocorrelation function relaxation time. At each timestep, atomic stresses on each monomer are calculated according to

1 X siab = −mivia vib − (ria Fijb + rja Fjia ), (5.22) 2 j6=i where a and b take on the values x, y, z to calculate the six components of the

symmetric stress tensor, siab is the stress on atom i in direction ab, mi is the bead

mass, vi is the bead velocity, ri is the position of bead i, and Fij is the force that

particle j exerts on particle i. The summation is performed over all other particles in

the system, including the nanoparticle. When a = b, the stress component is tensile,

and when a 6= b, the stress component is shear.

Our goal is to estimate local dynamic modulus using atomic stresses, but no

method to do this has been well-established in prior work. However, recall that bulk

dynamic modulus can be measured from MD simulations relatively easily using ex-

tensively studied non-equilibrium or equilibrium methods. In the non-equilibrium

67 method, cyclic strain of a given frequency is applied to a simulation box and the resulting stress is measured (or vice versa), and from the magnitude and phase shift of the response, the dynamic storage and loss modulus at that frequency are determined.103,151,152 Alternatively (and more relevant to this work) dynamic modulus can also be measured from an equilibrium MD simulation using ﬂuctuation dissipation relationships. Speciﬁcally, the complex modulus is calculated from the stress autocorrelation function (SACF):

Z +∞ 0 00 V −iωt G (ω) + iG (ω) = iω e hsab(0)sab(t)i dt, (5.23) kBT 0 where sab(t) is the bulk stress at time t in direction ab, hsab(0)sab(t)i is the SACF,

V is the system volume, and kBT is the thermal energy. Here, a Fourier transform has been used to decompose the complex modulus into the storage (G0) and loss (G00) moduli, which are functions of the frequency of excitation, ω. Compared to non- equilibrium methods, this method is more computationally eﬃcient since the modulus can be calculated for a continuous range of frequencies from a single simulation.

This technique has been used to compute dynamic moduli for a variety of polymer systems.153–159

The works referenced above measured the bulk SACF and calculated the bulk modulus. In contrast, our goal is to estimate local dynamic modulus as a function of distance from the nanoparticle. To do this precisely, it would be necessary to have a measurement of the SACF that could be deﬁned locally and would vary as a function of position. However, it is not obvious how to precisely deﬁne and calculate the

“local SACF” because the SACF is a function of time, and atoms are not stationary in time. Here, we have developed a method to approximate the SACF on a local scale

68 by dividing the system into concentric spherical shells centered on the nanoparticle

and calculating hs(0)s(t)iς , an estimate of the average stress autocorrelation function

of monomers in shell ς. The result is then used with Equation 5.23 to estimate the dynamic storage and loss modulus for each individual shell.

Speciﬁcally, for each shell and at each timestep t0, the stress on each bead, i,

whose center is in the shell at t0, is calculated according to Equation 5.22. Then

0 0 a stress autocorrelation si(t )si(t + ∆t) is calculated for each possible time window

length ∆t from 0.01τ to 8τ (1 to 800 timesteps). At each starting t0, this quantity is

0 0 averaged over all monomers in each shell, yielding [s(t )s(t + ∆t)]ς , the average stress

0 0 0 dissipation for atoms initially residing in shell ς at time t . [s(t )s(t + ∆t)]ς is further

0 averaged across all possible times, t , yielding hs(0)s(∆t)iς for each shell. Therefore,

0 0 our estimate of hs(t )s(t + ∆t)iς is the average of 10, 000 − ∆t/δt subwindows within

each trajectory. The results are further averaged across the ﬁve trajectories, ensuring

that the system is sampled at several conﬁgurations that are separated by timescales

longer than the end-to-end vector relaxation time.

Critically, atoms assigned to each shell are not necessarily the same at each au-

tocorrelation window start timestep, and moreover, atoms may move between shells

during the course of the autocorrelation time window, possibly leaving the shell to

which they were assigned. Therefore, this formulation of the SACF measures the

average stress dissipation of atoms that initially reside in a given shell, meaning that

it is an approximation of the true “local SACF”. As discussed above, some level of ap-

proximation is unavoidable; the SACF is necessarily measured using information from

particles that move over time and therefore does not have a precise local meaning.

To reduce the degree of approximation, the longest autocorrelation time window we

69 consider is ∆t = 8τ (800 timesteps). This time was selected such that the square root of the monomer mean squared displacement is less than 1.0, so the average monomer does not move signiﬁcantly more than the width of a single shell during the course of a single autocorrelation time window (over suﬃciently long timescales, our approximation of the local SACF would converge to the bulk SACF regardless of the location of the shell since the monomers would eventually explore the entire simulation box regardless of initial location). Bearing these caveats in mind, we hereafter refer to this approximation of the local SACF as the LSACF.

Since the system is isotropic, the three tensile components of the stress tensor are eﬀectively equivalent. Therefore, to improve the statistics of the measurement for each shell, we average the LSACF of the three on-diagonal (normal) components of the stress tensor. By the same logic, we also average the LSACF of the three oﬀ diagonal (shear) components. After this averaging, the LSACF is still quite noisy; other researchers have noted the bulk SACF is inherently noisy when calculated from

MD simulation.154 To further reduce the noise level, we apply a running average filter similar to the one applied by Sen et al., such that the final LSACF data at each time t is the average of all raw data points from 0.8∆t to 1.2∆t. This filter uses a wider time window than Sen and colleagues since our system is divided into shells and therefore has fewer atoms contributing to each ensemble. The effect of this filter on the magnitude of the final results is small, and a comparison between results obtained with and without the filter can be found in Section 5.3.2.

The ﬁnal challenge with our analysis arises from that fact that we truncate the autocorrelation data at 8τ. As a result of taking a Fourier transform of this ﬁnite time window with a discontinuity at its end, the calculated storage and loss moduli exhibit

70 a substantial amount of ringing. This effect could be reduced by lengthening the time window considered; however, we avoided this for two reasons: first, as discussed above, using a longer time window necessarily makes the LSACF a less localized description of stress fluctuations, and second, since the stresses on and location of every atom must be recorded and analyzed at every timestep, the data files required to perform this analysis become prohibitively large to store and analyze for more than an order of magnitude or two beyond 8τ. Instead, after filtering and prior to applying the Fourier transform, we artificially extend the raw LSACF data by performing a power law fit on the existing data above 1τ and then extrapolating the data out to

1000τ. This has the eﬀect of nearly eliminating the ringing in the dynamic modulus while leaving the overall magnitude relatively unchanged in the frequency range of interest, (which is above 2π/8 rad/τ and thus is primarily inﬂuenced by the LSACF data from times shorter than 8τ). Section 5.3.2 contains a comparative example of dynamic modulus data calculated with and without extending the raw LSACF data.

To characterize the mechanical properties of the interphase, we divide the system into a series of concentric shells centered on the nanoparticle, each of which has a thickness of 0.5σ. We apply the procedure described above to estimate the storage and loss modulus of each shell, and we also calculate the bulk storage and loss modulus in the standard fashion using the bulk stress autocorrelation data with Equation

5.23. We calculate both the shear modulus (from the oﬀ-diagonal component of the stress tensor) and the normal modulus (from the on-diagonal component of the stress tensor). From the storage modulus, G0, and the loss modulus, G00, we also calculate the magnitude of the complex modulus (|G∗|), and tan(δ) from Equations 5.8 and

5.21.

71 Before reporting detailed results (Section 5.4) for each of the systems studied, the following two sections provide additional analysis to contextualize those results.

In Section 5.3.1, we report on the dramatic inﬂuence of FENE bond interactions on the moduli calculated at high frequencies, and in Section 5.3.2, we demonstrate the impact of ﬁltering and extending the raw LSACF data (according to the procedure described above) on the modulus results.

5.3.1 Inﬂuence of Bond Vibrations on Modulus Results

Figure 5.1 reports |G∗| and tan(δ) for several shells around the nanoparticle in the

B homopolymer system as well as for the bulk (calculated from the standard SACF data of the entire system). These results will be discussed in greater detail in Section

5.4. Of primary interest here is the data above 20 rad/τ, where both the |G∗| and tan(δ) results exhibit significant fluctuations. Other researchers154 have attributed these fluctuations at high frequencies to FENE bond vibrations.‡

To test the hypothesis that the results above 20 rad/τ are heavily inﬂuenced by bond interactions, we have performed on additional simulation in which the FENE bonds are removed at the end of the NVE equilibration phase in a homopolymer

B system, producing a Lennard-Jones ﬂuid that otherwise retains the (non-bonded) monomer–monomer and monomer–nanoparticle interactions used in the bonded polymer simulations. After removing the bonds, the system was equilibrated for an additional 50,000τ before saving data for analysis. The results are shown in Figure

5.2.

‡The negative results in the tan(δ) data occur when δ exceeds π/2, causing tan(δ) to suddenly transition from large positive to large negative values. Further detail regarding exactly where and why this occurs is beyond the scope of this work, although we do note that δ never exceeds π/2 by more than 0.1%.

72 (a) (b)

Figure 5.1: Complex modulus (a) and hysteresis (b) of the B homopolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text.

(a) (b)

Figure 5.2: Complex modulus (a) and hysteresis (b) of the B monomer system with bonds removed as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the main text.

Note that the dramatic features above 20 rad/τ disappear when bonds are removed. However, the hysteresis peak between 4 and 20 rad/τ that occurs in shells near the nanoparticle is still present even in the system without bonds, suggesting

73 that this feature is primarily a result of monomer–nanoparticle interactions. For this reason, we have chosen to focus our investigation in Section 5.4 on this peak and leave analysis of the heavily bond-inﬂuenced features for future work.

5.3.2 Eﬀect of Pre-Processing Data

In Section 5.4, we will report viscoelastic properties as calculated from LSACF data that has been extended and ﬁltered according to the procedure described in

Section 5.3. To contextualize the results in that section, here we demonstrate the impact of this pre-processing on the homopolymer B system, as shown in Figure 5.3.

Figures 5.11a and 5.11b compare moduli calculated from raw LSACF data to moduli calculated from filtered and extended LSACF data. These results are dominated by a ringing with constant period in the (linear-scale) frequency domain that is consistent with taking a Fourier transform of a finite time window. On average, the results calculated from the extended and filtered LSACF data are slightly lower in magnitude than the results calculated from the raw LSACF.

Figures 5.3c and 5.3d consider moduli that are calculated from LSACF data that has been filtered but not extended. The results are still dominated by a ringing, but the average magnitude appears much more similar to the results calculated from extended and filtered LSACF data, suggesting that the filter may have been responsible for the slight reduction in magnitude that was seen in Figures 5.11a and 5.11b.

Figures 5.3e and 5.3f report results calculated from LSACF data that is extended but not ﬁltered. The ringing that dominated the results in Figures 5.3a-d is nearly eliminated, strongly suggesting that extending the LSACF data is responsible for

74 (a) (b)

(e) (f)

Figure 5.3: Complex modulus (a, c, e) and hysteresis (b, d, f) of the B homopolymer system with bonds removed as a function of distance from the nanoparticle, r. Monomers are assigned to shells as described in the text. Dashed lines report moduli calculated from LSACF data that is extended and filtered according to the procedure in Section 5.3. Solid lines report moduli calculated from raw LSACF data (a, b), LSACF data is filtered but not extended (c, d), and LSACF data that is extended but not filtered (e, f).

75 the change. The results from the extended but unﬁltered data also exhibit signiﬁ-

cantly more noise than the results calculated from extended and ﬁltered LSACF data,

demonstrating the positive impact of applying the ﬁlter.

5.4 Results

We now consider detailed results for several of the copolymer systems studied.

The results which follow will focus primarily on the systems’ behavior under shear,

although we will devote some analysis to ways in which the behavior diﬀers in re-

sponse to normal loading. Additionally, note that while the Figures 5.4 through 5.8

report results for all moduli that were calculated, the text comments primarily on

the magnitude of the complex modulus and tan(δ) since, combined, these contain

fundamentally the same information as the storage and loss moduli.

Figure 5.4 reports the various moduli as well as the time-domain LSACF data for

several shells around the nanoparticle in the B homopolymer system as well as for

the bulk (calculated from the standard SACF data of the entire system). For clarity,

we have truncated the frequency-domain plots on the high end at 20 rad/τ because

above this frequency, the results are dominated by FENE bond interactions, as was

demonstrated in Section 5.3.1. The truncation at 1 rad/τ is close the the minimum frequency that can be calculated from the window period of 0.8τ.

Measurement error was estimated by simulating five separate homopolymer B systems (each of which began equilibration at a different random initial configuration) and then performing the entire measurement process, including all stages of averaging, on each system. We then calculated the coefficient of variation (CV, the standard deviation divided by the mean) of the resulting datasets of five measurements in

76 (a) (b)

(e)

Figure 5.4: Magnitude of complex modulus (a) hysteresis (b), storage modulus (c), loss modulus (d), and time-domain LSACF data (e) of the B homopolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data.

77 each shell and at each frequency. The CV was found to be relatively independent of frequency and whether the property considered was tan(δ) or |G∗|; however, it did depend on shell, since the number of monomers contributing to the statistics of each shell depends on both the radius of the shell and the structural variations in local monomer density near the nanoparticle. The CV was approximately 3% in the shell nearest to the nanoparticle, 8% in the second shell, and 5% in the remaining three shells considered. The same process was also performed with respect to the bulk measurements, for which the CV was approximately 2%. Due to the computational cost of simulating each system four additional times, we did not estimate measurement error for the remaining systems, although the data for all systems appeared comparably smooth.

Generally, modulus increases with frequency and proximity to the nanoparticle, with the closest shell being approximately eight times stiﬀer than the bulk. The degree of mechanical reinforcement (the ratio of shell modulus to bulk modulus) is relatively independent of frequency. As expected, shells farther from the nanoparticle gradually approach the properties of the bulk, providing some evidence that the LSACF approach used in this work yields a good estimate of local viscoelastic properties.

The tan(δ) results show a substantial hysteresis peak that also increases with proximity to the nanoparticle. The peak in tan(δ) corresponds to a relative valley in the storage modulus rather than a relative peak in the loss modulus (which is qualitatively similar to the complex modulus). This peak occurs at approximately 9 rad/τ, a frequency that is relatively independent of nanoparticle proximity. We note that this frequency is very close to the reciprocal of the average time required for an

78 adsorbed monomer (whose center starts in the shell nearest to the nanoparticle) to

−1 move 1σ (approximately the same as the FS(k, t) relaxation time for |k| = 2πσ , as reported in Figure 4.11), which is the approximate spacing between peaks in the monomer–nanoparticle pair correlation function (Figure 4.4a). In a statistical dynamic process, hysteresis is maximized when the input frequency is close to the reciprocal of the process relaxation time.160 Therefore, we believe that the observed hysteresis peak is caused by the monomer–nanoparticle adsorption-desorption process wherein monomers move between the shell adjacent to the nanoparticle surface and the next closest structural peak 1σ away. Based on our discussion in Section

2.1 that τ > 2 × 10−10 s for a system at room temperature with length and mass scales similar to polybutadiene, the tan δ peak would occur at a frequency of less than 7 × 108 Hz. Experimentally, loss peaks have been measured around ∼106 Hz in polybutadiene-containing copolymers somewhat below room temperature using dielectric spectroscopy by Vo et al.120 The authors of that work attributed their loss measurements to relaxations in the polymer–nanoparticle interphase, so it is plausible that we are measuring a similar phenomenon.

For comparison, Figure 5.5 reports the same mechanical properties as function of distance from the nanoparticle in the A homopolymer system. Although the trends in |G∗| are similar, the homopolymer A system exhibits significantly less mechanical reinforcement than the homopolymer B system. Despite the fact that the monomer– nanoparticle interaction is slightly unfavorable relative to bulk monomer–monomer interactions, modulus still increases with proximity to the nanoparticle, likely because the hard surface still has a small stiffening effect on the local polymer. Relative to

79 the homopolymer B system, the interphase of increased modulus appears to be about

1σ thinner.

From Figure 5.5b, we see that the hysteresis follows the opposite trend from the modulus, actually decreasing slightly near the nanoparticle. Note also that this decrease occurs around the same frequencies where tan(δ) increased in the homopolymer

B system.

Next, we consider the copolymer systems, beginning with BL = 1, for which

interphase mechanical properties are reported in Figure 5.6. In this system, the

trends fall between those of the two homopolymer systems. Modulus increases more

sharply with proximity to the nanoparticle than in the homopolymer A system but

less sharply than in the homopolymer B system. The interphase hysteresis exhibits

a peak around the same frequency as in the homopolymer B system, but the peak is

lower and the interphase of increased modulus is narrower than in that system. The

fact that the dynamic mechanical properties of the BL = 1 system fall somewhere

between the homopolymer A and homopolymer B systems is fairly intuitive since the

chains are composed of alternating A and B monomers.

The interphase properties of the BL = 2 system, which are reported in Figure

5.7, are fairly similar to the BL = 1 system, with the notable exception that peak

hysteresis in the closest shell is much higher for BL = 2 than for BL = 1. Interestingly,

this hysteresis peak is signiﬁcantly higher than even the peak observed in the closest

shell of the homopolymer B system. This result is especially striking in light of

the fact that the homopolymer B system contains 50% more adsorbing monomers

than the BL = 2 system (although similar eﬀects for other molecular properties were

previously observed in Chapter 4).

80 (a) (b)

(e)

Figure 5.5: Magnitude of complex modulus (a) hysteresis (b), storage modulus (c), loss modulus (d), and time-domain LSACF data (e) of the A homopolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data.

81 (a) (b)

(e)

Figure 5.6: Magnitude of complex modulus (a) hysteresis (b), storage modulus (c), loss modulus (d), and time-domain LSACF data (e) of the BL = 1 copolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data.

82 (a) (b)

(e)

Figure 5.7: Magnitude of complex modulus (a) hysteresis (b), storage modulus (c), loss modulus (d), and time-domain LSACF data (e) of the BL = 2 copolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data.

83 To contrast with the previous two copolymer systems, which both contained very

short blocks, we now consider the BL = 25 system, in which chains are composed of

much longer blocks. The interphase dynamics of the BL = 25 system are reported in

Figure 5.8.

Overall, the trends for the BL = 25 system are fairly similar to those of the homopolymer B system. Both the modulus and peak hysteresis of the interphase are signiﬁcantly higher in the BL = 25 system than in the BL = 1 system or in most shells of the BL = 2 system, although hysteresis in the closest shell is lower than for BL = 2.

The width of the interphase of increased peak hysteresis is also greater for BL = 25.

Also of note is that the peak hysteresis of the two closest shells is higher than that of the homopolymer B system (although this diﬀerence is statistically signiﬁcant only in the closest shell).

The results for the other copolymer systems are qualitatively similar to those of the BL = 1, BL = 2, and BL = 25 systems, so we will not reproduce all of that data here. Instead, to understand the quantitative effect of blockiness in sequence, we focus on the two features that appear to most significantly differentiate the systems: each shell’s peak hysteresis and each shell’s degree of dynamic mechanical reinforcement relative to the bulk. We calculate peak hysteresis simply by recording the maximum tan δ in the studied range, and we calculate dynamic mechanical reinforcement by choosing a frequency and taking the ratio between the magnitude of the shell’s complex modulus and the magnitude of the bulk complex modulus. The results for mechanical reinforcement are relatively independent of the choice of frequency, and we select 9 rad/τ because it is close to the frequency at which the peak hysteresis

84 (a) (b)

(e)

Figure 5.8: Magnitude of complex modulus (a) hysteresis (b), storage modulus (c), loss modulus (d), and time-domain LSACF data (e) of the BL = 25 copolymer system as a function of distance from the nanoparticle, r. Shells are bounded as labeled, and monomers are assigned to shells as described in the text. Dotted lines in the LSACF data show the extension appended to the data.

85 occurs. In Figure 5.9, we plot peak hysteresis and reinforcement for each shell as a

function of distance from the nanoparticle.

(a) (b)

Figure 5.9: Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for the various copolymer and homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries).

Generally, the homopolymer A system exhibits the lowest interphase reinforcement, the homopolymer B system exhibits the highest, and the copolymer systems fall in between. In the closest shell, reinforcement is relatively independent of copolymer sequence, with all copolymer systems having reinforcement similar to the homopolymer B system. In the second and third shells, reinforcement usually increases slightly with block length, with the random copolymer system falling close to behavior of the BL = 5 and BL = 10 systems. There may be some correlation between the reinforcement and the percentage of B monomers residing in a given shell, as was reported in Figure 4.4b. The rank order of reinforcement in the ﬁrst three shells is very similar to the rank order in the percentage of B monomers residing in those shells,

86 and systems with a similar concentration of B monomers generally exhibit similar

reinforcement. However, it is hard to make a ﬁrm conclusion that modulus increases

with local B monomer density since, other than the systems with the shortest block

lengths, there is relatively little diﬀerence in the reinforcement between the diﬀerent

copolymer systems.

In contrast, the peak hysteresis varies signiﬁcantly between copolymer systems.

In the closest shell, all copolymer systems except the BL = 1 system exhibit more hysteresis than even the homopolymer B system, and hysteresis tends to decrease with increasing block length. This rank order largely reverses in the second shell, with hysteresis tending to increase with block length and all of the copolymer systems having interphase properties between that of the homopolymer systems, with the exception of the BL = 25 system. In contrast to the reinforcement results, these trends are not easily explained by local monomer composition. To explain this phenomenon, we note that, in all the copolymer systems, the region nearest the nanoparticle always has a higher relative concentration of B monomers (above 90% except in the BL = 1 system), and this B-dominant region is, in turn, surrounded by a region that has a higher relative concentration of A monomers. The longer the block length, the wider the initial B-rich region and the more gradual the B–A transition. When the B–A transition is sharp, it impacts the mobility of B monomers very near the nanoparticle surface, since it is energetically less favorable to swap an adsorbed B monomer with a nearby A monomer that it is to swap it with a nearby B monomer. Meanwhile, since a hysteresis peak is evident in the hompolymer B but not in the homopolymer A system, it is reasonable to conclude that hysteresis also increases to some degree with the relative concentration of B monomers in a copolymer system. We speculate that,

87 very near the nanoparticle (where monomers are directly adsorbed to the surface), the

eﬀect of the sharpness of the B–A transition, which tends to increase with decreasing

block length, dominates the hysteresis. Slightly farther from the nanoparticle, the

relative concentration of B monomers, which tends to increase with block length,

may dominate hysteresis. This may explain why the trend in hysteresis reverses with

respect to block length from the ﬁrst to second shell.

Some mechanisms contributing to this trend may be related to the results from

the Akcora group,148–150 which studied blends of adsorbing and nonadsorbing polymer. Notably, a portion of that work examined the eﬀect of adjusting adsorbed chain length, and their results indicate that in suﬃciently-loaded composites bulk hysteresis was higher in systems with shorter adsorbed chains,148,149 suggesting that shortening adsorbing chain length increased hysteresis in the polymer–nanoparticle interphase.

Although the authors do not directly discuss this eﬀect in terms of hysteresis, Senses and colleagues149 do comment that shorter adsorbed chains experience fewer interactions with nonadsorbed chains, thus decreasing interphase storage modulus. As we noted above, the tan(δ) peaks that we observed correspond mathematically to a relative reduction in storage modulus, rather than a change in the loss modulus, so it is possible that mechanism discussed in Senses et al. is relevant to our results. On the other hand, in Yang et al.148 decreasing adsorbed chain length actually increased

composite storage modulus, although tan(δ) still increased slightly because the loss

modulus increased as well. The authors of the latter work attributed the observed

eﬀect to increased mobility of the shorter adsorbed chains, which they speculated

allowed the chains to experience stronger dynamic coupling with the surrounding

88 matrix. Note that Senses and coworkers studied PMMA-adsorbed nanoparticles dispersed in PEO, while Yang et al. studied PVA-adsorbed nanoparticles in PEO, so the different effects observed are likely related to the chemical differences between the two systems.

Although the increase in hysteresis observed when shortening adsorbed chain lengths in the works discussed above may be analogous to our results that show hysteresis increasing with reduced adsorbed block length, we do note that since the prior work studied homopolymer blends whereas we study copolymers, some of the relevant dynamic mechanisms may be diﬀerent. In our case, segments of adsorbed monomers are necessarily coupled to nonadsorbing segments via bonds, which does not happen in blend systems. However, adsorbed segments in our systems may also couple dynamically with other segments via non-bonded interactions, a mechanism that is more relevant to the prior work on blends.

We have performed the same analysis of viscoelastic properties on the normal modulus results, which characterize a system’s response to normal stress. These results are reported in Figure 5.10. The relative reinforcement in terms of normal modulus of the interphase compared to the bulk is quantitatively very similar to the shear modulus results. However, though not reported in this ﬁgure, we do note that the absolute magnitude of the bulk normal modulus (and hence the magnitude of any given shell) is close to three times greater than that of the bulk shear modulus, which is consistent with a Poisson’s ratio close to 0.5.

In contrast to the reinforcement results, there are some notable diﬀerences in the peak hysteresis under normal stress compared to under shear stress. When subject to normal stress, the interphase region dissipates less than half as much energy as

89 (a) (b)

Figure 5.10: Dynamic mechanical reinforcement under normal stress at 9 rad/τ (a) and peak hysteresis under normal stress (b) for the various copolymer and homopolymer systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries).

when it is subject to shear stress. Farther from the nanoparticle, this diﬀerence is reduced, and it is least prominent in the polymer bulk. The qualitative trends in the peak hysteresis are relatively similar under normal and shear stress, although this is proportionally less variation between the systems in the shell nearest to the nanoparticle surface under normal stress compared to under shear.

As in earlier chapters, we also examine the apparent interphase width, which is known to depend on the molecular property used to deﬁne it.34,37 Here, we will deﬁne the interphase as extending from the nanoparticle surface (at 5σ) out to the edge of the last shell before the property of interest drops to within 10% of the bulk value. In terms of reinforcement, all the copolymer systems studied have the same interphase width of approximately 2.5σ, as does the homopolymer B system, but the homopolymer A system’s interphase is approximately 0.5σ narrower. In the case of the peak tan(δ), the BL = 1 and homopolymer A systems have interphases

90 approximately 1.0σ wide, the BL = 2 system has an interphase approximately 1.5σ

wide, and the interphase width of the remaining systems is approximately 2.5σ in response to shear stress and 2.0σ in response to normal stress.

Although not our focus in this investigation, we have also measured dynamic moduli in the homopolymer systems that were the focus of Chapter 3. Results are qualitatively similar to the A homopolymer (εNP = 1) and B homopolymer (εNP = 5) systems considered above. To compare the systems quantitatively, we use the same procedure and for the copolymer systems, reporting relative reinforcement and peak hysteresis in Figure 5.11.

(a) (b)

Figure 5.11: Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for each of the homopolymer systems studied, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries). Data points that lie outside the reported bounds are cases where δ is very close to π/2 (the system behaves nearly purely viscous), and tan(δ) is greater than 1,000.

In the homopolymer systems, reinforcement and peak hysteresis generally in-

creases with εNP and with proximity to the nanoparticle. The exceptions are for

91 εNP = 2, where peak hysteresis is relatively unchanged by proximity to the nanopar-

ticle, and for εNP = 1, where peak hysteresis decreases slightly with proximity to the nanoparticle, as was noted in the discussion of the A homopolymer system, above.

Above a threshold near εNP = 5, increasing εNP no longer appears to have a signiﬁcant

eﬀect on reinforcement within the range studies. Interestingly, hysteresis continues to

increase sharply with εNP beyond this threshold, showing no signs of having reached

a maximum by εNP = 8.

In terms of reinforcement, the interphase width of the εNP = 1 system is approx-

imately 2.0σ, and for all the other homopolymer systems, it is approximately 2.5σ.

In terms of peak hysteresis, all the homopolymer systems with εNP ≥ 4 have interphases that are 2.5σ wide, while the εNP = 1 and εNP = 3 systems have interphases approximately 1.0σ wide and the εNP = 2 system exhibits no signiﬁcant interphase.

5.5 Discussion

Using a localized version of the stress autocorrelation function, we have estimated

local dynamic modulus and hysteresis as a function of distance from a nanoparticle

in a simple coarse-grained simulation of a polymer nanocomposite. The method to

estimate local dynamic mechanical properties that was developed for this work has

the potential to be used for a large variety of heterogenous MD systems. However, it

is important to bear in mind that the validity of the estimation will be limited in cases

where the particles under consideration are able to move on length scales similar to the

discretization of the simulation box within the timescales considered. Therefore, the

method will yield more valid estimates at either high frequencies on in systems where

particle mobility is limited, such as glasses, highly-crosslinked elastomers, or crystals.

92 We also note that in this work, we benefited from the inherent symmetry of the system about the nanoparticle. Had we attempted to collect statistics on individual cubic subdomains with side lengths close to the width of our shells, the problem of collecting sufficient statistics would likely have been intractable. Therefore, in systems lacking any inherent symmetry, there will in some cases be a practical limit on how fine a scale this method can resolve heterogeneities.

By examining a series of simple AB copolymer sequences where the A and B monomer diﬀer only in that one type adsorbs more strongly to the nanoparticle, we have shown how copolymer sequence can have a dramatic impact on the interphase modulus even while the overall concentration of A and B monomers remains equal.

Interestingly, certain copolymer sequences with 50% adsorbing monomers can produce higher interphase hysteresis than even a system composed of 100% adsorbing monomers. This suggests that adjusting copolymer sequence can be a powerful tool for tuning interphase properties and, therefore, for tuning overall composite properties. Overall, these results provide a more complete understanding of the mechanisms that govern mechanical properties near copolymer–nanoparticle interfaces.

Ultimately this will inform new design strategies that seek to tune nanocomposite properties using copolymer sequence. Toward this end, in the next chapter, we will attempt to address the example design challenge in tire treads that was introduced in

Section 1.1.1. Speciﬁcally, we will examine the tan(δ) curve in more detail and simulate an additional series of copolymer sequences. These additional sequences aim to determine if it is possible to further increase hysteresis at high frequencies while preserving hysteresis at low frequencies. Additionally, the results in the next chapter

93 help us test our hypothesis that hysteresis very close to the nanoparticle depends on the sharpness of the B–A phase transition.

94 Chapter 6: Adjusting Copolymer Sequence to Modify Hysteresis

6.1 Motivation

In Section 1.1.1, we discussed a design challenge in tire tread compounds in which increasing traction typically requires increasing hysteresis at high frequencies, while decreasing rolling resistance to improve fuel economy typically requires reducing hysteresis at lower frequencies. Tire traction is often modeled as an integral of hysteresis across all frequencies of excitation experienced by a tire tread sliding across road surface asperities.76 Since road surfaces are typically fractal in nature,161 the hysteresis

across a wide range of frequencies between approximately 102 and 108 Hz contributes

to traction, and increasing hysteresis in any part of this range is usually beneﬁcial.

Meanwhile, although rolling resistance is a simpler phenomenon involving the tire

being deformed at only a single frequency at a time during free rolling, typical driv-

ing behaviors involves a range of tire rotation speeds between 100 and 102 Hz, so

decreasing hysteresis in any part of this range improves overall fuel economy.

As in the simulated systems covered in the previous chapter, experimental polymer

nanocomposites tested at a given temperature typically exhibit a notable hysteresis

peak at a particular frequency. This frequency is strongly related (inversely) to the

95 glass transition temperature (Tg) via the principle of time-temperature superposition.

Tire designers typically seek to optimize tire performance in a given temperature range

(summer or winter) by adjusting the glass transition temperature of tire composites so as to place the hysteresis peak within the frequency range associated with traction.

The Tg of polymer nanocomposites can be adjusted using several methods, and, in most circumstances, it is not overly challenging to adjust Tg in tire materials (most often by adjusting the relative ratios of styrene and butadiene used in the SBR).

However, a more formidable challenge arises since, as discussed above, the frequency ranges associated with traction and rolling resistance are essentially adjacent, meaning that when the hysteresis peak is placed in the traction region, it may also increase the hysteresis in the nearby rolling resistance region.

Therefore, optimizing the tradeoﬀ between traction and rolling resistance can be framed as the need to shape the hysteresis peak such that, when moving from low to high frequencies, the hysteresis rises as sharply as possible and as high as possible relative to the low-frequency baseline. Meanwhile, increasing peaks hysteresis should ideally not come at the cost of increasing hysteresis at frequencies slightly below the peak.

Note that while it is relatively easy to increase or decrease hysteresis across a very wide range of frequencies (most commonly in tire treads by adjusting crosslink density162), it is more diﬃcult to adjust the hysteresis curve such that hysteresis increases only in a narrow region while preserving or decreasing hysteresis slightly below the peak. Some prior innovations have made progress in this area; for example, the incorporation of silica nanoparticles in addition to carbon black has been shown

96 to decrease low-frequency hysteresis and sharpen the slope of the hysteresis peak as a function of increasing frequency.71,163

In this chapter, we will examine the tan(δ) peaks calculated from the systems considered in Chapter 5. We then simulate a small series of new sequences related to the sequences found to have the most promising hysteresis, and we seek to use these new sequences to increase peak hysteresis as high as possible while preserving hysteresis at frequencies slightly below the peak. Our exploration necessarily covers only a tiny fraction of possible copolymer sequences, with further exploration being planned for future work.

6.2 Comparison of Hysteresis in Original Systems

To provide direction in our search for sequences that improve the low- to high- frequency tradeoff, we first compare the tan(δ) peaks of each of the systems that were explored in Chapter 5. Here, we will focus primarily on hysteresis in the two shells closest to the nanoparticle because, in most systems, the hysteresis peak in the third shell is too low for comparative analysis. Hysteresis in each of the first two shells is compared in Figure 6.1. For brevity, in the analysis that follows, we will refer to the shell closest to the nanoparticle as the “first” shell, and the next closest shell as the

“second” shell.

Among the copolymer systems, peak hysteresis varies dramatically between sequences, but hysteresis in all the systems is nearly the same at frequencies lower than a factor of 2 below the frequency at which the hysteresis peaks. This is advanta- geous for our goal of determining promising sequences for improving tire properties.

Therefore, in the sections that follow, we will seek to maximize the height of the

97 (a) (b)

Figure 6.1: Hysteresis in the two closest shells to the nanoparticle, 5.5σ < r < 6.0σ (a) and 6.0σ < r < 6.5σ (b), for the various copolymer and homopolymer systems considered in Chapters 4 and 5, as labeled.

peak, thereby maximizing hysteresis in the frequency range relevant to traction. We note that, from the perspective of our motivating design challenge, it makes sense to compare the blocky copolymer systems to the random copolymer system. This is because random SBRs are the most common and most economical SBRs used in tires, so any system which has hysteresis lower than a purely random system is likely not worth considering for incorporation into treads.

In the second shell, both the BL = 25 and the B homopolymer systems exhibit higher peak hysteresis than the random copolymer system, albeit by a fairly small margin. In the ﬁrst shell, among the blocky copolymer systems studied so far, only the BL = 2 system exceeds the peak hysteresis of the random copolymer system, and it does so by a margin that is not statistically signiﬁcant. Interestingly, in the BL = 2 system, the hysteresis peak occurs at a lower frequency than in any other system studied. As discussed in the previous section, this is not of particular concern since

98 the frequency at which peak hysteresis occurs can typically be adjusted using other

methods.

6.3 Investigating Additional Sequences

6.3.1 Triblock and New Regular Multiblock Copolymers

In an attempt to ﬁnd sequences that increase peak hysteresis beyond that of the

random copolymer system (while still preserving low-frequency hysteresis), we will

analyze several new sequences motivated by the BL = 2 system, which was seen in

the previous section to have performance most comparable to the random copolymer

system. In Section 5.4, we theorized that the reason hysteresis is typically higher in

the multiblock copolymer systems with shorter blocks is that there is a sharper phase

transition near the nanoparticle, and the presence of an A-rich region just outside

the adsorbed B-rich region impacts monomer mobility and increases the hysteretic

losses associated with monomers moving between two adjacent shells close to the

nanoparticle. Therefore, in an attempt to maximize this eﬀect, we introduce a series

of new copolymer sequences in which short B blocks of deﬁned length are attached

to either end of a single long A block such that the total chain length is equal to

100. Thus, these new sequences are triblock copolymers with form BxAyBx, where

y = 100 − 2x, and x is the length of the terminal B blocks, set to 1, 2, 3, 4, 5, or

10 depending on the system. We refer to these new systems as BE = x. The goal of these sequences is to create as sharp a B–A phase boundary as possible near the nanoparticle in order to test the hypothesis that the sharpness of the phase boundary impacts hysteresis.

99 In parallel, we also examine two new regular multiblock copolymer systems with

block lengths of 3 and 4, reasoning that if peak hysteresis increases from the BL = 1

system to the BL = 2 system before decreasing again when the block length is in-

creased to 5, then it is possible that peak hysteresis actually reaches a maximum at a

block length of 3 or 4. In order to preserve the basic structure of our other multiblock

copolymer systems, all of which have an equal number of A and B blocks of equal

length, we increase the chain length to 102 in the system with a block length of 3

and to 104 in the system with a block length of 4. We refer to these two systems

∗ ∗ as BL = 3 and BL = 4 , including asterisks to serve as a reminder that the chain lengths diﬀer slightly from the multiblock copolymer systems introduced earlier.

A schematic of the additional copolymer sequences introduced in this section is shown in Figure 6.2. For brevity, we will hereafter refer to the regular multiblock copolymer systems and the triblock copolymer systems collectively as the “BL systems” and “BE systems”, respectively.

Figure 6.2: Schematic of the additional copolymer block sequences considered in this chap- ∗ ∗ ter. Chain length is 104 in the BL = 4 system, 102 in the BL = 3 system, and 100 in the triblock copolymer (BE) systems.

100 Before reporting hysteresis for the additional sequences, we will provide additional

context by brieﬂy characterizing the new systems using measures that were previously

hypothesized to be related to hysteresis. Figure 6.3 reports PB, the local percentage

of B monomers, as a function of distance from the nanoparticle for the new systems

as well as reference systems from earlier chapters. Note that PB is now calculated

as gBN(r)pB/(gAN(r)pA + gBN(r)pB), where gAN(r) and gBN(r) are the A monomer–

nanoparticle and B monomer–nanoparticle pair correlation functions, as before, and

pA and pB are the bulk fraction of A and B monomers, terms that are introduced since

the BE sequences do not have equal fractions of A and B, unlike all of the systems

that were examined in Figure 4.4b.

Figure 6.3: PB, the fraction of monomers at distance r that are type B, for the additional copolymer systems and reference copolymer systems from Chapters 4 and 5, as labeled.

∗ ∗ As expected, the structure of the BL = 3 and BL = 4 systems fall between that of the BL = 2 and BL = 5 systems. Close to the nanoparticle, the BE systems tend to have similar structure to the BL systems at the same B block length. Farther from the nanoparticle (at a distance that increases with BE), the structures of the BE

101 and BL systems diverge, and in the BE systems, PB trends sharply toward the bulk

concentration, which is predominantly A (PB(∞) = 2BE/100). As intended, the BE

systems exhibit signiﬁcantly sharper B–A phase transitions than the BL systems.

We also brieﬂy consider the BVACF relaxation times of the new systems, which

we hypothesized earlier may be related to the sharpness of the B–A phase transition.

The BVACF relaxation times for the new systems (as well as related systems for

comparison) are reported in Figure 6.4.

Figure 6.4: Bond vector relaxation times as a function of distance from the nanoparticle surface for the additional copolymer systems and selected copolymer systems from Chapters 4 and 5, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x- axis labels are located on the shell boundaries), and bonds are assigned to shells as described in the text. Measurement error increases with proximity to the nanoparticle because fewer bonds reside in closer shells.

The BE systems, despite having very sharp phase transitions, exhibit bond relaxation times that are signiﬁcantly shorter than the BL systems, which features

comparatively less sharp transitions. Therefore, we note that the sharpness of the

B–A phase transition is clearly not the only factor that determines the bond mo-

bility. It may be the case that the BE systems are more mobile because only the

102 chain ends adsorb to the nanoparticle rather than several segments from each chain,

perhaps meaning that entire chains in the BE systems are more mobile, thereby im- pacting bond mobility (a measurement of chain end-to-end relaxation times has not

∗ ∗ yet been performed for these systems). Meanwhile, the BL = 3 and BL = 4 systems do exhibit signiﬁcantly slower bond dynamics in the second shell, a trend that was previously observed in the BL = 2 and BL = 5 systems to a lesser extent.

Bearing these structural and dynamic results in mind, we now consider hysteresis in the first two shells of each of the new systems, as reported in Figure 6.5. Hysteresis in the first shell is significantly higher in each of the new systems (with the exception of BL = 1) than in any of the systems previous studied. Among the BL systems,

∗ hysteresis peaks at BL = 3. The hysteresis peak in the BL = 3 system also occurs at a slightly lower frequency than most of the systems and is similar to the BL = 2 system.

Hysteresis in the BE system is generally substantially higher than in the BL systems, and several systems exhibit nearly purely viscous behavior on a local scale in the ﬁrst shell. In the second shell, the BL = 3 and BE ≥ 3 systems exhibit higher hysteresis that than systems previously considered.

For more context, Figure 6.6 reports relative reinforcement and peak tan(δ) for the BE systems and reference BL systems. Returning to our earlier hypothesis that the sharpness of the B–A phase transition increases peak hysteresis in the shell closest to the nanoparticle, we see from Figures 6.3 and 6.6b that, as hypothesized, the BE systems have both sharper phase transitions and higher hysteresis in the ﬁrst shell than the BL systems. Additionally, among the BE systems, hysteresis in the ﬁrst shell increases with the sharpness of the B–A transition.

103 (a) (b)

Figure 6.5: Hysteresis in the two closest shells to the nanoparticle, 5.5σ < r < 6.0σ (a) and 6.0σ < r < 6.5σ (b), for the additional copolymer systems and selected copolymer systems from Chapters 4 and 5, as labeled. Data that fall outside the bounds of the chart indicate systems that are nearly purely viscous (δ ≈ π/2) on a local scale.

(a) (b)

Figure 6.6: Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for the additional copolymer systems and selected copolymer systems from Chapters 4 and 5, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries).

For the second shell, we had previously hypothesized that hysteresis would be dominated by the local percentage of B monomers, PB. However, this is not entirely

104 borne out in the data. For example, the BE = 10 system has by far the highest hysteresis in the second shell of any system studied, yet from Figure 4.4b, we see that it has a slightly lower PB than the BL = 10 system. The BE = 5 system also has a slightly higher hysteresis in the second shell than the BL = 5 system despite having lower PB. We do note that since monomers may move between adjacent shells during the time window used to calculate the stress autocorrelation, hysteresis in one shell may inﬂuence hysteresis measured in an adjacent shell, so highly granular analysis of local hysteresis is not necessarily appropriate. Overall, although these additional sequences have provided some additional evidence that hysteresis very near the nanoparticle is related to the sharpness of the B–A phase transition, more study is needed to fully understand the relationship of structure and sequence to hysteresis.

In terms of our motivating design challenge, we have found several BE sequences as well as a new BL sequence which produce signiﬁcantly higher hysteresis near the peak while not increasing hysteresis at frequencies slightly lower than the peak. However, we note that the BE sequences would be poor candidates for use in real tire tread compounds. This is because, in real tire compounds, the extremely long A blocks

(whether composed of styrene or butadiene) would tend to dramatically microphase separate to a degree that would adversely eﬀect tire properties. Therefore, in the following section, we study a small set of sequences that would be more reasonable for use in tire treads.

105 6.3.2 Modiﬁcations to Random Copolymers

In order to create a sequence that is feasible for use in tire treads, we seek to make modifications to a random copolymer sequence that are relatively minor while still increasing hysteresis near the peak relative to a purely random sequence. Motivated by the results from the BL systems (specifically the BL = 3 system, which had the highest hysteresis in the first two shells), we have simulated two new systems that are represented in the schematic shown in Figure 6.7. The first new system, which we refer to as RB, consists of two blocks of 3 B monomers attached to either end of a random sequence composed of 50 A and 46 B monomers (such that each entire chain is composed of 50% B monomers). Additionally, we simulate a system which consists of the two blocks of form B3A3 grafted to either end of an otherwise 50% B random copolymer sequence such that the A blocks bond to the random sequence and the

B blocks terminate the chain. We refer to this latter system RAB. Note that both systems have an overall fraction of 50% B monomers.

Figure 6.7: Schematic of the additional primarily-random copolymer sequences introduced in this section. Grey segments indicated randomized sequences composed 50 A and 44 B monomers (in the case of the RB system) or 44 A and 44 B monomers (in the case of the RAB system). The randomized components may vary between chains within the system.

The goal of these systems was to create a sharp B-A transition near the nanoparticle while still preserving the primarily random nature of the sequences. Figure 6.8 reports PB as a function of distance from the nanoparticle for the random copolymer

106 system as well as the two new predominantly random systems. The RB system has a marginally more significant phase transition than the purely random system, since at moderate distances from the nanoparticle, the concentration of B monomers is slightly lower than 50%, perhaps reflecting the concentration in the middle segments of the chains, which are slightly less likely to adsorb directly to the nanoparticle compared to B blocks on the chain ends. Meanwhile, the RAB system has a slightly steeper and more significant phase transition than either of the other systems, likely because the terminal BA blocks adsorb to the nanoparticle and form a phase with properties somewhat more similar to the BE = 3 system.

Figure 6.8: PB, the fraction of monomers at distance r that are type B, for the various predominantly random systems, as labeled.

Figure 6.9 plots hysteresis in the two shells nearest to the nanoparticle in the random, RB, and RAB systems. There appears to be no significant difference between the RB and the purely random system. On the other hand, the RAB system exhibits nearly 50% higher peak hysteresis in the first shell. Additionally, hysteresis half a decade below the peak is nearly identical to the purely random system, which,

107 combined with the higher peak hysteresis, suggests that such a modiﬁcation to a

random SBR sequence has the potential to improve performance when incorporated

into tire treads. In the second shell, there is relatively little diﬀerence between the

systems, with the RAB system having only slightly higher hysteresis.

(a) (b)

Figure 6.9: Hysteresis in the two closest shells to the nanoparticle, 5.5σ < r < 6.0σ (a) and 6.0σ < r < 6.5σ (b), for the various predominantly random systems, as labeled.

These trends are also confirmed in Figure 6.10, which plots peak hysteresis and reinforcement as a function of distance from the nanoparticle. From this figure, we also see that there is relatively little difference in reinforcement between the predominantly random systems.

6.4 Discussion

From the results in the previous section, we have seen that within the systems studied, one way to increase hysteresis is to increase the sharpness of the B–A microphase transition in the polymer–nanoparticle interphase. This may be because there is an

108 (a) (b)

Figure 6.10: Dynamic mechanical reinforcement under shear stress at 9 rad/τ (a) and peak hysteresis under shear stress (b) for the various predominantly random systems, as labeled. Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries).

energy cost associated with monomers moving back and forth across the microphase boundary, and this energy cost increases with the sharpness of the boundary. By deliberately constructing triblock copolymer sequences to maximize the sharpness of the phase boundary, we were able to dramatically increase interphase hysteresis. Al- though these triblock copolymers would be poor candidates for use in tire treads, we additionally examined some minor modiﬁcations to otherwise randoms copolymer chains and determined that, by placing short terminal AB blocks on both ends of otherwise random sequences, we were able to increase hysteresis relative to a purely random copolymer. This increase was not observed when using simple terminal adsorbing B blocks, suggesting that the additional nonadsorbing segment is critical to the observed increase in hysteresis. We also note that hysteresis at frequencies only slightly below that of the peak was not aﬀected by the addition of the terminal AB

109 blocks, suggesting that systems with this form could be a good candidate for improving the tradeoﬀ between traction and rolling resistance in SBR tire tread compounds.

This is especially true given that the systems represent only minor modiﬁcations to commonly-used random copolymer chains, suggesting the possibility that these systems could be relatively economical to produce. Finally, we note that since the observed eﬀect occurs on a highly-localized scale, we expect that it would still be relevant even in crosslinked systems.

Adsorbing functional groups have previously been used in SBR rubbers, including tire treads, to improve nanoparticle dispersion and hence overall material properties.164 However, we are unaware of any published work which has examined using bifunctional adsorbing/non-adsorbing oligomers to improve material properties in tire treads. Future work would be aimed at determining a reasonable interaction chemistry that could produce the desired short, regular adsorbing/non-adsorbing blocks on the end of random SBR chains. It is likely that those segments would not need to be composed specifically of styrene and butadiene, since the important mechanism appears to be the difference in the affinity of the two components of the terminal block for the nanoparticle. Therefore, future work could explore other interaction chemistries to produce the desired adsorbing and non-adsorbing behavior relative to the nanoparticle.

110 Chapter 7: Concluding Remarks

7.1 Summary of Results

Using coarse-grained molecular dynamics simulations of a series of simple polymer nanocomposites, we have explored the effect of polymer–nanoparticle interaction strength and copolymer sequence on properties of the polymer–nanoparticle interphase. In Chapter 3, we analyzed a series of homopolymer systems that differed only in the affinity of the polymer for the nanoparticle. We found that radial ordering in terms of local monomer density and bond alignment increased with both polymer–nanoparticle interaction strength and proximity to the nanoparticle. Mean- while, chain conformations depended on distance of the chain’s center of mass from the nanoparticle, with proximity to the nanoparticle tending to increase radius of gyration and the tendency for chains to align more parallel to the nanoparticle surface.

However, chain conformations were found to be relatively independent of polymer– nanoparticle interaction strength. The interphase region was also found to have different dynamics than the bulk, with end-to-end vector autocorrelation function, bond vector autocorrelation function, and self-intermediate scattering function relaxation times tending to increase with polymer–nanoparticle interaction strength and with proximity to the nanoparticle. The exception to this trend was in the case of the

111 lowest interaction strengths, for which relaxation times were found to decrease with nanoparticle proximity.

After establishing a solid understanding of the impact of polymer–nanoparticle interaction strength on interphase properties, the remainder of this work focused on the eﬀect of copolymer sequence. In Chapter 4, we examined a series of AB multiblock copolymer sequences where the only chemical diﬀerence between the A and

B monomers was their strength of attraction to a nanoparticle, with B monomers having a strong aﬃnity for the nanoparticle and A monomers interacting slightly un- favorably. Using this model, we examined the eﬀect of copolymer block length on the structure and dynamic properties of the polymer–nanoparticle interphase. Measures of interphase structural ordering, including local monomer density and bond orientations, were found to increase with block length at low block lengths before plateauing at higher blocks lengths. Interphase relaxation times tended to slow with increasing block length, with bond vector autocorrelation and self-intermediate scattering function relaxation times also plateauing at higher block lengths. We also compared the results from the multiblock copolymer systems to a random copolymer system as well as homopolymer A and homopolymer B systems. We found that, depending on the block length, interphase structure could be more or less ordered than that of the random copolymer system, and interphase dynamics could be faster or slower.

The block length at which interphase properties were most similar to the random copolymer system varied with the property being considered but was typically close to 5. Among the copolymers studied, adjusting sequence could tune the interphase to take on a range of properties between extremes bounded by the homopolymer A and homopolymer B systems. Moreover, we found that, depending on the interphase

112 property of interest, in some cases it is possible to produce interphase behavior that is outside the region bounded by homopolymer A and homopolymer B systems. This was most clear in the case of end-to-end relaxation times, where certain copolymer sequences with longer block lengths produce slower relaxations than even the pure B system.

In Chapter 5, we extended our analysis to include the eﬀects of copolymer sequence on interphase dynamic mechanical modulus. Using a localized version of the stress autocorrelation function in combination with a well-known method for calculating bulk dynamic modulus from equilibrium simulations, we estimated local dynamic modulus and hysteresis as a function of distance from the nanoparticle surface. Com- plex modulus was found to increase with proximity to the nanoparticle and with copolymer block length, and it appeared to be closely related to the local percentage of adsorbing monomers. Hysteresis increased with proximity to the nanoparticle and with block length in intermediate shells, but it tended to decrease with increasing block length in the shell closest to the nanoparticle. Compared to the magnitude of the complex modulus, the level of hysteresis in each shell was not as easily explained by the local percentage of adsorbing monomers. Interestingly, certain copolymer sequences, which contained 50% adsorbing monomers, were found to produce higher interphase hysteresis than even a system composed of 100% adsorbing monomers.

This suggests that adjusting copolymer sequence can be a powerful tool for tuning interphase dynamic mechanical properties and, therefore, for tuning overall composite properties.

To brieﬂy explore one way that copolymer sequence can be adjusted to produce a speciﬁc desired change in interphase viscoelastic properties, Chapter 6 explored a new

113 series of sequences aimed at maximizing interphase hysteresis. The new sequences included chains composed primarily of A but with short B blocks on either end, with lengths that varied between the systems. These sequences were found to dramatically increase interphase hysteresis despite the fact that they were composed of 10% or fewer adsorbing monomers, and we related the level of hysteresis to the sharpness of the B–

A interphase near the nanoparticle. Motivated by these sequences, we then explored a small set of modiﬁcations to primarily-random ccopolymer chains. We found that by bonding a short regular AB block to either end of longer random sequence (with the chains terminated by the B portion of the blocks) hysteresis could be increased signiﬁcantly relative to a purely random copolymer.

7.2 Future Work

In this work, we have explored only a small fraction of possible copolymer sequences in one simple nanocomposite model. The most straightforward possible future study would be to explore additional sequences. In Section 6.4, we were motivated by tire tread compounds and aimed to increase peak hysteresis, but different materials would of course yield different design goals. It is possible to conceive of a relatively straightforward neural-network-based approach for determining ideal sequences to meet particular sets of material property specifications; however, such an approach would be extremely computationally intensive, so there is still a significant advantage to be gained from developing a better understanding of the physical mechanisms underlying the relationships between sequence and properties.

114 Toward that end, there are many additional methods to measure interphase properties that could be employed to provide additional perspective on the role of sequence. For example, the self-intermediate scattering function relaxation time could be measured separately for A and B monomers, and bond orientations and bond vector relaxation times could be compared for A–A, A–B, and B–B bonds. Even the local stress autocorrelation could be analyzed separately for A and B monomers, and dynamic moduli could be calculated independently considering only A or only B monomers. Each of these quantities would provide further insight on the interdepen- dence of sequence, local monomer composition, and the inﬂuence of the nanoparticle.

Additionally, it would be compelling to examine, on a molecular level, the process of monomers moving back and forth across a sharp AB interface in order to see if hysteresis could be better predicted by combining information about the sharpness of the interphase with information about the frequency of A monomers moving across to the B side of the interphase and vice versa.

The nanoparticle’s eﬀect on local dynamic moduli could be further elucidated by measuring stress ﬂuctuations in spherical coordinates centered on the nanoparticle.

If we deﬁne a spherical coordinate system centered on the nanoparticle, then the six components of the symmetric spherical stress tensor can be calculated from the

Cartesian stress tensor according to

  sRR sRθ sRφ   sRθ sθθ sθφ  = (7.1) sRφ sθφ sφφ       sin θ cos φ sin θ sin φ cos θ sxx sxy sxz sin θ cos φ cos θ cos φ − sin φ       cos θ cos φ cos θ sin φ − sin θ sxy syy sxy sin θ sin φ cos θ sin φ − cos φ , − sin φ cos φ 0 sxz sxy szz − sin φ cos φ 0 where R, θ and φ are deﬁned in the usual fashion.

115 Since the system is rotationally-symmetric about the nanoparticle, the choice

of θ- and φ-axes are arbitrary, and stresses at a given radius along θ versus φ are expected to be equal. Therefore, we can average equivalent terms and describe the stress on each monomer with four parameters: sRR, the normal stress in the radial

direction, (sθθ + sφφ)/2, the average normal stress in the tangential direction, (sRθ +

sRφ)/2 the average shear stress in planes intersecting the nanoparticle, and (sθφ, the

shear stress in the plane normal the nanoparticle. Since θ and φ can be calculated

for each monomer relative to the nanoparticle-centered coordinate system at each

timestep, these stresses can each be evaluated from Equation 7.1 via a process that is

mathematically straightforward albeit computationally intensive. From these stresses,

four sets of dynamic moduli describing the system’s response to the four possible stress

modes can be calculated using the same procedure as was used to calculate Cartesian

moduli in Chapter 5.

We have so far obtained preliminary results for the BL = 25 system using this

method, and we have found that the four sets of moduli are qualitatively similar to

the Cartesian moduli from Chapter 5. Quantitatively, the system’s response to the

four stress modes diﬀers, as shown in Figure 7.1, which reports tan(δ) in response to

each stress modes.

As with stresses expressed in Cartesian coordinates, peak hysteresis is higher under

shear stress than normal stress. However, the use of spherical coordinates reveals how

hysteresis in response to stress in planes (in the case of shear stress) or along axes

(in the case of normal stress) that contain the nanoparticle is higher than when the

direction in which stress is applied does not contain the nanoparticle. An exploration

of how this diﬀerence varies with copolymer sequence would likely be illuminating,

116 Figure 7.1: Magnitude of complex modulus at 9 rad/τ (a) and peak hysteresis (b) for the BL = 25 system under various stress modes (as labeled). Shells are centered at the indicated distance and have a width of 0.5σ (x-axis labels are placed on the shell boundaries).

and these results could be analyzed along with directional measurements of monomer mobility such as the self-intermediate scattering function along diﬀerent k vectors, as was noted in Section 3.4.

Besides measuring additional interphase properties, the model itself can also be adjusted in order to provide perspective on additional types of systems. Additional nanoparticles could be introduced into the simulation, and interphase properties could be studied as a function of the separation distance between two nearby nanoparticles.

We additionally plan to investigate the effect of particle radius of curvature, including the limiting case where polymer interacts with a flat surface. Also, we hope to simulate blend systems to determine if we can reproduce some of the specific experimental results discussed in Chapter 5.

Systems could also be made more realistic by including the effects of unfavorable interactions between unlike monomers as well as the effect of modeling two polymers with different glass transition temperatures, which would allow us to examine how

117 copolymer sequence affects glass transition temperature in the interphase. As noted in Chapter 4, we also anticipate that this would expand the range of the interphase around the nanoparticle, since local properties would necessarily depend significantly on local monomer composition. In these circumstances, we expect that the nanoparticle will influence dynamic properties at least as far from the nanoparticle as there are structural differences relative to the bulk, and therefore, as seen in Figure 4.4b, the width of the dynamic interphase would depend more highly on sequence than it did in this investigation. All of these potential future directions will allow us to con- tinue to develop a better understanding of how copolymer sequence affects mechanical properties, which will ultimately provide additional methods to tune nanocomposite properties as well as a more complete understanding of the mechanisms that govern the properties of polymer–nanoparticle interphases.

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