RATIONAL DESIGN OF SOFT MATERIALS THROUGH

CHEMICAL ARCHITECTURES

A Dissertation

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Heyi Liang

December, 2019

RATIONAL DESIGN OF SOFT MATERIALS THROUGH

CHEMICAL ARCHITECTURES

Heyi Liang

Dissertation

Approved: Accepted:

______Advisor Department Chair Dr. Andrey V. Dobrynin Dr. Tianbo Liu

______Committee Member Interim Dean of the College Dr. Ali Dhinojwala Dr. Ali Dhinojwala

______Committee Member Dean of the Graduate School Dr. Matthew Becker Dr. Chand K. Midha

______Committee Member Date Dr. Mesfin Tsige

______Committee Member Dr. Hunter King

______Committee Member Dr. Kevin Cavicchi

ii ABSTRACT

Mimicking the mechanical behavior of biological tissues is crucial for tissue engineering, medical implants, and wearable devices. Many biological tissues—such as lung, skin and artery tissues—are supersoft at small deformation (Young’s Modulus E0 <

104 Pa) and stiffen with increasing deformation. This unique combination of mechanical softness and nonlinear elasticity cannot be easily duplicated by conventional synthetic elastomers composed of linear polymers.

Graft polymers, consisting of linear backbones densely grafted with short side chains, endow polymeric materials with two unique features owing to their special architecture: (i) the dilution of chain entanglement by side chains, and (ii) the stretching of the backbone due to steric repulsions of side chains. These distinct features pave the way for the design of supersoft elastomers with controllable nonlinear elasticity. Through a combination of theoretical calculations and coarse-grained molecular dynamics simulations, we have developed a general material design strategy which encodes the stress-strain curve of soft materials into the architecture of graft polymer networks. Such networks can be prepared through either chemical crosslinking of bottlebrush macromolecules or self-assembly of linear-bottlebrush-linear triblock copolymers. The mechanical response of the resultant materials is controlled by architectural parameters, including network strand length, side chain length, grafting density, as well as the chain length of blocks in triblock copolymers. By utilizing this design-by-architecture strategy,

iii replicas of jellyfish, artery and skin tissues based on bottlebrush poly(dimethylsiloxane)

(bbPDMS) are synthesized to test our approach.

iv ACKNOWLEDGEMENTS

As the five-year journey of graduate school study and research coming to an end,

I owe many people a debt of gratitude for their support. Most importantly, I would like to express my deepest appreciation to my advisor, Prof. Andrey Dobrynin. He guided me through this journey with great patience and paved the way to the completion of my dissertation. I enjoyed the everyday discussion with him when he always shared his invaluable experience and insightful idea about research without any reservation. Except doing research, he also provided me opportunities to mentor high school students and teach classes and generously supported me attend many conferences, from which I gain much professional experience crucial for my future career. After five-year training, I will take away not only knowledge and methodology of conducting research, but also the character of being considerable to others and the sense of humor towards life.

I am also grateful to my committee Prof. Ali Dhinojwala, Prof. Matthew Becker,

Prof. Mesfin Tsige, Prof. Hunter King and Prof. Kevin Cavicchi for helpful discussion on my research project. I would like to further thank Prof. Stephen Z. D. Cheng for his valuable advice on academic career development. I also had great pleasure working with

Prof. Sergei Sheiko from the University of North Carolina at Chapel Hill, Dr. Gary Grest from Sandia National Laboratories and Prof. Douglas Adamson at University of

Connecticut. Collaboration with them offered me valuable opportunities to explore different subject and get inspiration for my own research. Furthermore, I would like to

v express my gratitude to Prof. Lin Yao and Prof. Luyi Sun at University of Connecticut for their help when I spend the first year of my graduate study in the Polymer Program at

UConn.

I feel luck and thankful working together with my former and current labmates:

Dr. Zhen Cao, Dr. Zilu Wang, Yuan Tian, Michael Jacobs and Ryan Sayko. Without their heartwarming support and encouragement, I could not imagine how I would spend five year alone in the country thousand miles away from my homeland.

Last but not least, my greatest gratitude and love goes to my family. My parents have always been very supportive to any decision I made. Even though I have not been back home for four years, the weekly video chat with them is always the greatest time every week. My grandfather, who was a chemical engineer, shared with me his experience on study aboard and conducting research which is always encouraging and inspiring. My grandmother, who was a doctor, provided continuing care of my health. I dedicate this dissertation to my family. Without their unconditional love and support, I would have never gone this far.

vi TABLE OF CONTENTS

Page

LIST OF TABLES ...... x

LIST OF FIGURES ...... xii

CHAPTER

I. INTRODUCTION ...... 1

II. COMB AND BOTTLEBRUSH GRAFT POLYMERS IN A MELT ...... 6

2.1 Introduction ...... 6

2.2 Graft Homopolymers in a Melt ...... 8

2.2.1 Scaling Analysis ...... 8

2.2.2 Comparison with Simulations ...... 15

2.3 Graft Copolymers in a Melt ...... 24

2.3.1 Scaling Analysis ...... 24

2.3.2 Comparison with Simulations ...... 33

2.4 Comparison with Experiments ...... 40

2.5 Simulation Methods ...... 41

2.6 Conclusions ...... 44

III. SCATTERING FROM MELTS OF COMBS AND BOTTLEBRUSHES ...... 47

3.1 Introduction ...... 47

3.2 Simulation Results ...... 49

3.2.1 Diagram of States ...... 50

3.2.2 Structure Factor ...... 52 vii 3.3 Theoretical Analysis of the Static Structure Factor ...... 55

3.3.1 Comb regime ...... 58

3.3.2 Bottlebrush Regime ...... 63

3.4 Conclusions ...... 66

IV. ENTANGLEMENTS OF MELTS OF COMBS AND BOTTLEBRUSHES ...... 68

4.1 Introduction ...... 68

4.2 Entanglement Plateau Modulus of Graft Polymer Melts ...... 70

4.2.1 Scaling Analysis ...... 70

4.2.2 Comparison to Experiments ...... 77

4.3 Packing Number of Graft Polymer Melts ...... 86

4.3.1 Simulation Results ...... 88

4.3.2 Comparison with Experiments ...... 92

4.4 Simulation Methods ...... 94

4.5 Conclusions ...... 100

V. POLYMER NETWORKS WITH COMBS AND BOTTLEBRUSHES STRANDS 102

5.1 Introduction ...... 102

5.2 Scaling Analysis ...... 104

5.2.1 Combs ...... 106

5.2.2 Bottlebrushes ...... 107

5.2.3 Comparison with Linear Chain Networks ...... 108

5.3 Comparison with Simulations ...... 109

5.4 Comparison with Experiments ...... 116

5.5 Simulation Methods ...... 118

5.6 Conclusions ...... 123

viii VI. SELF-ASSEMBLED NETWORKS OF LINEAR-BOTTLEBRUSH-LINEAR TRIBLOCK COPOLYMERS ...... 125

6.1 Introduction ...... 125

6.2 Model of Self-Assembled Networks ...... 127

6.2.1 Self-Assembly and Equilibrium Properties ...... 128

6.2.2 Elastic Deformation of Self-Assembled Networks ...... 130

6.2.3 Yielding of Self-Assembled Networks ...... 132

6.3 Comparison with Simulations ...... 133

6.4 Comparison with Experiments ...... 138

6.5 Simulation Methods ...... 139

6.6 Conclusions ...... 141

VII. SUMMARY ...... 143

REFERENCES ...... 148

ix LIST OF TABLES

Table Page

Table 2.1 Effective Kuhn length of graft homopolymers in different regimes...... 15

Table 2.2 Summary of studied systems. Data sets with nbb = 100 are obtained from Macromolecules 2015, 48, 5006-5015...... 16

Table 2.3 Effective Kuhn length of graft copolymers in different regimes ...... 32

Table 2.4 Monomer volume, bond length and Kuhn length for different systems...... 34

Table 2.5 Summary of studied systems and symbol notations...... 35

Table 4.1 Ratio of the entanglement plateau shear modulus in a melt of graft polymers to that of linear chains in different conformation regimes...... 76

Table 4.2 Architectural parameters and rheological properties of graft polymers...... 78

Table 4.3 Molecular parameters of graft polymers.(1)...... 79

Table 4.4 Architectural parameters and rheological properties of PNB-g- PLA...... 84

Table 4.5 Molecular parameters of PNB-g-PLA ...... 84

Table 4.6 Architectural parameters, effective Kuhn length, entanglement length and packing number of linear and graft polymers ...... 90

Table 4.7 Degree of polymerization of the backbone strand between entanglements, ne,bb, calculated from primitive path analysis...... 96

Table 4.8 Tube diameter, dT, and DP of the backbone strand between entanglements, ne,bb, calculated from tube diameter approach ...... 97

Table 5.1 Structural shear modulus, G, and number of effective Kuhn segments per network strands, α − 1, in different conformational regimes...... 108

Table 5.2 Summary of studied systems ...... 110

Table 5.3 Summary of graft polymer networks...... 120 x Table 5.4 Summary of linear chain networks...... 120

Table 6.1 Summary of studied systems ...... 134

xi LIST OF FIGURES

Figure Page

Figure 1.1 (a) Diversity in materials’ mechanical properties illustrated by uniaxial tensile stress-deformation curves. (b) “Golden rule” of materials science establishes an inverse relationship between the Young’s modulus and elongation at break. To display materials with different elongation at break in a single plot, E0/ρ is shown as a function of λmax − λmax − 2, where ρ is the mass density. (Reproduced with permission from Nature 2017, 549, 497-501. Copyright 2017 Springer Nature.) ...... 2

Figure 2.1 A graft polymer chain and definition of architectural parameters ng and nsc. Backbone monomers are colored in red; side chain monomers are colored in blue...... 9

Figure 2.2 (a) Schematic representation of graft polymers as chains of blobs of size Rsc. Side chains and backbone of the test macromolecule are shown in blue and red, respectively. Surrounding macromolecules are colored in gray. (b) Conformations of graft polymers and the overlap between chains within the pervaded volume with size equal to that of the side chains, Rsc, in different regimes...... 9

Figure 2.3 Diagram of states of graft polymers in a melt. SBB – stretched backbone subregime, SSC – stretched side chain subregime, and RSC – rod-like side chain subregime. Logarithmic scales...... 12

Figure 2.4 Typical bond-bond correlation function for graft polymers with kspring = 30 kBT/σ2, ng = 4 and the DP of side chains nsc = 2 (red circles), 4 (green triangles), 8 (blue inverted triangles), 16 (magenta diamond) and 32 (cyan pentagons). Solid lines show the best fit curves of equation 2.18 ...... 17

Figure 2.5 Dependence of the normalized Kuhn length, bK/b, of the graft polymers on the crowding parameter Φ (see equation 2.1). Thin solid lines show scaling predictions in comb and bottlebrush regimes. Symbols are summarized in Table 2.2...... 19

Figure 2.6 (a) Dependence of the mean-square end-to-end distance of the section of the graft polymer backbone with n bonds on the number of bonds in a section, for macromolecules with ng = 4, values of the spring constants kspring = 30 kBT/σ2 (filled symbols) and 500 kBT/σ2 (open symbols) and the DP of side chains nsc = 2 (red circles), 4 xii (green triangles), 8 (blue inverted triangles), 16 (magenta diamond) and 32 (cyan pentagons). The solid and dash lines in this figure correspond to equation 2.19. (b) Dependence of the normalized mean-square end-to-end distance of the section of the graft polymer backbone with n bonds on the number of Kuhn segments in such sections. Symbols are summarized in Table 2.2...... 20

Figure 2.7 Dependence of the normalized mean square end-to-end distance of graft polymers as a function of the crowding parameter. Thin solid lines show scaling prediction in comb and bottlebrush regimes. Symbols are summarized in Table 2.2...... 21

Figure 2.8 (a) Dependence of the normalized mean-square end-to-end distance of the section of the side chain with n bonds on the number of Kuhn segments in such sections. Solid lines show simulation results for linear polymer chains with spring constant kspring = 30 kBT/σ2 (red) and 500 kBT/σ2 (blue) in a melt. (b) Normalized mean- square end-to-end distance of the section of the side chains with n bonds for graft polymers. The normalization factor Re,02(n) is the mean square end-to-end distance of the section of the linear chain staring from the point nsc from the chain end with n bonds in it and its end point locating between nsc and linear chain end. Symbols are summarized in Table 2.2...... 22

Figure 2.9 Diagram of states of graft polymers with values of the spring

constants kspring= 30 kBT/σ2 (a) and 500 kBT/σ2 (b) in a melt. SBB – stretched backbone subregime, SSC – stretched side chain subregime, and RSC – rod-like side chain subregime. Intersect point of crossover lines between different graft polymer regimes is set at nsc = 1. Symbols are summarized in Table 2.2...... 23

Figure 2.10 (a) Diagram of states of graft copolymers with bb > bs in a melt. SBB – stretched backbone subregime, SSC- stretched side chain subregime, and RSC – rod-like side chain subregime. Black solid lines are boundaries between comb and bottlebrush regimes, and red dashed lines show boundaries of different bottlebrush subregimes. The part of the diagram filled by red-white stripes corresponds to bottlebrushes with effective backbone Kuhn length bK ≈ bb. The upper boundary of the accessible region is given by φ-1 ≤ φmax-1 = nscvsngmaxvb+1, which is shown as the red solid line for ngmax = 1. (b) Diagram of states for graft polymers with identical backbones and side chains (b = bb = bs, l = lb = ls and v = vb = vs). Notations are the same as in panel (a). Logarithmic scales...... 29

xiii Figure 2.11 Bond-bond correlation functions G(s) of graft copolymer backbones with the DP of side chains nsc = 8, different intrinsic backbone stiffness K and side chain grafting density, 1/ng. Solid

lines represent the best fits to equation 2.18 using α, λ1 and λ2 as fitting parameters. Symbol notations are summarized in Table 2.5...... 36

Figure 2.12 Dependence of the reduced Kuhn length, bK/bb, of graft copolymers on the crowding parameter, Φ. The crowding parameters are calculated by equations 2.28~2.31. Colored symbols are summarized in Table 2.5. Gray circles represent graft polymers with K = 0 and identical backbones and side chains (i.e. graft homopolymers, see section 2.2.2.2 and Figure 2.5). Solid black lines show pure scaling regimes of the effective Kuhn length dependence on the crowding parameters for combs and bottlebrushes. The vertical dashed line corresponds to the crossover between combs and bottlebrushes, Φ ≈ Φ* ≅ 0.7...... 37

Figure 2.13 Diagrams of states of graft copolymers with backbone bending constants K = 1.5 (a), K = 4.0 (b) and K = 1.5 and backbone bead diameter 1.5σ (c). (d) Diagram of states of melts of graft polymers with identical side chains and backbones. Black solid lines are boundaries between comb and bottlebrush regimes, and red dashed lines show boundaries of different bottlebrush subregimes. The part of SSC subregime below the horizontal black dashed line and the part of the RSC subregime on the left from the vertical black dashed line correspond to bottlebrushes with the effective Kuhn length being on the order of backbone Kuhn length. Colored dotted curves represent combs and bottlebrushes with the same ng. Symbols in panel (a), (b), and (c) are listed in Table 2.5...... 39

Figure 2.14 Diagram of states of PNB-g-PLA graft copolymers. Red and blue filled circles represent bottlebrushes and combs respectively. Black solid line is the boundary between comb and bottlebrush regimes. Dashed red line separates different bottlebrush subregimes. Colored dotted curves represent graft copolymers with constant ng. The filled area corresponds to the forbidden region due to the chemistry limitation on the maximum number of side chains per backbone monomer, 1/ ngmax = 1...... 41

Figure 3.1 Snapshot of a bottlebrush macromolecule. Backbone monomers are colored in red; side chain monomers are colored in blue...... 47

Figure 3.2 Diagram of states of combs and bottlebrushes with bending constant K = 0.0 (a) and K = 1.5 (b) in a melt. Open symbols represent combs and filled symbols represent bottlebrushes. Dotted gray curves are constant ng curves calculated by equation 3.1...... 52 xiv Figure 3.3 (a) Static structure factor S(q) in melts of comb polymers with backbone bending constant K = 0, DP of spacer ng = 4 and different DPs of side chains. (b) Dependence of the peak position, q*, in the scattering function on the DP of side chains nsc and DP of spacer ng. Symbol shapes represent different backbone bending constant: K=0 (open circles), K=1.5 (open squares) and K=4.0 (open triangles). Different symbol colors show different ng as illustrated in Figure 3.2. (c) Static structure factor S(q) in melts of bottlebrush polymers with backbone bending constant K = 0, DP of spacer ng = 2 and different DPs of side chains. (d) Dependence of the peak position, q*, on the DP of side chains nsc for bottlebrushes. The dashed line corresponds to the best fit to function y = Ax-α with A = 2.79 ± 0.09 σ − 1 and α = 0.39 ± 0.01. Inset shows the characteristic length scale, d = 2π/q*, as a function of root-mean-square end-to-end distance of side chains, Rsc21/2. The dashed line is the best fit to the function y = ax + c with a = 0.95 ± 0.04 and c = 1.72 ± 0.2σ. Different symbol shapes represent graft polymers with backbone bending constant K=0 (filled circles), K=1.5 (filled squares) and K=4.0 (filled triangles). Different symbol colors show graft polymers with different ng as illustrated in Figure 3.2...... 53

Figure 3.4 Schematic diagram for the calculation of the structural correlation functions...... 57

Figure 3.5 (a) Comparison of the scattering function obtained in molecular dynamics simulations (points) and one calculated using the RPA method (solid lines). (b) Dependence of q*2Rnsc2 on the ratio nsc/ng . Symbol shapes represent graft polymers with different backbone bending constants: K = 0 (circles), K = 1.5 (squares) and K = 4.0 (triangles). Symbol colors represent combs with different ng as illustrated in Figure 3.2...... 59

Figure 3.6 Dependence of the normalized bottlebrush Kuhn length bK/bb on the dimensionless parameter q, where q = q*(bbbsl2ngnsc)1/4. The dash line is the best fit to function y = ax + c with a = 88 ± 4 and c = 0.77 ± 0.06. Symbol shapes represent graft polymers with different backbone bending constants: K = 0 (circles), K = 1.5 (squares) and K = 4.0 (triangles). Different symbol colors represent bottlebrushes with different ng as illustrated in Figure 3.2...... 66

Figure 4.1 Three distinct polymer melt systems. The equations above the cartoons correspond to the number of monomeric units in the corresponding entanglement strands...... 71

Figure 4.2 (a) Diagram of states of graft PBA. Solid black lines correspond to the boundary between comb and bottlebrush regimes. xv Red dashed lines mark boundaries between different bottlebrush subregimes. Red solid line corresponds to the crossover to the forbidden region. (b) Diagram of states of graft PBA, graft PS and graft PE. The solid black straight line corresponds to a crossover between comb and bottlebrush regimes for flexible (long) side chains. The solid, dashed and dotted black curved lines correspond to a crossover between comb and bottlebrush regimes with short (rod-like) side chains for PE, PS and PBA systems respectively. Dashed red lines mark boundaries between different bottlebrush subregimes. Boundaries of the corresponding “forbidden regions” are not shown. For both panels, symbol shapes correspond to different chemical structures: PBA (circles), PS (squares), and PE (triangles). Symbol colors represent different regimes: comb (blue) and bottlebrush (red)...... 79

Figure 4.3 Normalized entanglement modulus as a function of the compositional parameter φ − 1= 1+nscng for different graft-polymer systems: (a) PBA combs, PE bottlebrushes and PBA bottlebrushes. (b) PS combs and PE combs with entangled side chains. (c) Combined plot of the normalized entanglement modulus as a function of graft polymer composition for melts of combs and bottlebrushes from (a) and (b). (d) Normalized entanglement modulus as a function of crowding parameter. The crossover equation y = 1 + (x/0.7)3 between comb and bottlebrush regimes is shown by the dashed line. Symbol notations in all panels are the same as Figure 4.2...... 81

Figure 4.4 Normalized entanglement modulus as a function of the normalized crowding parameter for PBA combs (blue circles), PBA bottlebrushes (red circles), PNB-g-PLA combs (blue inverted triangles), and PNB-g-PLA bottlebrushes (red inverted triangles). Φ* = 0.4 for PNB-g-PLA and Φ* = 0.7 for PBA. The crossover equation y = 1 + x3 between comb and bottlebrush regimes is shown by the dashed line...... 85

Figure 4.5 Snapshots of graft polymers. (a) A comb macromolecule with dilute side chains shows chain-like behavior. (b) A bottlebrush macromolecule with densely grafted side chains shows filament-like behavior. Backbone bonds are shown in red and side chain bonds are colored in blue...... 87

Figure 4.6 Dependence of the normalized packing number Pe,gr/Pe,lin of graft polymers calculated using ne,bb obtained from PPA (filled symbols) and tube diameter (open symbols) as a function of (a) normalized Kuhn length, bK/b, and (b) the ratio describing overlap between side chains...... 91

xvi Figure 4.7 The ratio of packing number Pe,gr/Pe,lin as a function of the ratio of size of side chains and spacers Rnsc/Rng. Dashed and dotted lines highlight trends in simulation and experimental data respectively. Symbol shapes represent different data set: simulation (circles), graft PBA (squares) and PNB-g-PLA (inverted triangles). Symbol colors show different regimes of graft polymers: comb regime (comb) and bottlebrush regime (blue)...... 94

Figure 4.8 MSD of monomers g1t (red open circles) and of center of mass of molecules g3t (blue open circles) for different systems during long simulation runs lasting up to 3.7 ×107τLJ: (a) nbb = 401, ng = nsc = 8, (b) nbb = 401, ng = nsc = 16, (c) nbb = 1025, ng = nsc = 8, (d) nbb = 1025, ng = nsc = 16. Solid lines show expected time dependence of functions g1t ~ t1/4 and g3t ~ t1/2 in the reputation regime...... 98

Figure 4.9 MSD of monomers g1t (red open circles) and of center of mass of molecules g3t (blue open circles) for different systems with nbb = 401: (a) linear polymers, (b) ng = nsc = 16, (c) ng = nsc = 8, (d) ng = nsc = 4, (e) ng = nsc = 2. Black solid lines represent best fits to scaling laws, red dashed lines indicate the locations of crossover between the early Rouse regime and reptation regime of monomer MSD, g1t...... 99

Figure 5.1 (a) A graft polymer chain with the degree of polymerization (DP) of the backbone nbb, number of bonds between grafted side chains ng and DP of side chains nsc. (b) A graft polymer network with DP of the backbone between two crosslinks nx. Backbones and side chains are colored in red and blue, respectively. Green bonds in panel (b) indicate crosslinks...... 103

Figure 5.2 Dependence of the tensile stress σxx on the deformation ratio λ for networks of graft polymers with ng = 8, nx ≈ 16 and different side chain lengths nsc (a), with ng = 2, nsc = 8 and different network strand lengths nx (b), and with nsc = 8, nx ≈ 16 and different spacer lengths ng (c). The dashed lines are the best fit to equation 5.1 with structural shear modulus G and strand extension ratio β as fitting parameters. (d-f) Correlations between network mechanical properties and strand architecture illustrated by the linear relationship between reduced shear modulus and reduced density of stress supporting strands for networks in (a-c). Symbol notations are summarized in Table 5.2...... 112

Figure 5.3 (a) Universal relationship between mechanical properties of networks of graft polymers (structural shear modulus G and strand extension ratio β) and architectural parameters of network strands,

xvii described by nx and φ, illustrated by linear scaling between αG and βφ/nx. (b) Universality between the reduced deformation dependent network shear modulus G(I1)/G and the parameter βI1/3 for networks of graft polymers. The dashed lines are given by equation 5.17. Symbol notations in both panels (a) and (b) are summarized in Table 5.2...... 113

Figure 5.4 Breaking the “golden rule”, G ∝ λmax − 2, of the materials design. The “golden rule” is shown as the dash line with a slope -2 in logarithmic scales. Data for linear chain networks are shown by brown symbols: nx ≈ 4 (rhombs), 5 (triangles), 6 (inverted triangles), and 9 (squares). Symbol notations for comb and bottlebrush networks are summarized in Table 5.2. Insets show typical network strands with backbone shown in red and side chains colored in blue...... 115

Figure 5.5 (a) True stress-deformation curves of bottlebrush PDMS elastomers. Blue solid curves are experimental data, red dashed curves are best fit to equation 5.1 using G and β as fitting parameters. (b) Universality between the reduced deformation dependent network shear modulus G(I1)/G and the parameter β퐼1/3 for bottlebrush PDMS elastomers with ng = 1, side chain length nsc = 14 and different network stand length nx = 50 (purple squares), 67 (blue triangles), 100 (light blue inverted triangles), 200 (orange diamonds) and 400 (red hexagons). The dashed lines are given by equation 5.17. (c-d) Relationship between elastomer mechanical properties and their architectural parameters. Dashed lines are best linear fit...... 117

Figure 5.6 Dependence of the tensile stress σxx on the deformation ratio λ

of networks with kspring = 30 kBT/σ2, nx ≈ 16 and different values of nsc and (a) ng = 0.5, (b) ng = 2, (c) ng = 4, (d) ng = 8 and (e) ng = 16. Symbol notations are summarized in Table 5.2. Dashed lines are best fit to equation 5.1 using G and β as fitting parameters...... 121

Figure 5.7 Dependence of the tensile stress σxx on the deformation ratio λ

of networks with kspring = 500 kBT/σ2, nx ≈ 16 and different values of nsc and (a) ng = 0.5, (b) ng = 2, (c) ng = 4, (d) ng = 8 and (e) ng = 16. Symbol notations are summarized in Table 5.2. Dashed lines are best fit to equation 5.1 using G and β as fitting parameters...... 122

Figure 5.8 Dependence of the tensile stress σxx on the deformation ratio λ of linear chain networks with nx ≈ 4 (rhombs), nx ≈ 5 (triangles), nx ≈ 6 (inverted triangles), and nx ≈ 9 (squares). Dashed lines are best fit to equation 5.1 using G and β as fitting parameters...... 123

xviii Figure 6.1 (a) A triblock linear-bottlebrush-linear macromolecule with the degree of polymerization (DP) in the a linear block nL (shown in gray), DP in the bottlebrush backbone nbb (shown in red), DP of side chain nsc (shown in blue), and DP of the backbone spacer between neighboring grafting points ng. (b) Illustration of the linear chain domains of size RL consisting of Q linear chains connected by the bottlebrush blocks. Linear chain domains connected by highlighted bottlebrush block are shown in orange, other linear chain domains are shown in grey. Distance between spherical domains RL determines the size of the bottlebrush block in aggregates...... 126

Figure 6.2 (a) Probability distribution of the end-to-end distance of the bottlebrush block for LBL copolymers with nL = 10, nbb = 35, nsc = 4 and ng = 1. Insets show typical loop and bridge configurations of bottlebrushes. (b) Dependence of the fraction of bridges on the ratio nL/nbb. Dashed line shows the general trend. Symbols are given in Table 6.1...... 135

Figure 6.3 (a) Dependence of the tensile stress σtrue on the deformation ratio λ for self-assembled networks of LBL copolymers with similar volume fraction of linear blocks φL = 0.08. Solid lines are the best fit to equation 6.17 with structural shear modulus G and strand extension ratio  as fitting parameters. Dashed lines represent the best fit of yielding regime by equation 6.23. Insets show typical LBL copolymers configurations during deformation. (b) Dependence of the differential modulus dσtrue/dλ on the deformation ratio λ for self- assembled networks of LBL copolymers shown in panel (a). Solid lines correspond to dσtrue/dλ calculated using equation 6.17 with values of structural modulus G and strand extension ratio  obtained from fitting data in panel (a). Symbol notations are given in Table 6.1...... 136

Figure 6.4 (a) Dependence of G0β1/2 on parameters describing chemical structure of the LBL copolymers (equation 6.19). (b) Illustration of the nonlinear effects in determining B-block initial conformation and its dependence on the chemical structure of the LBL copolymers (equation 6.11). Symbol notations are given in Table 6.1. Dashed lines show expected linear correlations...... 138

Figure 6.5 (a) Dependence of the tensile stress σtrue on the deformation ratio λ for self-assembled networks of LBL copolymers with nL = 360, 480 and 810, nbb = 1065, nsc = 14 and ng = 1. Solid lines represent the best fit of elastic regime by equation 6.17. Dashed lines represent the best fit of yielding regime by equation 6.23. (b) Dependence of G0β1/2 on parameters describing chemical structure of the LBL copolymers (equation 6.19). (c) Illustration of the xix nonlinear effects in determining B-block initial conformation and its dependence on the chemical structure of the LBL copolymers (equation 6.11). Symbols for series of LBL copolymers: nbb = 302 (red circles), 938 (orange circles), and 1065 (blue circles)...... 139

Figure 7.1 (a) Flow chart of the procedure for replicating mechanical properties of gels in chemically crosslinked graft polymer network. (b) Stress-strain data for selected gels (white squares) with the best fit by equation 5.1 (red dashed lines), and stress-strain data for chemically crosslinked PDMS graft polymer networks (blue solid lines) synthesized to replicate the selected gels, with corresponding architectural triplet [nsc, ng, nx] as indicated. (Reproduced with permission from Nature 2017, 549, 497-501. Copyright 2017 Springer Nature) (c) Flow chart of the procedure for replicating mechanical properties of tissues in physical networks of self- assembled LBL triblock copolymers. (d) Stress-strain data for selected tissues (white squares) with the best fit by equation 5.1 (red dashed lines), and stress-strain data for PMMA-bbPDMS-PMMA physical networks (blue solid lines) synthesized to replicate the selected tissues, with corresponding architectural parameters [nbb, nL] as indicated. Each monomer on the bottlebrush PDMS block is grafted with one side chain with DP nsc = 14. (Reproduced with permission from Nature 2017, 549, 497-501. Copyright 2017 Springer Nature) ...... 147

xx CHAPTER I

INTRODUCTION

Materials, either synthetic or biological,1-7 show a rich variety of mechanical properties revealed by different shapes of their stress-deformation curves (see Figure

1.1a). For example, bones are rigid and brittle, leading to a rather steep and straight stress-deformation line ending at small deformation. A slowly rising line extending much longer in the stress-deformation plot indicates the softness and great extensibility of rubbers. Despite this variety of mechanical responses, there is a general trend correlating the value of the Young’s modulus at small deformation E0 with an elongation at break

λmax or strain at break εmax = λmax − 1 for materials undergoing uniaxial deformation

(Figure 1b).1 This relationship is called the “golden rule” of the materials science and identifies a general trend that more rigid materials are less deformable. For synthetic

−1 1/2 elastomers and rubbers both the modulus E0 ∝ nx and elongation at break λmax ∝ nx are uniquely determined by the degree of polymerization between crosslinks nx. This results

−2 in a universal relationship, E0 ∝ λmax, confining most polymeric networks into a single

8- trend line with a lower bound imposed by the entanglements modulus Eent (Figure 1b).

10 For solvent free elastomers an entanglement modulus is on the order of Eent ~ 1 MPa

11-15 and elongation at break λmax ~ 5 if no special network preparation methods are used.

−2 For brittle materials, the scaling relation E0 ∝ εmax follows from the fact that for a

2 material to break it requires a particular amount of energy density, E0εmax ≈ constant. 1 However, there is a large class of biological materials16 that occupies space below the “golden rule” trendline (Figure 1b). For example, skins feel very soft, but stiffens strongly as strain increases (and one can simply test it by stretching his/her skin on the neck). Such strain-stiffening (or strain-adaptive) behavior also leads to a small elongation at break since the breaking stress is reached under small deformation. The combination of softness and strong strain-stiffening, hence small elongation at break, poses a challenge of creating synthetic replicas capable of reproducing such mechanical properties. The conventional strategy is to use multicomponent systems and to apply an Edisonian approach to thoroughly explore the mixing of assorted polymers, cross-linking schemes, and solvents. However, this approach is imperfect in property control.16-23 For example, solvent can leak under the applied stress or evaporate when environmental conditions are changed.

Figure 1.1 (a) Diversity in materials’ mechanical properties illustrated by uniaxial tensile stress-deformation curves. (b) “Golden rule” of materials science establishes an inverse relationship between the Young’s modulus and elongation at break. To display materials with different elongation at break in a single plot, E0/ρ is shown as a function of λmax − −2 λmax, where ρ is the mass density. (Reproduced with permission from Nature 2017, 549, 497-501. Copyright 2017 Springer Nature.)

2 A new class of materials based on graft polymers24 has been developed recently to overcome the challenge of mimicking mechanical behaviors of soft biological tissues.1

Graft polymers consist of a long linear backbone grafted with shorter side chains. Grafted side chains play two important roles on the properties of polymeric materials: (i) dilute the backbone, (ii) stretch the backbone due to steric repulsion between side chains. The dilution effect suppresses the entanglement threshold of the backbone, making it possible to produce supersoft elastomers25-27 with Young’s modulus smaller than 1 kPa.8 The stretching of the backbone limits its extensibility and leads to strain-stiffening behavior in graft polymer networks1, 28-30. Through control over the architectural parameters, e.g., the degree of polymerization (DP) of side chains nsc and the backbone spacer between neighboring grafting points ng, one can rationally design polymeric materials with desired properties. In order to fully understand how the architecture of graft polymers affects the material properties and to enable the rational design of materials with controllable behaviors, this dissertation is devoted to study the structure-property relation of graft polymer materials using a combination of analytical calculations and molecular dynamics simulations.

In Chapter II, we start with an analysis on the conformation of graft polymers in a melt, which is the cornerstone of the structure-property relation. Graft polymers are classified into two regimes, combs and bottlebrushes, according to the crowding parameter Φ, which quantifies the extent of interpenetration between neighboring macromolecules. In the comb regime (Φ < 1), the grafting density is low, and two macromolecules can interpenetrate each other. In this regime, both backbones and side chains behave like unperturbed linear chains. In the bottlebrush regime (Φ > 1), steric

3 repulsion between side chains prevents interpenetration at high grafting density, and graft polymers behave like thick filaments. In this regime, the configuration of bottlebrushes can be described by the worm-like chain model with an effective Kuhn length proportional to the filament diameter.

In Chapter III, we calculate the static scattering function of graft polymer melts under the framework of random phase approximation and it is in good agreement with results from molecular dynamics simulations. We identify a peak in the scattering function and correlate its position with architectural parameters (nsc and ng ) of graft polymers. In particular, the scattering peak position of the bottlebrush is correlated with the effective Kuhn length, which provides the foundation for a new method of obtaining the Kuhn length of bottlebrushes from scattering data.

In Chapter IV, we establish the scaling relationship between the entanglement shear modulus and architectural parameters (nsc and ng) of graft polymer melts, which is verified in melts of graft poly(n-butyl acrylate) (PBA), graft polystyrene (PS), polyolefin and poly(norbornene)-graft-poly(lactide) (PNB-g-PLA). We also observe a breakdown of the Kavassalis−Noolandi conjecture31 which states that a universal number (called the packing number, Pe) of strands within a confined volume is required for the emerge of entanglement. Coarse-grained molecular dynamics simulation results show that the packing number is not universal for graft polymers with different architectures. This phenomenon is also observed in melts of graft PBA and PNB-g-PLA.

In Chapter V, we study how to control the mechanical behavior of chemically crosslinked graft polymer networks. According to the nonlinear network elasticity theory32, 33, two material parameters, structural shear modulus G and strand extension 4 ratio β, control the stress-deformation curve of polymer networks. Based on the graft polymer chain conformation studied in Chapter II, we find the correlation between the material parameters and architectural triplet [nsc, ng, nx] of graft polymer networks. This correlation is verified by the results of molecular dynamics simulations and experimental data of bottlebrush poly(dimethyl siloxane) (PDMS) elastomers.

In Chapter VI, we focus on the mechanical properties of self-assembled networks of linear-bottlebrush-linear (LBL) triblock copolymers. During the self-assembly, bottlebrush blocks form a continuum matrix physically crosslinked by small spherical domains of linear blocks. Governed by this microstructure, the deformation of the self- assembled network is a two-stage process. In the first stage, the network deforms in an elastic and nonlinear manner as a result of bottlebrush blocks stretching. In the second stage, the force acting on the bottlebrush blocks reach a critical value as deformation increases, and linear blocks are pulled out from the spherical domains leading to yielding at a constant instantaneous modulus. We establish a correlation between the two-stage deformation behavior and the architecture of the LBL copolymer characterized by the DP of the linear block nL and of the bottlebrush backbone nbb. Such correlation is validated by both molecular dynamics simulations and experimental data of physical networks formed by LBL copolymers where linear blocks are made of poly(methyl methacrylate) and bottlebrush blocks are made of poly(dimethyl siloxane) (PMMA-bbPDMS-PMMA).

Chapter VII summarizes the work covered in this dissertation and shows demonstrations on the rational design of graft polymer materials, in particular graft polymer networks.

5 CHAPTER II

COMB AND BOTTLEBRUSH GRAFT POLYMERS IN A MELT

2.1 Introduction

Graft polymers consisting of linear polymer backbones with grafted side chains are called either combs or bottlebrushes depending on grafting density of the side chains.24, 34-37 These macromolecules, depending on the grafting density and the degree of polymerization (DP) of side chains, can behave as linear backbone chains in an effective solvent of side chains or as flexible filaments where steric repulsion between the side chains controls the diameter and stiffness of the filaments.8, 38, 39 Therefore, side chains play a dual role of effective diluents and stiffeners of backbones. Such brush-like architecture allows for efficient control over materials properties through independent variation of the side chain length and their grafting density. For example, by diluting the backbones, side chains can significantly suppress the entanglement and reduce the viscosity of graft polymer melts, which make them easier to process.26, 40-46 The elimination of entanglement also enables the design of supersoft and superelastic materials25-27 with Young’s Modulus as low as 100 Pa and tensile train at break up to 800% in the solvent free state8. As another example, by stiffening the backbones through steric repulsion, side chains endow the graft polymer networks with strong strain-hardening behavior, which is used to develop dielectric elastomer for freestanding electroactuation under low applied electric field30.

6

Graft polymers are synthesized by one of the following three strategies: “grafting- through” (polymerization of macromonomers),47-49 “grafting-to” (attachment of the side chains to the backbone)50 and “grafting-from” (polymerization of the side chains from the backbone)36, 51-54. Thanks to the versatile repertoire of synthesis methods, chemists can precisely synthesize graft polymers with very different chemical (choice of monomers for backbones and side chains) and architectural (grafting density and DP of side chains) structures. However, due to the large number of structural (both chemical and architectural) parameters describing graft polymers, detailed mapping of structure- property relationships for these materials is extremely difficult. Despite substantial experimental8, 25-27, 37, 42-44, 55-67, theoretical8, 38, 57, 63, 68-70 and computational38, 39, 57, 63, 67, 71-

78 efforts to establish accurate correlations between the graft polymers structure and their physical properties, the complete solution of this problem still remains elusive.

In this chapter, we use a combination of the scaling analysis and coarse-grained computer simulations to provide general frameworks for classification of graft polymers into comb and bottlebrush classes that exhibit distinct conformational and physical properties. Specifically, we demonstrate that the Kuhn length and chain size of graft polymers in a melt state are universal functions of the crowding parameter, describing interpenetration between side chains and backbones belonging to different macromolecules. We also outline a diagram of conformational states of graft polymers in terms of two independently controlled parameters: DP of side chains nsc and volume fraction of the backbone monomers φ, which describes partitioning of monomers between backbone and side chains.

For the rest of the chapter, we first study a special case – graft homopolymers, where monomers of backbones and side chains are chosen to be identical. This allow us to only focus on how the graft polymer conformation is effected by the architectural parameters, including (i) the DP of side chains nsc, and (ii) the DP of backbone section between neighboring grafting points ng , which is also inversely proportional to the grafting density of side chains. Then we extend the results of graft homopolymers to graft copolymers, which are more commonly synthesized and utilized in experiments.

Especially, we study graft copolymers of which the backbone is intrinsically stiffer than the side chains, which describes most polynorbornene derived graft copolymers synthesized by ring-opening metathesis polymerization (ROMP) techinque79-82.

2.2 Graft Homopolymers in a Melt

In this section, we will first use a scaling approach to construct a diagram of

-1 conformational states of graft polymers in a melt in terms of nsc and φ . Then the scaling model predictions are compared with results of the molecular dynamics simulations for the effective Kuhn length and dimension of the backbone and side chain.

2.2.1 Scaling Analysis

Consider a graft polymer consisting of a linear chain backbone of the DP nbb with grafted side chains of the DP nsc (Figure 2.1). The side chains are equally spaced with ng bonds between two neighboring chains along the polymer backbone. Here we assume that both the backbone and side chain monomers are of the same type with the excluded volume v, bond length, l, and Kuhn length b.

8 (a)

Figure 2.1 A (b) graft polymer chain and definition of architectural parameters ng and nsc. Backbone monomers are colored in red; side chain monomers are colored in blue.

Figure 2.2 (a) Schematic representation of graft polymers as chains of blobs of size Rsc. Side chains and backbone of the test macromolecule are shown in blue and red, respectively. Surrounding macromolecules are colored in gray. (b) Conformations of graft polymers and the overlap between chains within the pervaded volume with size equal to that of the side chains, Rsc, in different regimes.

As shown in Figure 2.2a, each macromolecule occupies a pervaded volume V, which includes nbb(nsc/ng+1) monomers of its own and potentially monomers of the neighboring macromolecules. Herein, our classification of graft polymers as comb and bottlebrush macromolecules is based on the extend of mutual interpenetration (overlap) of neighboring molecules. To quantify the degree of interpenetration and establish how it

9 depends on the molecular architecture, the volume fraction of monomers of a test macromolecule within its own pervaded volume needs to be evaluated. At low grafting density, both the side chains and backbone display statistics of a random walk, whereby

1/2 the side chain size is described by Rsc ≈ (blnsc) (assuming flexible side chains with nsc ≥ b/l). Considering a test macromolecule as a chain of blobs with size Rsc , the

3 pervaded volume is estimated as V ≈ nbbRsc/nsc, which contains nsc backbone bonds. Now, we can define a crowding parameter, Φ, as a volume fraction of the monomers of test macromolecule within a pervaded volume

nnnvnn/1/1++ Vm bbscgscg( ) v ( )   33/21/2 , for nsc  b/l (2.1) VnRnbln bbscscsc/()

Following the same arguments, we can calculate the volume fraction of monomers belonging to a test macromolecule with rod-like side chains, nsc < b/l. In this case Rsc ≈ l nsc and equation 2.1 transforms to

v (nsc ng +1)   3 2 , for nsc  b/l (2.2) l nsc

If Φ < 1 , monomers of a test macromolecule occupy only a fraction of its pervaded volume. To maintain a constant monomer number density in a melt (ρ ≈ v-1), the pervaded volume of a test macromolecule “hosts” monomers from neighboring macromolecules (Figure 2.2b). As the crowding parameter Φ increases (with increasing

-1 nsc and/or grafting density ng ), the guest monomeric units are pushed out of the pervaded volume. At Φ ≈ 1, the pervaded volume is occupied only by the monomers belonging to a test macromolecule. This distinction between systems possessing different crowding parameters defines our classification of the comb ( Φ < 1 ) and bottlebrush ( Φ > 1 ) 10 polymers. Note that fractions cannot be larger than unity. The value of Φ > 1 corresponds to a hypothetical system, where bottlebrush macromolecules maintain ideal conformation of side chains and backbone even at infinitely (unreasonably) large grafting density. In real systems, however, the backbone and side chains will stretch to maintain the melt density (ρ ≈ v-1). The crossover between combs and bottlebrushes can be defined by setting Φ ≈ 1 and solving equations 2.1 and 2.2 for a composition parameter

n  = g (2.3) ng + nsc which describes partitioning of monomers between a side chain and backbone spacer between two neighboring side chains (i.e., “dilution” of the backbone). Note that the selection of nsc and  as variables for the diagram of state is more useful than nsc and ng in the previous representations,8 as it provides more distinct deconvolution of the inherent cross-correlations between the molecular dimensions and the architectural parameters of graft polymer as explained below. After some algebra, the crossover condition separating comb-like polymers from bottlebrushes is written as

 3/2 1/2 −−11(bl) n,/ for n b l   v sc sc (2.4)  32 l nsc,/ for n sc  b l

-1 Figure 2.3 summarizes different regimes of graft polymer in nsc and φ coordinates. In the comb regime (Φ < 1), the backbones can be considered as unperturbed ideal chains with the mean square end-to-end distance of the comb backbone

2 Re, bb n bb lb (2.5) and the effective Kuhn length defined as

11 2 Rebb, bbK  (2.6) nlbb

Figure 2.3 Diagram of states of graft polymers in a melt. SBB – stretched backbone subregime, SSC – stretched side chain subregime, and RSC – rod-like side chain subregime. Logarithmic scales.

In the interval of parameters for which Φ > 1 (bottlebrush regime in Figure 2.3),

3 the backbone stretches to decrease the number of the side chains within the volume Rsc and thus keep the constant monomer density ρ ≈ v−1. From the packing condition (ρv ≈ 1),

3 the number of backbone monomers nR within the volume Rsc decreases with increasing Φ

nR (nsc ng +1) v nR 3    1 (2.7) Rsc nsc

On the length scales larger than the side chain size, a bottlebrush can be considered as a flexible chain of blobs each of size Rsc

22nbb Re, bb R sc n bb lb , for nsc  b/l (2.8) nR

The effective Kuhn length of the bottlebrushes in this regime is defined as

12 2 Rebb, bbK  , for nsc  b/l (2.9) nlbb

In Figure 2.3, this regime is designated as stretched backbone (SBB) subregime to emphasize stretching of the backbone inside a cylindrical envelope of a bottlebrush macromolecule. Eventually, the section of the backbone with nR monomers becomes fully extended when nRl ≈ Rsc . This determines an upper boundary for the SBB subregime as

bl 2  −1  n , for n  b/l (2.10) v sc sc

Above this line, the side chains begin to stretch to satisfy the constant density condition

ρv ≈ 1 . Correspondingly, we designate this regime as stretched side shains (SSC) subregime. The packing condition in this subregime is given by

Rsc (nsc ng + 1) v v 3  2  1, for nsc  b/l (2.11) lRsc lRsc which can be solved for size of the side chains as

v R  , for n  b/l (2.12) sc l sc

Using equation 2.12, the mean square end-to-end distance of the bottlebrush is

22nlbb Re, bb R sc n bb vl /  , for nsc  b/l (2.13) Rsc

The Kuhn length of bottlebrushes in the SSC subregime is

v b  R  , for n  b/l (2.14) K sc l sc

The side chains become fully stretched when Rsc ≈ l nsc. This happens for 13 l 3  −1  n2 , for n  b/l (2.15) v sc sc

Above this line both side chains and backbone are fully stretched on the length scales smaller than the side chain size Rsc. The bottlebrush backbone remains flexible on the length scales larger than the side chain size resulting in the following expression for the mean square end-to-end distance

2 2nlbb 2 Re, bb R sc n bb n sc l (2.16) Rsc

The Kuhn length in this regime is equal to bK ≈ nscl. We call this regime rod-like side chain (RSC) subregime in the bottlebrush region of the diagram of states in Figure 2.3.

Note that our analysis of the properties of graft polymers in the RSC subregime

64, 74, 83 should be applicable to graft polymers with the rod-like side chains (nsc < b/l). In this case, equation 2.15 describes the crossover between combs and bottlebrushes, which is located in the bottom-left corner of the diagram in Figure 2.3. Unlike systems with flexible side chains, the backbone in a comb-like macromolecule in the crossover region

(Φ ≈ 1) is already almost fully stretched. Therefore, with increasing grafting density

(increasing crowding parameter ), the comb regime is directly followed by the RSC subregime, i.e. extension of the side chains. Since the side chains are always shorter than the bare Kuhn length of the backbone (Rsc ≈ nscl < b), the effective Kuhn length of the graft polymer will not be affected by the interaction of side chains, which leads to bK ≅ b.

The results for the effective Kuhn length of the graft homopolymers in different regimes of the diagram of state shown in Figures 2.3 and corresponding regime boundaries are summarized in Table 2.1.

14 Table 2.1 Effective Kuhn length of graft homopolymers in different regimes.

2.2.2 Comparison with Simulations

To verify predictions of the scaling model, we performed coarse-grained molecular dynamics simulations84, 85 of graft polymers in a melt. Macromolecule backbones and side chains are modelled as bead-spring chains composed of beads with diameter σ interacting through truncated shifted Lennard-Jones (LJ) potential. The connectivity of monomers into graft polymers is maintained by the combination of the

FENE and truncated shifted LJ potentials86. We performed simulations of

2 macromolecules with FENE potential spring constants (kspring) equal to 30 kBT/σ and 500

2 kBT/σ , where kB is the Boltzmann constant and T is the absolute temperature. The set of architectural parameters for studied systems is summarized in Table 2.2. For all studied systems, the monomer density is set to ρ = 0.8 σ -3. The simulation details are described in the Simulation Methods section of this chapter.

15 Table 2.2 Summary of studied systems. Data sets with nbb = 100 are obtained from Macromolecules 2015, 48, 5006-5015.

2.2.2.1 Bond-Bond Correlation Function

The effective Kuhn length of graft polymers in a melt is obtained from bond-bond correlation function of the backbone bonds. This function describes the decay of the orientational correlation between two unit bond vectors ni and ni+s pointing along backbone bonds and separated by s bonds and is defined as

1 nsbb −−1 Gs( ) = nni i+s (2.17) nsbb −−1 i=1 where the brackets 〈…〉 denote averaging over backbone configurations. To avoid chain end effects, we neglected 20 bonds on each backbone end when calculating bond-bond

16 correlation function. Figure 2.4 illustrates a typical bond-bond correlation function obtained in the simulations.

Figure 2.4 Typical bond-bond correlation function for graft polymers with kspring = 30 2 kBT/σ , ng = 4 and the DP of side chains nsc = 2 (red circles), 4 (green triangles), 8 (blue inverted triangles), 16 (magenta diamond) and 32 (cyan pentagons). Solid lines show the best fit curves of equation 2.18

Here we follow the approach developed in refs38, 87-89 to analyze the bond-bond correlation function. In the framework of this approach, the simulation data are fitted by the double-exponential decay function of the following form

 s   s  G(s) = (1−)exp−  + exp−  (2.18)  1   2  where α, λ1 and λ2 are fitting parameters. The existence of the two different correlation lengths λ1 and λ2 is the evidence of two different mechanisms of chain deformation. At short length scales, the decay of the orientational correlation is due to local chain tension, while at long length scales, it is a result of interactions between neighboring side chains.

Note that at even longer length scales, the bond-bond correlation function deviates from

17 the double-exponential function and should have a power law decay39, 78, 90. For this reason, the bond-bond correlation function was only fitted in the range 0 ≤ |s| ≤ 20.

2.2.2.2 Effective Kuhn Length

With the given bond-bond correlation function, the mean-square end-to-end distance of a backbone section with s bonds can be written as

2 nsbb − is+−1 221  Rsle () = n j nsbb − iji==1  (2.19)

2 =−+lgsgs((1)(,)(,)) 12 where l is the bond length and function g(λ,s) is defined as

−1/  −s /  1+ e −1/  1− e g(,s) = s −1/  − 2e 2 (2.20) 1− e (1− e−1/  )

Therefore, the effective Kuhn length bK of the comb or bottlebrush macromolecules can be calculated from fitting parameters of the bond-bond correlation function as

R2 (s) b = e = l((1−  )h( )+ h( )) (2.21) K sl 1 2 s→ where the function h(λ) is defined as

1+ e−1/  h() = (2.22) 1− e−1/ 

Figure 2.5 combines our simulation results for the Kuhn length obtained using equation 2.21 and values of the parameters α, λ1 and λ2 from fitting of the bond-bond correlation function. In accordance with the scaling model predictions (see equations 2.6 and 2.9), the reduced Kuhn length bK/b is plotted as a function of the crowding parameter

Φ. The value of the Kuhn length b for this plot was obtained from the analysis of the 18 simulation data for the bond-bond correlation function of linear chains. The value of the

2 bond length is equal to l = 0.985 for kspring = 30 kBT/σ and 0.837 for kspring = 500

2 −1 kBT/σ . The monomer excluded volume is estimated from the monomer density v = ρ .

As expected, all simulation data have collapsed into one universal plot. In a comb regime the effective Kuhn length saturates at b. With increasing the crowding parameter,  , the interaction between side chains results in stiffening of macromolecules which is manifested by the increase of the effective Kuhn length bK. In the range of crowding parameter Φ > 1, we recover a scaling dependence for the effective Kuhn length of bottlebrushes, bK ≈ bΦ (see equation 2.8). Note that the universal scaling relation bK ≈ bΦ indicates that all our simulation data in the bottlebrush regime could belong to the stretched backbone (SBB) subregime in Figure 2.3.

Figure 2.5 Dependence of the normalized Kuhn length, bK/b, of the graft polymers on the crowding parameter Φ (see equation 2.1). Thin solid lines show scaling predictions in comb and bottlebrush regimes. Symbols are summarized in Table 2.2.

19 Figure 2.6 (a) Dependence of the mean-square end-to-end distance of the section of the graft polymer backbone with n bonds on the number of bonds in a section, for 2 macromolecules with ng = 4, values of the spring constants kspring = 30 kBT/σ (filled 2 symbols) and 500 kBT/σ (open symbols) and the DP of side chains nsc = 2 (red circles), 4 (green triangles), 8 (blue inverted triangles), 16 (magenta diamond) and 32 (cyan pentagons). The solid and dash lines in this figure correspond to equation 2.19. (b) Dependence of the normalized mean-square end-to-end distance of the section of the graft polymer backbone with n bonds on the number of Kuhn segments in such sections. Symbols are summarized in Table 2.2.

2.2.2.3 Dimensions of Backbones and Side Chains

Figure 2.6a shows the simulation results for the mean square end-to-end distance

2 of the section of the backbone with n bonds, 〈Re,bb(n)〉. The stronger than linear growth of the mean square average end-to-end size of the section of the backbone with number of bonds n < 10 indicates that these sections of the backbone are stretched. Also, for this backbone sections the section size is independent on the DP of the side chains and is controlled by the local packing condition of beads. However, for larger backbone sections

2 approaching the DP of the full backbone, 〈Re,bb(n)〉 follows the linear scaling dependence.

This is exactly what one would expect for a linear chain with the effective Kuhn length

2 bK. This is confirmed in Figure 2.6b showing collapse of the data when 〈Re,bb(n)〉 is

20 normalized by the square of the effective Kuhn length. Note that the broadening of the crossover and data spreading in the crossover region in Figure 2.6b is a manifestation of the multiscale nature of the bond-bond correlation function (see equation 2.18 and Figure

2.4).

Figure 2.7 Dependence of the normalized mean square end-to-end distance of graft polymers as a function of the crowding parameter. Thin solid lines show scaling prediction in comb and bottlebrush regimes. Symbols are summarized in Table 2.2.

Figure 2.7 shows our data set for normalized size of the graft polymer backbone as a function of the crowding parameter, Φ, in both comb and bottlebrush regimes of the diagram of states shown in Figure 2.3. In the comb regime corresponding to interval of crowding parameter Φ < 1, the data points saturate indicating that the statistics of graft polymer backbones is the same as that of linear chains. However, in the interval of the crowding parameter Φ > 1, there is a linear increase of normalized graft polymer size

2 1/2 1/2 with crowding parameter, 〈Re,bb〉 ∝ Φ . This behavior is in agreement with the prediction of the scaling model (see equation 2.8). Note that the plots shown in Figures

2.5 and 2.7 look similar. This points out the fact that the statistics of the graft polymer 21 backbone is controlled by the local monomer packing and interactions between side chains.

Figure 2.8 (a) Dependence of the normalized mean-square end-to-end distance of the section of the side chain with n bonds on the number of Kuhn segments in such sections. Solid lines show simulation results for linear polymer chains with spring constant kspring = 2 2 30 kBT/σ (red) and 500 kBT/σ (blue) in a melt. (b) Normalized mean-square end-to-end distance of the section of the side chains with n bonds for graft polymers. The 2 normalization factor 〈Re,0(n)〉 is the mean square end-to-end distance of the section of the linear chain staring from the point nsc from the chain end with n bonds in it and its end point locating between nsc and linear chain end. Symbols are summarized in Table 2.2.

Figure 2.8a confirms that for almost all our systems side chains maintain their ideal chain conformations and these data sets correspond to comb regime and stretched backbone (SBB) subregime. Note that for bottlebrush systems with two side chains grafted to each backbone monomer (ng = 0.5) the crowdedness of the monomers forces stretching of the side chains to maintain the monomer volume fraction as shown in Figure

2.8b. These bottlebrushes belong to the stretched side chain (SSC) subregime. Note that side chains are nonuniformly stretched with stretching first increase for short sections of the side chains then begins to decrease as the number of bonds in the side chains

22 increases further. This classification of studied systems is further corroborated by diagram of states shown in Figure 2.9.

2.2.2.4 Diagram of States

Figure 2.9 shows the diagram of states of graft polymers. The crossover line from comb to bottlebrush regime is calculated by setting the value of the crowding parameter at crossover to Φ = 0.7 (see Figures 2.5 and 2.7). Note that it would be difficult to separate different bottlebrush regimes just looking at the dependence of the effective

Kuhn length or the mean square end-to-end distance of the backbone as a function of the crowding parameter Φ. In both SBB and SSC regimes, the side chain size has an identical

1/2 scaling dependence, Rsc ∝ nsc , for large nsc. This explains a good collapse of the data shown in Figures 2.5 and 2.7 even though data sets for ng = 0.5 are in SSC regime as follows from Figure 2.9.

Figure 2.9 Diagram of states of graft polymers with values of the spring constants kspring= 30 2 2 kBT/σ (a) and 500 kBT/σ (b) in a melt. SBB – stretched backbone subregime, SSC – stretched side chain subregime, and RSC – rod-like side chain subregime. Intersect point of crossover lines between different graft polymer regimes is set at nsc = 1. Symbols are summarized in Table 2.2. 23 2.3 Graft Copolymers in a Melt

In experiments, despite the flexibility in the choice of synthesis techniques, special care should be taken during the polymerization process to produce combs and bottlebrushes with chemically identical monomers. More commonly, however, the outcomes of the synthesis process are graft polymers with side chains and backbones consisting of chemically different monomers, e.g. poly(norbornene)-graft-poly(lactide)

(PNB-g-PLA) copolymers46, 79, 81. The difference in properties of backbones and side chains may significantly affect polymer conformations, particularly in the case of loosely grafted side chains. For example, side chains grafted to a relatively stiff backbone will have a marginal effect on the overall macromolecular conformation and related physical properties until the side-chain-induced stiffness overcomes the intrinsic backbone stiffness. To describe properties of graft copolymers with chemically different backbones and side chains one must explicitly account for the differences in their monomers’ excluded volumes, projection lengths and Kuhn lengths. Herein we present the results of scaling analysis and molecular dynamics simulations of static properties of such copolymers and their diagrams of states in a melt.

2.3.1 Scaling Analysis

Consider graft copolymers consisting of a linear chain backbone made of monomers of type A having the DP nbb with grafted side chains made of monomers of type B with DP nsc (see Figure 2.1). Each macromolecule consists of side chains that are equally spaced with ng bonds along the backbone between two neighboring grafting points. To account for two different types of monomers we assume that the backbone

24 monomers have excluded volume vb, monomer projection length lb, and Kuhn length bb and the monomers belonging to the side chains have excluded volume vs , monomer projection length ls, and Kuhn length bs. Here we only consider graft copolymers with backbones stiffer than the side chains, bb > bs, which are pertinent to the experimentally relevant situations46, 79, 81. It is important to point out that we are considering graft copolymers whose dissimilarity in molecular parameters does not lead to microphase separation of the side chains and backbones. This condition is satisfied if the product of the Flory-Huggins parameter  and net DP of the repeating motif of the graft polymer nsc

+ ng is smaller than the corresponding value of the critical point of the spinodal for

91 microphase separation, (nsc + ng )  . Partitioning monomers between side chains and backbones (composition of graft copolymers) is quantified by the volume fraction of backbone monomers, defined as

 n  = bg (2.23) bgsscnn+

2.3.1.1 Crowding Parameters.

Similar to the analysis in Section 2.2.1, the classification of the graft copolymers into “combs” and “bottlebrushes” is based on the concept of the crowding parameter, Φ, describing the mutual interpenetration of the graft copolymers in a melt. This parameter is defined as a ratio of the volume occupied by monomers of a test macromolecule

−1 Vvmb+= nv bbs n sc nnv bbgb n bb /  (2.24) to the pervaded volume

3 V nbb R sc / g (2.25)

25 of a chain of nbb/g backbone blobs with size equal to that of the side chains, Rsc, and each having g backbone monomers (see Figure 2.2a):

−1 Vgvmb  3 (2.26) VRsc

In order to write down an explicit expression of the crowding parameter in terms of the molecular parameters of graft copolymers we have to consider three different cases of the backbones and side chains conformations on the length scales of the blob size Rsc: (i) both backbones and side chains are flexible; (ii) rigid backbones and flexible side chains; and (iii) both backbones and side chains are rigid.

2 (i) Flexible backbones and side chains (nsc > bb/bsls). At low grafting densities, both the side chains and backbones display statistics of a random walk therefore the size of side chains is equal to

1/2 Rbscsssc ln ( ) (2.27)

Due to different rigidities of the backbones and side chains, there are g ≠ nsc backbone

1/2 monomers inside the blob with the size of the side chain Rsc ≈ (bblbg) . This results in the number of backbone monomers in the blob g ≈ nscbsls/bblb. Using the definition of the crowding parameter, Φ, (see equation 2.26) and the expression for the side chain size and number of backbone monomers in a blob we obtain

v  −1  b , for nbb l 2 / (2.28) 12/ n12/ scbs s (ls bl sb) b b sc

The boundary for applicability of this expression is given by the crossover condition to rigid backbones on the length scales of the blob size, Rsc≈bb. This takes place for the DP

2 of the side chains nsc ≈ bb/bsls. 26 2 (ii) Rigid backbones and flexible side chains (bs/ls < nsc ≤ bb/bsls). Decrease in

2 the DP of the side chains nsc below bb/bsls leads to a smaller blob size Rsc in comparison with the Kuhn length of the backbones, bb. For shorter flexible side chains, the number of the backbone monomers in the blob Rsc ≈ lbg is estimated as g ≈ √nscbsls/lb. Using this relationship, the volume fraction of monomers belonging to a test macromolecule with flexible side chains is estimated as

−1 vb  2  , for bs// l s n sc b b b s l s (2.29) l bssbsc l n

(iii) Rigid backbones and side chains (nsc ≤ bs/ls). Finally, when the side chains become shorter than their Kuhn length, both the backbones and side chains are rigid on the length scale of the blob size, such that Rsc ≈ lsnsc ≈ lbg. In this case, the crowding parameter is equal to

−1 vb   22, for nblscss / (2.30) llnsbsc

Note that the definition of the crowing parameter introduced above is only correct

in the interval of parameters where Rsc > Rng. If the opposite inequality holds, Rsc ≤ Rng, one should use the size of the spacer between grafting side chains with number of bonds ng for the blob size. In this case the crowding parameter is defined as

nv −1  gb , for RR (2.31) R3 sc ng ng

However, this correction in the blob size definition is only important in the small portion of the diagram of states as we will explain at the end of the next section. It is also interesting to point out that the value of the crowding parameter for linear chains (φ = 1) 27 −1/2 3/2 is estimated as Φ ≈ nbb vb/(bblb) , which corresponds to the volume fraction of monomers in a polymer coil.

2.3.1.2 Diagram of States of Graft Copolymers

Following our analysis of graft homopolymers in Section 2.2.1, we use the crowding parameter to separate different regimes of graft copolymers in the diagram of

−1 states in the (nsc, φ ) plane as shown in Figure 2.10a. For comparison in Figure 2.10b we present the diagram of states for melts of graft homopolymers. In particular, the crossover between combs and bottlebrushes is obtained by setting Φ ≈ Φ* ≈ 1.0, where

Φ* is the crossover value of the crowding parameter, and solving equations 2.28, 2.29 and 2.30 for φ−1 describing “dilution” of the backbone by the side chains as a function of the DP of the side chains nsc. After some algebra, the crossover conditions separating comb-like copolymers from bottlebrushes is written as

(b lb)1/2 l nfor nbb1/22,/ l   s sb b scscbs s −−112   vlbs b s l b nfor scssscbs blnbb,// s l (2.32)  l22 l nfor,/ nbl   s b scscss

In the comb regime, the backbones can be considered as unperturbed ideal chains with the mean square end-to-end distance of the comb backbone

2 Re, bb n bb l b b b (2.33) and the effective Kuhn length

2 Re, bb bbK b (2.34) nlbb b

28 (a) Graft Copolymers (b) Graft Homopolymers  −1

2 blvbbb /

blv2 / 2 bs lv bb/

1/+ vvsb 2 1 1 2 nsc vb bs bb bb v b 2 3 llsb ls ls blss l l

Figure 2.10 (a) Diagram of states of graft copolymers with bb > bs in a melt. SBB – stretched backbone subregime, SSC- stretched side chain subregime, and RSC – rod-like side chain subregime. Black solid lines are boundaries between comb and bottlebrush regimes, and red dashed lines show boundaries of different bottlebrush subregimes. The part of the diagram filled by red-white stripes corresponds to bottlebrushes with effective backbone Kuhn length bK ≈ bb. The upper boundary of the accessible region is given by -1 -1 max max φ ≤ φmax = nscvsΤng vb +1, which is shown as the red solid line for ng = 1. (b) Diagram of states for graft polymers with identical backbones and side chains (b = bb = bs, l = lb = ls and v = vb = vs). Notations are the same as in panel (a). Logarithmic scales.

In the bottlebrush regime, the value of the crowding parameter Φ > 1 (see Figure

2.10a). Note that these values of Φ correspond to a hypothetical system, where graft copolymers maintain ideal conformations of side chains and backbones even for large grafting densities. In real systems, for the range of parameters where Φ > 1, the backbone

3 should stretch to decrease the number of the side chains within the volume Rsc to keep the monomer number density constant. The number of backbone monomers g within the

3 volume Rsc is determined from the following packing condition

29 −1 gvb g l b b b 1 3   (2.35) Rsc n sc l s b s

On length scales larger than the side chain size, a bottlebrush macromolecule can be considered as a flexible chain of blobs each of size Rsc

n RRlbn22 bb , for n b b l 2 / (2.36) e, bbscbbbb g scbss

The effective Kuhn length of the bottlebrushes in this regime is

2 Rebb, bbK  b , for (2.37) lnbbb

In Figure 2.10a, this regime is designated as stretched backbone (SBB) subregime emphasizing stretching of the backbone.

Eventually, the section of the backbone with g monomers becomes fully extended when Rsc ≈ lbg. This determines an upper boundary for the SBB subregime in terms of molecular parameters of graft polymers

−1 bllssb   nsc , for nblscss / (2.38) vb

Above this line, the side chains begin to stretch to satisfy the packing condition.

Therefore, we named this regime stretched side chains (SSC) subregime. By considering the monomer packing condition, the size of the side chains in this subregime is

−1 Rvvvscbbb 32  1 Rsc , for nsc b s/ l s (2.39) lb Rl scb Rl scb 

Using equation 2.39, the mean square end-to-end distance of the bottlebrush can be written as

30 22lnvlbbbbb RRne, bbscbb , for Rbs c b (2.40) Rsc 

−1 2 Note that the condition Rsc ≥ bb is satisfied as long as φ ≥ bblb/vb. In this case the effective Kuhn length of bottlebrushes in the SSC subregime is

vb −12 bRK sc , for   bbbb l v / (2.41) lb

However, when the side chain size, Rsc, becomes smaller than the backbone Kuhn length

−1 2 bb, which takes place for φ < bblb/vb, the effective Kuhn length of the bottlebrush is equal to that of the backbone bK ≈ bb.

Finally, the side chains are fully stretched Rsc ≈ lsnsc when the graft polymer composition is on the order of

2 −12llsb   nsc , for nblscss / (2.42) vb

Thus, above this line both side chains and backbone are fully stretched on the length scales smaller than the side chain size Rsc. We call this regime rod-like side chain (RSC) subregime in the bottlebrush regime of the diagram of states in Figure 2.10a. The bottlebrush backbone remains flexible on the length scales larger than the side chain size as long as Rsc ≥ bb. This results in the following expression for the mean square end-to- end distance:

22lnb bb RRle, bbscs l n b nsc bb , for nsc b b/ l s (2.43) Rsc

The effective Kuhn length in this subregime is equal to

−1 2 2 bK  ls n sc , for   ls l b n sc/ v b (2.44)

31 The renormalization of the Kuhn length due to side chain interactions continues until the side chain size becomes on the order of the backbone Kuhn length, nsc ≈ bb/ls . For shorter side chains nsc < bb/ls the effective Kuhn length of a graft copolymer is equal to that of the backbone, bb.

Note that while there are no theoretical limitations on selecting ng, the chemical structure of the backbone dictates the maximum number of side chains possible to graft to

max −1 −1 a backbone monomer 1/ng . This constraint defines an upper boundary φ ≤ φmax =

max nscvsΤng vb +1 on chemically accessible regimes (forbidden region in Figure 2.10).

The results for the effective Kuhn length of the graft copolymers in different regimes of the diagram of state shown in Figures 2.10a and corresponding regime boundaries are summarized in Table 2.3. Analysis of the expressions for the effective

Kuhn length and regime boundaries obtained for graft copolymers indicates that they can be reduced to those derived for the graft polymers with identical side chains and backbones by setting bb = bs, lb = ls and vb = vs. (see Table 2.1, Figures 2.10b and 2.3).

Table 2.3 Effective Kuhn length of graft copolymers in different regimes

32 At the end of this section we would like to comment on using the side chain size,

Rsc , to define the blob dimensions for calculations of the crowding parameter and determining the crossover between comb and bottlebrush regimes. This definition of the blob size breaks down in the small region of the diagram of states where the volume fraction φ ≈ 1 and DP of the side chains nsc ≈ 1. Note that our boundary (see equation

2.32) between combs and bottlebrushes in this range of parameters should be viewed as a lower bound for an actual crossover. However, this substitution in the blob size definition does not change expressions for the chain size and the effective Kuhn length.

2.3.2 Comparison with Simulations

The scaling model of graft copolymers in a melt was tested in coarse-grained molecular dynamics simulations.84, 85 We modeled systems with different bending rigidities of the backbones and side chains and different sizes of the backbone and side chain monomers. In these simulations graft macromolecules were represented by bead- spring chains.86 All beads in a system interacted through modified truncated-shifted

Lennard-Jones (LJ) potentials. We performed two sets of simulations. In the first set of simulations both backbone and side chain beads had the same diameter db = 1.0σ, while in the second set of simulations the diameter of the backbone beads was increased to db =

1.5σ. The connectivity of the beads into backbones and side chains was maintained by the

2 FENE bonds with bond spring constant ksping = 30kBT/σ . The bending rigidity of the backbone was introduced through a bending potential with bending constant K.85 The side chains were modeled as flexible chains for which K = 0. We performed simulations of melts consisting of M graft copolymers. Simulations of graft copolymers with identical

33 beads were performed under NVT condition, with the monomer density set to ρ = 0.8σ−3.

Simulations of graft copolymers with larger backbone monomers were performed at constant pressure, while the value of which was set to that of the systems with identical beads. Monomer volume, bond length and Kuhn length of backbones and side chains in simulations are summarized in Table 2.4. Architectural parameters for studied systems are summarized in Table 2.5. The simulation details are described in the Simulation

Methods section.

Table 2.4 Monomer volume, bond length and Kuhn length for different systems.

34 Table 2.5 Summary of studied systems and symbol notations.

2.3.2.1 Bond-Bond Correlation Function

We begin our discussion of the simulation results by elucidating the effect of the graft copolymer structure on the stiffening of graft copolymers. The stiffening of backbones caused by steric repulsion between side chains can be analyzed by the bond- bond correlation function of the backbones, G(s) (see equations 2.17). Following the same approach used in section 2.2.2.1, we analyzed the bond-bond correlation function of

35 graft copolymer backbones and fitted them with a sum of two exponential decay (see equation 2.18). G(s) of some studied melts are shown in Figure 2.11. Note, that by increasing the backbone intrinsic bending rigidity and spacing between grafted side chains, the correlation function G(s) transforms into a single exponential function (see open symbols in Figure 2.11). This single exponential form of the bond-bond correlation function is a characteristic feature of semiflexible linear polymer chains.

Figure 2.11 Bond-bond correlation functions G(s) of graft copolymer backbones with the DP of side chains nsc = 8, different intrinsic backbone stiffness K and side chain grafting density, 1/ng. Solid lines represent the best fits to equation 2.18 using α, λ1 and λ2 as fitting parameters. Symbol notations are summarized in Table 2.5.

2.3.2.2 Effective Kuhn Length

Knowing the backbone bond-bond correlation function, we can calculate the effective Kuhn length of graft copolymers using the same approach in section 2.2.2.2.

Using equation 2.21 together with the fitting parameters obtained from fitting the bond- bond correlation functions by equation 2.18, the effective Kuhn length of graft copolymers can be obtained. The results of these calculations are summarized in Figure

36 2.12, showing the dependence of the effective Kuhn length of graft polymers on the crowding parameter Φ. To illustrate the universality of this representation we have also included simulation results obtained for graft homopolymers (see section 2.2.2.2).

Figure 2.12 Dependence of the reduced Kuhn length, bK/bb, of graft copolymers on the crowding parameter, Φ. The crowding parameters are calculated by equations 2.28~2.31. Colored symbols are summarized in Table 2.5. Gray circles represent graft polymers with K = 0 and identical backbones and side chains (i.e. graft homopolymers, see section 2.2.2.2 and Figure 2.5). Solid black lines show pure scaling regimes of the effective Kuhn length dependence on the crowding parameters for combs and bottlebrushes. The vertical dashed line corresponds to the crossover between combs and bottlebrushes, Φ ≈ * Φ ≅ 0.7.

It follows from this figure that for small values of the crowding parameter, Φ ≪ 1, interactions between side chains are too weak to change the backbone conformations, such that the effective Kuhn length is on the order of the bare Kuhn length of the backbone. At intermediate values of the crowding parameter (0.3 < Φ < 1), repulsion between side chains results in local backbone stretching and stiffening, lead to an increase of the effective Kuhn length. For densely grafted side chains such that the value of the crowding parameter becomes larger than unity, Φ > 1, the steric repulsion between side chains dominates backbone bending rigidity. In this regime, the effective Kuhn

37 length scales linearly with the crowding parameter, bK ≈ Φbb . This agrees with the scaling expression given by equation 2.37. Note that the analysis of different scaling regimes of the effective Kuhn length dependence allows us to establish the crossover value of the crowding parameter between comb and bottlebrush regimes. This crossover corresponds to the value of the crowding parameter Φ ≈ Φ* ≅ 0.7. It is important to point out, however, that not all points collapse into the universal curve. The deviation from the universal behavior is observed for two systems with bending constant of the backbone K

= 4.0. We will illustrate below that these two systems belong to the stretched side chain

(SSC) subregime with a weak renormalization of the effective Kuhn length.

2.3.2.3 Diagram of States

Having established the location of the crossover between combs and bottlebrushes from the analysis of the effective Kuhn length dependence on the crowding parameter, we can construct the diagram of states for the simulated systems by setting the crossover value of the crowding parameter Φ ≈ Φ* ≅ 0.7 (see Figure 2.12) and using this expression to rewrite equation 2.32 for φ−1 as a function of the side chain DP. Figure

2.13 shows diagrams of states for systems with the backbone bending rigidity K = 1.5 and

K = 4.0 as well as the backbone bending rigidity K = 1.5 and backbone bead diameter

1.5. For comparison in Figure 2.13d we show the diagram of states of graft polymers with identical backbones and side chains having K = 0 (i.e. graft homopolymers, see

Figure 2.9a). To construct these diagrams of states we used monomer volume, bond length and Kuhn length for different systems as summarized in Table 2.4.

38 Figure 2.13 Diagrams of states of graft copolymers with backbone bending constants K = 1.5 (a), K = 4.0 (b) and K = 1.5 and backbone bead diameter 1.5σ (c). (d) Diagram of states of melts of graft polymers with identical side chains and backbones. Black solid lines are boundaries between comb and bottlebrush regimes, and red dashed lines show boundaries of different bottlebrush subregimes. The part of SSC subregime below the horizontal black dashed line and the part of the RSC subregime on the left from the vertical black dashed line correspond to bottlebrushes with the effective Kuhn length being on the order of backbone Kuhn length. Colored dotted curves represent combs and bottlebrushes with the same ng. Symbols in panel (a), (b), and (c) are listed in Table 2.5.

It follows from these figures that the increase of backbone stiffness will broaden the range of parameters where graft copolymers are combs. For example, in order to synthesize bottlebrushes with the fixed spacer length ng = 2 (green dashed lines in Figure

2.13), the DP of the side chains should be larger than nsc ≈ 15 and nsc ≈ 117 respectively for graft copolymers with backbone bending constant K = 1.5 (Figure 2.13a) and K = 4.0 39 (Figure 2.13b). Increase of the intrinsic backbone stiffness opens a “window” in which the effective Kuhn length is controlled by the backbone bending rigidity. In this

“window” the steric repulsion between side chains is too weak to generate significant backbone stiffening. This is illustrated by systems of graft copolymers with K = 4.0, nsc = 8, ng = 0.5 and ng = 1.0 shown by open red and orange squares in Figure 2.13b. For these two systems, the effective Kuhn length is smaller than that of other systems with the same value of crowding parameter (see Figure 2.12). For bigger backbone beads (see

Figure 2.13c), the crossovers to SBB and SSC subregimes shift to the left corner of the diagram of states and shrink the comb regime in this interval of parameters.

2.4 Comparison with Experiments

We apply the scaling approach to construct a diagram of states of poly(norbornene)-graft-poly(lactide) (PNB-g-PLA) system.46 For these graft copolymers, the backbones are stiffer than the side chains. Molecular parameters for poly(norbornene)

3 backbone are lb = 0.61 nm, bb =1.7 nm, vb = 0.349 nm , and for poly(lactide) side chains

3 are ls = 0.37 nm, bs = 1.0 nm, vs = 0.095 nm . The value of the Kuhn length of the PNB backbone was estimated from the packing parameter and value of the shear modulus of

PNB.92 The crossover between combs and bottlebrushes was determined to be Φ ≈

Φ* ≅ 0.4.93 This value of the crossover crowding parameter was obtained by collapsing entanglement shear modulus data for poly(norbornene)-graft-poly(lactide) and poly(n- butyl acrylates) into one universal curve in the bottlebrush regime (see Chapter IV). Note, that at this time there is no undisputed explanation why one should change the crossover value of the crowding parameter for PNB-g-PLA systems from 0.7 to 0.4. Figure 2.14

40 presents the diagram of states of PNB-g-PLA graft copolymers. It follows from this figure that for majority of the diagram of states is occupied by combs and bottlebrushes with stretched backbones. By increasing the side chain DP, the forbidden region boundary approaches a crossover line between SSC and SBB subregimes. Thus, for these graft copolymers, in order to explore SSC subregime one should synthesize systems with shorter side chains.

Figure 2.14 Diagram of states of PNB-g-PLA graft copolymers. Red and blue filled circles represent bottlebrushes and combs respectively. Black solid line is the boundary between comb and bottlebrush regimes. Dashed red line separates different bottlebrush subregimes. Colored dotted curves represent graft copolymers with constant ng . The filled area corresponds to the forbidden region due to the chemistry limitation on the max maximum number of side chains per backbone monomer, 1/ ng = 1.

2.5 Simulation Methods

We performed coarse-grained molecular dynamics simulations of graft polymers in a melt. Graft polymer chain are represented by bead-spring chains, and each of them consists of a linear backbone chain with the number of monomer nbb = 129 and side chains with nsc = 2 ~ 40 monomers grafted to the backbone with ng = 0.5 ~ 32 backbone

41 bonds between nearest grafting points. In the case of ng = 0.5, two side chains were grafted to each backbone monomer. All macromolecules are central symmetric ended by two linear chain sections with the number of bonds in each being ng.

To model systems with different bead sizes we used the modified truncated- shifted Lennard-Jones (LJ) potential, which accounts for increased size of the beads through offsetting the interaction range by Δ. In this representation the interaction potential between two beads separated by a distance r is written as

 12 6 12 6              4  − −   +    r  rcut +  U (r) = LJ         (2.45) LJ   r −    r −    rcut   rcut      0 r  rcut + 

1/6 where the cutoff distance rcut = 2 σ. The value of the LJ interaction parameter was set to

εLJ = 1.5kBT (where kB is the Boltzmann constant and T is the absolute temperature). The connectivity of the beads into chains was described by the finite extensible nonlinear elastic potential (FENE) 86

UrkRrR = −−−0.5ln  1/ 222 FENEmaxmax( ) spring ( ( ) ) (2.46) with the maximum bond length Rmax = 1.5σ. The value of the spring constant was set to

2 2 kspring = 30kBT/σ or 500kBT/σ in one melt for graft homopolymers (section 2.2) and was

2 set to kspring = 30kBT/σ for graft copolymers (section 2.3). The repulsive part of the bond potential was represented by the truncated-shifted Lennard-Jones (LJ) potential with εLJ =

1/6 1.5kBT and rcut = 2 σ (see equation 2.45). For identical backbone and side chain beads, the parameter Δ = 0 such that their size was equal to 1.0σ. In simulations with the bulkier backbones, we set the value of the parameter Δ = 0.5σ for backbone-backbone pairs, Δ =

42 0.25σ for backbone-sidechain pairs and Δ = 0 for sidechain-sidechain pairs. With such parameter settings, the diameters of the backbone and side chain beads are 1.5σ and 1.0σ respectively.

The bending rigidity of the backbone was introduced by imposing the bending

potential controlling mutual orientations between consecutive unit bond vectors ni and

ni+1 along the polymer backbone

bend Ui, i++ 1= k B T K (1 −(nn i  i 1 )) (2.47) where K is the bending constant which values were equal to 0.0 (graft homopolymers),

1.5 or 4.0 (graft copolymers).

Simulations of graft homopolymers and graft copolymers with backbone and side chain beads of identical sizes were carried out in the canonical (NVT) ensemble. The constant temperature was maintained by coupling the system to a Langevin thermostat implemented in LAMMPS.84 In this case, the equation of motion of the i-th bead is given by the following equation

dv (t) mi =F( t) − v( t) + FR ( t) (2.48) dt i i i where m is the beads mass set to unity for all beads, vi(t) is the bead velocity, and Fi(t) is

R the net deterministic force acting on the i-th bead. The stochastic force Fi (t) acting on the

R R i-th bead has a zero average and a δ-function correlation 〈Fi (t) Fj (t') 〉 = 6kBTξδijδ(t

− t'). The friction coefficient ξ coupling a system to a Langevin thermostat is set to ξ =

1/2 0.1m/τLJ, where τLJ = σ(m/kBT) is the standard LJ-time. The velocity-Verlet algorithm

43 with a time step Δt = 0.005τLJ was used for integration of the equation of motion. All simulations were performed using LAMMPS under 3-D periodic boundary conditions.

Simulations were performed in accordance with the following procedure.

Macromolecules were randomly placed in a simulation box with monomer number

-3 density equals to 0.8σ . A simulation run lasting 250τLJ with non-bonded interactions between beads turned off was performed in order to relax the macromolecules’ initial conformations. The strength of the LJ interaction parameter εLJ between beads was then gradually increased to 1.5kBT. This was followed by a simulation run continued until the mean square end-to-end distance of the backbones reached an equilibrium (saturated as a

5 function of time). The equilibration run was followed by a product run lasting 5×10 τLJ which was used for data collection.

Simulations of graft copolymers with bigger backbone beads were first performed

3 at a constant pressure P = 4.5kBT/σ (NPT) ensemble to relax the conformation of macromolecules. The value of the pressure was set to that in the melt of graft copolymers with identical backbone and side chain beads. The constant pressure and temperature were maintained by coupling the system to Nose-Hoover barostat and thermostat85 with the pressure and temperature damping parameter both equal to 10τLJ. The relaxation run was performed until the mean square end-to-end distance of the backbones reached an equilibrium (saturated as a function of time). The production run was then performed in

5 the NVT ensemble as described above for duration 5×10 τLJ.

2.6 Conclusions

We developed a scaling model for graft polymers in the melt state. For both graft homopolymers and graft copolymers, we demonstrated that the classification into combs 44 and bottlebrushes could be carried out according to the crowding parameter, Φ, describing the mutual interpenetration between different macromolecules. In the comb regime (Φ < 1), the side chains play the role of a backbone diluent and graft polymers behave like linear chain backbones. In the bottlebrush regime (Φ > 1), steric repulsion between side chains expels monomers belonging to surrounding graft copolymers from the pervaded volume occupied by the test macromolecule. In this regime, graft polymers behave as mesoscopic filaments with effective Kuhn length proportional to the side chain size Rsc. In the framework of the scaling approach, we have constructed the diagram of states for graft polymers (see Figure 2.10).

Predictions of the scaling model were tested in molecular dynamics simulations of graft homopolymers as well as graft copolymers with semiflexible backbones and flexible side chains. In the comb regime ( Φ < 1 ), the backbone remains almost unperturbed, and the effective Kuhn length is similar to that of the backbone, bK ≈ bb. In the bottlebrush regime (Φ > 1), the repulsion between side chains stiffens the backbone and the effective Kuhn length increases linearly with the crowding parameter, bK ≈ bbΦ.

The results for effective Kuhn length were used to evaluate a crossover value of the crowding parameters between the comb and bottlebrush regimes, Φ* ≅ 0.7 , and to construct diagrams of states of graft polymers in a melt (see Figure 2.13). In contrast to graft homopolymers (Figure 2.13d), graft copolymers with backbones stiffer than side chains (bb > bs) display a broader range of molecular and architectural parameters where graft copolymers are combs (Figures 2.13a and 2.13b). While increasing the size of the backbone beads (vb > vs) widens the bottlebrush regime in the diagram of states (Figure

2.13c). 45 We have also applied the scaling approach to classify graft copolymers synthesized by the ROMP technique (Figure 2.14). Our analysis shows that the majority of the studied systems belong to the comb regime and the bottlebrush subregime with stretched backbones (SBB). The scaling model developed here will eliminate ambiguity in classification of graft polymer systems with assorted backbones and side chains.1

Parts of this chapter are reprinted with permission from Macromolecules 2017, 50 (8), 3430-3437, and Macromolecules 2019, 52 (10), 3942-3950. Copyright 2017, 2019 American Chemical Society 46 CHAPTER III

SCATTERING FROM MELTS OF COMBS AND BOTTLEBRUSHES

3.1 Introduction

Comb and bottlebrush graft polymers are macromolecules consist of linear chain backbones with grafted side chains as illustrated in Figure 3.1.24, 35-37 Through variation of the grafting density 1/ng of the side chains and their degree of polymerization (DP) nsc, one can control properties of materials at a synthesis stage for which combs and bottlebrushes serve as precursor polymers.8, 25, 30, 94 These polymers were used for the synthesis of networks with a unique combination of softness and elongation at break and thermoplastic elastomers combining photochromic and tissue-like mechanical properties.1,

29, 30 These unique features of comb and bottlebrush based materials are manifestations of the duality of the side chains, diluting the polymer backbones and meanwhile stiffening it through steric repulsion.8, 38, 95 Furthermore, this combination of backbone dilution and stiffening promotes the disentanglement of graft polymers.8, 43-46, 93

Figure 3.1 Snapshot of a bottlebrush macromolecule. Backbone monomers are colored in red; side chain monomers are colored in blue.

47

Scattering experiments can provide detailed information about the conformations of comb and bottlebrush macromolecules which is important for the optimization of properties of polymeric materials utilizing such polymer architectures. Recently the

WAXS technique was used to study molecular conformations of bottlebrushes in a melt and in self-assembled states.29, 96, 97 It was shown that there is a characteristic scattering peak in the range of wavenumbers q* corresponding to the length scales comparable with separation between backbones of the neighboring macromolecules. Specifically, for polyolefins with short side chains the length scale corresponding location of the peak was shown to scale linear with the degree of polymerization of the side chains suggesting a correlation with the backbone-backbone separation.96 This peak assignment in combination with the local packing constraints was used to describe the deformation of the bottlebrush block in self-assembled linear-bottlebrush-linear copolymers.97 The examples presented above illustrate how unique features of the scattering function from graft polymers can be used for the analysis of their conformations. However, until now there was no independent verification that would confirm the peak assignment in the scattering function and its relation to the backbone-backbone separation.

In this chapter, we use a combination of the molecular dynamics simulations and theoretical calculations to establish how architectural parameters of combs and bottlebrushes influence features of the scattering function in a melt. We show that the location of the peak in the scattering function depends on the degree of mutual interpenetration between neighboring macromolecules thus demonstrating a qualitatively different behavior in a melt of comb and bottlebrush macromolecules. The results of the computer simulations agree with calculations based on the melt scattering function

derived in the framework of the Random Phase Approximation (RPA) 91, 98-101 which explicitly accounts for macromolecular architecture and melt incompressibility.

3.2 Simulation Results

We use results of the coarse-grained molecular dynamics simulations of graft polymers (see Chapter II) to calculate the static structure factor in melts of combs and bottlebrushes and to establish how its features correlate with macromolecular conformations at different length scales. In our simulations, macromolecules were represented by bead-spring chains with diameter σ. Each comb and bottlebrush molecule consist of a linear chain backbone with the number of monomers nbb = 129 and grafted side chains having nsc monomers with ng backbone bonds between the nearest grafting points of the side chains. We have varied nsc between 2 and 32, and ng between 1 and 32.

The interactions between beads was modeled by pure repulsive truncated-shifted

1/6 Lennard-Jones (LJ) potential with the cutoff distance rcut = 2 σ and interaction parameter εLJ = 1.5kBT (where kB is the Boltzmann constant and T is the absolute temperature). The connectivity of beads into backbones and side chains was maintained by the bond potential described by a sum of the finite extensible nonlinear elastic (FENE)

2 potential with the spring constant kspring = 30kBT/σ and the maximum bond length Rmax =

1.5σ and pure repulsive truncated-shifted LJ potential. This set of parameters for the bond potential results in the bond length to be l = 0.985σ. To cover a broader class of combs and bottlebrushes with flexible and semiflexible backbones, an angular potential was introduced between neighboring backbone bonds with bending constant kθ = K kBT.

We have performed simulations of polymers with K = 0, 1.5 and 4.0 which corresponds

49 to backbone Kuhn lengths bb =1.93σ, 2.87σ and 7.22σ respectively. Note that the subscript b corresponds to parameters describing backbones while s represents those for side chains. In all our simulations side chains were flexible chains with the bending

3 constant K = 0 and bs = 1.93σ and the monomer density was set to ρσ = 0.8. The simulation details and equilibration procedures can be found in Chapter II, section 2.5.

3.2.1 Diagram of States

In order to establish how chemical structure influences the conformations of graft polymers, we begin our discussion with classification of macromolecules into combs and bottlebrushes. Following the results in Chapter II, we use the crowding parameter, Φ, which describe the degree of mutual interpenetration between side chains and backbones of neighboring macromolecules, for this classification. In our simulations, combs and bottlebrushes were made of monomers with the same excluded volume v which were connected by identical bonds with length l. The only difference between side chains and backbones was their Kuhn lengths. In this case, the crowding parameter depends on the monomer excluded volume v, bond length l, Kuhn lengths (bb, bs), DP of the side chains nsc and volume fraction of the backbone monomers

n  = g (3.1) nngsc+

Since in our simulations graft polymers have flexible side chains, the crowding parameter is written as102

v  −1  3/2 1/2 1/ 2 (3.2) l bs b b n sc

50 For values of the crowding parameter, Φ < Φ*, the side chains and backbones of graft polymers interpenetrate and remain in their unperturbed ideal chain conformations. We call this regime the comb regime. In the interval Φ > Φ* there is a significant renormalization of the effective backbone Kuhn length, bK, due to interaction between side chains. These graft polymers behave similar to filaments with the Kuhn length being proportional to the diameter (side chain size).38 This corresponds to the bottlebrush regime. The crossover value of the crowding parameter Φ*≈ 0.7 was determined from the analysis of the renormalization of the effective Kuhn length of graft polymers in a melt

(see Chapter II, section 2.3.2.2). It is important to point out that the unphysical value of the crowding parameter, Φ > 1, corresponds to a hypothetical system where backbones and side chains maintain their ideal conformations even in macromolecules with densely grafted side chains. In real systems, an increase of grafting density leads to the extension of both backbone and side chains in order to maintain constant density of monomers in a melt.

−1 Solving equation 3.2 for  as a function of the DP of the side chains, nsc, we obtain an expression describing the crossover boundary between comb and bottlebrush macromolecules

l3/2 b 1/2 b  −1 0.7 sbn 1/ 2 (3.3) v sc

In Figures 3.2, this crossover is shown as a solid line separating the comb and bottlebrush regimes for graft polymers with flexible (K = 0) and semiflexible (K = 1.5) backbones respectively. Note that we have not identified different subregimes in the bottlebrush regime characterizing specific backbone and side chain conformations as 51 described in detail in Chapter II. In the next section we will use this classification of graft polymers into combs and bottlebrushes to analyze scattering results.

(a) (b) Bottlebrush Bottlebrush

Comb Comb

Figure 3.2 Diagram of states of combs and bottlebrushes with bending constant K = 0.0 (a) and K = 1.5 (b) in a melt. Open symbols represent combs and filled symbols represent bottlebrushes. Dotted gray curves are constant ng curves calculated by equation 3.1.

3.2.2 Structure Factor

The static structure factor S(q) is defined as

1 NNbb S( qf )exp=−− fi qRR  ijij ( ) (3.4) V ij==11 where q is the scattering vector, fi is the form factor of the i-th bead located at point with the radius vector Ri. Summation in equation 3.4 is carried out over all beads Nb in a system and brackets 〈… 〉 denote the ensemble averaging. To calculate the structure factor of graft polymer melts, the monomer form factors were set to fb = 1.5 for the backbone beads and fs = 1.0 for the side chain beads. In our calculations of the structure factor we have adapted the FFT-based approach previously developed in our group89. In this representation the actual beads are assigned to the regular grid points.

52 Figure 3.3 (a) Static structure factor S(q) in melts of comb polymers with backbone bending constant K = 0, DP of spacer ng = 4 and different DPs of side chains. (b) Dependence of the peak position, q*, in the scattering function on the DP of side chains nsc and DP of spacer ng. Symbol shapes represent different backbone bending constant: K=0 (open circles), K=1.5 (open squares) and K=4.0 (open triangles). Different symbol colors show different ng as illustrated in Figure 3.2. (c) Static structure factor S(q) in melts of bottlebrush polymers with backbone bending constant K = 0, DP of spacer ng = 2 and different DPs of side chains. (d) Dependence of the peak position, q*, on the DP of side chains nsc for bottlebrushes. The dashed line corresponds to the best fit to function y = Ax-α with A = 2.79 ± 0.09 σ−1 and α = 0.39 ± 0.01. Inset shows the characteristic length scale, d = 2π/q*, as a function of root-mean-square end-to-end distance of side 2 1/2 chains, 〈Rsc〉 . The dashed line is the best fit to the function y = ax + c with a = 0.95 ± 0.04 and c = 1.72 ± 0.2σ. Different symbol shapes represent graft polymers with backbone bending constant K=0 (filled circles), K=1.5 (filled squares) and K=4.0 (filled triangles). Different symbol colors show graft polymers with different ng as illustrated in Figure 3.2. 53 Figure 3.3a shows examples of the static structure factors obtained in simulations of melts of combs. There are two clearly identifiable peaks. The large q-peak corresponds to monomer scattering of which location remains unchanged with increasing DP of side chains. The position of the second peak shifts towards the larger q-values as DP of side chains decreases. This is illustrated in Figure 3.3b representing scattering data for different comb systems. The value of q* decreases with increasing both DP of side chains nsc and spacer length ng between them. In the bottlebrush regime (Figure 3.3c) the function S(q) has a form similar to that in comb systems. The new feature, however, is the scaling relationship between the peak position q * and the DP of side chains nsc

(Figure 3.3d). The collapse of the data points corresponding to bottlebrushes with different values of ng and bare backbone Kuhn length bb indicates that the peak position is invariant of the bottlebrush backbone properties and grafting density side chains.

−0.39 Furthermore, the observed scaling dependence q* ∝ nsc indicates that the characteristic length scale corresponding to this peak d = 2π/q* is shorter than the size of the side

2 1/2 1/2 chains 〈Rsc〉 ∝ nsc . However, for short side chains, the ratio between two length scales

0.11 nsc could be difficult to identify as illustrated in the inset in Figure 3.3d showing a

2 1/2 linear correlation between d and 〈Rsc〉 . Furthermore, this correlation fails in melts of combs. In the next section, in order to understand the origin of this peculiar dependence of q* on architectural parameters (i.e. the DP of side chains nsc and spacer length ng), we will calculate the scattering function from the melts of combs and bottlebrushes under the framework of the RPA.

54 3.3 Theoretical Analysis of the Static Structure Factor

The scattering function S(q) can be expressed in terms of the elements Gαβ(q) of the matrix of pair correlation functions G(q) describing correlations in density fluctuations of monomers of types α and β 103, 104

SqffGqffqq( ) ==− ()()()  (3.5) where we use summation over repeating indexes. In rewriting equation 3.5 we take into account definition of the pair correlation function in terms of the monomer density fluctuations, δρα(q). Note that α, β = s corresponds to monomers of the side chains while

α, β = b represents those of the backbones. In the Random Phase Approximation (RPA) 98,

99, 101, 105 the matrix G(q) of the pair correlation functions is written in terms of the matrix of structural correlation functions g(q) describing the arrangement of monomers into macromolecules and the matrix of the direct correlation functions C(q) characterizing interactions between monomers98-101, 103

GgC−−11()()qqq=−( ) (3.6)

Elements gαβ(q) of the matrix of structural correlation function are defined as follows

N N gqi( )exp()=−− qRR (3.7)  Mij  ij==11 where ρM is the number density of macromolecules in a system, which for graft polymers is equal to ρM = ρΤ(nbb+ nsc(nbb − (ng − 1)) Τng ) ≈ ρφΤnbb . The summation in equation

3.7 is carried out over all monomers of types α and β located at points with radius vectors

R and R , and brackets 〈… 〉 denote averaging over conformations of graft polymers. iα jβ

The functions gαβ(q) are proportional to intra-chain correlation functions with 55 proportionality coefficient being number density of macromolecules. We will calculate the structural correlation functions in the case of comb and bottlebrush macromolecules below.

In the framework of the lattice model for a reference monomeric system of beads with equal excluded volume v and solubility parameters δα of bead type α, the elements of the matrix of the direct correlation functions are

2  v Cqv ()=− (3.8) kTvB 1−  where the last term accounts for the incompressibility of a melt with monomer volume fraction ρv = 1. Note that we consider a general case of graft copolymers for which solubility parameters of the backbone and side chains could be different. Good examples of such systems are poly(norbornene)-graft-poly(lactide) (PNB-g-PLA) copolymers.46, 65,

81, 93 This also allows us to establish connections with theoretical models describing microphase separation in multiblock copolymers.91, 101, 106, 107

For incompressible melts, the local density fluctuations should satisfy the incompressibility condition such that the density fluctuations of the monomers belonging to side chains are offset by those from the backbones, δρs(q)= − δρb(q). Taking this into account we can rewrite the expression for the scattering function as follows

2 S( q) =( fs − f b)  s()() q  s − q −1 (3.9) 2 −1 − 1 − 1 =( fs − f b) ( g ss( q ) + g bb ( q ) − 2 g sb ( q ) − 2 v )

−1 −1 2 where gαβ (q) are matrix elements of the inverse matrix g (q), and χ = v(δs − δb) /2kBT is the Flory-Huggins parameter for monomers belonging to side chains and backbones.

56 Thus in this approximation the form of the scattering function is identical to that of multiblock copolymers.91, 101

In order to write down an explicit form of the function S(q) for melts of combs and bottlebrushes, we have to know the structural correlation functions gαβ(q). There are two main building blocks in writing the expression for the structural correlation functions:

(i) correlation function for monomers belonging to the same block and (ii) correlation function for monomers belonging to different blocks as illustrated in Figure 3.4. Here we call blocks either side chains or sections of the backbone between two grafted side chains.

This representation of the correlation function holds for both the comb and bottlebrush regimes.

Figure 3.4 Schematic diagram for the calculation of the structural correlation functions.

57 3.3.1 Comb regime

In this regime, conformations of the side chains and backbones are close to those of the ideal (Gaussian) chains. Therefore, we can consider the contribution from each

2 91, 101 bond to correlation functions as that from a spring with spring constant 3kBT/l . This results in the following expression describing the contribution of each bond into the structural correlation function

22 gqiql( ) =−−=−exp()exp/( qRR 6 ii+1 ) ( ) (3.10)

The intrablock structural correlation function for a block with the degree of polymerization n is defined as

nnn  k gnij( qinn )exp()2()=−−=+−= k g q qRR ( ) ijk===111 (3.11) 21n+ exp1−+−q2 Rq 22 R 2 ng−−+2 qng ( qg)( )2 q ( ) 2 ( nn)  2n 2 (1(− gq )) 2 ql1 22 (qRn )

2 2 where we have introduced Rn ≡ l n/6. For the construction of the correlation functions we also need the intrablock correlation function with one point fixed at the grafting point of the block

n−1 n 22 j (1(− )gq ) 1exp−−( qRn ) n ()qg== qn ( ) (3.12)  ql1 22 j=0 (1(− ))g qq R n

The structural correlation functions of graft polymers with the number of side chains Nsc = (nbb − (ng − 1)) Τng are built on equations 3.11 and 3.12 modified with summations over the separating blocks. After some algebra the structural correlation function of graft polymers for an arbitrary number of side chains are written as

58 g( q )=  g ( q ); bb M nbb gq( )=+ Ngq ( ) 2 gq ( )22 ( qfqn ) ; (3.13) ss M( sc nsc n sc N sc ( g )) gq()= gq ()()1()  q( + gqhqn)  ()( qgqn + )(  qn ) sb M nsc N sc( g) n g g N sc g where we introduced two correlation functions to describe interblock correlations

m mmgqgq−++(1)()() m+1 hqmkgq()()();=−= k m  (1())− gq 2 k=0 (3.14) m m k mgqgq(1())()1−+− fqmkgqgqm ()()()()=−= 2 k=1 (1())− gq

Figure 3.5 (a) Comparison of the scattering function obtained in molecular dynamics simulations (points) and one calculated using the RPA method (solid lines). (b) 2 2 Dependence of q* Rnsc on the ratio nsc/ng . Symbol shapes represent graft polymers with different backbone bending constants: K = 0 (circles), K = 1.5 (squares) and K = 4.0 (triangles). Symbol colors represent combs with different n as illustrated in Figure 3.2. g

In Figure 3.5a we compare function S(q) obtained in simulations and one calculated analytically by the RPA method (equation 3.13). To map analytical and simulation results we take into account that in our simulations side chains and backbones have a persistence length, therefore in equations 3.10 ~ 3.14 we have substituted bond length l by √lbs for side chains and √lbb for backbones. This allowed us to match the

59 peak position. However, the maximum at the peak and its width were still different. In order to adjust the magnitude of the maximum and width, we minimized the difference between analytically calculated S(q) and one obtained in simulations in the vicinity of the peak by considering 2vχ as a fitting parameter. This optimization procedure converged for 2vχ = −0.35σ3. The negative value of the χ-parameter reflects mixing between side chains and backbones which one should expect for chains made of identical monomers.

Furthermore, this indicates entropic nature of the χ-parameter and breakdown of the random mixing approximation used in derivation of the lattice model describing reference monomer system. The value of the χ-parameter depends on ng , nsc and difference between Kuhn lengths of backbones and side chains. Thus, in simulated systems, the graft polymer structure influences local monomer packing in a melt which in turn changes chemical potentials of monomers belonging to side chains and backbones.

In this aspect, one could view χ-parameter as a correction factor, quantifying the entropic penalty for difference in local monomer packing around backbones and side chains.

It is important to point out that S(q) calculated in the framework of the RPA approximation adopted here does not describe monomer-monomer correlations on the length scales comparable with their bond length. Note that these correlations can be accounted for through q-dependence of the matrix of direct correlation functions C(q).

The good collapse between simulation and analytical results indicates that we can use the derived structural correlation functions to establish a scaling dependence of the peak position on architectural parameters of graft polymers. It is convenient to simplify the structural correlation functions (equation 3.13) by taking the limit of the large number

60 of side chains Nsc = nbbΤng ≫ 1 and the interval of wave numbers ql ≪ 1. In this approximation we have

2n2 gqN();   g bbMsc qR22 ng 2()exp qqR222 − nnscg ( ) gqNgqssMscn()();+ (3.15) sc 1exp−−qR22 ( ng ) 2()nq gqN()  gnsc sbMsc qR22 ng

The peak position q* in S(q) corresponds to the minimum of the function

gqgqgqssbbsb()()2()+− min 2 (3.16) gqssbbsb()()() gqgq − where we consider the explicit form of the elements of the inverse matrix g-1(q) .

Analytical results for the peak position q* can be obtained in two asymptotic regimes

2 1/2 corresponding to: (i) a dilute side chain regime, where the distance 〈Re(ng)〉 =√lbbng

2 1/2 between grafting points of the side chains is larger than the size 〈Re(nsc)〉 =√lbsnsc of

2 1/2 2 1/2 the side chains, 〈Re(ng)〉 > 〈Re(nsc)〉 , and (ii) an overlapping (semidilute) side chain

2 1/2 2 1/2 regime, where side chain overlap such that 〈Re(ng)〉 < 〈Re(nsc)〉 . (For identical side chains and backbones these conditions reduce to inequality condition between ng and nsc).

(i) Dilute side chains (Rnsc < Rng , 1 < q*Rng , 1 ≤ q*Rnsc ). In the case of dilute

2 2 side chains, exp( − q* Rng) ≪ 1 and correlations between side chains can be neglected.

Therefore, such combs can be treated as being made of unconnected symmetric triblocks

91 with a long backbone and macromolecular number density ρMNsc = ρΤ(ng + nsc). In the 61 interval of wave numbers qRnsc ≥ 1 , we can substitute corresponding approximations for side chain block functions g (q) and ξ (q) into equation 3.15 and obtain nsc nsc

2n2 gqN();   g bbMsc qR22 ng 2n2 gqN();   sc (3.17) ssMsc qR22 nsc 2nn gqN()  gsc sbMsc qRR422 nngsc

After some algebra which involves evaluation of the leading terms in equation 3.16 with matrix elements given by equation 3.17, we find that the peak position scales with the architectural parameters as

−1/2 −1/4 qRRn*  n (3.18) ( nngscgsc ) ( )

It follows from this expression that peak position shifts towards small q-region with side chain dilution (increasing the DP of the spacer between grafting points) and increasing

−1/4 DP of side chains. The scaling expression q* ∝ (nscng) was first derived by

Erukhimovich101 in the context of microphase separation in multiblock copolymer systems. This scaling dependence is verified in Figure 3.5b. Indeed, in the limit of the dilute side chains there is a good collapse of the simulation data. However, one observes a systematic deviation at the crossover to the regime of overlapping side chains.

−1 −1 (ii) Overlapping side chains ( Rng ≤ Rnsc , Rnsc < q* < Rng ). In the regime of overlapping side chains, inter-side-chain correlations become important. Therefore, to

2 2 2 2 account for interactions between side chains we expand exp( − q Rng) ≈ 1 − q Rng and use for functions g (q) and ξ (q) their approximations at qRn ≫ 1 which results in nsc nsc sc 62 2n2 g(); q  N g bb M sc qR22 ng

22 22nnsc sc 1 gss(); q+ M N sc 2 2 2 2 2 (3.19) q R q R 22 nnsc g qR ( nsc ) 2nn g() q  N g sc sb M sc q4 R 2 R 2 nng sc

The peak position in the function S(q) with structural correlation functions given by equation 3.19 is calculated to be

−−1/41/2 qnRnRnRRR* +1/21/22222− (3.20) scnscngnnnscgscgsc( ) ( ) nnscg

This expression accounts for deviation of the data points observed in Figure 3.5b in a crossover to the comb regime with overlapping side chains. Note that in this regime one should also expect weak renormalization of the backbone Kuhn length due to interactions between side chains.

3.3.2 Bottlebrush Regime

In this regime, interactions between side chains stiffen the backbone, increasing its Kuhn length from the bare value bb to bK ≈ bΦ/Φ* with increasing value of the crowding parameter above its crossover value Φ*.95 It is important to point out that equation 3.19 also describe structural correlations of bottlebrush macromolecules in the interval of wave numbers, qbK < 1 . In this q-range, the bottlebrushes can be approximated by flexible chains with effective Kuhn length equal to bK for which inter- side-chain correlation functions are given by equations 3.13 and backbone-backbone correlation function is approximated by the Debye function.108 (Note that in order to

63 correctly describe inter-side chain correlations and backbone-backbone correlations in the entire q-range one should use the Kholodenko’s expression109 for two point correlation

2 function of semiflexible chains.) Therefore, after substitution of Rng = lbKng/6 and

2 Rnsc = lbsnsc/6 into equation 3.19, the structural correlation functions of the bottlebrush systems are written as

B ng gbb( q ) 12 M N sc 2 ; q lbK  B nbsc36 s 1 gss( q )+ 12 M N sc 3 ; (3.21) 2 2 q lbs n g b K q lb ( s )

B bs 1 gsb( q ) 72 M N sc 2 b 2 K (q lbs ) where superscript index “B” stands for bottlebrush. Analysis of the equation 3.21 shows

B −4 that the leading term in det[gαβ(q)] ∝ q . In this approximation, the location of the peak is obtained by finding solution of the equation

d(q )4 gqB ss 4 36 00=−= 22q (3.22) dqn n b b l sc g K s which is

−1/4 −1/2 qn*6 n1/22 b b lRR (3.23) ( g sc s Knn ) ( scg )

Note that this expression is similar to equation 3.20 in the long side chain limit and to equation 3.18 where one has to substitute renormalized value of the backbone Kuhn

length bK instead of its bare value bb in the expression for Rng. Taking into account that in the bottlebrush regime, the effective backbone Kuhn length is proportional to the crowding parameter, bK ≈ bΦ/Φ*, one can show that the product ngbK is independent on 64 the DP of the spacer between grafting points and on the bare Kuhn length of the backbone (see equation 3.2) resulting in

2−1/8 1/4 −− 3/8 3/8 q*1( lbs v) ( − ) n sc  n sc (3.24)

Thus, in the bottlebrush regime, the position of the peak q* in the scattering function S(q) has a weaker dependence on the DP of the side chains than one would expect if it was

−1/2 directly related with the size of the side chains, q* ∝ nsc . This is in agreement with data shown in Figure 3.3d where we plotted q * as a function of the side chain DP, nsc. Indeed,

−0.39±0.01 simulation data are independent on ng and follow a scaling dependence nsc which

−0.375 is close to nsc predicted by equation 3.24.

Having established correlations between the peak position q* and DP of side chains (equation 3.23), we can use this result to evaluate the Kuhn length of bottlebrushes.

Rewiring equation 3.23 by expressing the effective Kuhn length bK in terms of the peak location q*, architectural and molecular parameters of side chains and backbones, one has

42 −1 bK/* b b= C( q b s b b l n g n sc ) (3.25) where C is a numerical constant. The value of this constant can be obtained by plotting the effective Kuhn length bK directly calculated from simulations as a function of the r.h.s of the equation 3.25 as shown in Figure 3.6. The linear fit, correlating directly measured bK with peak position and graft polymer’s molecular parameters, provides a calibration curve for extracting bottlebrush Kuhn length from scattering data.

65 Figure 3.6 Dependence of the normalized bottlebrush Kuhn length bK/bb on the 2 1/4 dimensionless parameter qො, where qො = q*(bbbsl ngnsc) . The dash line is the best fit to function y = ax + c with a = 88 ± 4 and c = 0.77 ± 0.06. Symbol shapes represent graft polymers with different backbone bending constants: K = 0 (circles), K = 1.5 (squares) and K = 4.0 (triangles). Different symbol colors represent bottlebrushes with different ng as illustrated in Figure 3.2.

3.4 Conclusions

We presented results of the coarse-grained molecular dynamics simulations for scattering function S(q) in melts of graft polymers with different macromolecular architectures. Our simulations have shown that the peak position q* in the scattering function has a different scaling behavior as a function of nsc and ng in comb and bottlebrush regimes. In melts of combs with dilute grafted side chains, ng > nsc , the peak

−1/4 position scales with the system architectural parameters as q* ∝ (nscng) (Figure 3.5b).

For melts of bottlebrushes one finds a stronger variation of q* with the DP of side chains

−3/8 −0.39±0.01 nsc such as, q* ∝ nsc , which is close to the best fit result q* ∝ nsc (Figure 3.3d).

These scaling relationships were derived by analyzing different asymptotic expressions for the function S(q) obtained in the framework of the Random Phase Approximation.

66 Furthermore, we showed that the information about the peak position can be used for the evaluation of the effective Kuhn length in bottlebrush melts (Figure 3.6). We hope that the analysis presented here will be useful for the interpretation of scattering data from melts of graft polymers and eliminate ambiguity in the peak assignment in the scattering function S(q).

At the end we would like to point out to what systems the developed scattering theory of combs and bottlebrushes in a melt can be applied. In calculating the scattering function, we assumed a contrast between backbones and side chains; therefore, our theoretical results should be directly applicable to bottlebrushes and combs consisting of chemically different monomers, e.g. poly(norbornene)-graft-poly(lactide) (PNB-g-PLA) copolymers.46, 65, 81, 93 Note that we also expect this approach to be valid for bottlebrushes with chemically identical monomers and densely grafted side chains for which scattering contrast is induced by difference in the electron density in the vicinity of the side chain grafting groups. 8, 29, 96, 97, 110 2

Parts of this chapter are reprinted with permission from Macromolecules 2019, 52 (15), 5555-5562. Copyright 2019 American Chemical Society. 67 CHAPTER IV

ENTANGLEMENTS OF MELTS OF COMBS AND BOTTLEBRUSHES

4.1 Introduction

Entanglements, topological constraints imposed by noncrossing restriction on the chains’ Brownian motion111-114, are responsible for unique viscoelastic properties observed in solutions and melts of polymers and biomacromolecules.10, 86, 115-120 Their macroscopic manifestation appears as a rubbery-like plateau in the time dependent stress relaxation modulus at the intermediate time scales. The magnitude of the plateau modulus, Ge, is a characteristic feature of polymers and is inversely proportional to the

10, 119 average number of monomeric units, ne,lin, in an entanglement strand Ge ∝ 1/ne,lin.

The relationship between ne,lin and polymer’s molecular parameters such as monomer projection length l, Kuhn length b and monomer number density ρ follows from the

Kavassalis-Noolandi conjecture,31 which states that for a linear polymer there should be a fixed number of entanglement strands, Pe,lin, within a confined volume occupied by an entanglement strand. The parameter Pe,lin is called the packing number and its value is

10, 31 almost a constant Pe,lin ≅ 20 for most flexible polymers. This results in an universal expression for the plateau shear modulus in terms of molecular parameters and the packing number.10, 31, 121 Universality of the packing number was also applied to describe dynamics of charged polymers,122, 123 solutions of associating polymers,124, 125 and biopolymers.126 68

Graft polymers with a long backbone and multiple grafted side chains constitute a unique class of polymers that combine the properties of chains and filaments. Loosely grafted comb-like macromolecules overlap and entangle similar to regular linear chains, whereas densely grafted bottlebrushes behave as semi-flexible filaments with negligible overlap and markedly lower entanglement density.8 Through independent variation of two architectural parameters, the degree of polymerization (DP) of side chains (nsc) and spacer between neighboring grafting points of side chains (ng), one can have substantial control over the entanglement plateau modulus and zero shear viscosity of polymer melts8, 40, 41, 43-45, 127 as well as Young’s modulus and elongation-at-break of polymer networks.1, 8, 25-30, 36-38, 56, 57, 128

In this chapter, we study the relationship between the entanglement plateau modulus and architectural parameters of graft polymer melts by the scaling analysis.

Using the classification of graft polymers developed in Chapter II, we demonstrate in the next section that graft polymers show distinct rheological behaviors in comb and bottlebrush regimes. Their entanglement plateau modulus in both comb and bottlebrush regimes can be explained by a universal packing number as suggested by the Kavassalis-

Noolandi conjecture. However, rheological data also indicate a systematic deviation of entanglement plateau modulus from the universal behavior in the crossover from combs to bottlebrushes. We propose in the third section that this is a manifestation of breakdown of the Kavassalis-Noolandi conjecture. We test our hypothesis by performing coarse- grained molecular dynamics simulations and calculating the packing number of graft polymers with different architectual parameters from the simulation results. The last section concludes with a short summary.

4.2 Entanglement Plateau Modulus of Graft Polymer Melts

Architectural disentanglement of graft polymers was first discovered by Fetters et al. in melts of poly(훼-olefins) by demonstrating distinct scaling relations for the decrease of the entanglement plateau modulus with linear mass density.40, 41 They obtained two empirical relationships that allowed for estimation of the entanglement plateau modulus

Ge of polyolefins from the average molecular weight per backbone bond (mb) as:

−3.49 Ge = 24.82mb for mb = 14 ~ 28 (loose grafts) (4.1)

−1.58 Ge = 41.84mb for mb = 35 ~ 56 (dense grafts) (4.2)

These equations have been used in the literature to describe various types of graft polymers with little explanation of the physics behind them. Herein we expand upon previously published theory on bottlebrush and comb polymer melts8, 38, 95 to validate equations 4.1 and 4.2. We also demonstrate universality of our model by testing it against different types of graft polymers synthesized here and reported previously. 8, 40, 41, 45, 110,

129

4.2.1 Scaling Analysis

Entanglements of polymer chains result in a plateau of storage modulus as a function of frequency.130 The plateau modulus depends on the average size (volume) of entanglement strands, which in turn is determined by the size and grafting density of side chains. Here we consider three distinct systems of entangled polymer melts: linear chain melts, graft polymer melts with unentangled side chains, and graft polymer melts with entangled side chains (see Figure 4.1).

70 Figure 4.1 Three distinct polymer melt systems. The equations above the cartoons correspond to the number of monomeric units in the corresponding entanglement strands.

In polymeric systems, the entanglement plateau modulus is of the order of the thermal energy kBT (kB is the Boltzmann constant and T is the absolute temperature) for each entanglement strand per unit volume

ρkBT Ge ≅ ρekBT ≅ (4.3) ne where ρe and ρ are respectively number densities of entanglement strands and monomers in a melt and ne is the average number of monomeric units in an entanglement strand. For melts of linear polymer chains (System I, Figure 4.1a), ne corresponds to ne,lin, the DP of the polymer strand between entanglements, which gives the entanglement plateau modulus as

ρkBT Ge,lin ≅ (4.4) ne,lin

The relationship between 푛푒,푙푖푛 and chain’s molecular parameters follows from the

Kavassalis-Noolandi conjecture stating that there is a fixed number of entanglement

3 8, strands, Pe ≅ 20, within a confinement volume a occupied by a strand with DP = ne,lin.

71 10, 31 For a melt of linear chains, the size and excluded volume of an entanglement strand are dT ≅ √ne,linbl and Ve = vne,lin . Therefore, the number of overlapping entanglement

3 strands inside the volume, dT, is estimated as

3 3Τ2 dT (bl) Pe,lin ≅ ≅ √ne,lin (4.5) Ve v which determines ne,lin and Ge,lin as a function of the molecular parameters l, b, and v as

2 2 v ne,lin ≅ Pe,lin (4.6) (bl)3

3 ρkBT (bl) Ge,lin ≅ 2 2 (4.7) Pe,lin v

Both properties exhibit minor variations with chemical composition (l, b, and v), which explains why majority of conventional linear polymers have the entanglement plateau modulus of the order of 105 Pa. In contrast, modification of polymer architecture results in significantly greater shifts in the entanglement properties as discussed below.

In melts of graft polymers (System II in Figure 4.1b), grafting side chains to a polymer backbone results in dilution of the backbone monomers by a factor

ng φ = (4.8) ng + nsc

Therefore, the entanglement plateau modulus (equation 4.3) of melt of graft polymers with entangled backbones decreases with increasing “swelling ratio” α ≡ φ−1 as

ρkBT Ge,gr ≅ −1 (4.9) ne,bbφ where ne,bb is DP of the backbone strand between entanglements. Note that for a melt of linear chains ( nsc = 0 and φ = 1 ), equation 4.9 is equivalent to equation 4.4, i.e.

72 Ge,gr = Ge,lin and ne,bb = ne,lin. In addition to dilution of the backbones, side chains lead to an increase of the effective Kuhn length bK of graft macromolecules (see Table 2.1 in

3 Chapter II). This in turn changes the tube diameter dT ≅ √ne,bbbKl and the excluded volume of a brush section with ne,bb backbone monomers between entanglements to

−1 3 8, 10, 31 Ve = vne,bbφ . In this case the number of the entangled chains within volume dT is

3Τ2 (bKl) P ≅ √n (4.10) e,gr vφ−1 e,bb

This expression determines ne,bb as a function of both the molecular (l, b, v) and architectural (nsc, φ) parameters. From equations 4.6 and 4.10, the entanglement DP of the backbone of graft polymers can be expressed in terms of that of linear chains as

P 2 3 3 e,gr b −2 b −2 ne,bb ≅ ( ) ( ) φ ne,lin ≅ ( ) φ ne,lin (4.11) Pe,lin bK bK

Note that the final form of the equation 4.11 is only correct if the number of entanglement strands is a universal number, Pe,lin ≈ Pe,gr . The entanglement plateau modulus of the melts of graft polymers with entangled backbones is obtained by substituting equation 4.11 into equation 4.9 and taking into account equation 4.4 for Ge,lin

φb 3 G ≅ G ( K) (4.12) e,gr e,lin b

The specifics of chain architecture enter through the graft polymer composition φ and dependence of the Kuhn length bK on nsc and φ (see Table 2.1 in Chapter II). It is important to point out that the φbKΤb factor in equation 4.12 includes two distinctly opposite effects of the side chains on the entanglement plateau modulus: (i) dilution of

3 the backbones by side chains (φ < 1) promotes plateau modulus decrease as Ge,gr ~ φ and

73 (ii) effective stiffening of the backbones due to steric repulsion between their side chains

3 (bKΤb > 1) causes the modulus to increase such as Ge,gr ~ (bKΤb) . In the comb regime

(bK ≅ b), the dilution effect prevails resulting in overall modulus decrease, whereas in the bottlebrush regime (bK > b), stiffening of the backbone weakens the decrease in modulus.

To develop a universal picture for all graft polymer melts, we separate the dilution and stiffening effect and rewrite equation 4.12 as

3 Ge,gr bK 3 ≅ ( ) (4.13) φ Ge,lin b which presents the normalized entanglement modulus as a function of the normalized

Kuhn length bKΤb. In polymer combs (Φ < 1, bK ≅ b), the DP of the backbone of an entanglement strand (equation 4.11) increases with dilution as

−2 ne,bb ≅ φ ne,lin (4.14) and the normalized entanglement modulus approaches unity:

Ge,gr ( ) ≅ 1 (Φ < 1) (4.15) φ3G e,lin comb

3 This equation demonstrates the dominating role of the dilution factor Ge,gr ~ φ in the comb regime. In the stretched backbone (SBB) sub-regime (Φ > 1), the steric repulsion of densely grafted side chains results in backbone stretching and an increase of the Kuhn length. According to Table 2.1 in Chapter II, the normalized Kuhn length of graft polymers in the SBB sub-regime increases linearly with the crowding parameter Φ as bKΤb ≅ Φ. Therefore, the normalized entanglement modulus can be expressed in terms of crowding parameter Φ as

74 Ge,gr ( ) ≅ Φ3 (Φ > 1) (4.16) φ3G e,lin SBB

3 This shows that the dilution effect (Ge,gr ~ φ ) is counterbalanced by the stiffening effect

3 (Ge,gr ~ Φ ), which weakens the decrease in modulus caused by the dilution of backbones by side chains. In the subsequent SSC sub-regime, the stiffening factor (bKΤb) can be calculated from the corresponding bK equations in Table 2.1 from Chapter II, which results in a different expression for the normalized modulus. However, this sub-regime takes place at high grafting densities (ng < 2), which, in practical terms, corresponds to bottlebrush systems with ng = 1. The RSC sub-regime also has marginal significance as it corresponds to even higher grafting densities with two and more side chains per backbone monomer. To summarize, equations 4.14 and 4.15 provide a universal representation of the entangled plateau modulus of conventional graft polymers with unentangled side chains.

For graft polymers with long side chains in the comb regime (System III, Figure

4.1c, Φ < 1, bK ≅ b), both backbones and side chains are entangled. Since there is no distinction between backbone and side chain strands, the system effectively behaves as a melt of linear chains with an entanglement plateau modulus defined by equation 4.4.

However, the DP of the entanglement strand is given by equation 4.14 as ne, comb ≅

−2 φ ne,lin, because the packing condition (equation 4.10) counts the total number of the other chain sections (including side chains and backbones) within pervaded volume of an entanglement strand. By substituting ne,lin in equation 4.4 with equation 4.14, we obtain the following expression for the entanglement plateau modulus of combs with entangled side chains: 75 2 Ge,ent comb ≅ Ge,linφ (4.17)

Note that we can also express the normalized plateau modulus of the entangled combs in terms of the crowding parameter such as

Ge,ent comb −1 3 ≅ φ ≅ √nscΦ (Φ < 1) (4.18) φ Ge,lin

Below we will test applicability of the equations 4.15 ~ 4.18 to rheological data of melts of graft polymers. Table 4.1 combines expressions of the plateau modulus of graft polymer melts in all regimes of the diagram of states in Figure 2.3 and Figure 2.10b from

Chapter II. As follows from this table, depending on the regime, the variation of the plateau modulus in melts of graft polymers can follow different scaling laws ranging

1.5 3 from Ge,gr ~ φ to Ge,gr ~ φ . This behavior is qualitatively similar to the plateau

a modulus change in semidilute polymer solutions Ge,lin ~ φ , where scaling exponent a depends on the solvent quality.10 However, in contrast to polymer solutions, disentanglement of macromolecules in melts of graft polymers is achieved without using any solvent.

Table 4.1 Ratio of the entanglement plateau shear modulus in a melt of graft polymers to that of linear chains in different conformation regimes.

(1) Regime boundaries are given in Table 2.1 from Chapter II. 76 4.2.2 Comparison to Experiments

To verify the scaling predictions, we have used rheological experiment data of graft polymers with different chemical structures, including graft poly(n-butyl acrylate)8,

93 (PBA), graft polystyrene45 (PS), graft polyolefin40, 41, 110, 129 (PE, the abbreviation of polyethylene is used because we regard ethylene as the repeat unit in our later analysis).

Table 4.2 summarizes architectural parameters and rheological properties of these graft polymers, and Table 4.3 summarizes molecular parameters of them (monomer length l,

Kuhn length b, and monomer volume v).

77 Table 4.2 Architectural parameters and rheological properties of graft polymers.

78 Table 4.3 Molecular parameters of graft polymers.(1)

(1)l is monomer length, and b is Kuhn length of the linear polymer strand. The monomer volume 푣 was calculated from the molecular weight of monomer M0, mass densities c and Avogadro’s number NA: v = M0/(cNA). PBA: Poly(n-butyl acrylate), PS: Polystyrene, PE: Polyethylene. (2)Data from Polym. Int. 2001, 50 (6), 625-634. (3)Data from Polymer Physics. Oxford University Press: New York, NY, 2003.

Figure 4.2 (a) Diagram of states of graft PBA. Solid black lines correspond to the boundary between comb and bottlebrush regimes. Red dashed lines mark boundaries between different bottlebrush subregimes. Red solid line corresponds to the crossover to the forbidden region. (b) Diagram of states of graft PBA, graft PS and graft PE. The solid black straight line corresponds to a crossover between comb and bottlebrush regimes for flexible (long) side chains. The solid, dashed and dotted black curved lines correspond to a crossover between comb and bottlebrush regimes with short (rod-like) side chains for PE, PS and PBA systems respectively. Dashed red lines mark boundaries between different bottlebrush subregimes. Boundaries of the corresponding “forbidden regions” are not shown. For both panels, symbol shapes correspond to different chemical structures: PBA (circles), PS (squares), and PE (triangles). Symbol colors represent different regimes: comb (blue) and bottlebrush (red).

Figure 4.2a shows the diagram of state of samples from graft PBA experiments8,

93. The crossover between comb and bottlebrush regimes (black solid lines) is calculated

−3/2 −1 −1Τ2 −3 −1 −2 by setting parameters v(bl) φ nsc = Φ* = 0.7 and vl φ nsc = Φ* = 0.7 for 79 flexible, nsc > bΤl and rod-like, nsc < bΤl , side chains, respectively. The coefficient

Φ* = 0.7 has been determined in computer simulations (see Chapter II). In this calculation we used l, b, and v values from Table 4.3. The location of the intersection

−1 point is obtained by substituting nsc = b/l = 6.8 into expression for φ at the crossover between comb and bottlebrush regimes: φ−1 = 0.7 b2lΤv = 2.6. The boundary between comb and bottlebrush regimes for nsc < b/l = 6.8 (rod-like side chains) is given by a

−1 3 −1 2 crossover expression φ = 1 + 0.43l v nsc, which is obtained by taking into account

−3 −1 −2 −1 scaling relation for the crowding parameter at a crossover Φ ≅ vl φ nsc ≈ 1 ⟹ φ ≈

3 −1 2 −1 l v nsc and requiring function φ to pass through the points with coordinates (6.8, 2.6) and (0, 1) corresponding to a linear chain limit. The upper boundary for the accessible

−1 regimes is given by φ = nsc + 1 (solid red line). Figure 4.2b shows a diagram of states of all samples from experiments. Unlike Figure 4.2a, the diagram in Figure 4.2b is plotted in terms of normalized φ and nsc , which accounts for the difference between graft polymers with rod-like (nscl < b) and flexible (nscl > b) side chains. In these new variables, graft polymers of different chemical compositions with flexible side chain can be described by one diagram of states, whereas the crossover between the comb regime and

RSC bottlebrush subregime for the systems with rod-like side chains depends on the molecular parameters (b, l and v). This is shown in Figure 4.2b as three curved lines in the interval nscl/b < 1. To obtain the diagram of states, we used l, b, and v values from

Table 4.3. Figure 4.2 shows that the experiment samples cover both comb and bottlebrush regimes. We will use this classification of the different systems for the interpretation of rheological data below.

80 Figure 4.3 Normalized entanglement modulus as a function of the compositional −1 parameter φ = 1+ nscΤng for different graft-polymer systems: (a) PBA combs, PE bottlebrushes and PBA bottlebrushes. (b) PS combs and PE combs with entangled side chains. (c) Combined plot of the normalized entanglement modulus as a function of graft polymer composition for melts of combs and bottlebrushes from (a) and (b). (d) Normalized entanglement modulus as a function of crowding parameter. The crossover equation y = 1 + (x/0.7)3 between comb and bottlebrush regimes is shown by the dashed line. Symbol notations in all panels are the same as Figure 4.2.

Figure 4.3a-c plot the normalized entanglement modulus in terms of the graft

−1 polymer composition φ = 1+ nscΤng for different graft polymers. For all PBA samples, the three polyolefin bottlebrushes (s-PPEN, s-PHEX, s-POCT) prepared by Fetters et al40 and all polyolefin samples (PH, PO, PN, PD, PDD, PTD, POD) prepared by Lopez-

Barron et al.,110 their side chains are shorter than the corresponding entanglement strands

131 41 of linear PBA (ne,lin = 219) and PE (ne,lin = 36). Therefore, these systems generally

81 a follow a scaling function Ge,gr ~ φ with the exponent varying between -3/2 and -3, which corresponds to the bottlebrush and comb regimes (Figure 4.3a) and agrees with the empirical relations by Fetters et al.3 (equations 4.2 and 4.1). However, the side chains in the PS45 and PE combs41 (PEC) are longer than the corresponding entanglement DPs of

45 41 linear chains for PS (ne,lin = 140) and PE (ne,lin = 36). These systems belong to System

III: comb polymers with entangled side chains (Figure 4.1c). In agreement with the equation 4.17, the data points for PS and PEC samples have collapsed into one universal curve with a slope of -2 in logarithmic scales (Figure 4.3b). However, due to the proximity of crossovers between the different graft polymer regimes, it is difficult to distinguish the corresponding scaling behaviors when plotting Ge,gr as a function of

1+ nscΤng (Figure 4.3c).

To address this uncertainty, we replotted the data as a function of the crowding parameter Φ (Figure 4.3d). The dashed line at Φ* = 0.7 corresponds to the crossover from the comb to bottlebrush regimes as discussed in Chapter II. The PBA combs and bottlebrushes with shorter side chains follow the respective scaling laws given by equations 4.15 and 4.16. The extension of the scaling dependence of the shear modulus

3 on the crowding parameter (Ge,gr ~ (φΦ) ) to SSC subregime (see Figure 4.2a) is due to the fact that for the densely grafted side chains with ng ≅ 1 the crowding parameter Φ ≅

−1 −1/2 −1/2 φ nsc ≅ φ , and the effective Kuhn length of the bottlebrush bK has the same scaling dependence on the crowding parameter Φ in both SBB and SSC subregimes (see

Table 2.1 in Chapter II). Note that a similar trend was observed in computer simulations from Chapter II. It is important to point out that there is a deviation of the experimental

82 data points from the universal curve close to the crossover value of the crowding parameter. The possible origin of this deviation will be discussed in the next section.

The normalized entanglement plateau modulus of the PS and PE combs with entangled side chains (data points in Figure 4.3d included in the red ovel) exhibits the scaling exponent of 1. This behavior is consistent with equation 4.18 for constant DP of the side chains. Note that the three polyolefin samples (s-PPEN, s-PHEX, s-POCT) by

Fetters et al40 belong to the RSC subregime and samples from Lopez-Barron el al110 cover both RSC and SBB subregimes (red triangles in Figure 4.2b). In the RSC subregime an

3 entanglement plateau modulus approaches Ge,gr ~ φ scaling behavior while in the SBB

3 subregime it is expected to follow Ge,gr ~ (φΦ) (Table 4.1). This is clearly seen in Figure

4.3d where the combined polyolefin data set (red triangles) approaches a universal

3 scaling dependence Ge,gr ~ (φΦ) with increasing DP of the side chains.

At last, we compare the behavior of graft homopolymers with graft copolymers with dissimilar backbone and side chains.46, 65 For graft block copolymers, the Kuhn length, monomer projection length, and excluded volume of the backbone and side chains are different, which shifts the crowding parameter at the comb-bottlebrush crossover. The effect of chemical heterogeneity can be explicitly considered by our model (see Chapter

II section 2.3) through graft polymer composition and crowding parameter as described below and demonstrated for recently studied poly(norbornene)-graft-poly(lactide) (PNB- g-PLA) systems (summarized in Table 4.4).46 To calculate the composition and crowding parameter of PNB-g-PLA, we use molecular parameters of the backbones (PDME, poly(dimethyl-5-norbornene-2,3-dicarboxylate)81) and side chains (PLA) listed in Table

83 4.5. The Kuhn length of the backbone PDME was estimated by the relationship between the packing parameter p and the entanglement plateau modulus92

−3 Ge = Ap (4.19) where the numerical coefficient A is temperature dependent and is 12.16MPa·Å3 for T =

413K, which is close to the reference temperature used to calculate Ge = 0.3MPa for linear PDME chains (396K).46 In terms of the packing parameter the Kuhn length of a polymer is expressed as follows

M b = 0 (4.20) clpNA where M0 is the monomer molar mass, l is the monomer projection length, c is the mass density and NA is the Avogadro number. Equations 2.23 and 2.28 from Chapter II was used to calculate the composition φ and crowding parameter Φ of PNB-g-PLA.

Table 4.4 Architectural parameters and rheological properties of PNB-g-PLA.

(1) z is the number ratio of backbone monomers being grafted with a side chain, i.e. grafting density 1/ng.

Table 4.5 Molecular parameters of PNB-g-PLA

(1) Monomer projection length is calculated by ChemDraw.

84 Figure 4.4 Normalized entanglement modulus as a function of the normalized crowding parameter for PBA combs (blue circles), PBA bottlebrushes (red circles), PNB-g-PLA combs (blue inverted triangles), and PNB-g-PLA bottlebrushes (red inverted triangles). Φ* = 0.4 for PNB-g-PLA and Φ* = 0.7 for PBA. The crossover equation y = 1 + x3 between comb and bottlebrush regimes is shown by the dashed line.

As shown in Figures 4.4, one can collapse both homopolymer and copolymer data by normalizing the crowding parameter by its crossover value. The crossover value is

Φ* = 0.7 for PBA and Φ* = 0.4 for PNB-g-PLA. An interesting feature of this plot is that both data sets demonstrate a deviation from the universal curve in the crossover region.

In this interval of parameters, side chains belonging to the same graft macromolecule begin to overlap and to push out the side chains belonging to neighboring macromolecules. Effectively, we deal with a transition from polymer chains to mesoscopic (brush-like) filaments. This may suggests a breakdown of Kavassalis-

31 Noolandi conjecture. Which means that the number of entanglement strands Pe within the confining entanglement volume depends on the grafting density in the crossover between comb and bottlebrush regimes. In the next section, we will study this peculiar behavior in detail.

85 4.3 Packing Number of Graft Polymer Melts

Dynamics of melts and solutions of high molecular weight polymers is controlled by topological constraints (entanglements) leading to chain reptation along an effective confining tube. For linear chains, the tube size is determined by a universal packing number Pe, which is the number of polymer strands within a confining tube required for chains to entangle. However, the important fundamental question, how the packing number Pe depends on the molecular architecture, remains unanswered. The difficulty in addressing this issue arises from the fact that there is no analytical model capable of calculating the packing number Pe directly from the chemical structure of macromolecules. For example, what number should be used to describe entanglements in melts of graft polymers (Figure 4.5) that consist of the linear chain backbone with grafted side chains.36, 40, 46, 65, 93, 110, 132, 133 In such macromolecules side chains play a dual role of the backbone diluents and stiffeners as the grafting density of the side chains increases.

At low grafting densities (Figure 4.5a), the backbones can be effectively treated as linear chains in a solvent of side chains (comb regime). However, in the case of densely grafted side chains (Figure 4.5b) when the interpenetration between side chains belonging to neighboring macromolecules is hindered by the steric repulsion, the properties of graft polymers are similar to those of flexible filaments which can be approximated by “fat” linear chains with the effective Kuhn length on the order of their diameter (bottlebrush regime).95, 102, 133

86 Figure 4.5 Snapshots of graft polymers. (a) A comb macromolecule with dilute side chains shows chain-like behavior. (b) A bottlebrush macromolecule with densely grafted side chains shows filament-like behavior. Backbone bonds are shown in red and side chain bonds are colored in blue.

A naïve extension of the Kavassalis-Noolandi conjecture to describe entanglements of such macromolecules is to assume that the packing number of graft polymers Pe,gr is identical to that of linear chains, Pe,lin. However, experimental data for the entanglement plateau modulus in melts of graft polymers shown in previous section indicate that the backbone dilution and stiffening effect are not sufficient to describe variations in the entanglement plateau modulus as the molecular architecture changes. To explain such observed trends, one should assume Pe,gr ≠ Pe,lin and that Pe,gr is dependent on the molecular architecture in some interval of system parameters. However, to prove this assumption requires knowledge of how exactly interactions between side chains renormalize the effective Kuhn length of graft polymers. This is difficult if not impossible to quantify experimentally. Therefore, we use coarse-grained molecular dynamics simulations to establish a relationship between packing numbers of graft polymers Pe,gr and linear chains Pe,lin and their molecular architecture in a melt.

87 The effect of the chain architecture on the packing number can be quantified by the normalized packing number defined as the ratio of the packing number of graft and linear polymers

1/2 3/2 Pe,gr n b = φ ( e,bb) ( K) (4.21) Pe,lin ne,lin b which is obtained from equations 4.5 and 4.10. Thus, in order to evaluate this ratio one needs to know how the backbone DP of the entanglement graft polymer strand, ne,bb, and the effective Kuhn length of the graft polymer, bK, depend on the chain architecture.

4.3.1 Simulation Results

We performed coarse-grained molecular dynamics simulations85 of graft polymers in a melt.95, 102 We studied graft polymers with the following sets of architectural parameters (nbb, ng, nsc): (417, 32, 8), (401, 16, 16), (1025, 16, 16), (401, 16, 8), (401, 16,

4), (401, 8, 8), (1025, 8, 8), (401, 4, 4), (401, 2, 2) and (1025, 1, 4). In these simulations graft polymer backbones and side chains were modelled as bead-spring chains composed of beads with diameter  interacting through truncated-shifted Lennard-Jones (LJ) potential with interaction parameter ε = 1.0 kBT (kB is the Boltzmann constant, T is the absolute temperature). The connectivity of monomers into graft polymers was maintained by the combination of the FENE and truncated-shifted LJ potentials86 with spring

2 constants to kspring = 30 kBT/σ and maximum bond length Rmax = 1.5σ. In addition, an angular potential was introduced between neighboring backbone bonds with the bending constant K = 1.5. The side chains were modelled as flexible chains without bending potential. Simulations are carried out under the NVT canonical ensemble, with the

88 monomer density set to ρσ3 = 0.85. For this set of system parameters, the bond length is l

= 0.96, bare backbone Kuhn length is b = 2.80 and side chain Kuhn length is bs =

1.97. Simulations were performed following the procedure described in Simulation

Methods section.

In our data analysis of the simulation results we have used two independent

121, 134-136 methods to obtain ne: (i) primitive path analysis (PPA) and (ii) tube diameter approach117, 137 by establishing the crossover between the early time Rouse dynamics and reptation regime of the mean-square displacement of backbone monomers. Details of both method are described in Simulation Methods (section 4.4). The effective Kuhn length of the grafted polymers as a function of their architecture was obtained from an analysis of the bond-bond correlation functions as described in Chapter II. The results of these calculations are summarized in Table 4.6. The PPA data for ne indicates that for graft polymers DP of the backbone strand between entanglements monotonically increases with increasing grafting density of the side chains or their length. The values of ne estimated from the tube diameter approach first show insignificant variations comparable with the error bars followed by a strong increase. However, the value of the packing parameter Pe,gr varies nonmonotonically with the molecular architecture. The value of Pe,gr obtained from PPA method changes between ~19 and ~16 as nsc increasing from 4 to 16 at constant distance between side branches ng = 16. These values are below

Pe,lin ≈ 20. For systems with equal composition φ = 0.5 and decreasing nsc from 16 to 2 we observe monotonic increase of the packing number. For nsc = ng =2, Pe,gr is close to its value for linear chains. Since the dilution effect is the same among graft polymer melts

89 with the constant composition, the change in the packing number cannot be explained by the side chain dilution. To illustrate this effect we plot Pe,gr/Pe,lin as a function of the ratio of the graft polymer Kuhn length bK to that of linear chains (Figure 4.6a). As the ratio of the Kuhn length bK increases, the ratio of the packing number decreases and reaches its minimum at bK/b ≅ 1.03. At the minimum, the decrease in Pe,gr is about 25% from the packing number in a melt of linear chains for results obtained from PPA method.

The ratio Pe,gr/Pe,lin begins to increase for bK/b > 1.03 and approaches 0.92 when bK/b > 1.12. This indicates that only small corrections to the Kuhn length (less than 12%) are required for graft polymers to behave as linear chains with the effective Kuhn length.

Table 4.6 Architectural parameters, effective Kuhn length, entanglement length and packing number of linear and graft polymers

A similar trend is observed for Pe,gr/Pe,lin calculated by the tube diameter analysis

(Table 4.6). However, this conclusion should be taken carefully. Comparison of the ne,bb values obtained from both methods indicates that these techniques give consistent results only when the number of the side chains per entanglement strand ne,bb/ng > 10. For the smaller number of the side chains per entanglement strand, ne,bb/ng < 10, one observes a

90 significant difference in the ne,bb values from different methods. It appears that the tube diameter method is insensitive to variations of the graft polymer structure for ne,bb/ng ~ 1.

This suggests that there should be a sufficient number of the grafted side chains per entangled strand (ne,bb/ng > 10) for the self-averaging of the backbone dynamics on the length scales of the tube diameter. If this condition breaks down the tube diameter method provides unreliable results.

Figure 4.6 Dependence of the normalized packing number Pe,gr/Pe,lin of graft polymers calculated using ne,bb obtained from PPA (filled symbols) and tube diameter (open symbols) as a function of (a) normalized Kuhn length, bK/b, and (b) the ratio describing overlap between side chains.

To investigate the physical reason for the observed nonmonotonic dependence of

the packing number, we replot the data (Figure 4.6b) in terms of the ratio Rnsc/Rngof the

size of the side chains Rnsc to that of the backbone spacers between neighboring grafting

points Rng. This parameter characterizes the degree of mutual interpenetration between side chains of the same macromolecule. The sizes of the side chains and backbone spacers for this plot were evaluated by using the expression for the mean-square end-to-

91 end distance of a semiflexible chain with the number of bonds n, bond length l and Kuhn length b 10

⟨R(n,b)2⟩ = nbl − b2(1 − exp(−2nl/b))/2 (4.22)

2 1/2 Note that evaluating the size of the spacer between grafted side chains Rng = ⟨R(ng,b) ⟩

2 1/2 and size of the side chains Rng = ⟨R(nsc,bs) ⟩ , we use the bare values of the corresponding Kuhn lengths b and bs in equation 4.22. Figure 4.6b shows that the

minimum of Pe,gr/Pe,lin achieved at Rnsc/Rng ≈ 0.85 corresponds to a crossover between

the regimes of dilute side chains (Rnsc < Rng) and of overlapping side chains (Rnsc > Rng).

This indicates that with increasing overlap between side chains a graft polymer begins to behave as a filament. This takes place in the comb regime before the crossover to the bottlebrush regime.95, 102 Note that the size of side chains and spacers can also be directly

measured from simulations, This will leads to a correction ~7% on the ratio Rnsc/Rng in the vicinity of the minimum in Figure 4.6b. However, for comparison with experiments

(see discussion below) it is preferable to calculate the ratio Rnsc/Rng using the bare backbone Kuhn length b and equation 4.22, since there are no experimental measurements of the size of side chains and spacers for graft polymers.

4.3.2 Comparison with Experiments

Experimentally the value of the ratio of packing number Pe,gr/Pe,lin can be obtained from the entanglement plateau modulus of melts of linear and graft polymers.

From equation 4.4 and 4.9, the ratio of the DP of the backbone strand between

92 entanglements to the DP of the entanglement linear polymer, ne,bb/ne,lin, can be written in terms of the entanglement plateau modulus of melts of linear and graft polymers as

n G e,bb = φ e,lin (4.23) ne,lin Ge,gr

By plugging this relationship into equation 4.21, one can obtain

−2 3 Ge,gr Pe,gr bK 3 = ( ) ( ) (4.24) φ Ge,lin Pe,lin b

In Chapter II, we have established that the effective Kuhn length is a universal function of the crowding parameter Φ. If the packing number is a universal constant as

31 proposed by the Kavassalis-Noolandi conjecture ( Pe,gr = Pe,lin ), the normalized

3 entanglement plateau modulus, Ge,gr/(φ Ge,lin) , will be a universal function of the normalized crowding parameter, Φ/Φ*, where Φ* corresponds to the crossover between comb to bottlebrush regimes. Rheological studies of graft PBA93 and PNB-g-PLA46 melts shown in the previous section have demonstrated such universality in comb and bottlebrush regimes. However, those data also display deviates from the universal curve in the crossover between combs and bottlebrushes, where graft polymers transition from chain-like to filament-like (Figure 4.4). This deviation is due to the difference between

Pe,gr and Pe,lin, and the ratio Pe,gr/Pe,lin can be estimated by a numerical factor describing the deviation of data (Figure 4.4) from the universal function (dashed line, y = 1 + x3) obtained under the assumption that Pe,gr = Pe,lin. In Figure 4.7, we combine the simulation

and experimental data together and replot them in terms of the ratio Rnsc/Rng describing the overlap between side chains. Both simulation and experimental data show consistent trends. However, there is a lag observed in experimental data sets approximately by a

93 factor of two for the ratio Rnsc/Rng . This could be due to the crudeness of the approximation based on the crossover function for the evaluation of the ratio Pe,gr/Pe,lin.

Figure 4.7 The ratio of packing number Pe,gr/Pe,lin as a function of the ratio of size of side chains and spacers Rnsc/Rng. Dashed and dotted lines highlight trends in simulation and experimental data respectively. Symbol shapes represent different data set: simulation (circles), graft PBA (squares) and PNB-g-PLA (inverted triangles). Symbol colors show different regimes of graft polymers: comb regime (comb) and bottlebrush regime (blue).

4.4 Simulation Methods

We performed molecular dynamics simulations of coarse-grained graft polymers in a melt. Melts of graft polymers were prepared by randomly generating 800 macromolecules with nbb = 401 or 417, 500 macromolecules for system with (nbb, ng, nsc)

= (1025, 16, 16) and (1025, 8, 8) and 300 macromolecules for system with (nbb, ng, nsc) =

(1025, 1, 4) at a monomer density = 0.85 −3 .The force field and simulation procedure used to prepare the graft polymer melts are described in Chapter II section 2.5.

94 6 Production run last 5×10 τLJ for graft polymer melts with nbb = 401 or 417, and

7 1~3.7×10 τLJ for graft polymer melts with nbb = 1025.

Primitive Path Analysis (PPA)121, 134-136: Five configurations of each sample were used for primitive path analysis to obtain the DP of the entanglement strand, ne. In this approach,121, 134 the chain ends of backbones were pinned, the intrachain repulsive LJ potentials were turned off, and the interchain repulsive LJ potentials were maintained.

The angular bending potentials of backbone bonds were also switched off. All side chain interactions and bonding potentials were switched off. This effectively resulted in solution of entangled backbones. A simulation run lasting 50τLJ was performed to relax the system. During this simulation run the integration time step was set to Δt = 0.005τLJ, friction coefficient was equal to ξ = 20m/τLJ and temperature was set to 0.001 ε/kB. This followed by simulation run continued for 1000τLJ, with friction coefficient set back to ξ =

0.5m/τLJ. The primitive path was approximated by a random walk characterized by a

Kuhn length bpp and a contour length Lpp = (nbb − 1)lpp, where lpp is the average bond length of the primitive path. Since the backbone ends are fixed, the mean-square end-to- end distance of the primitive path is identical to that of the graft polymer backbone,

2 2 ⟨Re⟩ = ⟨Rpp⟩ = bppLpp. The DP of the entanglement strand, ne, which is defined by the number of bonds per Kuhn segment of primitive path, is equal to

2 bpp ⟨Re⟩ ne = = 2 (4.25) lpp (nbb − 1)lpp

The results of the primitive path analysis are given in Table 4.7.

95 Table 4.7 Degree of polymerization of the backbone strand between entanglements, ne,bb, calculated from primitive path analysis.

(1) ne,bb calculated for different degree of polymerizations of the backbone, using 6 configurations obtained after 5×10 τLJ relaxation, standard deviations are calculated from (2) 5 samples. ne,bb calculated for different degree of polymerizations of the backbone, 7 using 5 configurations taken between 1~ 3.7 ×10 τLJ.

Tube Diameter Approach117, 137: In addition to the PPA approach we also used tube diameter calculations to obtain ne. To achieve this we calculated the mean square displacement (MSD) of monomers for the center 65 backbone beads, g1(t)=⟨[ri(t+τ) −

2 ri(τ)] ⟩, to suppress fluctuations from chain ends and to include enough grafting points,

2 and of the center of mass (cm) of molecules, g3(t)=⟨[rcm(t+τ) − rcm(τ)] ⟩ , where brackets 〈… 〉 correspond to the ensemble average over all τ. Figures 4.8 and 4.9 show time evolution of functions g1(t) and g3(t) for studied systems. There are three different

1 scaling regimes for g1(t) as function of time. In particular, one can identify t monomer diffusion regime with g1(t) ~ t at short time scales, Rouse regime for polymer strand

1/2 dynamics between entanglements for which g1(t) ~ t and finally chain reptation

1/4 regime with g1(t) ~ t . The tube diameter, dT , was estimated from location of the

* * 1/2 1/4 crossover (te, g1(te)) between g1(t) ~ t and g1(t) ~ t scaling regimes by setting dT =

96 * * 137 √3πg1(te)/2, where te is the corresponding time at the intersection point. The degree of polymerization of entangled backbone strand is estimated from the tube diameter as

2 dT ne,bb = (4.26) bKl where bK and l are Kuhn length and bond length of the backbone respectively. Table 4.8 summarizes the results from tube diameter analysis.

Table 4.8 Tube diameter, dT, and DP of the backbone strand between entanglements, ne,bb, calculated from tube diameter approach

Note that for our longest simulations of the graft polymers with (nbb, ng, nsc) =

(401, 16, 16) and (401, 8, 8) (Figures 4.8a and 4.8b), macromolecules have moved their

1/4 own size, confirming our estimate for ne,bb values. Also clear g1(t) ~ t regime observed in these simulations provides further justification that the Brownian motion of the graft polymers in a melt can be described by reptation motion of their backbones with an effective tube diameter.

97 Figure 4.8 MSD of monomers g1(t) (red open circles) and of center of mass of molecules g3(t) (blue open circles) for different systems during long simulation runs lasting up to 7 3.7 ×10 τLJ: (a) nbb = 401, ng = nsc = 8, (b) nbb = 401, ng = nsc = 16, (c) nbb = 1025, ng = nsc = 8, (d) nbb = 1025, ng = nsc = 16. Solid lines show expected time dependence of functions g (t) ~ t1/4 and g (t) ~ t1/2 in the reputation regime. 1 3

98 Figure 4.9 MSD of monomers g1(t) (red open circles) and of center of mass of molecules g3(t) (blue open circles) for different systems with nbb = 401: (a) linear polymers, (b) ng = nsc = 16, (c) ng = nsc = 8, (d) ng = nsc = 4, (e) ng = nsc = 2. Black solid lines represent best fits to scaling laws, red dashed lines indicate the locations of crossover between the early Rouse regime and reptation regime of monomer MSD, g (t). 1

99 4.5 Conclusions

We developed a scaling model for the analysis of the entanglement plateau modulus of graft polymer melts as a function of their molecular architecture. The model was tested for graft polymers of different chemical compositions (PBA, PS, PE and PNB- g-PLA) with different grafting densities and DP of the side chains. It is important to emphasize that backbones and side chains of the selected polymers have similar chemical compositions. Depending on the grafting density and DP of the side chains, the studied polymers exhibited distinct scaling laws for the entanglement plateau modulus as a function of the graft polymer composition φ = ng/(ng+nsc). In the case of unentangled

3 −3/2 3/2 side chains, the modulus follows Ge,gr ~ φ and Ge,gr ~ nsc ~ φ (for longer side chains nsc > ng with φ ≈ ng/nsc) in the comb and bottlebrush regimes, respectively (Figure 4.3a).

The entanglement plateau modulus of combs with long entangled side chains is given by

2 Ge,gr ~ φ (Figure 4.3b). In addition to the generic increase of modulus with φ, our model enabled a more distinct identification of the graft polymers regimes in terms of a crowding parameter Φ, which describes the degree of interpenetration of neighboring macromolecules. By plotting the normalized entanglement plateau modulus

3 Ge,grΤ(φ Ge,lin) as a function of Φ, we separated the conventional graft polymer regimes

(combs and bottlebrushes) from combs with entangled side chains and bottlebrushes with rigid or short side chains (Figure 4.3d).

We have shown that our model can be extended to graft copolymers with chemically different backbone and side chains (for example, PNB-g-PLA). A systematic deviation of the experimental data points from the universal scaling curve in the

100 crossover between the comb and bottlebrush regimes (Figure 4.4). This behavior could be explained by the variation of the number of entanglement strands Pe (packing number) within the confining volume of entanglement strands upon the chain-filament transition

(Figure 4.5). Detailed analysis of coarse-grained molecular dynamics simulations confirms the dependence of the packing number of graft polymers Pe,gr on the molecular

architecture. Pe,gr is shown to be a nonmonotonic function of the ratio Rnsc/Rng describing the overlap between neighboring side chains along the same backbone (Figure 4.6b). It first deviate from the packing number of linear polymers Pe,lin and decreases with

increasing ratio Rnsc/Rng, then begins to increase again and approaches the value expected for a melt of linear chains Pe,lin . The simulation results qualitatively agree with the dependence of the packing number obtained from plateau modulus in melts of graft poly(n-butyl acrylates) (PBA) and poly(norbornene)-graft-poly(lactide) (PNB-g-PLA)

(Figure 4.7).

Our result could play an important role in understanding viscoelastic properties of solutions and melts of complex graft polymers in which side chains have brush-like,138 dendrimer-like139-141 and charged bottlebrush-like 142, 143 structure. 3

Parts of this chapter are reprinted with permission from Macromolecules 2018, 51 (23), 10028-10039 and ACS Macro Lett. 2019, 8 (10), 1328-1333. Copyright 2018, 2019 American Chemical Society. 101 CHAPTER V

POLYMER NETWORKS WITH COMBS AND BOTTLEBRUSHES STRANDS

5.1 Introduction

Attachment of side chains to a linear backbone produces graft polymers with distinct physical properties from linear polymers.8, 34-37, 79, 82, 144-148 Depending on the grafting density and side chain length, we distinguish comb and bottlebrush macromolecules95. Increase of the grafting density and degree of polymerization (DP) of side chains suppresses the entanglement threshold of their melts8, 40, 43, 60, 62, 65, 127 which enables the creation of super-soft elastomers with modulus as low as 100 Pa without using any solvent1, 25-27, 30, 55, 56, 128, 149-151. When synthesizing graft polymer networks, chemists can manipulate three architectural parameters (see Figure 5.1): (i) the side chain length nsc , i.e., the DP of side chains, (ii) the spacer length ng , i.e., the number of backbone bonds between grafting points, and (iii) the network strand length nx, i.e., the

DP of backbones between crosslinks. Independent control over architectural parameters

−2 [nsc, ng, nx] allows for decoupling the correlation, G ∝ λmax, between the shear modulus

G and maximum elongation at break, λmax, established for linear chain networks. This ability in independent control over the stiffness and strain stiffening of graft polymer networks makes it possible to mimic the entire elastic deformation curve of biological tissues and synthetic gels in a single-component network just through a variation in molecular architecture of the network strands.1 102

Figure 5.1 (a) A graft polymer chain with the degree of polymerization (DP) of the backbone nbb, number of bonds between grafted side chains ng and DP of side chains nsc. (b) A graft polymer network with DP of the backbone between two crosslinks nx . Backbones and side chains are colored in red and blue, respectively. Green bonds in panel (b) indicate crosslinks.

Using the scaling approach and computer simulations, we present detailed analysis of how molecular architecture of network strands influences the mechanical response of graft polymer networks. Our study is based on the diagram of states of graft polymers in a melt developed in Chapter II, which determines different conformational

−1 regimes in terms of nsc and φ (see Figure 2.3 in Chapter II), where φ = ng/(ng+nsc) is the volume fraction of the backbone monomers (consider only graft homopolymers). In different graft polymer conformational regimes, the effective Kuhn length, bK, can be expressed in terms of the DP of side chains, nsc, the number of backbone bonds between grafting point of side chains, ng, and molecular parameters (monomer volume v, bond length l, bare Kuhn length of the backbone b) as summarized in Table 2.1 from Chapter II.

The rest of this chapter is organized as follows: (i) using expressions for the effective Kuhn length of graft polymers obtained for different conformational regimes, we develop a model of graft polymer networks, (ii) the model predictions are compared

with results of coarse-grained molecular dynamics simulations and (iii) the model is also applied to experimental results of deformations of the poly(dimethyl siloxane) (PDMS) graft polymer networks of the well-defined molecular architecture described by the architectural triplet [nsc, ng, nx].

5.2 Scaling Analysis

Consider networks of combs and bottlebrushes made by crosslinking their backbones in a melt state with the DP of the backbone between crosslinks nx . Graft polymers between crosslinks can be modelled as linear chains with effective Kuhn length,

38, 95 bK. For a broad range of synthetic and biological networks, it has been validated that the true stress in such networks undergoing a uniaxial deformation with elongation ratio λ at constant volume is32, 33, 57

−2 G β(λ2+2λ−1) σ(λ) = (λ2 − λ−1) (1+2 (1 − ) ) (5.1) 3 3 where the structural shear modulus G and strand extension ratio β are determined by the molecular architecture of the network strands and their concentration. The strand extension ratio, defined as

2 2 β = ⟨Rin⟩ΤRmax (5.2)

2 is the ratio of the mean square end-to-end distance of undeformed network strands ⟨Rin⟩ to the square of the contour length of a fully extended strand Rmax=nxl in as-prepared networks. By considering graft polymers as worm-like chains with the effective Kuhn

2 10 length bK, ⟨Rin⟩ can be written as

104 2 bK 2Rmax ⟨Rin⟩ = bKRmax (1 − (1 − exp (− ))) (5.3) 2Rmax bK

−1 Thus β is a function of the number of Kuhn segments per strand α ≡ RmaxΤbK

α 2 β = α (1 − (1 − exp (− ))) (5.4) 2 α

For flexible strands, bK ≪ Rmax, the extension ratio is equal to the inverse number of

Kuhn segments per strand, β ≈ α. However, for networks with either shorter or stiffer strands which is relevant for the case of the brush-like networks, β < α.

The structural shear modulus of the network, G, is proportional to the number density of the stress-supporting strands ρs

2 ⟨Rin⟩ G = C1kBTρs (5.5) bKRmax where C1 is a numerical constant that depends on the functionality of the crosslinks and network topology, kB is the Boltzmann constantan and T is the absolute temperature. In order to express ρs in terms of the architectural parameters of graft polymers, we must eliminate the contribution of dangling ends. Consider a network made by crosslinking graft polymers with DP of the backbone nbb. In this case, crosslinking each backbone strand will produce two chain ends of varying length. These dangling ends do not support stress upon deformation and effectively reduce the crosslink density. For a melt of graft polymers with monomer number density ρ, the density of monomers belonging to precursor graft polymer backbones is equal to ρφ and the density of stress-supporting strands can be written as

1 2 ρs = ρφ ( − ) (5.6) nx nbb 105 Note that for densely crosslinked networks, nx ≪ nbb , equation 5.6 reduces to a conventional expression for ρs ≅ ρφΤnx . Substituting equations 5.2, 5.4 and 5.6 into equation 5.5 we obtain

−1 −1 −1 G = C1ρkBTβα φ(nx − 2nbb ) (5.7)

Equation 5.7 is a general expression for the structural modulus of graft polymer networks of which the explicit form as a function of the architectural parameters of the graft polymers can be obtained by using corresponding expressions for β and α in different regimes of diagram of states (see Figure 2.3 in Chapter II).

Before moving forward, it is important to point out the difference between the structural shear modulus G and network shear modulus at small deformations G0

σ(λ) G −2 G0≡lim = (1+2(1 − β) ) (5.8) λ→1 λ2 − λ−1 3 that is commonly used in the case of linear chain networks. Both moduli (G and G0) have the same value only in the case of networks with coiled strands characterized by small extension ratio β ≪ 1 (β ≈ α). For networks with extended strands (like gels) these two network’s moduli are different as follows from equation 5.8.

Below we show how the model of nonlinear network deformation can be applied to networks made by crosslinking graft polymer melts.

5.2.1 Combs

The loose grafting of side chains in the comb regime does not perturb the ideal polymer chain conformation of either side chains or backbones in a melt and leads to bK ≈ b. The presence of side chains only results in the dilution of the stress-supporting backbones, which in turn lowers the structural shear modulus G. For networks with 106 flexible strands (β ≪ 1), the mean square end-to-end distance of the network strands and

2 deformation ratio can be written as ⟨Rin⟩ ≈ bRmax and β ≅ α, respectively. These relations give the following expressions for G and β:

−1 −1 G = C1ρkBTφ(nx − 2nbb ) (5.9)

−1 β = C2nx (5.10) where C2 is a numerical constant. It follows from equations 5.9 and 5.10 that the network structural shear modulus G and strands extension ratio β can be controlled independently, e.g., by maintaining constant nx and varying φ.

5.2.2 Bottlebrushes

We shall first focus on the stretched backbone (SBB) subregime. In this regime, steric repulsion between side chains results in backbone extension. This is manifested in larger effective Kuhn segments (see Table 2.1 in Chapter II) and alters the expression for the structural modulus G given by equation 5.7. The strand extension ratio β is given by equation 5.4 with parameter

−1 −1 −1/2 α = C2nx φ nsc (5.11) where C2 is a numerical constant. Thus, similar to comb systems, G and β can be controlled independently by coordinated variation of nx and φ. At higher grafting densities, the steric repulsion causes extension of side chains and results in two additional regimes of mechanical behavior. In the stretched side shains (SSC) and rod-like side chains (RSC) subregimes, the expression for the structural shear modulus G remains the same as in equation 5.7. However, a new equation for β (equation 5.4) is obtained by substituting the corresponding expressions for bK (see Table 2.1 in Chapter I) into the

107 definition of parameter α ≡ bK/Rmax. The expression for structural shear modulus and other parameters of networks of combs and bottlebrushes are summarized in Table 5.1.

Table 5.1 Structural shear modulus, G, and number of effective Kuhn segments per network strands, α−1, in different conformational regimes.

(1) The numerical constant C may be different in different conformational regimes. 2

5.2.3 Comparison with Linear Chain Networks

We can compare mechanical properties of the comb and bottlebrush networks with those of linear chains. For linear chain networks with the DP between crosslinks nx below the entanglement threshold, the structural shear modulus is10, 152

ρk T G ≈ B (5.12) nx

Note that in equation 5.12 we have neglected the dangling ends contribution, which corresponds to the case of the densely crosslinked networks with nx ≪ nbb (see equation

5.9). The strand extension ratio β is related to the elongation at break λmax of a network strand defined as the ratio of the maximum strand end-to-end distance Rmax, to the initial

2 1Τ2 size of the strand, ⟨Rin⟩ . For flexible network strands, we obtain the following scaling relation between λmax and nx

Rmax −1/2 nxl 1/2 λmax = = β = ≈ nx (5.13) 2 1Τ2 n bl ⟨Rin⟩ √ x

108 Comparing equations 5.12 and 5.13 we can conclude that for linear chain networks

−1 −2 G ∝ nx ∝ β ∝ λmax (5.14)

This equation represents a “golden rule” of the materials science, i.e., softer materials

(networks) are more deformable before they break. For networks of combs and bottlebrushes this golden rule can be broken through independent variation of the architectural parameters [nsc, ng, nx] as follows from equations 5.7~5.11 and one can design networks with more complex relationships between modulus and elongation at break (or extension ratio) depending on the molecular architecture of the graft polymers.1

5.3 Comparison with Simulations

Predictions of the scaling analysis of comb and bottlebrush networks are verified in coarse-grained molecular dynamics simulations85. In our simulations, graft polymer backbones and side chains are modelled as bead-spring chains153 composed of beads with diameter σ interacting through truncated shifted Lennard-Jones (LJ) potential.

Connectivity of the monomers into graft polymers and crosslinking bonds are represented by the combination of the FENE and truncated shifted LJ potentials. We performed simulations of macromolecules with spring constants of the FENE potential being equal

2 2 to 30 kBT/σ and 500 kBT/σ . The set of architectural parameters for studied systems is summarized in Table 5.2. For all studied systems, the monomer density is set to ρσ3 = 0.8.

The simulation details are described in Simulation Methods section. All simulations are performed using LAMMPS84 simulation package.

109 Table 5.2 Summary of studied systems

In our simulations, the graft polymers are crosslinked through the end monomers of the side chains, which corresponds to the experimentally studied systems1. Note that such crosslinking scheme results in a hybrid network composed of brush-like and linear network strands, which requires explicit consideration of the elastic response of both strands populations. However, in a wide range of the strands’ architectures the elastic response of graft polymer networks is dominated by the deformation of the graft polymer 110 strands. Therefore, such networks can be approximated by graft polymer networks crosslinked through their backbones as shown below. This approximation of the elastic properties of networks of graft polymers is also successfully used in analysis of the experimental data in refs 1, 30 and 57.

The mechanical properties of networks of graft polymers are obtained by analyzing stress as a function of the elongation ratio λ of networks undergoing a uniaxial elongation along the x-axis at a constant volume. The tensile stress σxx is calculated from

154-156 the diagonal elements of the pressure tensor Pii as follows

3 1 σ = P − ∑ P (5.15) xx 2 xx 2 ii i

The typical stress-deformation curves of graft polymer networks are shown in

Figure 5.2a-c (the remaining set of the stress-deformation curves can be found in

Simulation Methods section). In each figure, only one part of the architectural triplet [nsc, ng, nx] is varied.

In Figure 5.2a and 5.2b, the decrease of the side chain length and network strand length both leads to the higher shear modulus and lower extensibility. In Figure 5.2c, however, both shear modulus and extensibility increase with the decrease of grafting density. This demonstrates that elastomers made by crosslinking graft polymers could break the locked relationship between the shear modulus and elongation at break given by equation 5.14. It is important to point out that the trends in network deformations observed in Figures 5.2a-c are in a good qualitative agreement with experimental results for uniaxial deformation of the PDMS networks made of comb and bottlebrush strands.1

111 Figure 5.2 Dependence of the tensile stress σxx on the deformation ratio λ for networks of graft polymers with ng = 8, nx ≈ 16 and different side chain lengths nsc (a), with ng = 2, nsc = 8 and different network strand lengths nx (b), and with nsc = 8, nx ≈ 16 and different spacer lengths ng (c). The dashed lines are the best fit to equation 5.1 with structural shear modulus G and strand extension ratio β as fitting parameters. (d-f) Correlations between network mechanical properties and strand architecture illustrated by the linear relationship between reduced shear modulus and reduced density of stress supporting strands for networks in (a-c). Symbol notations are summarized in Table 5.2.

112 For each set of networks, the reduced shear modulus αG/β linearly increases with the increase of reduced density of strand φ/nx (Figure 5.2d-f). In Figures 5.2d and 5.2f, the intercept of linear fit is positive, because the network structure is not altered by the change of side chain length or grafting density. In Figure 5.2e, the reduced shear modulus vanishes at finite reduced density of strand. In this case, the length of strand increases due to the decrease in the number of crosslinks per chain ncr. The network structure breaks down when ncr,0 = 2, which corresponds to the intersection on reduced density of strand axis (φ/nx)0 = φ/(nbb/(ncr,0 + 1)) = 0.0047. To show the universality of the relationship between the molecular architecture of networks and their mechanical properties, we show the dependence of the reduced shear modulus αG of all studied graft polymer networks as a function of parameter βφ/nx in Figure 5.3a.

Figure 5.3 (a) Universal relationship between mechanical properties of networks of graft polymers (structural shear modulus G and strand extension ratio β) and architectural parameters of network strands, described by nx and φ, illustrated by linear scaling between αG and βφ/nx. (b) Universality between the reduced deformation dependent network shear modulus G(I1)/G and the parameter βI1/3 for networks of graft polymers. The dashed lines are given by equation 5.17. Symbol notations in both panels (a) and (b) are summarized in Table 5.2.

113 The universality in the network mechanical properties is not only seen in their correlations with network architectural parameters but also in an excellent fit of the simulation data to the universal function describing the stress in deformed networks undergoing uniaxial elongation (see equation 5.1). To highlight this fact we introduce the deformation dependent shear modulus33

−2 σxx(λ) G βI1(λ) G(I1) ≡ = (1+2 (1 − ) ) (5.16) λ2 − λ−1 3 3

2 where I1(λ) is the first invariant for uniaxial network deformation I1(λ) = λ +2/λ. Figure

5.3b combines data for the reduced deformation dependent network shear modulus

G(I1)/G as a function of βI1/3. The dash lines in Figure 5.3b is given by the following equation33

G(I ) 1 βI (λ) −2 1 = (1+2 (1 − 1 ) ) (5.17) G 3 3

The good collapse of simulation data provides further confirmation that graft polymer networks can be described as networks of linear chains with the effective Kuhn length in which the concentration of the stress supporting strands should be corrected by the fraction of the backbone monomer φ. The deviation of simulation data from the universal plot is due to bond stretching under larger deformation.

114 −2 Figure 5.4 Breaking the “golden rule”, G ∝ λmax, of the materials design. The “golden rule” is shown as the dash line with a slope -2 in logarithmic scales. Data for linear chain networks are shown by brown symbols: nx ≈ 4 (rhombs), 5 (triangles), 6 (inverted triangles), and 9 (squares). Symbol notations for comb and bottlebrush networks are summarized in Table 5.2. Insets show typical network strands with backbone shown in red and side chains colored in blue.

The ability to control the network mechanical properties by complex polymer

−2 architecture result in the breakdown of “golden rule” G ∝ λmax ≈ β for the linear chain networks that is preordained by the DP of network strands between crosslinks for unentangled networks as illustrated in Figure 5.4. This figure shows data sets from

Figures 5.2a-c together with simulation data for networks of linear chains (brown symbols). To make connection with experimental studies of the graft polymer networks we used experimentally established correlation between strands extension ratio and

−1/2 1, 30 elongation at break λmax = β . In the case of graft polymer networks we have data points located in the range of smaller values of the shear modulus along the “golden rule” line (blue symbols). This is due to a combination of the backbone dilution effect and effective stiffening of the polymer backbone due to increase in the DP of the side chains.

115 −2 There are also data sets showing a clear deviation from G ∝ λmax ≈ β scaling dependence

(see filled green and open symbols). The green symbols correspond to a crossover to a stiff network regime with decreasing DP between crosslinks nx (decrease in the number of the effective Kuhn lengths per network strand between crosslinks). There is a new

4 −2 peculiar scaling relation, G ∝ λmax ≈ β , demonstrating an increase of the structural

−1/2 modulus with elongation at break λmax ≈ β (open symbols). This scaling reflects a crossover from the bottlebrush regime to the comb regime through variation of the

−1 grafting density ng of the side chains while keeping the DP between crosslinks nx almost constant.

5.4 Comparison with Experiments

In this section, we apply the model of nonlinear network deformation to mechanical experiment data of elastomers of bottlebrush poly(dimethyl siloxane)

1, 30 (PDMS). Each backbone monomer is grafted with a side chain with DP nsc = 14, while the DP of backbone network strand nx varies from 50 to 400. Figure 5.5a shows the true stress-deformation curve of bottlebrush PDMS elastomers. Unlike elastomers of linear polymers (e.g., rubbers) which deform linearly, these elastomers strain-stiffen at relatively small deformation. This is also demonstrated by the universal plot of the reduced deformation dependent network shear modulus G(I1)/G as a function of βI1/3 in Figure 5.5b. The structural shear modulus G and strand extension ratio β of theses elastomers are obtained by fitting the true stress-deformation curve with equation 5.1.

116 Figure 5.5 (a) True stress-deformation curves of bottlebrush PDMS elastomers. Blue solid curves are experimental data, red dashed curves are best fit to equation 5.1 using G and β as fitting parameters. (b) Universality between the reduced deformation dependent network shear modulus G(I1)/G and the parameter β 퐼1 /3 for bottlebrush PDMS elastomers with ng = 1, side chain length nsc = 14 and different network stand length nx = 50 (purple squares), 67 (blue triangles), 100 (light blue inverted triangles), 200 (orange diamonds) and 400 (red hexagons). The dashed lines are given by equation 5.17. (c-d) Relationship between elastomer mechanical properties and their architectural parameters. Dashed lines are best linear fit.

As expected from the scaling analysis (Table 5.1), the mechanical properties (G and β) of bottlebrush elastomers are related to the network architectural parameters [nsc, ng, nx]. Figure 5.5c shows the linear relationship between the reduced shear modulus

117 and reduced density of stress supporting strands, which is also observed in simulation results (Figure 5.2d-f). These bottlebrush PDMS network strands belong to the stretched side chain (SSC) subregime, so the number of Kuhn segments per network strand is

−1 −1/2 described by α ≈ C2nx φ (Table 5.1). This relationship is proved in Figure 5.5d. The good collapse of the data provides further corroboration of our model of nonlinear deformation of graft polymer elastomers.

5.5 Simulation Methods

We performed molecular dynamics simulations of graft polymer networks. Each graft polymer chain consists of a backbone with the number of monomers nbb= 129, and side chains with nsc monomers being grafted to the backbone with ng backbone bonds between adjacent grafting points. To study the effect of chain architecture on the network properties, nsc is changed from 2 to 32, and ng is varied between 0.5 to 16 (see Table 5.3).

In the case of ng = 0.5, two side chains are grafted to each backbone monomer. Both chain ends of graft macromolecules are capped by linear chain segments with ng bonds.

In the case of ng = 0.5, each ends of macromolecules are capped by one backbone monomer.

Graft polymer networks are prepared by crosslinking graft polymer chains in a melt which includes 100 graft polymer chains. The force field and simulation procedure used to prepare the graft polymer melts are described in Chapter II section 2.5. The crosslink bond is created between a randomly selected pair of beads in accordance with the following algorithm: (1) the beads are on the end of the side chains belonging to different macromolecules, (2) the distance between the beads should be smaller than

118 1.15σ (for ng ≤ 8) or 1.5σ (for ng = 16), (3) there is no crosslink bond between the macromolecules which the beads belonging to, (4) only one crosslinking bond can be formed between two end beads. In a crosslinking step, all pairs of qualified beads are connected by the FENE bonds, this is followed by a 5τLJ simulation run. Crosslinking steps are carried out recursively until 350 crosslink bonds are created (i.e. on average 7 crosslink bonds per macromolecules). For ng= 16, it is almost impossible to create 350 crosslink bonds (which requires all side chain ends to be crosslinked), so the crosslink step is carried out 200 times to create as many crosslink bonds as possible. In the case of linear chain networks, we crosslinked linear chains with DP N = 65 in melts with 787 chains by creating FENE bonds between monomers belonging to different chains following the procedure for monomer selection as described above. The final number of crosslinks, as well as the average number of backbone bonds between two adjacent crosslink points nx , are summarized in Table 5.3 and Table 5.4. After crosslinking

5 process is complete, the network is relaxed for 5×10 τLJ.

To obtain the stress-deformation curve, a set of uniaxial deformation simulations is performed. The final deformation state is obtained by a series of small affine

-1/2 -1/2 deformation [xi, yi, zi] →[(1+Δλ)xi, (1+Δλ) yi, (1+Δλ) zi] with an increment Δλ =

0.025. Each small incremental deformation is achieved by deforming the network at a

3 4 constant rate within 2.5×10 τLJ, followed by a 10 τLJ run for equilibration and a

4 1.5×10 τLJ run for calculation of the average stress. The stress-deformation curves for all networks are shown in Figure 5.6, Figure 5.7 and Figure 5.8.

119 Table 5.3 Summary of graft polymer networks.

Table 5.4 Summary of linear chain networks.

120 Figure 5.6 Dependence of the tensile stress σxx on the deformation ratio λ of networks 2 with kspring = 30 kBT/σ , nx ≈ 16 and different values of nsc and (a) ng = 0.5, (b) ng = 2, (c) ng = 4, (d) ng = 8 and (e) ng = 16. Symbol notations are summarized in Table 5.2. Dashed lines are best fit to equation 5.1 using G and β as fitting parameters.

121 Figure 5.7 Dependence of the tensile stress σxx on the deformation ratio λ of networks 2 with kspring = 500 kBT/σ , nx ≈ 16 and different values of nsc and (a) ng = 0.5, (b) ng = 2, (c) ng = 4, (d) ng = 8 and (e) ng = 16. Symbol notations are summarized in Table 5.2. Dashed lines are best fit to equation 5.1 using G and β as fitting parameters.

122 Figure 5.8 Dependence of the tensile stress σxx on the deformation ratio λ of linear chain networks with nx ≈ 4 (rhombs), nx ≈ 5 (triangles), nx ≈ 6 (inverted triangles), and nx ≈ 9 (squares). Dashed lines are best fit to equation 5.1 using G and β as fitting parameters.

5.6 Conclusions

We use a combination of analytical calculations and computer simulations to show that networks of graft polymers can be described as networks of linear chains with effective Kuhn length of which the value is controlled by the DP of the backbone between side chains ng and DP of side chains nsc. This representation of the network strands follows from the analysis of the comb and bottlebrush macromolecules in a melt95.

The partition of monomers between side chains and backbones results in effective dilution of the stress supporting strands manifested in decrease in the structural shear modulus of networks by a factor . This simple backbone dilution correction works well in the case of the networks made by crosslinking comb macromolecules. In this comb regime, the shear modulus of the graft polymer networks is  times smaller than corresponding shear modulus of the networks of linear chains of the same DP between crosslinks. However, in the case of bottlebrushes, there is an additional contribution to the structural shear modulus due to the stiffening and extension of network strands. The 123 interplay between the dilution and stiffening effects allows breaking down the “golden

−2 rule” G ∝ λmax ≈ β for linear chain networks with simultaneously increasing stiffness and extensibility (Figure 5.4). By changing the molecular architecture of the network strands, it is possible to make more rigid networks that at the same time are super-elastic. This peculiar behavior of networks of graft polymers observed in our simulations is in agreement with recently published experimental data on PDMS networks and is utilized to design elastomers mimicking the mechanical behavior of soft biological tissues which show a combination of softness and strong strain-stiffening.1 4

Parts of this chapter are reprinted with permission from Macromolecules 2018, 51 (2), 638-645. Copyright 2018 American Chemical Society. 124 CHAPTER VI

SELF-ASSEMBLED NETWORKS OF LINEAR-BOTTLEBRUSH-LINEAR

TRIBLOCK COPOLYMERS

6.1 Introduction

Biological tissues are soft on touch and show strain adaptive stiffening upon deformation to prevent break down and resulting in injury.157, 158 In the case of skin-like tissue, the effective elastic modulus could increase by several orders in magnitude in a narrow interval of deformations.157-159 This response of skin reflects its composite structure made of a scaffold of stiff collagen fibers interwoven with elastin networks.

Collagen fibers are responsible for strain adaptive stiffening while elastin networks ensures elastic recoil.157, 160 Such mechanical behavior is hard to replicate in chemically crosslinked networks.158, 161 It was shown recently that this task can be achieved in linear- bottlebrush-linear (LBL) triblock copolymers (Figure 6.1a) with one poly(dimethyl siloxane) (PDMS) bottlebrush block (B-block) and two poly(methyl methacrylate)

(PMMA) linear chain blocks (L-block).29 The two types of blocks have different physical and chemical properties which results in microphase separation and formation of a network with linear blocks forming network nodes (spherical domains) and bottlebrushes filling the space between them (Figure 6.1b). The bottlebrush blocks are responsible for low modulus at small deformations and strong strain adaptive stiffening, while linear blocks provide energy dissipation pathway enhancing extensibility.29

125

Figure 6.1 (a) A triblock linear-bottlebrush-linear macromolecule with the degree of polymerization (DP) in the a linear block nL (shown in gray), DP in the bottlebrush backbone nbb (shown in red), DP of side chain nsc (shown in blue), and DP of the backbone spacer between neighboring grafting points ng. (b) Illustration of the linear chain domains of size RL consisting of Q linear chains connected by the bottlebrush blocks. Linear chain domains connected by highlighted bottlebrush block are shown in orange, other linear chain domains are shown in grey. Distance between spherical domains RL determines the size of the bottlebrush block in aggregates.

To understand the relationship between the strain-adaptive mechanical behavior and the chemical and architectural structure of the self-assembled network, we have used a combination of analytical calculations and coarse-grained molecular dynamics simulations to study the deformation of physical networks formed by self-assembly of linear-bottlebrush-linear (LBL) triblock copolymers (Figure 6.1). We developed a model of microphase separation of LBL copolymers which explicitly considers nonlinear deformation of the bottlebrush blocks in self-assembled structures. This nonlinear deformation of the bottlebrush blocks is an integral part of the model describing strain- adaptive behavior of the self-assembled network. First, the deformation of such networks is controlled by reversible elongation of the bottlebrush blocks bridging spherical domains of linear chains. This elastic network deformation regime is followed by a network yielding regime. Crossover to this regime takes place when a typical force acting on the bottlebrush blocks exceed the force required to pull the linear chains out of spherical domains creating a new interface between linear and bottlebrush blocks. The prediction of the copolymer self-assembly model and strain-adaptive network deformation model are tested in coarse-grained molecular dynamics simulations of LBL copolymers and verified by experiments on PMMA-bbPDMS-PMMA copolymers (“bb” emphasizes the bottlebrush architecture of the middle PDMS block).29

6.2 Model of Self-Assembled Networks

Consider an LBL triblock copolymer (Figure 6.1a) consisting of linear L-blocks with the degree of polymerization (DP) nL and bottlebrush B-block with backbone DP nbb.

In the bottlebrush block, side chains with DP nsc are grafted to the backbone such that the

DP of the backbone spacer between neighboring grafting points is ng. The properties of the linear chains are characterized by the bond length lL, Kuhn length bL and monomer pervaded volume vL. For bottlebrush block, we assume that backbones and side chains are made of identical monomers with bond length lB , Kuhn length bB and monomer pervaded volume vB . The partitioning of monomers between linear and bottlebrush blocks is described by the volume fraction of monomers belonging to the L-blocks

22vL nv LL n L L == (6.1) 2/vL n L++ v B( n bb n sc n g n bb ) 2/vL n L+ v B n bb 

In simplifying equation 6.1 we introduce the volume fraction of the backbone monomers

ng  = (6.2) nngsc+

Interactions between side chains stiffen the bottlebrush blocks such that the effective Kuhn length bK increases with increasing value of the crowding parameter, Φ, 127 describing the degree of mutual interpenetration between side chains and backbones of neighboring bottlebrush blocks.95, 102 The crowding parameter depends on the molecular parameters of the bottlebrush block, fraction of the backbone monomers as follows

v  −1  B (6.3) 3/2 n12/ (lbBB) sc

Computer simulations of the melts of combs and bottlebrushes in Chapter II have shown that the effective Kuhn length scales linearly with the crowding parameter in the bottlebrush regime

bbKB   /* (6.4)

The crossover value of the crowding parameter Φ* ≈ 0.7 was determined from analysis of the simulation results of the effective Kuhn length of graft polymers in a melt. (see

Chapter II)

6.2.1 Self-Assembly and Equilibrium Properties

Immiscible L- and B-blocks microphase separate, forming L- and B-domains of which the morphology is determined by the chemical structure of individual blocks. To obtain the equilibrium domain sizes as a function of the copolymer structure, we develop a Flory-like approach by assuming the strong segregation limit with narrow interface between L- and B-domains and considering the nonlinear deformation of the bottlebrush blocks.

Consider spherical aggregates consisting of Q L-blocks with radius RL and the distance between L-block aggregates being equal to 2RB (Figure 6.1b). Parameters Q, RL and RB are related through packing conditions of monomers belonging to L- and B-blocks,

128 such that the volumes occupied by the L- and B-blocks in an aggregate are equal to the sum of the pervaded volume of all monomers:

3 4 (RRLB+ ) vB n bb v L n L =Q vLL n + = Q (6.5) 32f L where f is a packing fraction of aggregate lattice (f=0.74 for HCP lattice) accounting for additional volume to be filled by B-blocks to maintain uniform density in a system.

Considering the packing condition for volume occupied by spherical L-domains, we have

1/3 3 1/3 RvnQRQLLLgl== (6.6) 4

1/3 where Rgl = (3vLnLΤ4π) is the size of the globule formed by one collapsed linear block.

Comparing equations 6.5 and 6.6, we can write down an explicit expression for the bottlebrush block size as a function of the aggregation number Q

RRfR=−= Q /11/3 1/3 (6.7) BLLdgl (( ) ) 3

1/3 where parameter δ3d = ( fΤφL) − 1 describes domain packing.

The equilibrium size of the domains is determined by balancing the elastic free energy of the bottlebrush block and surface energy per L-block in the aggregate with surface tension γLB of the L-B interface. Note that the surface tension of the L-B interface has been shown to demonstrate a weak dependence on the molecular architecture of the

B-blocks.162 In this approximation the free energy per LBL macromolecule is

22 1 4RRRBmax 8   LB L  +2 2 + 1/3 (6.8) kBKKBB T2 b Rmax b (1− 4 R / R max ) Q k T

The first two term on the r.h.s. of the equation 6.8 accounts for nonlinear deformation of the bottlebrush block by approximating it as a semiflexible chain with the effective Kuhn 129 32, 163, 164 length bK and maximum bottlebrush backbone elongation, Rmax = lBnbb. The last term on the r.h.s. of equation 6.8 describes the contribution from the surface energy of L-

B interface. We can eliminate Q-dependence in the equation 6.8 by equation 6.7

2 3  148 RRBdmax3  LBglR ++ 22 (6.9) kTbRbLRRkTBKKBBB2(14/)maxmax −

Minimization of this free energy with respect to the spacing between L-domains 2RB results in the nonlinear expression for the strand extension ratio

22  = 4/RRB max (6.10) of the B-block as a function of the structural parameters of the LBL copolymers

3 3/22 −  LB Rbgl K (12(1)16+−= ) 3d 2 (6.11) kB TR max

Note that in the limit β ≪ 1, this expression reduces to the classical expression for the

1/3 165, 166 domain size dependence on the DP of L- and B-blocks, RB ∝ (nLnbb) .

6.2.2 Elastic Deformation of Self-Assembled Networks

To describe deformation of the LBL networks where L-domains are playing a role of the multifunctional crosslinks with functionality Q ≫ 1, we will use the affine network model.10, 152 In the framework of this approach each bottlebrush strand with the initial

x x y y z z size Rin = 2RB deforms affinely such that Rs = λxRin, Rs = λyRin, Rs = λzRin with λi being the deformation ratios along the principal axis and the free energy of a network is equal to the sum of contributions from Nch bottlebrush strands connecting aggregates of L- blocks in a system32, 163 38

130 NNch ch 2 RRs max Fnet( i) = F elast( R s) = k B T  + 22 (6.12) ss==112bK Rmax b K (1− R s / R max )

2 Taking into account that for isotropic affinely deformed networks, ⟨Rs ⟩ =

−1 Nch 2 2 Nch ∑s=1 Rs ≈ I1({λi})⟨Rin⟩/3 and using pre-averaging approximation for ⟨(1 − x2)−1⟩ ≈ (1 − ⟨x2⟩)−1 we have163

−1 II  11( ii) −1 ( ) FGVneti() +− 1 (6.13) 63 

2 2 2 where V is the system volume, I1({λi}) = λx+λy+λz is the first strain invariant of the deformation matrix. Structural shear modulus of the network, G, is proportional to the number density of the stress-supporting strands ρs

R2 in −1 Gk TkBsBs T  (6.14) bRK max where parameter  is related to the number of the effective Kuhn segments per

−1 bottlebrush block, α ≡ RmaxΤbK. For a network of LBL copolymers the density of the stress supporting strands (bottlebrush blocks) is evaluated as

−1 B SBbb++=−( scgbbLvn( nnnv LL /2(1) n ) ) (6.15) nbb

Combining equations 5.14 and 5.15, we can write down the following expression for the structural shear modulus

B −1 G− kBL T (1 )  (6.16) nbb

131 In the case of the uniaxial network deformation at a constant volume, the network extends in one direction, λx = λ, while it contracts in two others, λy = λz = 1/√λ. The true stress generated in a network under uniaxial deformation is

−2  Fnet ( ) G 212 − true =−+−+ ( ) 12 12 // 3 ( ) (6.17) V  3 ( ( ) )

It is important to point out that the structural modulus (equation 6.16) is different from the shear modulus at small deformations (λ → 1)

 −2 GG=+− true 121/ 3  0 21− ( ( ) ) (6.18) − =1

This modulus, G0, also includes information about bottlebrush extensibility determined by the parameter  (see Chapter V). Using equation 6.11 we can write down the following scaling relation for the shear modulus at small deformations in terms of the structural parameters as follows

3    −2  R Gk−+−− T BB(1)1 2  1(1)−−11/2  LB gl 03BLdL ( ( ) ) (6.19) nRnbbbb max

It is important to point out that this expression correlating the chemical structure of bottlebrush block with network mechanical properties is qualitatively different from one derived for chemically crosslinked bottlebrush networks.1, 28, 38

6.2.3 Yielding of Self-Assembled Networks

The main difference between the deformation of chemical networks and self- assembled networks of LBL copolymers is that the stress-deformation relationship given by equation 6.16 breaks down when the sections of the linear L-blocks begin to be pulled out from the spherical aggregates. This takes place in the nonlinear bottlebrush strand 132 deformation regime where each bottlebrush strand can be considered as a nonlinear spring with an effective spring constant32

−2 2kTB 2 Kbb () (1 − (  + 2 /  ) / 3) (6.20) Rbmax K

At the onset of the yielding process the pulling force, fp, in the bottlebrush strand is equal to that required to generate a new interface with surface tension γLB. This balance results in

1/2 2 flKRpLBbbppin ( ) ( /3) (6.21) where λp is the extension ratio at the onset of yielding (pulling out of linear blocks from

2 1/2 aggregates) and factor λp(⟨Rin⟩/3) accounts for the deformation of bottlebrush strands from their initial state. Accounting for the concentration of stress supporting strands and using the expression for the stress in network (equation 6.17) and the structural modulus

G (equation 6.16) we obtain the following expressions for the yielding stress as a function of the yielding strain:

221/21/2 pS−  LBinpLBBLp l Rl (1 ) (6.22)

In the entire yielding regime, the true stress can be approximated as

true  p +E y(  −  p ) (6.23)

2 where Ey ≈ ρSγLBl is the effective yielding Young’s modulus.

6.3 Comparison with Simulations

To test our model of cascading network deformation we performed coarse-grained molecular dynamics simulations85 of self-assembled LBL networks. In these simulations

133 macromolecule backbones, side chains and linear chain blocks were modelled as bead- spring chains composed of beads with diameter  interacting through truncated shifted

Lennard-Jones (LJ) potential. All bonds in copolymers were represented by the combination of the FENE and truncated shifted LJ-potentials. 86 The linear chains making up the end blocks consists of nL monomers. The middle bottlebrush block has number of monomers in the backbone nbb, side chains with nsc = 4 and number of bonds between grafted side chains ng = 1 (one side chain per monomer of the backbone). The list of the studied systems is given in Table 6.1. The system consisted of 2600~8500 triblock macromolecules at final density equal to  ≈ 0.86-3. Details of the system preparation and durations of the simulation runs are summarized in the Simulation Methods section.

Table 6.1 Summary of studied systems

134 Figure 6.2 (a) Probability distribution of the end-to-end distance of the bottlebrush block for LBL copolymers with nL = 10, nbb = 35, nsc = 4 and ng = 1. Insets show typical loop and bridge configurations of bottlebrushes. (b) Dependence of the fraction of bridges on the ratio n /n . Dashed line shows the general trend. Symbols are given in Table 6.1. L bb

We begin our discussion of simulation results with analysis of the structure of the self-assembled networks. The bottlebrush blocks connecting spherical domains of the linear chain blocks are partitioned into two groups corresponding to loops and bridges.

This is illustrated in Figure 6.2a showing the probability distribution of the end-to-end distance of the bottlebrush blocks. There are two well separated peaks in this figure. The first narrow peak corresponds to a loop configuration of the bottlebrush block when both ends of the block end up in the same spherical domain. The second broad peak represents bridge configuration of the bottlebrush connecting two spherical domains made of linear chain blocks. The fraction of the bridges monotonically decreases with increasing the ratio of the DP of the linear chain block nL to the DP of the bottlebrush backbone as sown in Figure 6.2b. This linear decrease in fraction of the bridges can be explained by increasing the spherical domain size with increasing the DP of linear block, nL , accompanied by the increase of the area per bottlebrush block. These two effects lower 135 the configurational entropy penalty for bottlebrush block to form a loop. Note that only bridges support stress and contribute to the physical network mechanical response.

Figure 6.3 (a) Dependence of the tensile stress σtrue on the deformation ratio λ for self- assembled networks of LBL copolymers with similar volume fraction of linear blocks φL = 0.08. Solid lines are the best fit to equation 6.17 with structural shear modulus G and strand extension ratio  as fitting parameters. Dashed lines represent the best fit of yielding regime by equation 6.23. Insets show typical LBL copolymers configurations during deformation. (b) Dependence of the differential modulus dσtrue/dλ on the deformation ratio λ for self-assembled networks of LBL copolymers shown in panel (a). Solid lines correspond to dσtrue/dλ calculated using equation 6.17 with values of structural modulus G and strand extension ratio  obtained from fitting data in panel (a). Symbol notations are given in Table 6.1.

Figure 6.3a shows results of the uniaxial deformations of the self-assembled networks (see Simulation Methods section for details). There are two qualitatively different regimes. In the first deformation regime the spherical domains formed by the linear L-blocks play a role of the physical crosslinks and the elastic response of the network is determined by the nonlinear deformation of bottlebrush blocks (see inset in

Figure 6.3a). This regime is called elastic regime. In this deformation regime simulation data are described by approximating bottlebrush blocks by semiflexible filaments in accordance with equation 6.17. Below we establish correlations of mechanical properties

136 of the self-assembled networks and molecular architecture of copolymers. As deformation of the bottlebrush strands increases the force generated in the bottlebrush backbone becomes sufficient to pull an extra monomer from the aggregate (L-domain)

(see inset in Figure 6.3a). The pulling force is associated with the creation of a new interface. This process occurs at a constant force such that the network true stress scales linearly with network deformation ratio  σtrue ∝ λ. The crossover between two different deformation regimes is more clearly seen by plotting the differential stress dσtrue/dλ as a function of deformation ratio  (Figure 6.3b). In this representation one see characteristic S-shape curves highlighting two different mechanisms of the physical network deformation.29

Having identified two different regimes in self-assembled network deformation, we can directly test model predictions for elastic regime where network elasticity is controlled by the elongation of the bottlebrush block. Figure 6.4a shows the correlation between macroscopic network properties characterized by shear modulus at small deformations G0 and bottlebrush strand extension ratio  and chemical structure of the

LBL copolymers (equation 6.19). The collapse of the data confirms the expected linear scaling relationship. Furthermore, in Figure 6.4b we demonstrate that for studied networks the deformation of bottlebrush blocks connecting spherical domains of the L- blocks is a result of a fine interplay between nonlinear bottlebrush elasticity and surface energy of the L-B interface (equation 6.11). Note that for this plot we approximated the

1/2 effective Kuhn length of the bottlebrush block as bK ∝ nsc which is expected for

38, 95, 102 bottlebrushes with densely grafted side chains, ng = 1 (see Chapter II).

137 1/2 Figure 6.4 (a) Dependence of G0β on parameters describing chemical structure of the LBL copolymers (equation 6.19). (b) Illustration of the nonlinear effects in determining B-block initial conformation and its dependence on the chemical structure of the LBL copolymers (equation 6.11). Symbol notations are given in Table 6.1. Dashed lines show expected linear correlations.

6.4 Comparison with Experiments

In this section we show that our analytical model and simulation results are in a good agreement with mechanical data for self-assembled networks of LBL copolymers with one poly(dimethyl siloxane) (PDMS) bottlebrush block (B-block) and two poly(methyl methacrylate) (PMMA) linear blocks (L-blocks).29 These copolymers have

PDMS side chains with the same DP, nsc = 14, grafted to each monomeric unit of the backbone, ng = 1. The DP of the PMMA block nL was varied between 50 and 800 while the DP of the bottlebrush backbone nbb was equal to 302, 938 and 1065.

Figure 6.5a shows typical deformation curves for this type of copolymers. There are two different deformation regimes of self-assembled networks. First, the network of

LBL copolymers undergoes elastic deformation which is described by the nonlinear network deformation model (equation 6.17). This is confirmed in Figures 6.5b and 6.5c

138 showing the universal scaling for shear modulus at small deformations G0 and bottlebrush strand extension ratio  on the structural parameters of LBL copolymers in accordance with equations 6.19 and 6.11 respectively. The good collapse of the data provides further corroboration of our model of the deformation of self-assembled networks. The elastic deformation regime is followed by the crossover to the yielding regime. Note that experimental data show a much broader crossover to this regime than coarse-grained molecular dynamics simulations. This may due to the polydispersity of synthesized triblock copolymers.

Figure 6.5 (a) Dependence of the tensile stress σtrue on the deformation ratio λ for self- assembled networks of LBL copolymers with nL = 360, 480 and 810, nbb = 1065, nsc = 14 and ng = 1. Solid lines represent the best fit of elastic regime by equation 6.17. Dashed 1/2 lines represent the best fit of yielding regime by equation 6.23. (b) Dependence of G0β on parameters describing chemical structure of the LBL copolymers (equation 6.19). (c) Illustration of the nonlinear effects in determining B-block initial conformation and its dependence on the chemical structure of the LBL copolymers (equation 6.11). Symbols for series of LBL copolymers: nbb = 302 (red circles), 938 (orange circles), and 1065 (blue circles).

6.5 Simulation Methods

Coarse-grained molecular dynamics simulation approach was used to model nonlinear deformation of the self-assembled networks of linear-bottlebrush-linear copolymers. In these simulations copolymer macromolecules are represented as bead-

139 spring chains of beads with diameter  connected by the FENE bonds.86 Force field and equation of motion of beads being used in the simulation are described in Chapter II section 2.5. A self-assembled network consisted of 2681 triblock macromolecules was prepared by first placing these macromolecules randomly in a cubic 100  100  100 simulation box with a monomer density  = 0.85σ−3. Initially the non-bonded interaction

1/6 potentials between all beads were set to be the same having ε = 1.0kBT and rcut = 2 σ.

Chain conformations were relaxed in accordance with methodology described in Chapter

II section 2.5. To force microphase separation the strength of the interaction parameter between monomers belonging to linear and bottlebrush blocks (L-B pairs) εLB was

3 gradually changed from 1.0kBT to 6.0kBT with an increment of 0.1kBT during 5×10 τLJ for each increment change. The interaction parameters between L-L and B-B pairs were kept constant and equal to εLL = εBB = 1.0 kBT . This simulation run continued for

5 2.5×10 τLJ . During this simulation run, linear chains slowly aggregate into small spherical domains and both linear and bottlebrush segments remain in an amorphous state.

The structure of aggregates was frozen by performing a NPT85 simulation run under

3 4 pressure P = 5.0kBT/σ for duration of 5×10 τLJ is carried. During this simulation run the interaction potential between L-L pairs is changed to an attractive truncated-shifted LJ- potential with ε = 3.0kBT and rcut = 2.5σ. At the end of this simulation run, the average density of the self-assembled network increases to  = 0.86σ−3 due to the slight decrease in the equilibrium distance between monomers belonging to linear chains. To complete

85 4 relaxation of the self-assembled network a NVT simulation run lasting 5×10 τLJ was

140 carried out. For this simulation run the interaction parameters (ε, rcut) were set to (3.0kBT,

1/6 1/6 2.5) for L-L pairs, (1.0kBT, 2 ) for B-B pairs and (6.0kBT, 2 ) for L-B pairs.

The stress-deformation curves are obtained by performing sets of uniaxial deformation simulations. 32, 38, 154-156, 163 A new deformation state was obtained by a series

-1/2 -1/2 of small affine deformations [xi, yi, zi] →[(1+Δλ)xi, (1+Δλ) yi, (1+Δλ) zi] with an increment Δλ = 0.025. Each small incremental deformation was achieved by deforming

3 4 the network at a constant rate with duration 2.5×10 τLJ, followed by a 10 τLJ equilibration

4 run and a 1.5×10 τLJ production run.

6.6 Conclusions

In conclusion we have shown that the deformation of self-assembled networks of linear-bottlebrush-linear copolymers is a process with two stages. First bottlebrush strands undergo an elastic deformation. This process is reversible. The deformation of the network of bottlebrush strands can be described by the nonlinear network deformation model developed for networks of semiflexible chains.32, 163 This follows by the yielding regime when the linear blocks are pulled out from spherical domains. This deformation could be irreversible if one reaches a stage of the complete removal of the linear chains from the domains. In simulations this process is manifested in the drop of the stress. The results of the computer simulations are in very good qualitative agreement with experimental data on PMMA-bbPDMS-PMMA copolymer systems.29

It is important to point out that the approach developed here for accounting nonlinear deformation of the bottlebrush blocks can also be used in describing similar effects in copolymers with different chemical architecture of the blocks (linear chains,

141 combs or bottlebrushes).24, 36, 37, 167-171 In particular, such nonlinear corrections could result in shifting of the phase boundaries between phases with different morphologies.107,

165, 167, 171 Furthermore, our model of the strain-adaptive deformation of LBL networks can be directly applied to describe deformation of self-assembled networks made of linear chain copolymers.168, 172-174 5

Parts of this chapter are reprinted with permission from Macromolecules [Online early access]. DOI: 10.1021/acs.macromol.9b01859. Published online: Nov 4, 2019. Copyright 2019 American Chemical Society. 142 CHAPTER VII

SUMMARY

To summarize, this dissertation proposes a material design platform based on graft polymers, which are composed of a linear backbone grafted with many short side chains. The structure-property relationship of melts, chemically crosslinked networks and self-assembled networks based on graft polymers are studied with a combination of analytical calculations and coarse-grained molecular dynamics simulations. Material properties, such as the entanglement shear modulus of melts, stiffness and nonlinear response to deformations of networks, are correlated with the structural parameters of graft polymers, including the degree of polymerization (DP) of side chains nsc, DP of spacers between neighboring grafting points ng, DP of backbone of network strands nx, as well as the chemical structure of monomers which influences the monomer volume v, monomer projection length l, and bare Kuhn length b of backbones and side chains.

In Chapter II, the chain conformation of graft polymers is thoroughly studied. The complex architectural structures ( nsc , ng ) and chemical structures (v, l, and b of backbones and side chains) determine the chain conformation of graft polymers through one parameter: the crowding parameter Φ, which describes the extent of interpenetration between different macromolecules. According to the crowding parameter, graft polymers are classified into two regimes, combs and bottlebrushes, with different chain

143

conformations. When Φ < 1, different macromolecules overlap within the excluded volume of side chains and the graft polymers are called combs. In the comb regime, both backbones and side chains behave like ideal linear chains in a melt. When Φ > 1, densely grafted side chains expel monomers of other macromolecules out of its excluded volume and graft polymers are called bottlebrushes. In the bottlebrush regime, backbones will be stretched due to the steric repulsion of side chains and the macromolecule behaves like a semiflexible filament, of which the effective Kuhn length is linearly proportional to the crowding parameter Φ.

Chapter III provides detailed analysis on the static scattering function of graft polymer melts. Under the framework of random phase approximation, the analytical expression of the static scattering function is derived. A peak in the static scattering function is identified and correlated with intrachain correlations. A scaling relationship between the location of this peak q* and the architectural parameters (nsc, ng) of graft polymers is derived and verified in coarse-grained molecular dynamics simulations.

Furthermore, the peak position of bottlebrushes is also related to their effective Kuhn

−4 −1 length, bK ∝ q* (nscng) . This provides a convenient way to characterize the effective

Kuhn length of bottlebrushes from scattering experiments and allows experimentalists to test the prediction of the bottlebrush’s effective Kuhn length derived in Chapter II.

Chapter IV shows the effect of the graft polymer architecture on the rheological behaviors of melts. Melts of graft polymers in different configurational regimes (as studied in Chapter II) show distinct dependences on the graft polymer architecture. In the comb regime, side chains dilute the concentration of long linear backbones and suppress entanglements like what solvent molecules do in polymer solutions. In the bottlebrush regime, besides the dilution effect, side chains also stiffen the backbone due to steric repulsion between them. Since stiffer chains are more prone to entanglements, the stiffening effect counteracts the dilution effect. Therefore, side chain length and grafting density have weaker effects on reducing the entanglement shear modulus on the melts of bottlebrushes than on the melts of combs. This can help polymer engineers to find the

“sweet spot” between the cost of grafting more and longer side chains to linear backbones and the ease of polymer melt processing.

Chapter V presents the structure-property relationship of chemically crosslinked graft polymer networks. Based on the chain conformation (Chapter II) and nonlinear elasticity theory32, 33, the network stress in response to both linear and nonlinear deformation is encoded with the architectural parameter triplet [nsc, ng , nx ]. The rich parameter space of graft polymer networks enables the independent control of both stiffness and elongation at break of networks, which is a mission impossible for traditional linear polymer elastomers. In particular, bottlebrush networks show great potential in mimicking the mechanical behavior of biological tissues which show a combination of super softness and strain-stiffening. In this case, the dilution effect of side chains lowers the density of stress supporting backbones, thus reducing the modulus of the network. Meanwhile, the stiffening of backbone due to the steric repulsion of side chains endows the network with strain-stiffening behavior at small deformations.

However, to mimic some biological tissues with strong strain-stiffening like skins, the pre-stretch of backbone is not enough owing to the difficulty in synthesis of bottlebrush networks with densely grafted long side chains. To overcome this limitation, a new strategy is proposed in Chapter VI.

145 In Chapter VI, physical networks formed by self-assembly of linear-bottlebrush- linear (LBL) triblock copolymers is proposed as the strategy to mimic the mechanical behavior of skin-like tissues. The chemical and physical dissimilarity between the linear and bottlebrush blocks leads to microphase separation, which forces further extension of the bottlebrush backbone in addition to the stiffening effect by steric repulsion of grafted side chains. The extent of pre-stretching, as well as the shear modulus at small deformation, can be controlled by the DP of linear blocks nL and the bottlebrush backbone nbb.

The structure-property relationship derived and verified in this dissertation allows the rational design of materials base on graft polymers. For example, chemical crosslinked networks and physical networks can be designed to mimic the mechanical behavior of soft gels1 and biological tissues1, 29. The design strategy is outlined as followed and summarized in Figures 7.1a and 7.1c: (i) fit the stress-strain curve to the nonlinear elasticity model (equations 5.1) in terms of the structural shear modulus, G, and strand extension ratio, β; (ii) use the established structure-property relationship of graft polymer networks to determine the desired chemical architecture [nsc , ng , nx ] of the network strand (or [nbb, nL] of the LBL triblock copolymer); (iii) synthesize the graft polymer network with the given chemical architecture (or synthesize the LBL triblock copolymer and prepare the physical network by self-assembly); and (iv) measure mechanical properties of the replica and verify the replication procedure. Such replications by solvent-free graft polymer networks prevent the solvent evaporation and freezing issue of synthetic gels under harsh working environments and provide good candidates for the application of soft robotics. 146 Figure 7.1 (a) Flow chart of the procedure for replicating mechanical properties of gels in chemically crosslinked graft polymer network. (b) Stress-strain data for selected gels (white squares) with the best fit by equation 5.1 (red dashed lines), and stress-strain data for chemically crosslinked PDMS graft polymer networks (blue solid lines) synthesized to replicate the selected gels, with corresponding architectural triplet [nsc , ng , nx ] as indicated. (Reproduced with permission from Nature 2017, 549, 497-501. Copyright 2017 Springer Nature) (c) Flow chart of the procedure for replicating mechanical properties of tissues in physical networks of self-assembled LBL triblock copolymers. (d) Stress-strain data for selected tissues (white squares) with the best fit by equation 5.1 (red dashed lines), and stress-strain data for PMMA-bbPDMS-PMMA physical networks (blue solid lines) synthesized to replicate the selected tissues, with corresponding architectural parameters [nbb, nL] as indicated. Each monomer on the bottlebrush PDMS block is grafted with one side chain with DP nsc = 14. (Reproduced with permission from Nature 2017, 549, 497-501. Copyright 2017 Springer Nature)

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