Computational Study of Chain Transfer Reactions in Self-Initiated High-

Temperature Polymerization of Alkyl Acrylates

A Thesis

Submitted to the Faculty

of

Drexel University

by

Nazanin Moghadam Maragheh

in partial fulfillment of the

requirements for the degree

of

Doctor of Philosophy

March 2015 © Copyright March 2015

Nazanin Moghadam Maragheh. All Rights Reserved. ii

Acknowledgements

I want to thank my advisor, Prof. Masoud Soroush, for advising me and being supportive throughout my graduate studies. I also wish to thank my co-advisor, Prof. Andrew Rappe, for his invaluable inputs and suggestions on my research.

I would also like to acknowledge all of my committee members Prof. Giuseppe Palmese,

Prof. Raj Mutharasan, Dr. Michael Grady, Prof. Masoud Soroush, Prof. Andrew

Rappe, and Dr. Sriraj Srinivasan for their valuable time, input and advice. A special thank is extended to Pat Corcoran for providing me with monomer conversion measurements. I appreciate Dr. Michael Grady for his insightful suggestions on the experiments performed. I thank Dr. Sriraj Srinivasan for his views and constructive criticism. I am thankful to my lab members, Pat

Corcoran, Taha Mohseni, Yuriy Smolin, and students in Prof. Rappe’s group at

University of Pennsylvania, especially Shi Liu and Lai Jiang, for their help and interesting discussions. Acknowledgement is also made to the donors of the

American Chemical Society Petroleum Research Fund and National Science

Foundation for funding my research. Last, but not the least, I want to thank my husband, Erfan Mostafid, my parents, Farah Pourferdows and Parviz

Moghadam, and family for their love, understanding and unconditional support, without whom none of this would ever have been possible. iii

Table of Contents

List of Tables...... vii

List of Figures...... xi

Abstract...... xiv

1. Introduction ...... 1

2. Background ...... 4

2.1. Kinetics of High-Temperature Free-Radical Polymerization ...... 4

2.2. Chain Transfer Reactions ...... 5

2.3. Ab Initio Molecular Orbital Theory ...... 6

2.4. Transition State Theory ...... 8

2.5. Computational Quantum Chemistry ...... 9

3. Computational Methods and Procedures Used ...... 12

3.1. Computational Method ...... 12

3.2. Application of ORCA in Chemical Shift Calculations ...... 13

3.3. Design of Reaction Scheme ...... 15

3.4. File Types ...... 18

3.5. Types of Calculations ...... 18

3.5.1. Geometry Optimization...... 19

3.5.2. Hessian for Optimized Geometry ...... 19 iv

3.5.3. Saddle Point ...... 20

3.5.4. Hessian for Saddle Point ...... 20

3.5.5. Intrinsic Reaction Coordinate (IRC) ...... 20

4. Chain Transfer to Monomer Reactions in Polymerization of Alkyl Acrylates ...... 21

4.1. Introduction ...... 21

4.2. Computational Methods ...... 24

4.3. Results and Discussion ...... 25

▪ 4.3.1. Chain Transfer to Monomer Mechanisms for M2 ...... 25

▪ 4.3.2. Chain Transfer to Monomer Mechanisms for M1 ...... 42

4.3.3. Effect of Live Polymer Chain Length ...... 44

4.3.4. Effect of Radical Location in a Live Chain ...... 46

4.3.5. Chain Transfer to Monomer Reaction for MA in the Presence of Initiator ...... 49

4.3.6. Replacement of MA Unit with MMA and St ...... 51

4.4. Concluding Remarks ...... 52

5. Theoretical Study of Chain Transfer to Polymer Reactions of Alkyl Acrylates ...... 54

5.1. Introduction ...... 54

5.2. Computational Methods ...... 56

5.3. Results and Discussion ...... 57

5.3.1. Mechanisms of CTP in MA, EA and n-BA ...... 57

5.3.2. Effect of the Type of the Radical that Initiated a Live MA-Polymer Chain ...... 62

5.3.3. CTP Mechanism for EA and n-BA ...... 65 v

5.3.4. Effect of the Live Polymer Chain Length ...... 69

5.3.5. Different Structures of the Dead Polymers ...... 72

5.3.6. Continuum Solvation Models: IEF-PCM and COSMO ...... 76

5.4. Concluding Remarks ...... 82

6. Chain Transfer to Solvent Reactions of Alkyl Acrylates ...... 84

6.1. Introduction ...... 84

6.2. Computational Methods ...... 86

6.3. Results and Discussion ...... 87

6.3.1. Most Likely CTS Mechanisms for MA, EA, and n-BA ...... 87

6.3.2. Chain Transfer to n-Butanol, sec-Butanol, and tert-Butanol ...... 93

6.3.3. Continuum Solvation Models: PCM and COSMO ...... 97

6.3.4. Effect of the Type of Initiating Radical on CTS ...... 102

6.3.5. Effect of Live Polymer Chain Length ...... 106

6.4. Concluding Remarks ...... 109

7. Entropy Effect on Activation Energy and Gibb’s Free Energy of Activation ...... 111

7.1. Large Contribution of Entropy in Gibb’s Free Energy of Activation ...... 111

7.2. Correct Reactant for a Bimolecular Reaction ...... 112

7.3. QM/MM Hybrid Method ...... 113

8. Macroscopic Modeling of Spontaneous Polymerization of n-Butyl Acrylate ...... 118

8.1. Introduction ...... 118

8.2. Mathematical Modeling ...... 120 vi

8.2.1. Reaction Mechanisms ...... 120

8.2.2. Rate Equations ...... 122

8.2.3. Batch Reactor Model ...... 127

8.3. Experimental and Analytical Procedures ...... 127

8.4. Parameter Estimation ...... 129

8.5. Concluding Remarks ...... 133

9. Conclusions and Future Directions ...... 134

9.1. Conclusions ...... 134

9.2. Future Directions ...... 136

Bibliography...... 141

Vita...... 156 vii

List of Tables

Table 3-1: Assignment of 13C-NMR Spectra for MA, EA and n-BA (Chemical Shifts are Relative to TMS with Absolute Isotropic Shielding of 187 ppm for the 13C) ...... 16

‡ Table 4-1: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Four CTM Reactions in MA at 298 K ...... 28

Table 4-2: H-R Bond Dissociation Energies (kJ mol-1) at 298 K ...... 29

‡ Table 4-3: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of ‡ -1 -1 -1 Activation (∆G ) in kJ mol ; Frequency Factor (A) and Rate Constant (ktrM) in M s , for the Two CTM Reactions in EA at 298 K ...... 32

‡ Table 4-4: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Four CTM Reactions in n-BA at 298 K ...... 34

Table 4-5: Assignment of 13C-NMR Spectra for MA, EA and n-BA (Chemical Shifts are Relative to TMS with Absolute Isotropic Shielding of 187 ppm for the 13C) ...... 39

Table 4-6: The Energy Differences between the Reactants and the Products in kJ mol-1 for the Four CTM Reactions in MA at 298 K ...... 41

Table 4-7: The Energy Differences between the Reactants and the Products in kJ mol-1 for the Two CTM Reactions in EA at 298 K ...... 41

Table 4-8: The Energy Differences between the Reactants and the Products in kJ mol-1 for the Four CTM Reactions in n-BA at 298 K ...... 42

‡ Table 4-9: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the MA-1, ▪ EA-1 and n-BA-1 Reactions Involving Two-Monomer-Unit Live Chains Initiated by M1 at 298 K...... 45

‡ Table 4-10: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the MA-1, ▪ EA-1 and n-BA-1 Reactions Involving Three-Monomer-Unit Live Chains Initiated by M2 at 298 K Calculated Using B3LYP/6-31G(d) and M06-2X/6-31G(d,p) ...... 48

‡ Table 4-11: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the MA-1 ▪ Reaction Involving Three-Monomer-Unit Live Chains Initiated by M1 at 298 K Calculated Using M06-2X/6-31G(d,p) ...... 50 viii

‡ Table 4-12: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the MA-1 ▪ Reaction Involving Two-Monomer-Unit Live Chain Initiated by M2 in the Presence of Initiator (tert-Butyl peroxyacetate) at 298 K ...... 50

‡ Table 4-13: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the MA-1- MMA and MA-1-St Mechanisms at 298 K ...... 51

-1 Table 5-1: Energy Differences of the Reactants (ER) and Products (EP), in kJ mol for Three CTP Mechanisms for MA at 298 K ...... 59

-1 Table 5-2: Energy Differences of the Reactants (ER) and Products (EP), in kJ mol for the Four CTP Mechanisms for EA at 298 K ...... 60

-1 Table 5-3: Energy Differences of the Reactants (ER) and Products (EP), in kJ mol for the Six CTP Mechanisms in n-BA at 298 K ...... 60

Table 5-4: H-R Bond Dissociation Energies (kJ mol-1) at 298 K ...... 61

‡ Table 5-5: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of ‡ -1 Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for the Most Probable CTP Mechanisms for MA at 298 K; and Rate Constant without Tunneling at 413 K (k413) ...... 64

‡ Table 5-6: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of ‡ -1 Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for the Most Probable CTP Mechanisms for EA (EA2-D1-1) at 298 K; and Rate Constant without Tunneling at 413 K (k413) ...... 66

‡ Table 5-7: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of ‡ -1 Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for the n-BA2-D1-1 Mechanism at 298 K; and Rate Constant without Tunneling at 413 K (k413) ...... 67

‡ ‡ Table 5-8: Bond Length in Å Activation Energy (Ea), Enthalpy (∆H ), and Free Energy (∆G ) in -1 kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate -1 -1 Constant (k: without tunneling and kw: with tunneling) in M s by Considering the Radicals with Different Monomer Units for MA and EA, Using B3LYP/6-31G(d,p) ...... 71

Table 5-9: H-R Bond Dissociation Energies (kJ mol-1) for D2 and D3 Dead Polymers at 298 K, Using B3LYP/6-31G(d,p) ...... 73

‡ Table 5-10: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of ‡ -1 Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for the MA2-D2-1 and MA2-D3-1 Mechanisms at 298 K; and Rate Constant without Tunneling at 413 K (k413) ..... 78 ix

‡ Table 5-11: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of ‡ -1 Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for the Most Probable Mechanisms of Chain Transfer to D2 and D3 Dead Polymers of EA and n-BA at 298 K; and Rate Constant without Tunneling at 413 K (k413) ...... 80

‡ Table 5-12: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; and Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Most Probable CTP Mechanisms for MA at 298 K, Calculated Using IEF-PCM and COSMO ... 81

‡ Table 5-13: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; and Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Most Probable CTP Mechanisms for EA and n-BA at 298 K, Calculated Using COSMO and IEF- PCM ...... 82

Table 6-1: H-R Bond Dissociation Energies (kJ mol-1) at 298 K ...... 88

‡ Table 6-2: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of ‡ -1 Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of MA at 298 K ...... 90

‡ Table 6-3: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of ‡ -1 Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of EA at 298 K ...... 94

‡ Table 6-4: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of ‡ -1 Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of n-BA at 298 K ...... 95

Table 6-5: H-R Bond Dissociation Energies (kJ mol-1) at 298 K ...... 97

‡ Table 6-6: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of ‡ -1 Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for CTB1-2, CTBsec1-2, and CTBtert1-2 Mechanisms of MA at 298 K...... 99

‡ Table 6-7: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1s-1, for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of MA, EA, and n-BA at 298 K, Using PCM ...... 100

‡ Table 6-8: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1s-1, for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of MA, EA, and n-BA at 298 K, Using COSMO ...... 101

‡ Table 6-9: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of ‡ -1 Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor x

-1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for CTB1-1, CTM1-1, and CTX1-1 Mechanisms of MA, EA, and n-BA at 298 K ...... 104

‡ Table 6-10: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1s-1, for CTB1-1, CTM1-1, and CTX1-1 Mechanisms of MA, EA, and n-BA at 298 K, Using PCM ...... 105

‡ Table 6-11: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1s-1, for CTB1-1, CTM1-1, and CTX1-1 Mechanisms of MA, EA, and n-BA at 298 K, Using COSMO ...... 106

‡ Table 6-12: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of ‡ -1 Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s , for CTB1-2´, CTM1-2´, and CTX1-2´ Mechanisms of MA, EA, and n-BA at 298 K ...... 108

‡ Table 6-13: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for CTB1-2´, CTM1-2´, and CTX1-2´ Mechanisms of MA, EA, and n-BA at 298 K, Using PCM ...... 109

Table 7-1: Contribution of Translational, Rotational, and Vibrational Entropies in the Total Entropy (jmol-1K-1)of Activation for the Most Likely Mechanisms of Chain transfer to MEK (CTM1-2), Chain Transfer to Monomer (MA-1), and Chain Transfer to Polymer (MA2-D2-1) for MA, Using B3LYP/6-31G(d,p) ...... 111

-1 -1 Table 7-2: Entropy of Activation in J mol K , Activation Energy (Ea), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; for the most likely chain transfer to MEK mechanism (CTM1-2) at 298 K, Using B3LYP/6-31G(d,p) ...... 112

‡ Table 7-3: Energy Barrier (E0), Activation Energy (Ea), Enthalpy of Activation (∆H ), Zero-Point Vibrational Energy (ZPVE), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the Cyclobutane Formation from Two Ethylenes, at 298 K...... 117

Table 7-4: Change of Entropy Decomposed into Translational, Rotational, and Vibrational in J mol-1, Using B3LYP/ 6-31G ...... 117

Table 8-1: Polymerization Reactions ...... 121

Table 8-2: Reaction Rate Constant Values ...... 129

Table 8-3: Reaction Rate Constant Definitions and Correlations [9, 167] ...... 130

Table 8-4: Dimensionless Kinetic Parameter Values Needed to Calculate Rate Constants Given in Table 8-3 [9, 167] ...... 130

-1 Table 8-5: Activation Energy (Ea), Energy Barrier (E0) in kJ mol ; Frequency Factor (A) and Rate -1 -1 Constant (ki) in M s for Monomer Self-Initiation Reaction ...... 133 xi

List of Figures

Figure 3-1: Carbon Atoms Considered in MA, EA, n-BA, and p-Xylene for 13C-NMR Calculations...... 14

Figure 3-2: Input Template File for NMR Chemical Shifts Calculations...... 15

Figure 3-3: Output Template File for NMR Chemical Shifts Calculations...... 17

Figure 4-1: Two Types of Mono-radical Generated by Self-Initiation...... 26

Figure 4-2: Possible End-chain Transfer to Monomer Reactions for MA...... 26

Figure 4-3: Transition State Geometry of the Three Mechanisms: (a) MA-1, (b) EA-1, and (c) n- BA-1...... 30

Figure 4-4: Possible End-chain Transfer to Monomer Reactions for EA...... 31

Figure 4-5: Possible End-chain Transfer to Monomer Reactions for n-BA...... 33

Figure 4-6: Dead Polymer Chains Generated via the MA-1, EA-1 and n-BA-1 Mechanisms (a: MA-1, b: EA-1, c: n-BA-1)...... 37

Figure 4-7: 13C-NMR Spectra of the Dead Polymer Chains Generated via the MA-1, EA-1, and n- BA-1 Mechanisms at 298 K (δ is relative to tetramethylsilane [TMS] with an isotropic 13C nuclear magnetic shielding value of 187 ppm)...... 38

Figure 4-8: The Most Probable Mechanisms for CTM Reactions Involving a Two-Monomer-Unit ▪ Live Chain Initiated by M1 ...... 43

Figure 4-9: Transition State Geometries of MA-1 (a), EA-1 (b) and n-BA-1 (c) Mechanisms ▪ Involving a Two-Monomer-Unit Live Chain Initiated by M1 ...... 44

Figure 4-10: The Most Probable Mechanisms for CTM Reactions Involving a Three-Monomer- ▪ Unit Live Chain Initiated by M2 ...... 46

Figure 4-11: Transition State Geometries of the MA-1, EA-1 and n-BA-1 Mechanisms Involving ▪ a Three-Monomer-Unit Live Chain Initiated by M2 (a: MA, b: EA, c: n-BA)...... 47

▪ ▪ ▪ Figure 4-12: CTM Reaction Involving the Live Chain Q3 and P3 Initiated by M1 ...... 48

▪ Figure 4-13: Transition State Geometry of MA-1 Reaction Involving the Live Chain Q3 ...... 49

Figure 4-14: Transition State Geometries of the MA-1-MMA and MA-1-St Mechanisms ▪ Involving a Two-Monomer-Unit Live Chain Initiated by M2 (a: MA-1-MMA, b: MA-1-St)...... 52 xii

Figure 5-1: Possible Chain Transfer to Polymer Mechanisms for MA...... 58

Figure 5-2: Possible Chain Transfer to Polymer Mechanisms for EA ...... 58

Figure 5-3: Possible Chain Transfer to Polymer Mechanisms for n-BA...... 59

Figure 5-4: Most Probable CTP Mechanism Involving Two-MA-Unit Live Chain Initiated by ▪ M1 ...... 62

Figure 5-5: Transition State Geometry for CTP Reaction; a: Methyl Acrylate (MA2-D1-1), b: Ethyl Acrylate (EA2-D1-1), c: n-Butyl Acrylate (n-BA2-D1-1)...... 63

Figure 5-6: Transition State Geometry for CTP Reaction via Hydrogen Abstraction from D1 ▪ Dead Polymer by M1 in Methyl Acrylate (MA1-D1-1)...... 65

Figure 5-7: Intrinsic Reaction Coordinate Path for MA2-D1-1, EA2-D1-1, and n-BA2-D1-1 Mechanism in the Alkyl Acrylates; The Energies are Relative to Reactants...... 68

Figure 5-8: Most Probable CTP Mechanism Involving a Three (MA2-D1-1(a)) or Four (MA2- ▪ D1-1(b)) MA-Unit Live Chain Initiated by M2 ...... 69

Figure 5-9: Transition State Geometry for CTP Reaction in Methyl and Ethyl Acrylate; a: MA2- D1-1(a), b: MA2-D1-1(b), c: EA2-D1-1(a)...... 70

Figure 5-10: Most Probable CTP Mechanism Involving a Three EA-Unit Live Chain Initiated by ▪ M2 ...... 71

Figure 5-11: Different Structures of Dead Polymer; a) Termination by Coupling of Two Mono- radicals, b) Termination by Hydrogen Abstraction, c) Termination by Coupling of a Live Chain and Mono-radical...... 72

Figure 5-12: Possible Chain Transfer to D2 and D3 Dead Polymers Mechanisms for MA, EA, and n-BA...... 74

Figure 5-13: Most Probable CTP Mechanism Involving a Two MA-Unit Live Chain Initiated by ▪ M2 ...... 75

Figure 5-14: Most Probable CTP Mechanism Involving a Two EA-Unit Live Chain Initiated by ▪ M2 ...... 76

Figure 5-15: Most Probable CTP Mechanism Involving a Two BA-Unit Live Chain Initiated by ▪ M2 ...... 77

Figure 5-16: Transition State Geometry for CTP Reaction; a: Methyl Acrylate (MA2-D2-1), b: Methyl Acrylate (MA2-D3-1)...... 79

Figure 5-17: Transition State Geometry for CTP Reaction; a: Ethyl Acrylate (EA2-D2-1), b: Ethyl Acrylate (EA2-D3-1)...... 80 xiii

Figure 6-1: Possible End-chain Transfer to Solvent Reactions Involving a Two-Monomer-Unit ▪ Live Chain Initiated by M2 . CTB = Chain Transfer to n-Butanol, CTM = Chain Transfer to Methyl Ethyl Ketone, CTX = Chain Transfer to p-Xylene...... 87

Figure 6-2: Transition State Geometry of the CTB1-2 Mechanisms for (a) MA, (b) EA, and (c) n- BA...... 91

Figure 6-3: Transition State Geometry of the CTM1-2 Mechanisms for (a) MA, (b) EA, and (c) n- BA...... 92

Figure 6-4: Transition State Geometry of the CTX1-2 Mechanisms for (a) MA, (b) EA, and (c) n- BA...... 93

Figure 6-5: Possible End-chain Transfer to sec-Butanol and tert-Butanol Reactions for MA ▪ Involving a Two-Monomer-Unit Live Chain Initiated by M2 ...... 96

Figure 6-6: The Most Likely Mechanisms for CTS Reactions Involving a Two-Monomer-Unit ▪ Live Chain Initiated by M1 ...... 103

Figure 6-7: The Most Likely Mechanisms for CTS Reactions Involving a Three-Monomer-Unit ▪ Live Chain Initiated by M2 ...... 107

Figure 7-1: Modified Format of PDB File for Ethylene ...... 114

Figure 7-2: The Input File Template for Optimization of Ethylene...... 116

Figure 8-1: Model Prediction and Measurements of Conversion as well as the Sensitivity of -1 -1 Predicted Conversion to 10% Changes in ki (ki*=4.39E-15 L.mol .s ) at 413 K...... 131

-1 -1 Figure 8-2: Model Prediction and Measurements of Conversion (ki=2.09E-14 L.mol .s ) at 433 K...... 131

-1 -1 Figure 8-3: Model Prediction and Measurements of Conversion (ki=3.50E-13 L.mol .s ) at 433 K...... 132 xiv

Abstract Computational Study of Chain Transfer Reactions in Self-Initiated High-Temperature Polymerization of Alkyl Acrylates Nazanin Moghadam Maragheh Advisor: Prof. Masoud Soroush Co-advisor: Prof. Andrew Rappe

A better understanding of the kinetics of intermolecular chain transfer reactions, such as chain transfer to solvent, monomer and polymer, allows one to produce higher quality polymer resins. In this research project, kinetics of chain transfer to monomer (CTM), chain transfer to polymer (CTP), and chain transfer to solvent (CTS) reactions in self- initiated high-temperature homo-polymerization of alkyl acrylates (methyl, ethyl and n- butyl acrylate) were studied. Possible mechanisms of the chain transfer reactions were studied using density functional theory (DFT) calculations. Transition state theory was used to estimate rate constants of the reactions. Effects of live polymer chain length, type of mono-radical that initiated the live polymer chain, and type of live polymer chain radical (tertiary vs. secondary) on the energy barriers and rate constants of the involved reaction steps were investigated theoretically.

The results indicated that abstractions of a hydrogen atom (by a live polymer chain) from the methyl group in methyl acrylate (MA), the methylene group in ethyl acrylate (EA), and methylene groups in n-butyl acrylate (n-BA) are the most likely mechanisms of

CTM.

Among all possible CTP reaction mechanisms, hydrogen atom abstraction from a tertiary carbon atom, which leads to the formation of a stable tertiary radical, was identified as the most favorable mechanism for CTP reaction in alkyl acrylates. The CTP reactivity of

Integral equation formalism (IEF) and conductor-like screening model (COSMO) were applied to study solvent effects on CTP reactions. While the application of IEF showed strong solvent effects on the kinetic parameters of CTP reactions, the application of COSMO showed no such remarkable effects. The CTP reactivity of

MA, EA, and n-BA was also compared. The difference in the end-substituent group

(MA, EA, or n-BA) does not affect the kinetics of the CTP reactions remarkably.

Chain transfer reactions to several solvents such as butanol (polar, protic), methyl ethyl ketone (MEK) (polar, aprotic), and p-xylene (nonpolar) were studied. The estimated lower activation energies and higher rate constants of chain transfer to n- butanol compared to those of MEK and p-xylene are indicative of the higher CTS reactivity of n-butanol. Among n-butanol, sec-butanol, and tert-butanol, tert-butanol has the highest CTS energy barrier and the lowest rate constant. Polarizable continuum model (PCM) and COSMO were applied to explore chain transfer to n- butanol, MEK and p-xylene from polymer chains of MA, EA and n-BA. Application of PCM resulted in huge changes in the kinetic parameters of chain transfer to n- butanol. The type of mono-radical generated via self-initiation has little or no effect on the capability of MA, EA and n-BA live polymer chains to undergo chain transfer reactions. Energy barriers of the CTS reactions do not change significantly with the length of the live polymer chain.

Before this work, although a general mechanism for chain transfer reactions had been presented, no specific mechanisms for different types of chain transfer reactions were proposed. Moreover, before this study it was inconclusive which dead polymer structure was most likely to provide the hydrogen atom during CTP reactions.

A macroscopic first-principles mathematical model that represents a batch polymerization reactor in which spontaneous thermal free-radical polymerization of n-butyl acrylate takes place was developed. The model considered polymerization reactions that are most likely to occur. Reaction rate equations were derived using the method of moments. The model was used to gain a better understanding of the role of the monomer in spontaneous initiation of the polymerization. The monomer self-initiation reaction rate constant was estimated from batch-polymerization conversion-measurements at elevated temperatures (>100ºC). The rate constant is estimated to be 4.39E-15 M-1 L-1 at 413K, which is comparable with the value

(1.04E-14 M-1 L-1) obtained via quantum chemical calculations. xv

MA, EA, and n-BA was also compared; the difference in the end-substituent group (MA, EA, or n-BA) does not affect the kinetics of the CTP reactions remarkably.

Tertiary hydrogens of dead polymers formed by disproportionation reactions are most likely to be transferred to live polymer chains in CTP reactions.

Chain transfer reactions to several solvents such as butanol (polar, protic), methyl ethyl ketone (MEK) (polar, aprotic), and p-xylene (nonpolar) were studied theoretically. The estimated lower activation energies and higher rate constants of chain transfer to n- butanol compared to those of MEK and p-xylene are indicative of the higher CTS reactivity of n-butanol. Among n-butanol, sec-butanol, and tert-butanol, tert-butanol has the highest CTS energy barrier and the lowest rate constant. Polarizable continuum model (PCM) and conductor-like screening model (COSMO) solvation models were applied to explore chain transfer to n-butanol, MEK and p-xylene from polymer chains of MA, EA and n-BA. The type of mono-radical generated via self-initiation has little or no effect on the capability of MA, EA and n-BA live polymer chains to undergo chain transfer reactions. Energy barriers of the chain transfer to solvent reactions do not change significantly with the length of the live polymer chain. Before this work, although a general mechanism for chain transfer reactions had been presented, no specific mechanisms for different types of chain transfer reactions were proposed. Moreover, before this study it was inconclusive which dead polymer structure was most likely to provide the hydrogen

atom during CTP reactions.

A macroscopic first-principles mathematical model that represents a batch xvi polymerization reactor in which spontaneous thermal free-radical polymerization of n- butyl acrylate takes place was developed. The model considered polymerization reactions that are most likely to occur. Reaction rate equations were derived using the method of moments. The model was used to gain a better understanding of the role of the monomer in spontaneous initiation of the polymerization. The monomer self-initiation reaction rate constant was estimated from batch-polymerization conversion-measurements at elevated temperatures (>100ºC). The rate constant was estimated to be 4.39E-15 M-1 L-1 at 413K, which is comparable with the value (1.04E-14 M-1 L-1) obtained via quantum chemical calculations.

1

1. Introduction

Acrylates are used to produce acrylic binder resins, which are the key component of paint and coating formulations. US demand for automotive coating, adhesive, and sealant will increase 9.4% annually through 2014 [1]. Environmental regulations, which require the volatile organic content of the resins to be less than 300 ppm, have caused changes in the basic design of resins used in automobile coating formulations [2, 3]. High temperature

(>100 oC) free-radical polymerization allows production of acrylic resins with lower solvent content, while keeping the resin viscosity within the desired range [4-6].

However, polymerization at high temperatures is accompanied by more dominant secondary reactions [7, 8] such as β-scission [9], monomer self-initiation [10-12], and intra-molecular and intermolecular chain transfer reactions [9, 13]. It has been reported that secondary reactions have a strong influence on the nature and end-use properties of the polymer product [14]. These reactions reduce the polymer chain length and broaden the molecular weight distribution of the final product at higher temperatures [13].

Previous studies of self-initiated polymerization of methyl acrylate (MA), ethyl acrylate

(EA) and n-butyl acrylate (n-BA) using electro-spray ionization-Fourier transform mass spectrometry (ESI-FTMS) [7] and matrix-assisted laser desorption ionization (MALDI)

[15] showed abundant polymer chains with end-groups formed by chain transfer reactions. Nuclear magnetic resonance (NMR) analysis of these polymers indicates the possible presence of end groups from chain transfer reactions at various temperatures

(100-180oC) [7, 13, 16, 17].

Pulsed-laser polymerization/size exclusion chromatography (PLP/SEC) experiments were carried out at low and high temperatures for determination of chain transfer and radical 2 propagation rate coefficients of acrylates [18-20]. While reliable rate constants and narrow polymer molecular weight distribution were obtained at less than 30oC, broad molecular-weight distribution and inaccurate rate constants were reported at temperatures above 30oC [21, 22]. Chain transfer to monomer [18, 23, 24] and to polymer (specifically backbiting) [21, 25-28] were identified as the main reactions that were causing the discrepancies in the molecular-weight distributions and reaction rate coefficients. While these analytical techniques have been very useful in characterizing acrylate polymers generated from thermal free-radical polymerization, they have been unable to conclusively determine specific reaction mechanisms.

Macroscopic kinetic modeling has been extensively used [8] to estimate the rate constants of initiation, propagation, chain transfer and termination reactions in thermal polymerization of acrylates. Kinetic parameters have been predicted based on monomer conversion and polymer average molecular weights [8, 29]. However, reliability of the estimated rate coefficients depends on the validity of the postulated reaction mechanisms and the certainty of the measurements. Due to these concerns, macroscopic mechanistic modeling is sometimes incapable of determining either the reaction mechanisms or the individual reaction step rates conclusively.

Computational quantum chemistry has been used successfully to identify most likely reaction mechanisms in free-radical polymerization systems [30-37]. Quantum mechanical modeling enables the identification of transient species that cannot be seen in experiments and the prediction of reaction pathways with high accuracy and low computational cost. Density functional theory (DFT) has been widely applied to calculate the rate constants of initiation, propagation, transfer and termination reactions in free and 3 controlled radical polymerization of various monomers. The forward and reverse reaction rate coefficients have been calculated for a series of reversible addition-fragmentation chain transfer (RAFT) reactions using high-level wave function based quantum chemistry calculations [38]. DFT-based methods are computationally less intensive than wave function based methods, and are therefore preferred to study chain transfer reactions in polymerization of alkyl acrylates. DFT has been used to accurately predict molecular geometries and rate constants of various reactions in polymerization of alkenes and acrylates [31, 37]. DFT calculations, using the B3LYP functional and the 6-31G(d) basis set, have been used to explore self-initiation mechanisms of styrene [39], MA, EA, n-BA

[10, 11], and methyl methacrylate (MMA) [12], and cyclohexanone-monomer co- initiation reaction in thermal homo-polymerization of MA and MMA [40]. The B3LYP functional has been also used in the study of free-radical polymerization of alkenes [31,

36] and propagation of MA and MMA [41].

This research project is aimed at understanding chain transfer reactions, in particular, chain transfer to monomer, polymer, and solvent to be used to control high-temperature polymerization of alkyl acrylates. In this research, mechanisms of chain transfer reactions in high-temperature polymerization of methyl, ethyl, and n-butyl acrylate and structures of individual species (i.e. reactants, transition states and products) were identified using quantum chemical calculations. The kinetics of these reactions was determined using transition state theory. 4

2. Background High functional acrylic resins can be obtained by generating uniform chain length

polymers. Although, application of various transfer agents to control the growth of

propagating chains is a viable process; the use of self-regulating chain transfer radicals

seems attractive. In self-initiated polymerization of alkyl acrylates, generation of self-

regulating chain transfer radicals from β-scission, chain transfer to solvent, polymer and

monomer have been reported [42]. To identify the formation mechanisms of these

radicals and design a self-regulating process, application of computational quantum

chemistry is important.

2.1. Kinetics of High-Temperature Free-Radical Polymerization

Primary reactions, i.e. initiation, propagation, and termination, take place at low and high

temperatures. It has been reported [7-9, 13, 17] that at elevated temperatures secondary

reactions occur at higher rates in polymerization of alkyl acrylates. The primary reactions

are:

௞೏ ܫ ሱሮʹܴǤ Initiation

௞ Ǥ ೔ ܴ ൅ܯ՜ܲଵ

௞೛ ܲ௡ ൅ܯሱሮܲ௡ାଵ Propagation

௞೟೎ ܲ௡ ൅ܲ௠ ሱሮܦ௡ା௠ Termination

௞೟೏ ܲ௡ ൅ܲ௠ ሱሮܦ௡ ൅ܦ௠

where M is the monomer, Pn and Pm are secondary radicals with chain length n and m, Dn

and Dm are dead polymer chains of length n and m 5

The secondary reactions include backbiting, short chain branching formation, β-scission,

chain transfer to monomer, polymer, and solvent, and self-initiation:

௞್್ ܲ௡ ሱሮܳ௡ Backbiting

௞೛೜ ܳ௡ ൅ܯሱሮܲ௡ାଵሺܵܥܤሻ Formation of Short Chain Branching

௞ഁ ܳ௡ ሱሮܦ௡ିଶሺܶܦܤሻ ൅ܲଶ β-Scission

௞೟ೝಾ ௡ ൅ܲଵ Chain Transfer to Monomerܦሱۛሮܯ௡ ൅ܲ

௞೟ೝು ௡ ൅ܳ௠ Chain Transfer to Polymerܦ௠ ሱۛሮܦ௡ ൅ܲ

௞೟ೝೄ ௡ ൅ܲ଴ Chain Transfer to Solventܦ௡ ൅ܵሱۛሮܲ

௞ೞ೔ ͵ܯ ሱሮܲଵ ൅ܲଶ Self-Initiation

The occurrence of backbiting can lead to the formation of tertiary or mid-chain radicals

Qn. This radical can either propagate to form short chain branches (SCB) or undergo β-

scission reaction to form dead polymer chain with a terminal double bond (TDB).

2.2. Chain Transfer Reactions

Chain transfer is a chain breaking reaction which results in a decrease in the size of the

propagating polymer chain. Its effect on the polymerization rate is dependent on whether

the rate of reinitiating is comparable to that of the original propagating radical. The effect

of the reduction in molecular weight of final polymer is due to premature termination of a

growing polymer by the transfer of a hydrogen or other atom or species to it from some

compound present in the system (monomer, initiator or solvent). These radical

displacement reactions are termed chain transfer reactions and may be defined as:

–” Ǥ൅ƐȂ൅Ǥ 6 where XA may be monomer, initiator, solvent, or other substance and X is the atom or species transferred (Ktr is the chain transfer rate constant).

Chain transfer to polymer backbone principally can occur via two various pathways: (i) through an inter-molecular H abstraction, (ii) through an intra-molecular H abstraction

(backbiting) [43]. Transfer to polymer results in the formation of a radical site on a polymer chain. The polymerization of monomer at this site leads to the production of a branched polymer. Inter-molecular chain transfer to polymer leads to the formation of long-chain branches, but the intra-molecular reaction may cause branches of any length, which depends on where along the polymer chain the back-biting H-abstraction occurs

[44]. Chain transfer to monomer and initiator can involve in decreasing the generated polymer molecular weight. In some instances, the chain transfer agent is the solvent, while in others it is an added compound. The control of the concentrations of the monomer, initiator, and solvent or the use of deliberately added chain transfer agents to control the molecular weight of a polymerization, is of prime importance.

2.3. Ab Initio Molecular Orbital Theory

The objective of molecular orbital theory is to provide methods to solve the stationary, time independent multi electron Schrodinger wave equation:

ܪߖ ൌ ܧߖ (2.1) where H is the Hamiltonian operator, ψ is the wave function and E is the electronic energy, related to a multi nuclear configurations.

In ab initio molecular orbital theory [39, 45], one can estimate the wave function using one electron functions or “spin orbitals” (χ) , which is a product of spatial orbital ψi

(x,y,z), and the spin orbital (α or β). 7

The spatial orbitals can be defined in terms of basis functions as equation (2.2):

௞ (ሻ (2.2ݎఓ௜ ߶ఓሺܥ ߖ௜ሺݔǡ ݕǡ ݖሻ ൌσఓୀଵ

where Cμi is the molecular expansion coefficient, and Фμ (r) is the basis function. The

problem in application of wave equation is related to Eigen value problem whose exact

solution can be feasible if we process infinite basis sets.

A basis set is combination of several basis functions to determine the shape of the

wavefunction, therefore choosing the correct basis set makes occurring the computation

of the wave function for a finite set of orbitals.

The Hamiltonian operator (H) for a generalized case of N electrons and M nuclei is

represented by:

ଵ ଶே ெ ଵ ଶ ே ெ ௓಺ ேିଵ ே ଵ ெିଵ ெ ௓಺௓಻ ܪൌെσ௜ୀଵ ߘ௜ െ σ௜ୀଵ ߘ௜ െ σσ௜ୀଵ ூୀଵ ൅ σσ௜ୀଵ ௝ୀ௜ାଵ ൅ σσூୀଵ ௃ୀூାଵ ଶ ଶெ಺ ௥೔಺ ௥೔ೕ ோ಺಻ (2.3) In the equation (2.3), i and j are used for electrons, and I and J are for nuclei. The first

term on the left hand side in equation (3) is the kinetic energy equation of the electron,

and the second term is the kinetic energy equation of the nucleus. The remaining three

terms are equations for the columbic interactions that occur between nuclei-electrons,

electron-electron, and nucleus-nucleus. In Born-Oppenheimer approximation the motion

of the nuclei is negligible in comparison to the motion of an electron. So, for the system

of N electrons the nuclei can simplify equation (3) to:

ଵ ଶே ே ெ ௓಺ ேିଵ ே ଵ ܪൌെσ௜ୀଵ ߘ௜ െ σσ௜ୀଵ ூୀଵ ൅ σσ௜ୀଵ ௝ୀ௜ାଵ  (2.4) ଶ ௥೔಺ ௥೔ೕ Rewriting equation (2.1) for motion of electrons only, one obtains:

ܪ௘ߖ௘ሺݎ௜ǡܴூሻൌܧ௘ߖ௘ሺݎ௜ǡܴூሻ (2.5)

where RI represents the nuclear coordinates, and ri denotes the electronic coordinates.

Through various numerical techniques, equation (2.5) can be solved to obtain the 8 equilibrium geometry of the molecule. Second derivative of the wave function and the saddle point on the reaction coordinate can be applied to obtain vibrational frequencies of these molecules, and the transition state geometry and energy barrier, respectively.

Even in the simplest form of free radical polymerization, it comprises discreet propagation, initiation and bimolecular termination reactions, and in most practical systems a variety of additional reactions, also occur. Since, the rates of these individual reactions are often chain length dependent, and in industrially important processes more than one type of monomer and/or additional reagents are often present, it becomes clear that obtaining accurate and precise measurements of the rate coefficients of the various individual reactions can be a challenge. Model-free determination of the individual rate coefficients in more complex systems, such as copolymerization or controlled radical polymerization, remains elusive.

Computational quantum chemistry allows one to solve these problems due to the rate coefficients of isolated individual reactions can be calculated directly, without recourse to kinetic model based assumptions. However, for accurate results in radical reactions high levels of theory are required, and their cost scales exponentially with the size of the system, so accurate procedures are impractical for polymeric molecules. Nonetheless, rapid and continuing increase in computer power, computational chemistry is rapidly establishing itself as an accurate and useful kinetic tool for the radical polymer field.

2.4. Transition State Theory

It is a simple approach [36, 46, 47] that aids in understanding chemical reactions and calculating the kinetic parameters of the reacting system based on the following assumptions: 9 a) In the space represented by the coordinates and momentum of the reacting particles, it is possible to define a dividing surface such that all reactants crossing this plane go on to form products, and do not re-cross the dividing surface. b) The transition state is in statistical equilibrium with the reactants. c) Motions through the transition state can be treated as a classical translation.

The following equation relates the rate coefficient at a specific temperature k(T) to the properties of the reactant (s) and transition state:

௞ ் ொ ିா (ଵି௠ ಳ శ ݁ݔ݌ሺ బሻ (2.6ܭሺܶሻ ൌ݇ ௛ ςೝ೐ೌ೎೟ೌ೙೟ೞ ொ೔ ோ் where k(T) is the rate constant at temperature T, kB is the Boltzmann constant, h is the

Plancks constant, E0 is the energy barrier, Q+ and Qi are the molecular partition functions of transition state and reactants, respectively. R is the universal gas constant. The overall molecular partition function includes translational, rotational, and vibrational partition functions.

2.5. Computational Quantum Chemistry

Computational quantum chemistry has been applied to predict the reaction kinetics in recent years [11, 12, 40, 48-50]. It has gained importance to explore and obtain the lowest energy molecular geometries of reactants, products, intermediates, and transition state structures that cannot be obtained via experimental measurements. The non-relativistic, electronic Schrodinger wave equation is solved by choosing a particular level of theory

(Method+basis set) to calculate the electronic wave functions and energies related to different arrangements of nucleus. The equilibrium geometries, reaction barriers, enthalpy, and rate coefficients of reactions can be derived via quantum chemical information. Computational quantum chemistry can not only be used for better 10 understanding of the mechanisms of the reactions, but also it can improve kinetic models for describing free-radical polymerization.

Both density functional theory (DFT) and wavefunction-based quantum chemical methods have been used to study free-radical polymerization reaction mechanisms such as self-initiation and propagation reactions in thermal polymerization of alkyl acrylates

[10-12, 41, 49]. DFT is in principle an exact ground-state technique. The accuracy of

DFT depends on the approximation of exchange-correlation functionals. Generally, DFT can predict molecular geometries with high accuracy [51], but energy barriers with less accuracy compared to high-level wavefunction-based methods such as MP2 [45, 52]. It is important to note that modern exchange-correlation functionals such as M06 [53] and

ωB97x-D [54] have increased the reliability and accuracy of DFT-predicted energy barriers. These functionals have allowed exploration of the molecular and kinetic properties of larger molecules such as polymer chains and chain transfer reactions with lower computational cost and reasonably good accuracy [41]. For large (greater than 30 atoms) polymer systems, DFT [55] has been shown to be a practical and reliable substitute for wavefunction-based methods [31]. One should ascertain the adequacy of a level of theory for a system of interest prior to estimating the kinetics and thermodynamics of reaction. Exchange-correlation functionals can be classified based on their functional form [56-58]. Local functionals (depending only on charge density ρ(r) at that point r), generalized gradient-approximation (GGA) (depending on ρ(r) and |’

ρ(r)|), and meta-GGA functionals (depending on ρ(r), |’ ρ(r)|, and ’2 ρ(r)) can be combined with Hartree-Fock exchange functionals to increase accuracy. These combined functionals are known as hybrid functionals. B3LYP, which is a hybrid GGA functional, 11 has been used extensively due to its attractive performance-to-cost ratio [10, 11]. For example, B3LYP/6-31G(d) has been used to study self-initiation of styrene [39], MA, EA and n-BA [10, 11], and MMA [12]. B3LYP/6-31G(d) was found [59] to overestimate the reaction rate constants for self-initiation of MA in comparison to MP2/6-31G(d). Due to high computational costs and inherent size limitations of MP2/6-31G(d), this level of theory is not used to study chain transfer reactions of large polymer chains. In addition,

B3LYP/6-31G(d) has also predicted alkyl acrylate self-initiation rate constants different from those estimated from polymer sample measurements. Meta-GGA and hybrid meta-

GGA functionals such as M06-2X provide more accurate prediction of barrier heights, as they can adequately account for van der Waals interactions [53, 60-63]. Propagation rate constants of alkyl acrylates and methacrylates were determined using different DFT- based hybrid functionals [41]. 12

3. Computational Methods and Procedures Used

3.1. Computational Method

A large number of software packages, such as ACES II [64], ADF [65], DALTON [66],

GAMESS [67], MOLPRO [68], and GAUSSIAN [69], can be applied to perform quantum chemical calculations using ab initio and density functional methods.

All calculations in this project were performed using GAMESS [67]. The functionals

B3LYP, X3LYP, M06-2X with the basis sets 6-31G(d), 6-31G (d,p), 6-311G(d), and 6-

311G(d,p) were used to determine the molecular geometries of reactants, products, and transition states in gas phase. Optimized structures were characterized with hessian calculations. A rate constant k(T) was calculated using transition state theory [46] with

௞ ் οுșି்οௌș (ሻଵି௠ ಳ ‡š’ ቀെ ቁ (3.1סሺܶሻ ൌሺܿ݇ ௛ ோ் where c0 is the inverse of the reference volume assumed in translational partition function calculation, kB is the Boltzmann constant, T is temperature, h is Planck’s constant, R is the universal gas constant, m is the molecularity of the reaction, and ∆S‡ and ∆H‡ are the entropy and enthalpy of activation, respectively. ∆H‡ is given by:

ș οܪ ൌሺܧ଴ ൅ ܼܸܲܧ ൅ οοܪሻ்ௌିோ (3.2) where ∆∆H is a temperature correction; the difference in enthalpy of the transition state

(TS) and the reactants (R), ZPVE is the difference in zero-point vibrational energy between the transition state and the reactants, and E0 is the difference in electronic energy of the transition state and the reactants. The activation energy (Ea) was calculated using:

ș ܧ௔ ൌοܪ ൅ܴ݉ܶ (3.3) and the frequency factor (A) by:

௞ ் ௠ோାοௌș (ሻଵି௠ ಳ ‡š’ ቀ ቁ (3.4סൌሺܿܣ ௛ ோ 13

∆∆H, ZPVE, and ∆S‡ calculations were carried out using the rigid rotor harmonic oscillator (RRHO) approximation [70]. The RRHO approximation is the most basic model to estimate thermal and entropic parameters. It treats the whole molecule as a rigid rotor (RR) and each vibrational mode as a harmonic oscillator (HO). This approximation allows the rotational and vibrational entropies calculated separately. It has been shown in the previous studies [10-12, 40] that the RRHO approximation exhibits reasonable computational cost and accuracy for studying large polymer system. Scaling factors of

0.960, 0.961, 0.966, and 0.967 were used for the B3LYP functional with the 6-31G(d), 6-

31G(d,p), 6-311G(d), and 6-311G(d,p) basis sets, respectively. These factors were obtained from the National Institute of Standards and Technology (NIST) scientific and technical database [71].

In this work, the NMR spectra of dead polymer chains and end groups were computed using various levels of theory. We used ORCA program [72] with B3LYP and X3LYP functionals to predict chemical shifts. B3LYP has been reported to be one of the best functionals for 1H chemical shift prediction and provide adequately accurate 13C chemical shifts [73].

3.2. Application of ORCA in Chemical Shift Calculations

Performance of ORCA as an efficient program for chemical shift calculations has been reported in the recent years [74, 75]. To evaluate performance of ORCA for chemical shift calculations, B3LYP/6-31 G(d,p) was applied to compare calculated C-NMR spectra for MA, EA, n-BA, and p-Xylene Figure 3-1 with those obtained through polymer sample measurements. Different functionals (B3LYP/6-31G (d,p), B3LYP/6-311G (d),

X3LYP/6-31G (d,p) and X3LYP/6-311G (d)) were used to validate the chemical shift 14 results. These results are given in Table 3-1. Table 3-1 shows that the computational results are in good agreement with those reported previously based on polymer sample measurements [76-78].

1 1 4 MA EA 2 O 2 O CH3 CH CH3 CH 2 2 3 3 4 5 O O

8 CH3 n-BA p-Xylene 1 7 1 4 2 2 6 O CH3 2CH 3 6 5 3 5 4 O 7 CH3

Figure 3-1: Carbon Atoms Considered in MA, EA, n-BA, and p-Xylene for 13C-NMR Calculations.

An input file template for chemical shift calculations for tetramethylsilane (TMS) is as shown in Figure 3-2. The output file, which contains detailed information about the orientation of the tensor, the eigenvalues, and its isotropic part, is shown in Figure 3-3.

According to the output file given in Figure 3-3, the absolute, isotropic shielding for the

1H nuclei is predicted to be 31.3 ppm and for the 13C it is 187.2 ppm. In order to compare the computed results with experiment, a standard molecule (for example CH4 or TMS) for the type of nucleus of interest should be selected as the reference and then obtain the chemical shift by subtraction of the reference value from the computed value. In this 15 study TMS was selected as the reference molecule. The same functional and basis set should be applied for both reference and target calculations.

! RHF B3LYP 6-31G* SmallPrint TightSCF Grid4 IGLOII * xyz 0 1 SI -0.0015819444 -0.0002458491 0.0013039774 C 0.9529107571 1.5444659113 -0.5482106855 C -0.2147169260 0.0239239634 1.8859720511 C -1.7047652517 -0.0183687220 -0.8339148306 C 0.9684958910 -1.5496244978 -0.5055376750 H 1.9520513500 1.5765162491 -0.0990048096 H 1.0792897037 1.5664137303 -1.6366064054 H 0.4296505415 2.4623146849 -0.2574003990 H -0.7668943434 -0.8551497472 2.2369571021 H -0.7632596683 0.9134736958 2.2155779053 H 0.7566078022 0.0276504881 2.3935306236 H -2.2809177791 -0.9055523410 -0.5478011299 H -1.6097604642 -0.0238939532 -1.9256388336 H -2.2940469508 0.8632768750 -0.5576484849 H 0.4539191636 -2.4647570763 -0.1915798454 H 1.0974903656 -1.5989213057 -1.5927397691 H 1.9668702570 -1.5592695029 -0.0537078760 * %eprnmr ori IGLO # alternative OwnNuc LocMet FB # localization method for IGLO # FB=Faster-Boys PM=Pipek-Mezey (default) Nuclei = all C { shift } Nuclei = all H { shift } end

Figure 3-2: Input Template File for NMR Chemical Shifts Calculations.

3.3. Design of Reaction Scheme

Individual reaction mechanisms in thermal free-radical polymerization of alkyl acrylates can be studied via computational quantum chemistry. Reasonable mechanisms design which can satisfy describing the reaction of interest is important. The reaction 16 mechanisms should be designed using polymerization literature or fundamental organic chemistry principles.

Table 3-1: Assignment of 13C-NMR Spectra for MA, EA and n-BA (Chemical Shifts are Relative to TMS with Absolute Isotropic Shielding of 187 ppm for the 13C)

13C- B3LYP B3LYP X3LYP X3LYP Experiment[7, 100] NMR 6-31G(d,p) 6-311G(d) 6-31G(d,p) 6-311G(d) MA 128.5 1 136.5 136.9 136.5 137 166.7 2 172.4 172.8 172.8 173.3 130.6 3 139.2 139.6 139.3 139.8 51.6 4 55.6 56 55.3 55.8 EA 128.9 1 138.4 138.8 138.3 138.8 166.2 2 171 171.4 171 171.5 60.5 3 65.6 66 65.3 65.8 14.3 4 17 17.4 16.8 17.3 130.3 5 136.3 136.7 136.3 136.8 n-BA 128 1 138.2 138.6 138.1 138.6 166 2 170.6 171 170.7 171.2 60 3 63.9 64.3 63.5 64 30 4 36.4 36.8 36.1 36.6 130 5 136.2 136.6 136.2 136.7 21 6 22.6 23 22.2 22.7 14 7 14.4 14.8 14 14.5 p-Xylene 134.7 1 141.6 142 141.6 142.1 129 2 134.8 135.2 134.8 135.3 129 3 134.7 135.1 134.8 135.3 135 4 141.6 142 141.7 142.2 129 5 134.4 134.8 134.4 134.9 129 6 134.5 134.9 134.5 135 20.9 7 23.1 23.5 22.8 23.3 20.9 8 23.1 23.5 22.7 23.2 17

Probability of occurrence of various mechanisms proposed by organic chemistry should be evaluated in thermal free-radical polymerization of acrylates for identifying the most likely mechanism.

Nucleus 4C : ------Raw-matrix : 0.0001866 -0.0000039 -0.0000011 -0.0000039 0.0001904 0.0000021 -0.0000012 0.0000021 0.0001847

Diagonalized sT*s matrix: sDSO 0.0002556 0.0002556 0.0002620 iso= 0.0002577 sPSO -0.0000716 -0.0000713 -0.0000685 iso= -0.0000705 ------Total 0.0001840 0.0001842 0.0001934 iso= 0.0001872

Orientation: X 0.3035662 0.8041977 -0.5109928 Y 0.4567101 0.3478633 0.8187838 Z -0.8362197 0.4819307 0.2616857

------Nucleus 5H : ------Raw-matrix : 0.0000363 0.0000013 0.0000033 0.0000030 0.0000290 0.0000005 0.0000022 -0.0000003 0.0000288

Diagonalized sT*s matrix: sDSO 0.0000252 0.0000271 0.0000396 iso= 0.0000306 sPSO 0.0000024 0.0000017 -0.0000020 iso= 0.0000007 ------Total 0.0000276 0.0000288 0.0000377 iso= 0.0000313

Orientation: X 0.3675639 0.0261262 0.9296312 Y -0.4877924 -0.8456529 0.2166327 Z -0.7918051 0.5330934 0.2980873

Figure 3-3: Output Template File for NMR Chemical Shifts Calculations. 18

3.4. File Types

An input file “*.inp” generally includes definition of molecular structure and coordinate representing the structure, type of calculations, level of theory, number of iteration and memory allocation. One can use a visualization software such as MacMolPlt or Molekel to construct the preliminary structures or make it manually based on available knowledge of bond-distance, bond angle, and dihedral angle between atoms. Output files, i.e.

“*.log”, “*.dat”, and “*.irc”, are generated after running a calculation. Convergence or error messages are shown in a “*.log” file. The “*.dat” file contains electronic structures which can be used for further calculations in following steps. A “*.irc” file generally obtained from Hessian calculations, including vibrational frequency modes, or intrinsic reaction coordinate (IRC) calculations which provides geometry of species on the IRC path.

3.5. Types of Calculations

The quantum chemical calculations performed to investigate the chain transfer reaction mechanisms in the present study were followed several steps (for the reactants only the first two steps should be performed):

1. Geometry Optimization

2. Hessian for Optimized Geometry

3. Saddle Point

4. Hessian for Saddle Point

5. Intrinsic Reaction Coordinate Calculation 19

3.5.1. Geometry Optimization

Structures and energies of reactants, products and transitions states are determined through geometry optimization. Three degrees of freedom represented by bond length, bond angle, and dihedral angle are considered for each atom. In the case of transition state, one or two of the degrees of freedom should be constrained depends on their importance in describing the reaction mechanism and chemical intuition. The rest of them

(all of them in the case of reactants) are relaxed and optimized to determine the local minima from some initial starting structure (educated guess) on potential energy surface

(PES). In this step the first derivative of energy (known as gradient) with respect to the geometric coordinates is calculated. When this value reaches a set convergence tolerance value (close to zero), the corresponding geometry can be considered as the stationary point. Reactants, products, and transition sates are called stationary points on PES.

3.5.2. Hessian for Optimized Geometry

Second order derivative of energy with respect to coordinates is calculated in Hessian calculations to provide the frequency modes of the stationary points on PES. For reactants and products all the frequencies should be positive to represent the local minima. However, for transition state structure one single negative frequency (singlet negative force constant) should be achieved representing a geometry (or point on PES) which is a maxima in only one direction and a minima in all other directions. Geometries having more than one negative force constant are re-optimized until only one negative force constant is obtained. 20

3.5.3. Saddle Point

In this step an unconstrained optimization is performed for the stationary point having only one imaginary frequency. The results obtained through Hessian calculations in previous step are included in the saddle point input file. When incorrect vibrational mode is followed the calculation terminates abnormally and all the calculations should be performed for another stationary point. The output file from normally terminated calculations can be viewed using MacMolPlt to make sure that the structure is neither a reactant nor a product.

3.5.4. Hessian for Saddle Point

To ensure that the saddle point has only one negative force constant under unconstrained conditions Hessian calculations should be performed for the successful saddle point structure in previous step. If one imaginary frequency is obtained from Hessian calculations, then the saddle point can be considered as the transition state structure for the mechanism of interest.

3.5.5. Intrinsic Reaction Coordinate (IRC)

The intrinsic reaction coordinate calculations are carried out to verify that the determined transition structure connects two local minima (reactant and product) on PES. These calculations are performed in forward and reverse directions starting at saddle point.

Moreover, it can be determined whether the pathway from reactants to products is concerted or non-concerted via IRC calculations. 21

4. Chain Transfer to Monomer Reactions in Polymerization of Alkyl Acrylates

4.1. Introduction

Previous studies of self-initiated polymerization of methyl acrylate (MA), ethyl acrylate

(EA) and n-butyl acrylate (n-BA) using electro-spray ionization-Fourier transform mass spectrometry (ESI-FTMS) [7] and MALDI [15] showed abundant polymer chains with end-groups formed by chain transfer reactions. Nuclear magnetic resonance (NMR) analysis of these polymers indicates the possible presence of end groups from chain- transfer-to-monomer (CTM) reactions at various temperatures (100-180oC) [7]. CTM reactions are capable of limiting the maximum polymer molecular weight that can be achieved for a given monomer [79, 80] and strongly influence the molecular weight distribution of dead polymer chains [81-84]. Determination of most likely reaction mechanisms leads to design and optimize polymerizatio processes more accurate, yielding polymer products with desired properties. Controlled radical polymerization processes, such as nitroxide-mediated polymerization (NMP), atom transfer radical polymerization (ATRP) and reversible addition-fragmentation chain transfer

(RAFT), involve the use of agents to control the growth of propagating chains, which leads to the formation of uniform chain-length polymers [85-90]. It has been reported that in thermal polymerization of alkyl acrylates, in the absence of these agents, self- regulation and consequently uniform chain-length polymers can be achieved [91].

These suggest that some of the chain transfer mechanisms are capable of being self- regulatory. Therefore, a good understanding of the underlying mechanisms is needed to develop controlled thermal polymerization processes. To the best of our knowledge, this work is the first study of all likely mechanisms of CTM in three alkyl acrylates (MA, EA and n-BA), revealing the most 22 likely CTM mechanisms in the homo-polymerization. Before this study, only a general description of a chain transfer reaction was available as a reaction of a live polymer chain with a transfer agent (monomer, polymer, solvent, or initiator), without providing any reaction mechanisms [80, 92]. Pulsed-laser polymerization/size exclusion chromatography (PLP/SEC) experiments were carried out at low and high temperatures for determination of chain transfer and radical propagation rate coefficients of acrylates

[18, 19, 23]. While reliable rate constants and narrow polymer molecular weight distribution were obtained at less than 30oC, broad molecular-weight distribution and inaccurate rate constants was reported at temperatures above 30oC [21, 22, 24]. Chain transfer to monomer [18, 23, 24] and to polymer (specifically backbiting) [21, 25, 26] were identified as the main reactions that were causing the discrepancies in the molecular-weight distributions and reaction rate coefficients. Reaction rate constants in high-temperature polymerization of alkyl acrylates have typically been estimated from polymer sample measurements such as monomer conversion and average molecular weights. The measured monomer conversion and molecular weights have been fitted to a macroscopic mechanistic model to determine the kinetics of polymerization [8, 29, 59].

The rate constants of CTM reactions in methyl methacrylate (MMA), styrene, and α- methylstyrene polymerization have been determined with little difficulty [81]. Gilbert et al. [83] estimated rate constants of CTM reactions in emulsion polymerization of n-BA from polymer molecular weight distributions. Previous reports have shown that hydrogen abstraction by tertiary poly-n-BA live chains contributes to the rate of CTM [28, 82, 93].

However, the reliability of the estimated rate coefficients is dependent on the validity of the postulated reaction mechanisms and the certainty of the measurements. Due to these 23 concerns, macroscopic mechanistic modeling is sometimes incapable of determining either the reaction mechanisms or the individual reaction step rates conclusively.

CTM rate constants for free-radical polymerization of acrylates have been reported to be difficult to estimate due to large uncertainties in experimental measurements [93]. The presence of trace impurities, which can act as chain transfer agents, was mentioned as the cause for the variation in estimated rate constant values [83, 94]. The temperature dependence of the rate constant of transfer to monomer reaction in styrene was estimated from polymer sample measurements [95]. Considering the difficulties in estimating CTM rate constants, computational quantum chemistry has been considered as an alternative way of estimating these parameters.

Reports indicate that solvent molecules participate in chain transfer reactions [45]. They tend to act as chain transfer agents by reacting with live polymer chains, and consequently affecting the microstructure and polydispersity of polymer chains.

Experimental studies [15] have shown that the polarity of solvents can impact the rate of initiation reactions in thermal polymerization of alkyl acrylates. Energy barriers predicted using the polarizable continuum model (PCM) and gas phase quantum chemical calculations for polymerization of acrylic acid and acrylates in inert and non-polar solvents have been found to be in agreement [96, 97]. The use of PCM for polar solvents has been less successful [98]. This may be attributed to the fact that PCM only takes into account electrostatic interactions, while neglecting non-electrostatic interactions, such as hydrogen bonding and van der Waals interactions. The solvent effect is described better by the hybrid quantum mechanical/molecular mechanical (QM/MM) method, in which a small (more relevant) portion of the system is treated quantum mechanically and the rest 24 of the system (with explicit solvents) is described by classical force fields [99, 100]. The

QM/MM method has been applied to study the molecular behavior of proteins [101].

However, the application of QM/MM is limited by the scarcity of accurate MM parameters and the difficulty to describe an accurate QM/MM interface [99-101]. In this study, all calculations are performed in the gas phase to make the computations affordable, and the rigid-rotor harmonic oscillator (RRHO) approximation is used to include vibrational entropy.

C-NMR has been used to study EA and n-BA homo-polymerization [7, 44]. The peaks assigned to the carbon atoms in different positions along the main chain or the side chain can help determine which molecular structures are formed during the polymerization process. Based on these results, one can validate proposed reaction mechanisms. In this chapter, the NMR spectra of dead polymer chains and end groups have been computed using various levels of theory. We use ORCA program [72] with B3LYP and X3LYP functionals to predict chemical shifts. B3LYP has been reported to be one of the best functionals for 1H chemical shift prediction and provide adequately accurate 13C chemical shifts [73]. The proposed mechanisms of CTM have been characterized using the computed rate constants and NMR spectra.

4.2. Computational Methods

This part presents a computational study of CTM reactions in self-initiated high- temperature homo-polymerization of alkyl acrylates (methyl, ethyl and n-butyl acrylate).

Several mechanisms of CTM were studied. The effects of the length of live polymer chains and the type of mono-radical that initiated the live polymer chains on the energy barriers and rate constants of the involved reaction steps were investigated theoretically. 25

All calculations were carried out using DFT. Three types of hybrid functionals (B3LYP,

X3LYP, and M06-2X) and four basis sets (6-31G(d), 6-31G(d,p), 6-311G(d), and 6-

311G(d,p)) were applied to predict energy barriers and the molecular geometries of the reactants, products, and transition states. Transition state theory was used to estimate rate constants. The results indicated that abstraction of a hydrogen atom (by live polymer chains) from the methyl group in MA, the methylene group in EA, and methylene groups in n-BA are the most likely mechanisms of CTM. Also, the rate constants of CTM reactions, calculated using M06-2X, were in good agreement with those estimated from polymer sample measurements using macroscopic mechanistic models. The rate constant values did not change significantly with the length of live polymer chains. Abstraction of a hydrogen atom by a tertiary radical had a higher energy barrier than abstraction by a secondary radical, which agrees with experimental findings. The calculated and experimental NMR spectra of dead polymer chains produced by CTM reactions were comparable. The major finding of this section was that CTM occurs most likely via hydrogen abstraction by live polymer chains from the methyl group of MA and methylene group(s) of ethyl (n-butyl) acrylate. Herein, the computed results for MA have been reported [48].

4.3. Results and Discussion

▪ 4.3.1. Chain Transfer to Monomer Mechanisms for M2

4.3.1.1. Methyl Acrylate

Recent studies using quantum chemical calculations [10, 11] and matrix-assisted laser desorption ionization (MALDI) [15] showed that monomer self-initiation is a likely mechanism of initiation in spontaneous thermal polymerization of alkyl acrylates. The 26 mono-radicals generated by self-initiation [10, 11] are shown in Figure 4-1. As shown in

▪ Figure 4-2, we have studied four CTM mechanisms for M2 radical of methyl acrylate.

Figure 4-1: Two Types of Mono-radical Generated by Self-Initiation.

Figure 4-2: Possible End-chain Transfer to Monomer Reactions for MA. 27

The mechanisms are: abstraction of a methyl hydrogen atom from the monomer by a live chain (MA-1), abstraction of a vinylic hydrogen atom from the monomer (MA-2), abstraction of a methine hydrogen atom from the monomer (MA-3), and transfer of a hydrogen atom from a live polymer chain to the monomer (MA-4). These mechanisms are studied by choosing r(H2-C3) and r(C1-H2) as reaction coordinates. The potential energy surface is sampled by varying C-H bond lengths between 1.19 Å and 1.59 Å.

The activation energies, activation enthalpies, frequency factors, and rate constants of the four mechanisms are given LQ 7DEOH  It was determined using different levels of theory that MA-1 is the most kinetically favorable mechanism, and MA-2, MA-3 and

MA-4 mechanisms have higher activation energies. This agrees with the fact that the bond-dissociation energy of a methyl hydrogen is lower than that of a methine hydrogen and a vinylic hydrogen, as given in 7DEOH7Ke difference in the bond-dissociation energies can be attributed to the lower stability of the vinyl radical [102, 103]. In this study, bond-dissociation energy is defined as the energy difference between a monomer molecule and bond cleavage products (hydrogen radical and monomer monoradical)

[104]:

‘†‹••‘ ‹ƒ–‹‘‡”‰› ൌ  ܧሺBond Cleavage Productsሻ െܧሺMonomerሻ (4.1)

It represents the energy required to break a carbon-hydrogen bond of a molecule; it is a measure of the strength of a C-H bond.

Mulliken charge analysis also reveals that the methyl carbon atom (-0.117) is more positive than vinylic carbon atoms (-0.247 and -0.126), and therefore, more likely to release a hydrogen atom. 28

‡ Table 4-1: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Four CTM Reactions in MA at 298 K

31G(d) 311G(d) 311G(d) 311G(d) 31G(d,p) 31G(d,p) 31G(d,p) B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP 311G(d,p) 311G(d,p) 311G(d,p) M06-2X M06-2X M06-2X 6- 6- 6- 6- 6- 6- 6- 6- 6- 6-

Hydrogen Abstraction via MA-1 Ea 71 68 74 71 62 68 65 56 61 58

∆H‡ 66 63 69 66 57 63 60 51 56 54

∆G‡ 114 111 117 114 110 116 113 108 112 110

logeA 15.4 15.46 15.56 15.5 13.3 13.28 13.36 11.77 11.93 12.07

k 1.8E-06 5.3E-06 5.4E-07 1.8E-06 9E-06 7.8E-07 2.9E-06 1.7E-05 3E-06 8.1E-06

Hydrogen Abstraction via MA-2

Ea 100 98 102 100 95 94 95 90 93 92

∆H‡ 95 93 97 95 90 89 90 85 88 87

∆G‡ 137 134 138 137 127 145 137 135 137 134

logeA 17.6 18.05 18.11 18 19.96 12.42 16 14.34 15 15.47

k 1.5E-10 5.4E-10 9.9E-11 1.7E-10 9.7E-09 6.9E-12 1.7E-10 3.1E-10 1.6E-10 4.2E-10

Hydrogen Abstraction via MA-3 Ea 93 91 94 95 87 93 91 80 84 83

∆H‡ 88 86 89 90 82 88 86 75 79 78

∆G‡ 129 127 139 130 122 127 126 125 130 128

logeA 18.22 18.10 14.72 18.51 18.77 18.95 18.51 14.33 13.95 14.19

k 4.2E-09 8.3E-09 7.7E-11 2.3E-09 7.6E-08 1.0E-08 1.3E-08 1.7E-08 2.5E-09 4.9E-09

Hydrogen Transfer via MA-4 Ea 79 75 81 79 73 79 78 73 75 71

∆H‡ 74 70 76 74 68 74 73 68 70 66

∆G‡ 124 118 125 121 116 124 119 120 124 128

logeA 14.41 15.34 15.2 15.5 15.34 14.37 15.92 13.61 13.07 9.67

k 3.2E-08 2.9E-07 2.2E-08 7.9E-08 6.8E-07 2.9E-08 2.0E-07 1.6E-07 2.9E-08 6.3E-09 29

Table 4-2: H-R Bond Dissociation Energies (kJ mol-1) at 298 K

B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP 6-31G(d, p) 6-311G(d) 6-311G(d, p) 6-31G(d, p) 6-311G(d) 6-311G(d, p)

Methyl Hydrogen (MA-1) 432 425 425 433 426 426 Vinylic Hydrogen (MA-2) 483 474 478 484 475 479 Methine Hydrogen (MA-3) 490 480 484 491 482 485 Methylene Hydrogen (EA-1)

H-R 421 414 415 422 415 417 Methyl Hydrogen (EA-2) 452 443 445 453 445 446 Methylene Hydrogen (n-BA-1) Bond-dissociation energy 420 412 413 422 414 416 Methylene Hydrogen (n-BA-2) 433 424 426 435 426 428 Methylene Hydrogen (n-BA-3) 430 421 423 431 423 424 Methyl Hydrogen (n-BA-4) 447 438 440 448 439 441

)LJXUH D shows the transition-state geometry for the MA-1 mechanism, which has

H2-C3 and C1-H2 bond lengths of 1.32 and 1.40 Å, respectively. We found that the computed thermodynamic quantities (activation energy and rate constants) are not sensitive to the size of basis sets (6-31G(d), 6-31G(d,p), 6-311G(d)), but vary significantly depending on the type of density functionals. For a given density functional, the difference in calculated energy barrier using different basis sets (6-31G(d), 6-

31G(d,p), 6-311G(d)) is generally below 6 kJ mol-1. Changing the type of functional, 30 however, can result in ≈15 kJ mol-1 difference in barrier height and two orders of magnitude difference in rate constants. The functional has more impact on the predicted barriers than the basis set for studying chain transfer reactions of all alkyl acrylates.

Figure 4-3: Transition State Geometry of the Three Mechanisms: (a) MA-1, (b) EA-1, and (c) n-BA-1. 31

4.3.1.2. Ethyl Acrylate

▪ For EA, two CTM mechanisms for the M2 radical of ethyl acrylate in Figure 4-4 are considered. These are the abstraction of a methylene hydrogen atom (EA-1) and abstraction of a methyl hydrogen atom (EA-2). Abstraction of methine and vinylic hydrogens from EA are not studied, as our studies on MA already showed that these reactions have higher energy barriers.

Figure 4-4: Possible End-chain Transfer to Monomer Reactions for EA.

Abstraction of hydrogen from EA-1 and EA-2 is investigated by constraining C1-H2 and

H2-C3 bond lengths between 1.19 Å and 1.59 Å. The optimized geometry of the transition-state structure for the EA-1 mechanism has H2-C3 and C1-H2 bond lengths of

1.34 Å and 1.37 Å, respectively (Figure 4-3 (b)). The activation energies and rate constants for these mechanisms are given in Table 4-3. The barrier of the EA-1 mechanism was found to be lower than that of EA-2 mechanism. It can be seen from the

Mulliken charges of the ethyl acrylate carbon atoms (methylene carbon atom = 0.01, 32

‡ Table 4-3: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy ‡ -1 -1 -1 of Activation (∆G ) in kJ mol ; Frequency Factor (A) and Rate Constant (ktrM) in M s , for the Two CTM Reactions in EA at 298 K ) ) ) ) ) ) ) ) ) d d d d ) d,p d,p d,p d,p d,p d,p B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP M06-2X M06-2X M06-2X 6-31G( 6-311G( 6-311G( 6-311G( 6-31G( 6-31G( 6-31G( 6-311G( 6-311G( 6-311G(

Hydrogen Abstraction via EA-1

Ea 56 54 60 57 50 56 54 41 40 44

∆H‡ 51 49 55 52 45 51 49 36 35 39

∆G‡ 106 103 110 107 100 107 104 92 92 95

logeA 12.57 12.8 12.37 12.78 12.54 12.37 12.25 11.75 11.74 11.78

k 4.1E-05 1.4E-04 7.2E-06 3.2E-05 4.6E-04 3.1E-05 7.7E-05 9.9E-03 1.4E-02 3.1E-03

Hydrogen Abstraction via EA-2

Ea 74 72 79 78 67 71 73 45 55 51

∆H‡ 69 67 74 73 62 66 68 40 50 46

∆G‡ 125 122 129 119 119 132 115 108 110 106

logeA 12.23 12.26 12.35 16.19 11.44 8.12 15.86 7.27 10.47 10.36

k 2.1E-08 6E-08 4E-09 2.4E-07 1.9E-07 1.3E-09 1.3E-06 2.2E-05 7.5E-06 3.5E-05

methyl carbon atom = -0.383, and vinylic carbon atom = -0.12 and -0.26) that the methylene carbon is the most electrophilic, indicating its higher tendency to donate a proton. This also agrees with the fact that the bond-dissociation energy of the side-chain methylene group in EA-1 was lower than that of the methine group 7DEOH  .

Therefore, the occurrence of the EA-1 chain transfer mechanism in thermal polymerization of EA is highly probable. 33

4.3.1.3. n-Butyl Acrylate

▪ We studied four CTM mechanisms for M2 radical of n-BA, as shown in)LJXUH These are: abstraction of a hydrogen atom from three different methylene groups (n-BA- 1, n-

BA-2 and n-BA-3), and abstraction of a methyl hydrogen atom (n-BA-4). These mechanisms are investigated by constraining the C1-H2 and H2-C3 bond lengths between 1.19 Å and 1.59 Å. The transition-state structure for the n-BA-1 reaction is shownLQ)LJXUH F ,Ws geometry has H2-C3 and C1-H2 bond lengths of 1.34 Å and

1.38 Å, respectively.

Figure 4-5: Possible End-chain Transfer to Monomer Reactions for n-BA. 34

Table 4-4 lists the activation energies and rate constants for the reactions.

‡ Table 4-4: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Four CTM Reactions in n-BA at 298 K 31G(d) 311G(d) 311G(d) 311G(d) 31G(d,p) 31G(d,p) 31G(d,p) B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP 311G(d,p) 311G(d,p) M06-2X M06-2X M06-2X 311G(d,p) 6- 6- 6- 6- 6- 6- 6- 6- 6- 6- Hydrogen Abstraction via n-BA-1

Ea 51 47 55 53 42 51 49 31 36 43

∆H‡ 46 42 50 48 37 46 44 26 31 38

∆G‡ 106 111 107 110 105 108 106 83 89 96 logeA 10.48 6.73 11.68 9.62 7.25 9.49 9.56 11.9 11.09 11.37

k 4.1E-05 5.7E-06 3.2E-05 8.3E-06 6.6E-05 1.5E-05 4.2E-05 0.5 0.04 2.3E-03

Hydrogen Abstraction via n-BA-2 Ea 66 64 68 70 52 60 60 26 33 30

∆H‡ 61 59 63 65 47 55 55 21 28 25

∆G‡ 110 107 120 109 115 125 116 89 87 92 logeA 14.86 14.98 11.73 17.05 7.01 6.43 10.08 7.57 10.88 7.65

k 8.4E-06 2.3E-05 1.7E-07 1.2E-05 9E-07 1.9E-08 8.2E-07 4.5E-02 7.9E-02 9.6E-03

Hydrogen Abstraction via n-BA-3 Ea 63 61 66 66 56 65 63 40 45 45

∆H‡ 58 56 61 61 51 60 58 34 40 40

∆G‡ 129 126 111 104 121 115 122 100 98 100 logeA 6.11 6.27 14.41 17.4 6.40 12.46 8.62 8.26 11.2 10.64

k 4E-09 1.1E-08 6E-06 9.4E-05 8.2E-08 1.1E-06 5.6E-08 5.2E-04 9.8E-04 5.2E-04

Hydrogen Abstraction via n-BA-4 Ea 70 68 72 72 63 70 68 53 53 55

∆H‡ 65 62 67 67 58 65 63 48 48 50

∆G‡ 130 129 131 127 125 126 125 110 117 111 logeA 8.01 7.8 8.58 10.28 7.52 10.21 9.49 9.74 6.51 9.96

k 2E-09 3.9E-09 1.4E-09 7.7E-09 1.7E-08 1.4E-08 1.6E-08 9.1E-06 4E-07 4.8E-06

-1 -1 -1 Exp. [82] Ea =31kJ mol k = 0.6 M s 35

We determined that the rate constant of n-BA-1 is significantly higher than that of n-BA-

2, n-BA-3 and n-BA-4 with different levels of theory. It was identified that the abstraction of a hydrogen atom from a methylene group adjacent to methyl group in the butyl side chain has higher activation energy than that of the other methylene hydrogenatoms (Table 4-4). Mulliken charge analysis reveals that the methylene group carbon next to the ester oxygen of the end-substituent butyl group (0.022) is more electrophilic in comparison to the other methylene carbon atoms (-0.269 and -0.18) and methyl carbon atom (-0.37). Therefore, the hydrogen atom is more likely released from that methylene group. The bond-dissociation energy given in Table 4-2 also indicates that the C-H bond in a methylene group is much weaker than that in a methyl group. We found that the calculated activation energy and rate constant of hydrogen abstraction from the methylene group adjacent to the ester oxygen of the butyl side chain using M06-2X/6-

31G(d,p) was quite similar to experimental values [82]. The calculated activation energy and the rate constant are 31 kJ mol-1 and 0.5 L.mol-1.s-1, and the experimental values are

31 kJ mol-1 and 0.6 L.mol-1.s-1 [82, 83]. We observe no significant difference in the activation energies and rate constants for abstracting a hydrogen atom from the three methylene groups of n-BA, while a previous study [82] had reported that the two middle methylene groups of the side chain have lowest barriers for releasing a hydrogen atom.

The lowest energy mechanisms in MA, EA and n-BA were compared, and it was identified that the activation energy of MA-1 (Table 4-1) is higher than that of EA-1

(Table 4-3) or n-BA-1 (Table 4-4). This indicates that the type of the hydrogen atom influences the height of the barrier. As reported in Tables 1, 3 and 4, the activation energy for the most favorable CTM mechanism in MA (MA-1) is much higher (≈ 20 36 kJ/mol) than those for the most favorable ones in EA (EA-1) and n-BA (n-BA-1, n-BA-2 and n-BA-3). More precisely, the trend in reaction barrier relates to the type of hydrogen atoms available in the monomer; the bond dissociation energy of a methyl hydrogen atom in MA is higher than that of methylene hydrogen atoms in EA and n-BA.

We compute the C-NMR chemical shifts of the dead polymer chains formed by the most probable CTM mechanisms (MA-1, EA-1, n-BA-1) using ORCA [72].

Figure 4-6 shows these polymers. We have applied different methods (B3LYP and

X3LYP) and basis sets (6-31G(d), 6-31G(d,p), 6-311G(d), and 6-311G(d,p)) for calculating the chemical shifts. All the functionals give similar results for the chemical shifts Figure 4-7. The calculated NMR chemical shifts for the dead polymer chains generated via the MA-1, EA-1, and n-BA-1 mechanisms, given in Table 4-5 are comparable with experimental values reported in spontaneous polymerization of n-BA [7,

44].

This agreement suggests that the mechanisms found to kinetically favorable are most likely occurring in thermal polymerization of alkyl acrylates. No experimental result in spontaneous polymerization was found to be similar with the calculated results for the chemical shifts of the product generated from the MA-4 mechanism. Based on these comparisons, one can conclude that the MA-4 mechanism seems not to be the most favorable one for CTM reactions. The energy differences of the optimized reactants and products for the CTM mechanisms in MA, EA, and n-BA are reported in Table 4-6,

Table 4-7, and Table 4-8, respectively. 37

a

b

c

Figure 4-6: Dead Polymer Chains Generated via the MA-1, EA-1 and n-BA-1 Mechanisms (a: MA-1, b: EA-1, c: n-BA-1). 38

Figure 4-7: 13C-NMR Spectra of the Dead Polymer Chains Generated via the MA-1, EA- 1, and n-BA-1 Mechanisms at 298 K (δ is relative to tetramethylsilane [TMS] with an isotropic 13C nuclear magnetic shielding value of 187 ppm). 39

Table 4-5: Assignment of 13C-NMR Spectra for MA, EA and n-BA (Chemical Shifts are Relative to TMS with Absolute Isotropic Shielding of 187 ppm for the 13C)

Description Details Calculated Peaks Experimental Peaks[7, 100] Carbonyl C’s COOX next to the 177.6-179.2 - terminal saturation X=CH3 177.2-179 172.7-172.9 X=CH2CH3 177.1-179 - X=CH2CH2CH2CH3 COOX next to the 171.1-173 - terminal unsaturation X=CH3 170-172.5 166.4-166.7 X=CH2CH3 170.6-172.5 - X=CH2CH2CH2CH3 CH’s in the main chain Methine group in the 130.8-131.6 - main chain, CH next X=CH3 to the COOX 130.8-131.5 125.4-126 X=CH2CH3 130.8-131.4 - X=CH2CH2CH2CH3 Methine group in the 155.2-156.1 - main chain of MA, CH Methine group in the 155.2-156.2 137.7-137.9 main chain of EA, CH Methine group in the 155.4-156.3 - main chain of BA, CH CH2’s on the side chain, close to terminal saturation CH2 on the ethyl side 66.3-67 60.4-60.9 chain CH2 on the butyl side 70.5-71.6 - chain, next to the ester oxygen CH2 in the middle of 36.1-36.4 30.5 the butyl side chain CH2 on the butyl side 24.3-24.8 30.5 chain, prior to methyl group CH2’s on the side chain, close to terminal unsaturation CH2 on the ethyl side 65.6-66.4 60.4-60.9 chain CH2 on the butyl side 70.4-71 - chain, next to the ester oxygen CH2 in the middle of 36.3-36.7 30.5 the butyl side chain 40

Table 4-5 (Continued)

CH2 on the butyl side 24.5-25 30.5 chain, prior to methyl group CH2’s in the main chain CH2 next to the 35.3-36.1 - COOX X=CH3 35.5-36.7 33.2-37 X=CH2CH3 35.9-36.9 33.7-37.4 X=CH2CH2CH2CH3 CH2 in the main 31.5-32.6 - chain of MA, close to the double bond CH2 in the main 31.6-32.8 28-29.6 chain of EA, close to the double bond CH2 in the main 32.3-33.5 33.7-37.4 chain of BA, close to the double bond CH3’s C in O-CH3, close to 55-55.7 - terminal saturation C in O-CH3, close to 54.6-55.2 - terminal unsaturation C in the end group 15.8-16.3 14.20 CH3, on the ethyl side chain, close to terminal saturation C in the end group 16-16.5 14.20 CH3, on the ethyl side chain, close to terminal unsaturation C in the end group 16.9-17.4 - CH3, on the butyl side chain, close to terminal saturation C in the end group 17-17.5 - CH3, on the butyl side chain, close to terminal unsaturation 41

Table 4-6: The Energy Differences between the Reactants and the Products in kJ mol-1 for the Four CTM Reactions in MA at 298 K ) ) ) ) ) ) ) ) ) d d d d, p d, d, p d, d, p d, d, p d, p d, p B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP M06-2X M06-2X M06-2X 6-311G( 6-311G( 6-311G( 6-31G( 6-31G( 6-31G( 6-311G( 6-311G( 6-311G(

MA-1

∆E(P-R) 25 25 22 25 25 23 22 23 21

MA-2

∆E(P-R) 83 81 82 83 80 82 72 70 71

MA-3

∆E(P-R) 76 74 76 76 74 76 73 70 72

MA-4

∆E(P-R) -42 -41 -40 -42 -40 -39 -32 -31 -30

Table 4-7: The Energy Differences between the Reactants and the Products in kJ mol-1 for the Two CTM Reactions in EA at 298 K ) ) ) ) ) ) ) ) ) d d d d, p d, d, p d, d, p d, d, p d, p d, p B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP M06-2X M06-2X M06-2X 6-311G( 6-311G( 6-311G( 6-31G( 6-31G( 6-31G( 6-311G( 6-311G( 6-311G(

EA-1

∆E(P-R) 14 15 13 14 15 13 13 9 13

EA-2

∆E(P-R) 46 44 44 46 44 43 37 30 35 42

Table 4-8: The Energy Differences between the Reactants and the Products in kJ mol-1 for the Four CTM Reactions in n-BA at 298 K ) ) ) ) ) ) ) ) ) d d d d, p d, p d, d, p d, d, p d, p d, p B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP M06-2X M06-2X M06-2X 6-311G( 6-311G( 6-311G( 6-31G( 6-31G( 6-31G( 6-311G( 6-311G( 6-311G(

n-BA-1

∆E(P-R) 14 13 12 15 14 13 12 12 11

n-BA-2

∆E(P-R) 27 25 25 27 25 25 19 18 17

n-BA-3

∆E(P-R) 24 22 22 24 22 21 16 15 13

n-BA-4

∆E(P-R) 41 39 38 41 39 38 30 28 26

▪ 4.3.2. Chain Transfer to Monomer Mechanisms for M1

As shown Table 4-8, the chain transfer reactions, MA-1, EA-1 and n-BA-1, of a 2-

▪ monomer-unit live polymer chain initiated via M1 are investigated using B3LYP,

X3LYP and M06-2X functionals. The calculated transition states have C1-H2 and H2-C3 bond lengths of 1.31 Å and 1.40 Å in MA, 1.33 Å and 1.38 Å in EA, and 1.34 Å and 1.38

Å in n-BA,. The transition state structures for these mechanisms are shown in Figure 4-9, and activation energies and rate constants are given in Table 4-9. Comparing these results

▪ with those calculated in the previous section for M2 initiated chains shows little difference in activation energy and rate constants, which indicates that the type of initiating radical of the live polymer chain has little effect on the rate of the CTM reactions. 43

Figure 4-8: The Most Probable Mechanisms for CTM Reactions Involving a Two- ▪ Monomer-Unit Live Chain Initiated by M1 . 44

Figure 4-9: Transition State Geometries of MA-1 (a), EA-1 (b) and n-BA-1 (c) ▪ Mechanisms Involving a Two-Monomer-Unit Live Chain Initiated by M1 .

4.3.3. Effect of Live Polymer Chain Length

To understand the effect of polymer chain length on the kinetics of CTM, we investigated

▪ MA-1, EA-1 and n-BA-1 mechanisms with a 3-monomer-unit live chain initiated via M2

(Figure 4-10). We constrained the C1-H2 and H2-C3 bond lengths between 1.19 Å and

1.59 Å in the three mechanisms. We found the bond lengths of H2-C3 and C1-H2 to be 45

‡ Table 4-9: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the MA-1, EA-1 and n-BA-1 Reactions Involving Two-Monomer-Unit Live Chains ▪ Initiated by M1 at 298 K ) ) ) ) ) ) ) ) ) ) d d d d d,p d,p d,p d,p d,p d,p B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP M06-2X M06-2X M06-2X 6-31G( 6-311G( 6-311G( 6-311G( 6-31G( 6-31G( 6-31G( 6-311G( 6-311G( 6-311G(

MA-1

Ea 78 76 81 78 68 75 73 63 67 65

∆H‡ 73 71 76 73 63 71 68 58 62 60

∆G‡ 120 117 124 120 124 126 123 115 119 118

logeA 15.58 15.9 15.28 15.65 9.89 12.27 12.43 11.63 11.5 11.35

k 1.2E-07 4.3E-07 2.7E-08 1.3E-07 2.7E-08 1.3E-08 4.9E-08 1.3E-06 1.9E-07 3.8E-07

EA-1

Ea 63 60 69 69 62 65 65 48 49 52

∆H‡ 58 56 64 64 57 60 60 43 44 48

∆G‡ 115 113 114 107 97 110 101 101 91 94

logeA 11.68 11.66 14.30 17.43 18.47 14.56 17.96 10.97 15.55 15.76

k 1.1E-06 2.9E-06 1.5E-06 3.3E-05 1.7 E-03 8.6E-06 2.6E-04 2.7E-04 1.8E-02 4.4E-03

n-BA-1

Ea 67 65 71 71 64 68 66 57 59 59

∆H‡ 62 60 66 66 59 63 61 52 54 54

∆G‡ 111 109 112 103 100 110 108 90 99 103

logeA 14.75 14.9 15.97 19.83 18.28 15.56 15.41 19.46 16.34 14.9

k 4.5E-06 1.1E-05 3.1E-06 1.3E-04 4.9E-04 7.3E-06 1.5E-05 2.9E-02 5.7E-04 1.3E-04

1.31 Å and 1.4 Å in MA, 1.33 Å and 1.38 Å in EA, and 1.34 Å and 1.38 Å in n-BA. It was observed that the transition-state geometries of 3-monomer-unit live chains (Figure

4-11) are similar to those of 2-monomer-unit live chains discussed in Section 3.1. It can 46 be concluded that the end-substituent group does not significantly affect the geometry of the reaction center. As given in Table 4-10, the calculated activation energies of the CTM reactions using B3LYP/6-31G(d) vary very little with the length of the polymer chain.

However, M06-2X/6-31G(d,p) did show increase in the energy barrier for n-BA-1 with increasing chain length. This is likely due to the long-range exchange interactions in the hybrid meta-functional M06-2X.

Figure 4-10: The Most Probable Mechanisms for CTM Reactions Involving a Three- ▪ Monomer-Unit Live Chain Initiated by M2 .

4.3.4. Effect of Radical Location in a Live Chain

Previous studies [7, 8, 17] have shown that inter- and intra-molecular chain transfer reactions occur at high rates at high temperatures (>100oC), which can lead to the 47

Figure 4-11: Transition State Geometries of the MA-1, EA-1 and n-BA-1 Mechanisms ▪ Involving a Three-Monomer-Unit Live Chain Initiated by M2 (a: MA, b: EA, c: n-BA).

formation of tertiary radicals [7] and that both the secondary and tertiary radicals are

▪ capable of initiating polymer chains. We also studied the CTM reactions of an M1 -

▪ initiated three-monomer-unit live chain with a secondary radical, denoted by P3 , and an

▪ ▪ M1 -initiated three-monomer-unit live chain with a tertiary radical, denoted by Q3 , as shown in Figure 4-12. The H2-C3 and C1-H2 bond lengths for the transition state of the

▪ MA-1 mechanism involving the Q3 live chain was found to be 1.28 Å and 1.43 Å, respectively (Figure 4-13). The calculated energy barriers and rate constants of these two 48

‡ Table 4-10: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M- 1 s-1 of the MA-1, EA-1 and n-BA-1 Reactions Involving Three-Monomer-Unit Live ▪ Chains Initiated by M2 at 298 K Calculated Using B3LYP/6-31G(d) and M06-2X/6- 31G(d,p)

‡ ‡ Ea ∆H ∆G logeAk

MA-1 B3LYP 69 64 120 12.07 1.2×10-7 M06-2X 55 50 102 13.63 2×10-4

EA-1 B3LYP 55 50 108 11.2 1.5×10-5 M06-2X 35 30 93 9.53 9.2×10-3

n-BA-1 B3LYP 52 48 125 3.5 2.1×10-8 M06-2X 54 49 102 13.22 1.7×10-4

▪ ▪ ▪ Figure 4-12: CTM Reaction Involving the Live Chain Q3 and P3 Initiated by M1 . 49

▪ Figure 4-13: Transition State Geometry of MA-1 Reaction Involving the Live Chain Q3 .

mechanisms are given in Table 4-11. The activation energy of hydrogen abstraction by a tertiary radical is higher than that of hydrogen abstraction by a secondary radical. This suggests that the tertiary radical center probably prefers to react with a monomer and form chain branches or undergo β-scission reaction, rather than abstracting hydrogen atoms. This agrees with previous reports [7, 9], where NMR and mass spectrometry have shown the formation of chain branches on tertiary radicals.

4.3.5. Chain Transfer to Monomer Reaction for MA in the Presence of Initiator

In order to explore the effect of initiator on CTM reactions, the most likely CTM

▪ mechanism for MA, MA-1, of a 2-monomer-unit live polymer chain initiated via M2 were investigated in the presence of tert-Butyl peroxyacetate, as the initiator, using

B3LYP (6-31G(d) and 6-31G(d, p) basis sets). In other words, the transition state was 50 determined considering the initiator in the system. The kinetic parameters of the reaction are given in Table 4-12.

‡ Table 4-11: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M- 1 -1 ▪ s of the MA-1 Reaction Involving Three-Monomer-Unit Live Chains Initiated by M1 at 298 K Calculated Using M06-2X/6-31G(d,p)

‡ (C1-H2) Å (C3-H2) Å Ea ∆H logeAk

Secondary Live Chain 1.40 1.32 55 50 12.89 8.3×10-5

Tertiary Live Chain 1.43 1.28 67 62 9.93 3.5×10-8

‡ Table 4-12: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M- 1 -1 ▪ s of the MA-1 Reaction Involving Two-Monomer-Unit Live Chain Initiated by M2 in the Presence of Initiator (tert-Butyl peroxyacetate) at 298 K

In the Presence of Initiator In the Absence of Initiator ) ) ) ) d d d,p d,p 31G( 31G( B3LYP B3LYP B3LYP B3LYP -31G( -31G( 6- 6- 6 6

Ea 75 73 71 68 ∆H‡ 70 68 66 63 ∆G‡ 138 134 114 111 logeA 7.12 8.07 15.4 15.46 k 8.8E-11 5.98E-10 1.8E-06 5.3E-06

The computational results declare that the activation energy in the presence of the initiator is higher (5 kJ mol-1) than that estimated for spontaneous polymerization of MA

(section 4.3.1). Although the rise of activation energy in the presence of initiator is small, 51 there is a large reduction (4 orders of magnitude) in the rate constant of the reaction for the system including the initiator molecule. It can be justified through that the initiator is more competitive than live chain radical in getting involved in the reaction with the monomer. Also, the presence of initiator affects the entropy of the system and consequently the frequency factor.

4.3.6. Replacement of MA Unit with MMA and St

The most likely mechanism of CTM reaction for MA (MA-1) were explored considering different chain configurations of the live chain radical composed of MA, MMA, and St.

In this section, the MA unit next to the radical in the live chain was replaced with MMA

(MA-1-MMA) and St units (MA-1-St). Figure 4-14 shows the geometries of the transition states for MA-1 mechanism in which the live chain radical contains MMA

[Figure 4-14 (a)] and St [Figure 4-14 (b)] units. Kinetic parameters of these reactions are given in Table 4-13.

‡ Table 4-13: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M- 1 s-1 of the MA-1-MMA and MA-1-St Mechanisms at 298 K

MA-1-MMA MA-1-St ) ) ) ) P d,p d,p d,p d,p 2X 2X B3LYP B3LY M06- -31G( -31G( -31G( -31G( M06- 6 6 6 6

Ea 107 95 93 78 ∆H‡ 103 90 88 73 ∆G‡ 141 133 138 128 logeA 19.04 17.22 14.44 12.41 k 2.6E-11 6.7E-10 9.9E-11 5.2E-09 52

Figure 4-14: Transition State Geometries of the MA-1-MMA and MA-1-St Mechanisms ▪ Involving a Two-Monomer-Unit Live Chain Initiated by M2 (a: MA-1-MMA, b: MA-1- St).

According to these results the rate constants of MA-1-MMA and MA-1-St is lower (4-5 orders of magnitude) than that of MA-1. The replacement of MA with MMA and St units made an increase in the activation energies of CTM reaction. It was found that the activation energy of MA-1 is lower than that of MA-1-MMA (40 kJ mol-1) and MA-1-St

(25 kJ mol-1). It seems that MMA and St units enhance stability of the radical and reduce its reactivity for undergoing CTM reaction. These results are in agreement with those reported previously for backbiting reaction in acrylate-based copolymers [50].

4.4. Concluding Remarks

The mechanisms for chain transfer to monomer in self-initiated high-temperature polymerization of three alkyl acrylates were studied using different levels of theory. 53

Abstraction of a methylene group hydrogen by a live polymer chain in EA and n-BA, and a methyl group hydrogen in MA were found to be the favorable mechanisms for chain transfer to monomer reaction. The kinetic parameters calculated using M06-2X/6-

31G(d,p) are closest to those estimated from polymer sample measurement. The C-H bond lengths of transition state structures for CTM reactions in MA, EA, and n-BA were found to be insensitive to the choice of functionals. NMR spectra of various reacting species in CTM reactions in MA, EA and n-BA were predicted. The NMR chemical shifts of dead polymer chains from MA-1, EA-1 and n-BA-1 are comparable to those from polymer sample analyses, which confirms that the products of the chain transfer reactions proposed in this study are formed in polymerization of MA, EA and n-BA. It was found that the polymer chain length had little effect on the activation energies and rate constants of the mechanisms of CTM reaction when applying B3LYP/6-31G(d) functional. However, M06-2X/6-31G(d,p) showed differences between the kinetic parameters of CTM reactions involving live chains with two and three monomer units.

▪ ▪ MA, EA and n-BA live chains initiated by M2 and those initiated by M1 showed similar hydrogen abstraction abilities, which indicates the lesser influence of self-initiating species on CTM reaction. Hydrogen abstraction by a tertiary radical has a much larger energy barrier than abstraction by a secondary radical. 54

5. Theoretical Study of Chain Transfer to Polymer Reactions of Alkyl Acrylates

5.1. Introduction

It has been reported that chain transfer to polymer (CTP) reactions affect the molecular architecture of the final polymer due to the formation of long chain branches [16, 44].

Therefore, better understanding of CTP mechanisms is required for producing desired products. The long chain branches formed through CTP reaction can be terminated by coupling and consequently gel formation occurs during the polymerization process.

Control of CTP reactions is a great assist to optimize polymerization process. Based on the results obtained through polymer sample measurement, the mechanism of intermolecular chain transfer reaction involves abstraction of hydrogen atom by a live polymer chain from a dead polymer chain [13, 44]. However, the location of dead polymers (formed through different types of termination reactions) hydrogen abstracted is still ambiguous. These tertiary radicals then propagate to form long-chain branches, causing gel formation [13]. In intra-molecular chain transfer (backbiting) reactions, the propagating secondary radical abstracts a hydrogen atom from its backbone and renders formation of branches of various lengths [44, 50, 105, 106]. Chiefari et al. have suggested that at lower monomer concentration, intra-molecular CTP is the predominant pathway for hydrogen abstraction from the backbone. However, intermolecular CTP becomes more important at higher monomer concentration [7, 13, 43, 44, 107, 108].

Numerous experimental [25-27, 44, 109-114] and theoretical [13, 20, 115-118] investigations have pointed out that CTP reactions can strongly impact the overall rate of polymerization. It is recognized that a fundamental understanding of the mechanism of these reactions is highly valuable to design and optimize high-temperature 55 polymerization processes [13, 119, 120].

Secondary reactions, such as CTP and β-scission reactions, in thermal polymerization of n-BA and n-butyl methacrylate (BMA) have been studied using nuclear magnetic resonance (NMR) spectroscopy and electrospray ionization/Fourier transform mass spectroscopy (ESI/FTMS) [13, 16, 17]. NMR analysis of these polymers indicated the presence of end groups from CTP reactions at temperatures lower than 70oC [16, 27, 44].

Experimental studies have shown the important role of intra-molecular chain transfer and scission reactions in the reduction of average molecular weight and rate of the polymerization reaction [9, 13, 20, 121]. NMR and ESI/FTMS analyses of samples from spontaneous (no thermal initiator added) high-temperature homo-polymerization of EA and n-BA have shown that different chain types are generated during the polymerization

[7]. The NMR and ESI/FTMS analysis revealed the presence of branch points, indicating the propagation of mid-chain tertiary radicals. Previous studies have used pulsed-laser polymerization/size exclusion chromatography to study intra-molecular CTP reactions in polymerization of alkyl acrylate [19, 20]. While pulsed-laser polymerization has been used to estimate the rate constant of the propagation reactions (Kp), for monomers such as styrene [122] and methyl methacrylate (MMA) [123], and chain transfer reaction in polymerization of BMA [124] at temperatures higher than 30oC, no consistent value of propagation rate constant has been determined for alkyl acrylates. The molecular weight distributions of pulsed-laser polymerization generated polymers depicted peak broadening at temperatures higher than 30oC. The broadening of the peak was attributed to the occurrence of chain transfer reactions such as inter- and intra-molecular chain transfer to polymer reactions [27, 28]. At temperatures above 30oC, intermolecular CTP 56 reactions in free-radical polymerization of n-BA [13, 44] and 2-ehylhexyl acrylate [16] have also been studied using NMR spectroscopy. While these analytical techniques have been very useful in characterizing acrylate polymers generated from thermal free-radical polymerization, they have been unable to conclusively determine specific reaction mechanisms and estimate kinetic parameters of the reactions, individually.

5.2. Computational Methods

The mechanisms of CTP reactions in self-initiated high-temperature homo- polymerization of alkyl acrylates were studied using first-principles calculations. Several possible mechanisms have been investigated using three types of hybrid functionals

(B3LYP, X3LYP and M06-2X) and four basis sets (6-31G(d), 6-31G(d, p), 6-311G(d) and 6-311G(d, p)). The energy barrier and rate constant of each reaction was calculated using transition state theory and the rigid rotor harmonic oscillator (RRHO) approximation. Reactants and transition states are validated by performing Hessian calculations. Three different structures for the dead polymers, formed through different termination reactions, which are involved in the CTP reactions have been considered in this study. It was found that the abstraction of a hydrogen atom from a tertiary carbon atom of a dead polymer chain by a live polymer chain is the most probable mechanism of

CTP in self-initiated polymerization of alkyl acrylates. Implicit solvent models, integral equation formalism (IEF) which is a version of polarizable continuum model (PCM) and conductor-like screening model (COSMO) are applied to account for solvent effects.

Minimum-energy pathways for several reactions of interest are determined using intrinsic reaction coordinated (IRC) calculations. The CTP reactivity of MA, EA, and n-BA has also been compared. The length of the polymer chain was found to have little effect on 57 the calculated activation energies and transition state geometries in all the CTP mechanisms explored in this study. Moreover, CTP reactions involving homo-polymer chains possessing methyl, ethyl, and butyl side chains have similar energy barriers and rate constants. The chemical shifts of the species produced through chain transfer to dead polymers of MA, EA, and n-BA were calculated with the ORCA program [72]. Quantum tunneling is considered in CTP reactions [125-128]. The Wigner tunneling [128] correction is calculated in this chapter using:

ଵ ௛ఔș ߢൎͳ൅ ሺ ሻଶ (5.1) ଶସ ௞ಳ் where ߥș is the imaginary frequency of the transition state.

The experimentally measured chemical shifts are within a couple of ppm of the calculations. All of these results provide evidence that CTP reactions produce branched acrylate polymers via hydrogen abstraction by a live polymer chain from a dead-polymer- chain tertiary carbon. We found that dead polymers with different structures have different CTP reactivities. The mechanisms of these reactions, in which various types of dead polymers are involved, were ambiguous through experimental studies. The following section provides some parts of these results.

5.3. Results and Discussion

5.3.1. Mechanisms of CTP in MA, EA and n-BA

We studied CTP mechanisms for MA, EA and n-BA, which are shown in Figure 5-1,

Figure 5-2, and Figure 5-3, respectively. The energy differences of optimized reactants and products involved in each of these mechanisms were calculated by applying B3LYP and X3LYP functionals and four different basis sets (6-31G(d), 6-31G(d,p), 6-311G(d) and 6-311G(d,p)). These results are reported in Table 5-1, Table 5-2, and Table 5-3, 58

CH3 O

. H O CH3 MeOOC C + O 3CH O COOMe CH3

. C H CH3 O 2 O . O C CH3 O CH3 COOMe O CH O MeOOC + 3CH 3 O O CH3 CH3 MA2-D1-1 MA2-D1-2 CH3 O

O CH3 . O H2C O MA2-D1-3 CH3

Figure 5-1: Possible Chain Transfer to Polymer Mechanisms for MA.

CH3 O O . + 3CH EtOOC C H O CH3 O COOEt CH3 . C H2 CH3

. COOEt C COOEt EtOOC COOEt EtOOC EtOOC + CH3 EA2-D1-1 CH3 EA2-D1-2

CH CH3 3 O O . O H2C O 3CH . O CH C O CH3 3 H O O CH CH3 EA2-D1-3 3 EA2-D1-4

Figure 5-2: Possible Chain Transfer to Polymer Mechanisms for EA 59

CH 3 O

. + O BuOOC C H 3CH O CH3 O COOBu CH3

. H C 3CH 2

. C COOBu COOBu COOBu BuOOC BuOOC + BuOOC CH CH3 3 n-Bu2-D1-2 n-Bu2-D1-1

CH H 3CH 3 O O . O C O H7C3 . O C H C O C3H7 H5C2 3 7 O O CH H CH 3 n-Bu2-D1-3 3 n-Bu2-D1-4

3CH CH O 3 O CH O . O 3 . O C H H2C C 3 7 O C3H7 O CH O H 3 n-Bu2-D1-5 CH3 n-Bu2-D1-6

Figure 5-3: Possible Chain Transfer to Polymer Mechanisms for n-BA.

-1 Table 5-1: Energy Differences of the Reactants (ER) and Products (EP), in kJ mol for Three CTP Mechanisms for MA at 298 K ) ) ) ) ) ) ) ) d d d d d,p d,p d,p d,p B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP X3LYP 6-31G( 6-31G( -311G( -311G( 6-311G( 6-311G( 6-31G( 6-31G( 6 6

MA2-D1-1

∆E(R-P) 17 17 17 17 21 22 17 18 MA2-D1-2

∆E(P-R) 42 41 40 39 41 41 40 39 MA2-D1-3

∆E(P-R) 31 29 28 27 29 28 28 27 60

-1 Table 5-2: Energy Differences of the Reactants (ER) and Products (EP), in kJ mol for the Four CTP Mechanisms for EA at 298 K ) ) ) ) ) ) ) ) d d d d d,p d,p d,p d,p B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP X3LYP 6-31G( 6-31G( -311G( -311G( 6-311G( 6-311G( 6-31G( 6-31G( 6 6

EA2-D1-1

∆E(R-P) 17 17 16 17 17 17 16 17 EA2-D1-2

∆E(P-R) 42 42 40 39 42 42 40 39 EA2-D1-3

∆E(P-R) 18 17 18 16 18 17 17 16 EA2-D1-4

∆E(P-R) 47 47 46 45 47 46 45 44

-1 Table 5-3: Energy Differences of the Reactants (ER) and Products (EP), in kJ mol for the Six CTP Mechanisms in n-BA at 298 K ) ) ) ) ) ) ) ) d d d d d,p d,p d,p d,p B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP X3LYP 6-31G( 6-31G( 6-311G( 6-311G( 6-31G( 6-31G( 6-311G( 6-311G(

n-BA2-D1-1

∆E(R-P) 17 17 16 17 17 16 16 17 n-BA2-D1-2

∆E(P-R) 42 42 40 40 42 42 40 40 n-BA2-D1-3

∆E(P-R) 18 18 17 16 18 18 17 16 n-BA2-D1-4

∆E(P-R) 30 30 29 28 30 30 24 28 n-BA2-D1-5

∆E(P-R) 26 26 24 24 26 26 24 24 n-BA2-D1-6

∆E(P-R) 43 43 41 40 43 43 40 39 61 respectively. We found (Table 5-1, Table 5-2, and Table 5-3) that MA2-D1-1, EA2-D1-1, and n-BA2-D1-1 mechanisms are exothermic in comparison to the other mechanisms, which are endothermic. The products formed through MA2-D1-1, EA2-D1-1, and n-

BA2-D1-1 mechanisms have lower energy (higher stability) than those formed through other mechanisms. Since more stable radical center is the one which is more substituted, the higher stability of the generated tertiary radicals vs. secondary radicals enhances the likelihood of the occurrence of the MA2-D1-1, EA2-D1-1 and n-BA2-D1-1 mechanisms.

.

Table 5-4: H-R Bond Dissociation Energies (kJ mol-1) at 298 K

B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP X3LYP 6-31G(d) 6-31G(d,p) 6-311G(d) 6-311G(d,p) 6-31G(d) 6-31G(d,p) 6-311G(d) 6-311G(d,p)

MA2-D1-1 387 390 383 385 385 387 384 387 MA2-D1-2 446 448 440 441 447 450 441 443 MA2-D1-3 435 436 428 429 435 437 429 431 EA2-D1-1 387 390 383 385 388 391 385 387 EA2-D1-2 445 448 440 441 447 450 441 443 EA2-D1-3 422 424 417 419 423 425 418 420

H-R EA2-D1-4 451 453 445 447 452 455 446 448 n-BA2-D1-1 387 390 383 385 388 392 385 387 Bond-dissociation energy Bond-dissociation n-BA2-D1-2 446 449 440 442 447 450 441 443 n-BA2-D1-3 422 424 417 418 423 426 418 419 n-BA2-D1-4 433 436 429 430 435 438 425 432 n-BA2-D1-5 429 432 424 426 431 434 425 427 n-BA2-D1-6 447 449 440 442 448 451 441 443

While in agreement with previous results [63], they indicate that the bond-dissociation energies of hydrogen atoms attached to the tertiary carbon atoms (which are abstracted 62 via MA2-D1-1, EA2-D1-1, and n-BA2-D1-1 mechanisms) are much lower (about 50 kJ/mol) than those of other hydrogen atoms of dead polymers. This suggests that such a tertiary carbon atom has a higher tendency to release a hydrogen atom.

5.3.2. Effect of the Type of the Radical that Initiated a Live MA-Polymer Chain

• • Two types of mono-radicals (M1 and M2 ) are produced in monomer self-initiation of alkyl acrylates (Figure 4-1). These are proposed to propagate polymerization via formation of live polymer chains. The hydrogen atom abstractions from a tertiary carbon

• atom of a dead polymer chain that had been initiated by M1 (MA1-D1-1, shown in

• Figure 5-1) and M2 (MA2-D1-1, shown in Figure 5-1) are investigated in this chapter.

. M1 Monomer COOMe COOMe . . C 2CH COOMe C COOMe CH H + 3CH 3 H H

MA1-D1-1

CH3 O COOMe COOMe . CH COOMe C COOMe + O CH3 2 CH O 3CH 3CH 3 H + O CH3 CH3 O . O C CH3 O 3CH O CH3

Figure 5-4: Most Probable CTP Mechanism Involving Two-MA-Unit Live Chain ▪ Initiated by M1 .

Three different functionals, B3LYP, X3LYP, and M06-2X, and several basis sets are applied. The MA1-D1-1 and MA2-D1-1 mechanisms are explored by constraining C1- 63

H2 and H2-C3 bond lengths between 1.19 and 1.59 Å. Figure 5-5(a) shows the transition- state geometry for the MA2-D1-1 mechanism with C1-H2 and H2-C3 bond lengths of

1.37 and 1.35 Å, respectively.

a b

c

Figure 5-5: Transition State Geometry for CTP Reaction; a: Methyl Acrylate (MA2-D1- 1), b: Ethyl Acrylate (EA2-D1-1), c: n-Butyl Acrylate (n-BA2-D1-1).

The activation energies, enthalpies of activation, frequency factors, and rate constants of the MA2-D1-1 mechanism are provided in Table 5-5. These results show that the activation energies and rate constants calculated using different basis sets vary by ± 10 kJ/mol and 2 orders of magnitude. 64

‡ Table 5-5: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy ‡ -1 of Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1 s-1, for the Most Probable CTP Mechanisms for MA at 298 K; and Rate Constant without Tunneling at 413 K (k413) ) ) ) ) ) ) ) ) ) ) ) ) B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP X3LYP M06-2X M06-2X M06-2X M06-2X 6-31G( d 6-31G( d 6-31G( d 6-311G( d 6-311G( d 6-311G( d 6-31G( d,p 6-31G( d,p 6-31G( d,p 6-311G( d,p 6-311G( d,p 6-311G( d,p

Hydrogen Abstraction via MA2-D1-1

Ea 57 55 63 60 52 47 59 54 30 28 31 29 ∆H‡ 52 50 58 55 47 42 54 49 25 23 26 24 ∆G‡ 108 108 117 111 108 106 105 104 88 85 85 84

logeA 12.08 11.32 11.17 12.16 10.16 8.76 14.17 12.39 9.29 9.56 10.97 10.67 k 1.6E-5 1.6E-5 5.4E-7 5.9E-6 2.0E-5 4.1E-5 5.5E-5 7.8E-5 5.8E-2 1.9E-1 1.9E-1 3.4E-1

ߢw 3.58 3.49 3.64 3.6 3.83 3.73 3.98 3.78 3.11 3.08 3.19 3.07

kw 5.7E-5 5.6E-5 1.9E-6 2.1E-5 7.6E-5 1.5E-4 2.2E-4 2.9E-4 1.8E-1 5.8E-1 6.1E-1 1.04

k413 9.9E-4 1.6E-3 3.3E-5 3.5E-4 1.3E-3 3.2E-3 2.6E-3 3.7E-3 2.08E0 1.1E+1 8.1E+1 1.9E+1

Hydrogen Abstraction via MA1-D1-1

Ea 62 61 68 64 56 53 63 57 34 28 33 29 ∆H‡ 57 56 63 59 51 49 58 52 29 23 28 24 ∆G‡ 116 115 123 118 111 107 109 106 84 87 88 87

logeA 10.99 10.84 10.52 11.15 10.49 11.15 13.97 12.7 12.35 9.11 10.49 9.1 k 7.2E-7 1.1E-6 4.9E-8 3.8E-7 6.2E-6 2.9E-5 9.9E-6 3.3E-5 2.7E-1 9.6E-2 6.2E-2 8.9E-2

ߢw 3.79 3.70 3.93 3.73 3.84 3.75 3.94 3.79 3.15 3.11 3.23 3.18

kw 2.7E-6 4.1E-6 1.9E-7 1.4E-6 2.4E-5 1.1E-4 3.9E-5 1.3E-4 8.5E-1 3.0E-1 2.0E-1 2.8E-1

k413 3.7E-5 7.5E-5 2.2E-6 1.6E-5 2.8E-4 2.0E-3 2.5E-4 2.6E-3 1.9E+1 4.7E0 3.1E0 3.5E0

It is important to note that M06-2X functional could be more accurate than the other functionals since it accounts for van der Waals (vdW) interactions [60, 61]. The calculated rate constants using M06-2X have been reported to be in good agreement with experiments for the mechanisms of chain transfer to monomer in MA, EA and n-BA [48].

No significant change in activation energies and rate constants was observed with different basis sets. The transition-state structure of the MA1-D1-1 mechanism has C1- 65

H2 and H2-C3 bond lengths of 1.37 and 1.36 Å, respectively (Figure 5-6). Table 5-5 gives the kinetic parameters for the MA1-D1-1 mechanism. A comparison of the activation energies and rate constants calculated for the MA2-D1-1 and MA1-D1-1 mechanisms indicates that the type of the radical that initiated the live chain has little or no effect on the capability of the live chain for undergoing CTP reaction.

Figure 5-6: Transition State Geometry for CTP Reaction via Hydrogen Abstraction from ▪ D1 Dead Polymer by M1 in Methyl Acrylate (MA1-D1-1).

5.3.3. CTP Mechanism for EA and n-BA

The mechanism of CTP reactions for EA (EA2-D1-1) and n-BA (n-BA2-D1-1) are also examined (Figure 5-2 and Figure 5-3). The hydrogen atom abstraction from a tertiary

• carbon atom by a live polymer chain initiated by M2 is studied by constraining C1-H2 and H2-C3 bond lengths between 1.19 Å and 1.59 Å. The C1-H2 and H2-C3 bond lengths of the transition-state structure for the EA2-D1-1 mechanism are 1.38 Å and 1.35 66

Å, respectively [Figure 5-5(b)]. The activation energies, enthalpies of reaction, frequency factors, and rate constants of the EA2-D1-1 mechanism are given in Table 5-6.

‡ Table 5-6: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy ‡ -1 of Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1 s-1, for the Most Probable CTP Mechanisms for EA (EA2-D1-1) at 298 K; and Rate Constant without Tunneling at 413 K (k413) ) ) ) ) ) ) ) ) )

) ) )

d,p M06-X B3LYP B3LYP B3LYP M062X M062X M062X B3LYP X3LYP X3LYP X3LYP X3LYP 6-31G( d 6-31G( d 6-31G( d -311G( d,p -311G( d,p -311G( d,p 6-311G( d 6-311G( d 6-311G( d 6-31G( d,p 6-31G( d,p 6-31G( 6 6 6

Ea 59 56 66 63 54 49 62 59 24 21 24 23 ‡ ∆H 54 51 61 58 49 44 57 54 19 16 19 19 ‡ ∆G 110 107 116 113 101 102 104 101 83 81 84 83 logeA 11.96 12.22 12.31 12.39 13.81 11.35 15.61 15.51 8.71 8.7 8.6 8.7 k 8.4E-6 2.8E-5 6.4E-7 2.5E-6 3.2E-4 1.9E-4 8.5E-5 2.7E-4 4.2 1.1 3.2E-1 4.5E-1 ߢw 3.66 3.59 3.80 3.78 3.86 3.76 4.00 3.80 3.15 3.12 3.23 3.16 kw 3.1E-5 1.0E-4 2.4E-6 9.5E-6 1.2E-3 7.1E-4 3.4E-4 1.0E-3 1.3E+1 3.4 1.0 1.4 k413 1.4E-3 5.9E-3 9.6E-5 7.0E-4 5.1E-2 3.3E-2 1.4E-2 2.7E-2 7.1E+1 1.3E+2 4.6E+1 5.3E+1

The activation energies and rate constants calculated using the methods B3LYP and

X3LYP and the basis sets (6-31G(d), 6-31G(d,p), 6-311G(d), and 6-311G(d,p)) are different at most by 13 kJ/mol and 2 orders of magnitude. The sensitivity of the predicted kinetic parameters using M06-2X to the applied basis sets is lower than that using

B3LYP or X3LYP functionals; no or little difference in activation energies (3 kJ/mol) and rate constants (one order of magnitude) can be observed by applying different basis sets. The transition-state structure of the n-BA2-D1-1 mechanism is shown in Figure

5-5(c), and the kinetic parameter values are given in Table 5-7. Our finding that n-BA2-

D1-1 is the most probable mechanism of CTP is in agreement with previous studies [20] .

The calculated activation energy, using B3LYP /6-31G(d, p), is 20 kJ/mol higher than a reported experimental value of 29 kJ/mol, and the calculated rate constant (7.4 × 10 -6) is 67 lower by about 4 orders of magnitude. This disagreement may be due to solvent effects, not accounted for in our gas-phase studies. Solvent molecules, because of their dielectric screening, can increase the stability of the reactants and transition states and consequently change the barriers. Other interactions such as hydrogen bonding or complex formation may also affect chain transfer rates. These solvation behaviors can affect the reaction entropy.

‡ Table 5-7: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy ‡ -1 of Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1 s-1, for the n-BA2-D1-1 Mechanism at 298 K; and Rate Constant without Tunneling at 413 K (k413)

B3LYP/6-31G(d) B3LYP/6-31G(d,p) Ea 56 53 ‡ ∆H 51 48 ‡ ∆G 114 110 logeA 9.27 9.71 k 1.70E-6 7.40E-6 ߢw 3.75 3.68 kw 6.37E-6 2.72E-5 k413 1.30E-3 4.50E-3

Calculated 13C-NMR chemical shifts of the products of CTP reactions (MA2-D1-1, EA2-

D1-1 and n-BA2-D1-1) are shown in Figure 4-6 and Figure 4-7. Table 4-5 compares the calculated NMR chemical shifts with experimental values reported for spontaneous polymerization of n-BA [7, 44]. The computed and experimental chemical shifts are similar (within the error range of 10%) suggesting that the dead polymers could be produced via the proposed mechanisms of CTP reaction. The kinetic parameters 68 estimated for the most likely CTP mechanisms of MA, EA and n-BA (MA2-D1-1, EA2-

D1-1, and n-BA2-D1-1) indicate that the end-substituent groups (methyl, ethyl, and butyl acrylate side chains) do not affect the kinetics of CTP reaction in the alkyl acrylates. This can be explained through the similarity of the reactive sites involved in the CTP reaction of MA, EA, and n-BA. The pathways for the CTP mechanisms in the alkyl acrylates

(MA2-D1-1, EA2-D1-1 and n-BA2-D1-1) were determined through intrinsic reaction coordinate (IRC) calculations (Figure 5-7).

Figure 5-7: Intrinsic Reaction Coordinate Path for MA2-D1-1, EA2-D1-1, and n-BA2- D1-1 Mechanism in the Alkyl Acrylates; The Energies are Relative to Reactants. 69

5.3.4. Effect of the Live Polymer Chain Length

The effects of the length of a live polymer chain on the activation energies and the geometries of transition states are explored for MA and EA. Hydrogen atom abstraction

• from a dead polymer chain by a live chain, which is initiated by M2 and has 3 or 4 monomer units, is investigated for MA (Figure 5-8).

MA2-D1-1(a)

CH3 O COOMe H + O CH3 . O MeOOC C 3CH COOMe O CH3 CH3 O COOMe . O C CH3 CH + O MeOOC 2 3CH COOMe O CH3

MA2-D1-1(b)

COOMe CH3 COOMe O

+ O CH3 . O MeOOC C H 3CH

O CH COOMe 3

COOMe CH COOMe 3 O . O C CH3 CH + O MeOOC 2 3CH O COOMe CH3

Figure 5-8: Most Probable CTP Mechanism Involving a Three (MA2-D1-1(a)) or Four ▪ (MA2-D1-1(b)) MA-Unit Live Chain Initiated by M2 .

Transition-state geometry for hydrogen atom abstraction by a MA live chain consisting of

3 or 4 monomer units has C1-H2 and H2-C3 bond lengths of 1.37 Å and 1.35 Å [Figure

5-9 (a)], and 1.39 Å and 1.36 Å [Figure 5-9(b)], respectively. The activation energies and 70

a b

c

Figure 5-9: Transition State Geometry for CTP Reaction in Methyl and Ethyl Acrylate; a: MA2-D1-1(a), b: MA2-D1-1(b), c: EA2-D1-1(a).

rate constants of these mechanisms are given in Table 5-8. They indicate that the rate constants change at most 2 orders of magnitude, as the length of the live chain changes.

The activation energies of these reactions nearly do not change (at most 4 kJ/mol) by increasing the length of the live chain.

The mechanism of CTP for EA (EA2-D1-1(a)) was explored as shown in Figure 5-10. In this mechanism the live chain has 3 monomer units. The transition-state structure with

C1-H2 and H2-C3 bond lengths of 1.37 Å and 1.36 Å can be observed in Figure 5-9(c).

The rate constants of CTP reactions of EA do not change, as the length of the live polymer chain increases (Table 5-8). Ab initio calculations for propagation reactions of 71

‡ Table 5-8: Bond Length in Å Activation Energy (Ea), Enthalpy (∆H ), and Free Energy ‡ -1 (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor -1 -1 (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M s by Considering the Radicals with Different Monomer Units for MA and EA, Using B3LYP/6-31G(d,p)

‡ ‡ (C1-H2) Å (C3-H2) Å Ea ∆H ∆G logeA k ߢw kw MA2-D1-1 Three Monomer Units 1.37 1.35 59 54 114 10.63 1.70E-6 3.76 6.39E-6 MA2-D1-1 Four Monomer Units 1.39 1.36 56 51 118 7.56 3.00E-7 3.86 1.16E-6 EA2-D1-1 Three Monomer Units 1.37 1.36 54 49 123 4.52 3.80E-8 3.79 1.44E-7

EA2-D1-1(a)

CH3 O COOEt CH O . H + 3 EtOOC C O CH3 COOEt O CH3

CH3 COOEt . + C COOEt EtOOC CH2 EtOOC COOEt CH3

Figure 5-10: Most Probable CTP Mechanism Involving a Three EA-Unit Live Chain ▪ Initiated by M2 .

MA and MMA [41] have shown the insensitivity of the propagation rate constants to the live chain length. This insensitivity has also been reported for homo-termination rate coefficients in free-radical polymerization of acrylates [129]. These findings indicate that it is appropriate to use a dimer (trimer) model system to study the chain transfer to 72 polymer reaction. These previous findings are in agreement with our results presented above.

5.3.5. Different Structures of the Dead Polymers

Dead polymers with different structures may be formed via different termination

• reactions (Figure 5-11). Two mono-radicals (M1 ) can undergo termination (by coupling) reaction to form a dead polymer (D1) [Figure 5-11(a)]. Chain transfer to D1 reaction has

• been discussed in sections 5.3.1-5.3.4. Mono-radicals M1 after one step propagation can be terminated by hydrogen abstraction [Figure 5-11(b)].

a) COOX COOX COOX . . C C CH3 H + 3CH 3CH H 3CH

COOX D1

b) COOX COOX . . C COOX + H 3CH CH COOX H 3 D2

c) COOX COOX COOX COOX . . C C COOX + CH CH H 3 3CH 3 CH H 3 D3 COOX

X=CH3,C2H5,C4H9

Figure 5-11: Different Structures of Dead Polymer; a) Termination by Coupling of Two Mono-radicals, b) Termination by Hydrogen Abstraction, c) Termination by Coupling of a Live Chain and Mono-radical. 73

These mono-radicals (including two monomer units) can also react with another mono- radical (including one monomer unit) to form D3 through termination reaction [Figure

5-11(c)]. Live polymer chains can also abstract a hydrogen atom from these dead polymers (D2 and D3). B3LYP/6-31G(d,p) functional was used to calculate the bond- dissociation energies of hydrogen atoms of dead polymers (D2 and D3) which can be abstracted by the live polymer chain (Figure 5-12). The bond energies of these hydrogen atoms are reported in Table 5-9.

Table 5-9: H-R Bond Dissociation Energies (kJ mol-1) for D2 and D3 Dead Polymers at 298 K, Using B3LYP/6-31G(d,p)

H-R Bond-dissociation energy Y-D2-1 Y-D2-2 Y-D2-3 Y-D3-1 Y-D3-2 Y-D3-3 Y-D3-4 363 417 387 357 364 366 405

These results indicate that those hydrogen atoms being abstracted via Y-D2-1 and Y-D3-

1 mechanisms are the most labile ones for abstraction. The most probable mechanisms of chain transfer to D2 and D3 dead polymers are shown for MA (Figure 5-13), EA (Figure

5-14), and n-BA (Figure 5-15). Table 5-10 gives the kinetic parameters of these mechanisms for MA, using B3LYP and M06-2X (6-31G(d,p), 6-311G(d), and 6-

311G(d,p)) functionals. A comparison of these results with those obtained for chain transfer to D1 dead polymer of MA, using B3LYP functional, indicates that the rate constant of the MA2-D2-1 mechanism is about 3 orders of magnitude higher than that calculated for MA2-D1-1, and its energy barrier is lower by about 6 kJ/mol. Although using M06-2X functional shows the same difference in the rate constants of MA2-D1-1 74

COOX

. + XOOC C H 3CH COOX

COOX COOX

. C COOX COOX 3CH XOOC + Y-D2-1

COOX COOX H

. C . COOX 3CH C COOX 3CH H Y-D2-2 Y-D2-3

COOX COOX

. XOOC C H + CH3 3CH

COOX COOX

COOX COOX

. C CH3 3CH COOX XOOC + COOX Y-D3-1

COOX COOX COOX COOX

. C CH . CH3 3 CH 3CH 3 C

COOX COOX Y-D3-2 Y-D3-3 COOX COOX

. CH3 3CH C

H COOX X=CH3,C2H5,C4H9 Y-D3-4

Y=MA,EA,n-BA

Figure 5-12: Possible Chain Transfer to D2 and D3 Dead Polymers Mechanisms for MA, EA, and n-BA. 75

MA2-D2-1 COOMe

. H MeOOC C + 3CH COOMe COOMe COOMe

. COOMe + C MeOOC 3CH COOMe

MA2-D3-1

MeOOC COOMe

H . CH3 MeOOC C + 3CH

COOMe COOMe MeOOC COOMe

. COOMe C CH MeOOC + 3 3CH

COOMe

Figure 5-13: Most Probable CTP Mechanism Involving a Two MA-Unit Live Chain ▪ Initiated by M2 .

and MA2-D2-1 (3 orders of magnitude), it results in a much lower difference in energy barriers. The geometries of transition-state structures of MA2-D2-1 and MA2-D3-1 mechanisms are shown in Figure 5-16. We applied B3LYP functional to explore these mechanisms for EA (EA2-D2-1 and EA2-D3-1) and n-BA (n-BA2-D2-1 and n-BA2-D3-

1). Figure 5-17 shows the transition state structures for EA2-D2-1 and EA2-D3-1. The activation energies and rate constants of the most probable chain transfer to D2 and D3 dead polymers of EA and n-BA are provide in Table 5-11. These results indicate that the activation energies for EA2-D2-1 and n-BA2-D2-1 mechanisms are lower (and the rate 76 constants are higher) than those calculated for EA2-D1-1, EA2-D3-1, n-BA2-D1-1, and n-BA2-D3-1 mechanisms.

EA2-D2-1 COOEt

. EtOOC C H + 3CH COOEt

COOEt COOEt

. COOEt + C EtOOC 3CH COOEt

EA2-D3-1 COOEt COOEt

. + CH3 EtOOC C H 3CH

COOEt COOEt COOEt COOEt

COOEt + . EtOOC C CH3 3CH

COOEt

Figure 5-14: Most Probable CTP Mechanism Involving a Two EA-Unit Live Chain ▪ Initiated by M2 .

These studies reveal the higher reactivity of D2 for undergoing CTP reactions (relative to

D1 and D3). It is concluded that the other tertiary carbon atom adjacent to the one whose hydrogen is being abstracted by the live polymer chain affects the reactivity of the structure for getting involved in CTP reaction.

5.3.6. Continuum Solvation Models: IEF-PCM and COSMO

The solvent effects on the kinetics of the most likely CTP mechanisms identified with gas-phase calculations are studied using two different solvation models, IEF-PCM and 77

n-BA2-D2-1 COOBu

. + BuOOC C H 3CH COOBu

COOBu COOBu

. COOBu + C BuOOC 3CH COOBu

n-BA2-D3-1 COOBuCOOBu

. + CH3 BuOOC C H 3CH

COOBu COOBu COOBuCOOBu

COOBu + . BuOOC C CH3 3CH

COOBu

Figure 5-15: Most Probable CTP Mechanism Involving a Two BA-Unit Live Chain ▪ Initiated by M2 .

COSMO, and in two types of solvents, n-butanol and p-xylene. As shown in Table 5-12, the application of IEF-PCM, using B3LYP and M06-2X functionals (6-31G(d,p) and 6-

311G(d,p) basis sets), shows strong solvent effects on the activation energy and rate constant of CTP reactions of the alkyl acrylates in n-butanol but weakly affects those in p-xylene, compared to gas-phase values. The IEF-PCM-calculated activation energies in n-butanol are higher than those obtained via gas-phase calculations, resulting in lower rate constants. IEF-PCM calculations are further carried out to study CTP mechanisms for EA and n-BA, using B3LYP/6-31G(d) and 6-31G(d,p) (Table 5-13). Again, the application of IEF-PCM shows strong solvent effects on the activation energies and rate 78

‡ Table 5-10: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free ‡ -1 Energy of Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1 s-1, for the MA2-D2-1 and MA2-D3-1 Mechanisms at 298 K; and Rate Constant without Tunneling at 413 K (k413)

B3LYP B3LYP B3LYP M06-2X M06-2X M06-2X 6-31G(d,p) 6-311(d) 6-311G(d,p) 6-31G(d,p) 6-311(d) 6-311G(d,p) Hydrogen Abstraction via MA2-D2-1 E a 49 56 53 27 30 28 ∆H‡ 44 51 48 22 25 23 ∆G‡ 92 97 95 67 72 71 log A e 15.31 15.74 15.73 16.38 15.65 15.53 k 1.07E-2 1.26E-3 4.07E-3 2.36E+2 3.69E+1 6.52E+1 ߢ w 3.44 3.68 3.66 2.98 3.11 2.97 kw 3.68E-2 4.63E-3 1.49E-2 7.03E+2 1.15E+2 1.94E+2 k 413 4.3E-1 5.3E-2 2.2E-1 5.9E+3 1.7E+3 2.1E+3 Hydrogen Abstraction via MA2-D3-1 E a 62 69 66 23 26 24 ∆H‡ 57 64 61 18 21 19 ∆G‡ 112 118 116 86 88 88 log A e 12.32 12.78 12.34 7.47 7.51 6.58 k 3.18E-6 3.34E-7 6.38E-7 1.47E-1 5.74E-2 6.18E-2 ߢ w 3.85 4.13 3.93 3.32 3.42 3.29 kw 1.22E-5 1.38E-6 2.51E-6 4.88E-1 1.96E-1 2.03E-1 k 413 2.3E-4 1.9E-5 3.2E-5 6.0E0 2.9E0 3.3E0

constants of CTP reactions in n-butanol, while its impact on the kinetic parameters of the reactions in p-xylene is negligible. Moreover, Table 5-12 and Table 5-13 suggest that the effects that IEF-PCM predicts do not depend on the end substituent group. The reduction in the rate of the CTP reactions in n-butanol agrees with the “inhibiting effect” of n- butanol reported by Liang et al., [130] who investigated the effect of n-butanol on the rate of intra-molecular chain transfer to polymer reactions. They reported that n-butanol 79

a b

Figure 5-16: Transition State Geometry for CTP Reaction; a: Methyl Acrylate (MA2-D2- 1), b: Methyl Acrylate (MA2-D3-1).

inhibits backbiting reactions and consequently reduces the rate of branching during the polymerization of n-BA and increases the average molecular weights of the polymer product. However, as shown in Table 5-12 and Table 5-13, the application of COSMO does not result in significant change of the (gas-phase-calculated) kinetic parameters of

CTP reactions in n-butanol and p-xylene. The insignificant effects of COSMO on CTP reactions are in agreement with previous theoretical results reported for CTS reactions of acrylates [131] and propagation reactions of acrylonitrile and vinyl chloride [45]. On the basis of these results, we suggest that the IEF-PCM is a more appropriate solvation model for studying the free-radical polymerization of acrylates than COSMO. 80

a b

Figure 5-17: Transition State Geometry for CTP Reaction; a: Ethyl Acrylate (EA2-D2-1), b: Ethyl Acrylate (EA2-D3-1).

‡ Table 5-11: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free ‡ -1 Energy of Activation (∆G ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1 s-1, for the Most Probable Mechanisms of Chain Transfer to D2 and D3 Dead Polymers of EA and n-BA at 298 K; and Rate Constant without Tunneling at 413 K (k413)

B3LYP B3LYP B3LYP 6-31G(d) 6-31G(d,p) 6-31G(d) EA2-D2-1 EA2-D3-1 EA2-D2-1 EA2-D3-1 n-BA2-D2-1 n-BA2-D3-1 Ea 52 63 49 60 49 60 ‡ ∆H 47 58 44 55 44 55 ‡ ∆G 98 111 95 108 102 115 logeA 13.21 12.53 14.19 13.56 11.19 10.5 k 4.19E-4 2.5E-6 3.49E-3 1.95E-5 1.86E-4 1.1E-6 ߢw 3.46 3.90 3.51 3.93 3.95 4.24 kw 1.45E-3 9.75E-6 1.22E-2 7.66E-5 7.35E-4 4.66E-6 k413 5.3E-2 1.1E-4 4.8E-1 3.1E-3 2.6E-2 6.3E-4 81

‡ Table 5-12: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; and Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Most Probable CTP Mechanisms for MA at 298 K, Calculated Using IEF-PCM and COSMO

B3LYP B3LYP B3LYP B3LYP B3LYP 311G(d,p) 311G(d,p) 311G(d,p) 311G(d,p) M06-2X M06-2X M06-2X M06-2X 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6- 6- 6- 6-

Hydrogen Abstraction via MA2-D2-1 Hydrogen Abstraction via MA2-D3-1

Ea 53 55 30 31 64 67 24 26 ∆H‡ 48 50 25 26 59 62 19 21 ∆G‡ 97 98 82 85 116 119 88 92 -xylene p

logeA 14.94 15.30 11.81 10.73 11.45 11.69 6.88 6.04 k 1.55E-3 1.01E-3 7.4E-1 1.68E-1 5.69E-7 2.16E-7 6.04E-2 1.16E-2

Hydrogen Abstraction via MA2-D2-1 Hydrogen Abstraction via MA2-D3-1 IEF-PCM IEF-PCM Ea 56 59 34 36 70 75 31 33 ∆H‡ 51 54 29 31 65 70 26 28 ∆G‡ 105 104 87 92 125 128 98 102 -butanol -butanol n logeA 13.08 14.21 11.09 9.88 10.49 11.09 5.44 4.84 k 6.8E-5 6.81E-5 7.2E-2 9.60E-3 1.93E-8 4.68E-9 8.50E-4 2.07E-4 Hydrogen Abstraction via MA2-D2-1 Hydrogen Abstraction via MA2-D3-1

Ea 49 52 24 25 57 60 23 25 ∆H‡ 44 47 19 20 52 55 18 20 ∆G‡ 94 99 78 79 109 113 87 88 -xylene p

logeA 14.61 13.62 11.12 10.61 11.81 11.21 6.76 7.12 k 5.37E-3 6.29E-4 3.34 1.68 1.38E-5 2.25E-6 7.2E-2 5.1E-2 Hydrogen Abstraction via MA2-D2-1 Hydrogen Abstraction via MA2-D3-1 COSMO

Ea 50 51 24 25 56 59 23 24 ∆H‡ 45 46 19 20 51 54 18 19 ∆G‡ 95 97 78 79 107 111 87 86 -butanol -butanol n logeA 14.62 13.86 11.12 10.13 11.93 11.57 6.76 7.72 k 5.4E-3 1.2E-3 3.34 1.56 2.32E-5 4.8E-6 7.3E-2 1.4E-1 82

‡ Table 5-13: Activation Energy (Ea), Enthalpy of Activation (∆H ), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; and Frequency Factor (A) and Rate Constant (k) in M-1 s-1, for the Most Probable CTP Mechanisms for EA and n-BA at 298 K, Calculated Using COSMO and IEF-PCM

B3LYP B3LYP B3LYP 6-31G(d) 6-31G(d,p) 6-31G(d)

EA2-D2-1 EA2-D3-1 EA2-D2-1 EA2-D3-1 BA2-D2-1 BA2-D3-1

Ea 50 62 49 61 48 57 ∆H‡ 45 57 44 56 43 52 ∆G‡ 98 112 96 109 100 111 COSMO COSMO logeA 13.13 12.65 13.50 13.38 11.81 10.73 k 8.7E-4 4.25E-6 1.87E-3 1.31E-5 5.20E-4 4.66E-6 EA2-D2-1 EA2-D3-1 EA2-D2-1 EA2-D3-1 BA2-D2-1 BA2-D3-1

Ea 60 70 56 69 57 65 ∆H‡ 55 65 52 64 52 60 ∆G‡ 113 125 111 125 115 125 IEF-PCM logeA 11.09 10.61 10.73 10.01 9.17 8.32 k 1.99E-6 2.18E-8 4.66E-6 1.79E-8 9.76E-7 1.67E-8

5.4. Concluding Remarks

The mechanisms of intermolecular CTP reactions in self-initiated high-temperature polymerization of alkyl acrylates were studied theoretically. The abstraction of a hydrogen atom from a tertiary carbon atom was found to be the most favorable CTP

• mechanism in alkyl acrylates. This study indicated that the mono-radical M2 is as

• reactive as M1 in CTP reactions of methyl acrylate. Four different basis sets (6-31G(d),

6-31G(d,p), 6-311G(d), and 6-311G(d,p)) were applied to validate the calculated transition states and energy barriers. These basis sets predicted similar transition state geometries for the CTP mechanisms, activation energies with at most 10 kJ/mol difference, and rate constants with at most 2 orders of magnitude difference. The end 83 substituent groups of the monomers were found to have little effect on the energy barriers of the CTP reactions. The study indicated that tertiary hydrogens of dead polymers formed by disproportionation reactions are most likely to be transferred to live polymer chains in CTP reactions. The levels of theory applied in this study are accurate enough to predict the mechanistic pathways and transition state structures, but further investigation with higher levels of theory is recommended. While the application of IEF-PCM showed strong solvent effects on the kinetic parameters of the CTP reactions of MA, EA, and n-

BA in n-butanol, the application of COSMO showed no such remarkable effects. 84

6. Chain Transfer to Solvent Reactions of Alkyl Acrylates

6.1. Introduction

Occurrence of secondary reactions, such as β-scission, CTS and radical transfer to solvent from initiator radical, in high-temperature polymerization of n-butyl acrylate (n-BA) have been identified using liquid chromatography-electrospray ionization-tandem mass spectrometry (LC-ESI-MS) [132]. In thermal polymerization of ethyl acrylate (EA), methyl acrylate (MA), and ethyl methacrylate (EMA), transfer constants for various solvents, such as hydrocarbons, alcohols, ketones, acids, and esters, have been reported at

80oC via polymer sample measurements [133, 134]. Moreover, the effect of solvent in homo-polymerization of n-BA has been investigated [135]. These experimental studies

0.5 estimated lumped rate constants (kp/kt ), which showed that as the solvent concentration increases, the rate of CTS reactions and the rate of formation of shorter chains increase and subsequently these shorter chains undergo termination reactions faster than longer chains. While the influence of different solvents concentration on overall rate of polymerization (Rp) has been reported [135], investigation of solvent effects on the individual rate constants (kp and kt) is challenging. Application of chain transfer agents

(CTAs) during controlled radical polymerization processes, such as nitroxide-mediated polymerization (NMP), atom transfer radical polymerization (ATRP) and reversible addition-fragmentation chain transfer (RAFT) has been reported [85-90]. CTAs control the growth of propagating chains and lead to the formation of uniform chain-length polymers. However, in thermal polymerization of alkyl acrylates, in the absence of these agents, self-regulation and polymers with uniform chain lengths have been observed [91].

These observations can be attributed to the self-regulatory capability of chain transfer 85 mechanisms. Therefore, a fundamental understanding of these mechanisms becomes important to control, design, and optimize thermal polymerization processes.

Solvents increase the stability of transition state geometries in solution phase polymerization [39]. Different solvent continuum models have been applied to explore the solvent effects on the solute [36, 55]. In the continuum model, solvent is treated as a dielectric continuum mean field polarized by the solute which is placed in this continuum in an inner cavity. While the self-consistent reaction field (SCRF) method defines the solute in a spherical cavity [136], polarizable continuum model (PCM) introduces molecular shape for the cavity [137, 138]. PCM has been applied to predict the propagation rate coefficient of acrylic acid in the presence of toluene [55]. However, microscopic structure of the solvent-solute interaction cannot be described through these models. Conductor-like screening model (COSMO), originally developed by Klamt and

Shuurmann [139], is another approach for polarized continuum calculations in which the surrounding medium (solvent) is assumed as a conductor rather than a dielectric in order to simplify the electrostatic interactions between solvent and solute. The effects of solvents (having different dielectric constants) on the initiation and propagation reactions in polymerization of ethylene have been explored using COSMO [140]. Also, COSMO has been applied to predict non-equilibrium solvation energies of biphenyl-cyclohexane- naphthalene [141]. In this chapter we compare the performance of PCM and COSMO.

Different CTM mechanisms for MA, EA, and n-BA have been explored using quantum chemical calculations previously [48]. Chain transfer to dead-polymers (formed via several termination reactions) mechanisms have also been studied [142]. 86

To the best of our knowledge, as of yet, computational investigation of the most likely mechanisms of CTS in homo-polymerization of MA, EA and n-BA has not been reported. Although the existence of CTS reactions has been known for many decades [80,

92], prior to this study, no individual CTS reaction mechanisms have been reported.

6.2. Computational Methods

This section presents a computational/theoretical study of CTS reactions of MA, EA and n-BA polymerization in butanol (polar, protic), methyl ethyl ketone (MEK) (polar, aprotic), and p-xylene (nonpolar). Three types of hybrid functionals (B3LYP, X3LYP, and M06-2X) and three different basis sets (6-31G(d,p), 6-311G(d), and 6-311G(d,p)) are applied to predict energy barriers, and molecular geometries of reactants, products and transition states. We have explored the abstraction of hydrogen from n-butanol, MEK, and p-xylene by a live polymer chain to identify the most likely mechanisms of CTS reactions in MA, EA, and n-BA polymerization. The activation energy and rate constants of CTS mechanisms have been computed using transition state theory. Quantum tunneling is considered in CTP reactions [125-128]. The Wigner tunneling [128] correction is calculated in this chapter using:

ଵ ௛ఔș ߢൎͳ൅ ሺ ሻଶ (6.1) ଶସ ௞ಳ் PCM and COSMO solvation models are applied to explore chain transfer to n-butanol,

MEK and p-xylene from polymer chains of MA, EA and n-BA. The effect of self- initiating monoradicals on CTS is also investigated. All calculations are performed using

GAMESS [67]. PCM and COSMO, which are implemented in GAMESS, are applied for treatment of solvent effects. These two continuum solvation methods characterize the solvent molecule by physical properties such as dielectric constant and solvent radius. 87

6.3. Results and Discussion

6.3.1. Most Likely CTS Mechanisms for MA, EA, and n-BA

Different mechanisms of CTS reactions for n-butanol, MEK, and p-xylene are shown in

Figure 6-1. The hydrogen atom is abstracted via various mechanisms (Figure 6-1). The bond dissociation energies of these hydrogen atoms are given in Table 6-1.

H COOX XOOC C OH XOOC + 3CH COOX H + C OH CH O 3CH 3 CTB1-2 CTB2-2 H

C OH OH CH OH 2CH 3CH C 3

CTB3-2 H CTB4-2 CTB5-2

O H COOX XOOC XOOC C + CH3 3CH COOX O O + O CH 3 CH CH2 C 3 3CH 3CH 2CH H CTM1-2 CTM2-2 CTM3-2

CH3

H XOOC C COOX + XOOC COOX CH CH2 + 3 CH3 C

CTX1-2 CTX2-2 X=CH , C H , C H 3 2 5 4 9 CH CH3 3

Figure 6-1: Possible End-chain Transfer to Solvent Reactions Involving a Two- ▪ Monomer-Unit Live Chain Initiated by M2 . CTB = Chain Transfer to n-Butanol, CTM = Chain Transfer to Methyl Ethyl Ketone, CTX = Chain Transfer to p-Xylene. 88

Table 6-1: H-R Bond Dissociation Energies (kJ mol-1) at 298 K

B3LYP B3LYP B3LYP M06-2X M06-2X M06-2X 6-31G(d,p) 6-311(d) 6-311(d,p) 6-31G(d,p) 6-311(d) 6-311(d,p) CTB1-2 405 401 403 414 411 412 CTB2-2 428 416 434 442 430 439 CTB3-2 421 417 419 430 425 427 CTB4-2 420 417 419 430 426 427 CTB5-2 437 434 435 442 438 448 CTM1-2 389 387 390 401 398 400 CTM2-2 413 410 413 421 419 420 CTM3-2 441 437 439 451 447 447 CTX1-2 389 384 387 403 400 401 CTX2-2 467 472 476 480 476 479

Two functionals, B3LYP and M06-2X, and three basis sets, 6-31G(d,p), 6-311G(d) and

6-311G(d,p), are used to calculate the bond-dissociation energies. According to Table

6-1, C-H breaking bonds in CTB1-2, CTM1-2, and CTX1-2 mechanisms are weaker than those in other mechanisms. Cleavage of methylene group hydrogen bonds forms radicals which are more stable than those formed through methyl group hydrogen bonds cleavage.

It can confirm the ease of hydrogen abstraction from methylene group rather than methyl group in n-butanol. Mulliken charge analysis also shows that the methylene carbon atom

(0.047) next to the oxygen atom is more positive than the oxygen atom (-0.573), and consequently, more likely to release a hydrogen atom. The same conclusion can also be obtained for MEK via comparing the Mulliken charge of methylene carbon atom (-0.309) and two other methyl carbon atoms (-0.351 and -0.435). However, due to the presence of delocalised molecular orbitals (stable ring) in p-xylene abstraction of methyl group hydrogen is more favorable. We apply three functionals (B3LYP, X3LYP, and M06-2X) 89 and several basis sets (6-31G(d,p), 6-311G(d) and 6-311G(d,p)) to estimate the thermodynamic and kinetic parameters (activation energies, enthalpies of reaction, Gibbs free energies, frequency factors, and rate constants) of the most likely mechanisms of

CTS reactions of MA, EA, and n-BA. Table 6-2 presents the kinetic parameters of the most likely mechanisms of chain transfer to n-butanol, MEK and p-xylene for MA. The transition-state geometries for CTB1-2, CTM1-2, and CTX1-2 are shown in Figure 6-2,

Figure 6-3, and Figure 6-4, respectively. It was found that the activation energy of chain transfer to n-butanol is lower than that of MEK and p-xylene reactions (Table 6-2). The rate constant for chain transfer to n-butanol is higher than that of MEK and p-xylene. We attribute this to the polar and protic nature of n-butanol , which can readily donate a hydrogen to the polymer chain to facilitate chain transfer. p-xylene and MEK lack a labile hydrogen atom to perform transfer. It was determined that M06-2X, a hybrid meta-

GGA functional, provides lower activation energies and higher rate constants than

B3LYP and X3LYP. This agrees with the findings reported for CTM reactions of alkyl acrylates [48]. The estimated kinetic parameters for chain transfer to n-butanol, MEK, and p-xylene for EA and n-BA are given in Table 6-3 and Table 6-4, respectively. The similarities in the predicted results for MA, EA, and n-BA show a little effect of the end substituent groups of the live chains. The activation energy of chain transfer to p-xylene estimated using M06-2X/6-31G(d,p) functional is in agreement with those reported through polymer sample measurements in high-temperature n-BA polymerization process

[9]. 90

‡ Table 6-2: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy ‡ -1 of Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1 s-1, for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of MA at 298 K ) ) ) ) ) ) ) ) ) d d d d,p d,p d,p d,p d,p d,p 2X 2X 2X 2X B3LYP B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP -31G( 6-311( -31G( 6-311( M06- -31G( M06- 6-311( M06- 6-311( 6-311( 6-311( 6 6 6

CTB1-2 E a 45.50 52.70 50.50 42.30 51.00 48.10 22.90 27.10 25.20 ∆H‡ 40.60 47.70 44.60 37.10 45.70 43.30 17.90 22.10 20.10 ∆G‡ 95.10 102.60 100.00 90.70 100.20 96.60 78.40 79.60 80.00 log A e 12.65 12.52 12.29 13.01 12.65 13.14 10.25 11.47 10.49 k 3.30E-03 1.60E-04 3.07E-04 1.73E-02 3.60E-04 1.90E-03 2.78E+00 1.72E+00 1.38E+00 ߢ w 3.27 3.50 3.49 3.28 3.48 3.49 3.41 3.50 3.40 k w 1.07E-02 5.60E-04 1.07E-03 5.67E-02 1.25E-03 6.63E-03 9.48E+00 6.02E+00 4.70E+00 CTM1-2 E a 68.20 75.30 72.00 63.30 71.20 69.10 50.30 52.10 51.50 ∆H‡ 63.30 70.30 67.20 57.90 66.20 64.20 45.40 47.10 46.20 ∆G‡ 107.40 114.00 110.40 101.70 109.10 106.50 100.50 107.80 101.90 log A e 16.85 17.15 17.23 16.98 17.35 17.59 12.40 10.17 12.17 k 2.26E-05 1.80E-06 7.24E-06 1.90E-04 1.13E-05 3.35E-05 3.60E-04 1.90E-05 1.80E-04 ߢ w 3.61 3.83 3.78 3.63 3.81 3.79 3.17 3.27 3.15 kw 8.15E-05 6.89E-06 2.73E-05 6.89E-04 4.30E-05 1.27E-04 1.14E-03 6.21E-05 5.67E-04 CTX1-2 E a 65.90 71.60 70.50 63.40 70.20 68.30 54.70 56.00 55.20 ∆H‡ 61.00 66.60 65.00 58.20 65.50 62.70 49.70 51.00 50.50 ∆G‡ 107.50 113.00 111.80 104.40 111.40 107.70 100.60 107.40 102.40 log A e 15.89 16.04 15.78 16.02 16.14 16.50 14.14 11.90 13.74 k 2.19E-05 2.60E-06 3.13E-06 7.00E-05 5.07E-06 1.57E-05 3.60E-04 2.30E-05 1.95E-04 ߢ w 3.57 3.77 3.76 3.58 3.75 3.77 3.32 3.42 3.33 k w 7.81E-05 9.80E-06 1.18E-05 2.50E-04 1.90E-05 5.90E-05 1.19E-03 7.86E-05 6.49E-04 91

Figure 6-2: Transition State Geometry of the CTB1-2 Mechanisms for (a) MA, (b) EA, and (c) n-BA. 92

Figure 6-3: Transition State Geometry of the CTM1-2 Mechanisms for (a) MA, (b) EA, and (c) n-BA. 93

Figure 6-4: Transition State Geometry of the CTX1-2 Mechanisms for (a) MA, (b) EA, and (c) n-BA.

6.3.2. Chain Transfer to n-Butanol, sec-Butanol, and tert-Butanol

Different mechanisms of chain transfer to n-butanol, sec-butanol, and tert-butanol are

shown in )LJXUH$Oive polymer chain can abstract a hydrogen atom from several

locations in these solvents. We calculated bond-dissociation energies of all available

hydrogen atoms in these VROYHQWV )LJXUH XVing two distinct functionals (B3LYP and

M06-2X) and three basis sets (6-31G(d,p), 6-311G(d,p) and 6-311G(d)), as shown in

7DEOH  7KH Uesults indicate that the weakest C-H bonds are those broken in the

CTBsec1-2 and CTBtert1-2 mechanisms. It agrees with the higher capability of 94 methylene carbon atoms for releasing hydrogen in comparison to methyl carbon atoms and oxygen.

‡ Table 6-3: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy ‡ -1 of Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1s-1, for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of EA at 298 K ) ) ) ) ) ) ) ) ) d d d d,p d,p d,p d,p d,p d,p -2X B3LYP B3LYP B3LYP X3LYP X3LYP X3LYP 6-311( 6-311( M06-2X M06-2X 6-311( 6-311( 6-311( 6-31G( 6-31G( 6-31G( -311( M06 6

CTB1-2 E a 45.60 53.30 50.30 40.60 49.00 48.40 19.40 23.80 21.20 ∆H‡ 40.60 48.30 44.70 35.70 43.60 43.40 14.50 18.90 16.40 ∆G‡ 94.30 102.00 98.30 88.70 96.90 95.60 74.20 79.10 76.90 log A e 13.00 13.08 13.02 13.26 13.14 13.62 10.56 10.36 10.25 k 4.50E-03 2.20E-04 6.90E-04 4.40E-02 1.30E-03 2.70E-03 1.53E+01 2.11E+00 5.43E+00 ߢ w 3.29 3.51 3.48 3.28 3.52 3.51 3.49 3.59 3.49 k w 1.48E-02 7.72E-04 2.40E-03 1.44E-01 4.57E-03 9.47E-03 5.34E+01 7.57E+00 1.89E+01 CTM1-2 E a 68.20 75.60 72.30 61.20 70.20 68.50 45.50 49.90 48.30 ∆H‡ 63.30 70.60 66.90 56.50 65.30 62.70 40.60 44.90 42.80 ∆G‡ 105.10 111.30 109.20 98.20 107.90 104.40 98.10 103.90 100.90 log A e 17.80 18.24 17.59 17.83 17.47 17.83 11.45 10.85 11.21 k 5.90E-05 4.70E-06 9.20E-06 1.03E-03 1.90E-05 5.40E-05 9.90E-04 9.30E-05 2.50E-04 ߢ w 3.60 3.82 3.80 3.61 3.85 3.81 3.17 3.27 3.20 kw 2.12E-04 1.79E-05 3.49E-05 3.71E-03 7.30E-05 2.06E-04 3.13E-03 3.04E-04 8.00E-04 CTX1-2 E a 66.10 72.90 70.30 62.00 69.30 67.50 53.20 55.50 51.10 ∆H‡ 61.20 68.00 64.80 57.40 64.10 62.20 48.30 50.60 45.90 ∆G‡ 103.30 113.00 106.50 100.30 105.80 103.30 102.00 104.10 102.20 log A e 17.67 16.51 17.83 17.35 17.83 18.06 13.02 12.89 11.93 k 1.20E-04 2.40E-06 2.63E-05 4.60E-04 3.93E-05 1.03E-04 2.10E-04 7.40E-05 1.68E-04 ߢ w 3.59 3.79 3.84 3.57 3.82 3.83 3.30 3.40 3.28 kw 4.30E-04 9.09E-06 1.00E-04 1.64E-03 1.50E-04 3.94E-04 6.93E-04 2.52E-04 5.50E-04 95

‡ Table 6-4: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy ‡ -1 of Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1s-1, for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of n-BA at 298 K ) ) ) ) ) - ) ) ) ) d d d d,p d,p d,p d,p d,p d,p B3LYP B3LYP X3LYP X3LYP X3LYP 311( 6-311( M06-2X M06-2X 6-311( M06-2X 6-311( B3LYP6 6-311( 6-311( 6-31G( 6-31G( 6-31G(

CTB1-2 E a 45.10 53.30 52.40 40.30 46.60 46.10 20.40 21.00 20.30 ∆H‡ 40.20 48.20 46.60 34.70 41.80 41.40 15.40 16.50 15.00 ∆G‡ 94.10 101.50 101.10 86.60 95.40 94.40 76.50 77.00 76.70 log A e 12.89 13.14 12.65 13.74 13.02 13.26 10.00 10.25 9.77 k 5.00E-03 2.30E-04 2.10E-04 8.00E-02 3.00E-03 4.70E-03 5.90E+00 5.90E+00 4.83E+00 ߢ w 3.28 3.52 3.34 3.32 3.54 3.36 3.50 3.60 3.49 kw 1.64E-02 8.09E-04 7.01E-04 2.65E-01 1.06E-02 1.58E-02 2.06E+01 2.12E+01 1.68E+01 CTM1-2 E a 63.10 69.20 69.60 60.00 67.30 66.60 41.00 45.50 43.10 ∆H‡ 58.20 63.80 65.20 55.50 62.00 61.90 36.10 40.40 38.50 ∆G‡ 99.90 106.40 106.60 98.40 103.40 103.60 95.10 98.50 97.20 log A e 17.83 17.47 17.95 17.35 17.95 17.83 10.85 11.21 10.97 k 4.80E-04 2.85E-05 3.93E-05 1.04E-03 9.94E-05 1.20E-04 3.40E-03 7.80E-04 1.60E-03 ߢ w 3.59 3.82 3.62 3.58 3.86 3.66 3.17 3.29 3.19 kw 1.72E-03 1.09E-04 1.42E-04 3.72E-03 3.83E-04 4.39E-04 1.07E-02 2.56E-03 5.10E-03 CTX1-2 E a 64.30 67.50 70.40 58.50 65.20 65.20 54.70 57.00 59.10 ∆H‡ 58.90 61.70 65.30 53.50 60.50 60.30 49.70 51.60 53.60 ∆G‡ 99.10 104.90 107.90 94.90 103.40 102.30 100.10 104.90 107.20 log A e 18.43 17.23 17.47 17.95 17.35 17.71 14.34 13.14 13.02 k 5.40E-04 4.45E-05 1.76E-05 3.40E-03 1.30E-04 1.80E-04 4.40E-04 5.17E-05 1.96E-05 ߢ w 3.58 3.78 3.59 3.62 3.82 3.63 3.34 3.42 3.35 kw 1.93E-03 1.68E-04 6.31E-05 1.23E-02 4.96E-04 6.53E-04 1.47E-03 1.76E-04 6.57E-05

Based on the Mulliken charge analysis, methylene carbon atom next to the oxygen

(0.152), in sec-butanol is more reactive for undergoing hydrogen transfer than the other 96 methylene carbon atom (-0.228). Although the capability of oxygen for donating hydrogen is lower than methylene carbon atom, due to the symmetrical structure of tert- butanol, hydrogen abstraction from oxygen atom is more favorable than that from methyl carbon atoms.

OH H COOMe MeOOC C + MeOOC 3CH CH3 COOMe + OH O OH CH CH C 3CH 3 3 CH C CH CH3 3 3 H

CTBsec1-2 CTBsec2-2 CTBsec3-2

H OH MeOOC C COOMe + MeOOC CH CH COOMe 3 3 CH + 3 OH O

3CH CH2 3CH CH3 3CH 3CH CTBtert1-2 CTBtert2-2

Figure 6-5: Possible End-chain Transfer to sec-Butanol and tert-Butanol Reactions for ▪ MA Involving a Two-Monomer-Unit Live Chain Initiated by M2 .

The calculated kinetic parameters of these mechanisms for MA are given in Table 6-6, which indicates that tert-butanol has the lowest chain-transfer rate constant, among n- butanol, sec-butanol, and tert-butanol. These findings are in agreement with previous studies for chain transfer to tert-butanol in MA polymerization at 80oC [133]. 97

Table 6-5: H-R Bond Dissociation Energies (kJ mol-1) at 298 K ) ) ) ) ) ) d d d,p d,p d,p d,p 2X 2X 2X 2X B3LYP B3LYP B3LYP 6-311( 6-311( -31G( M06- -31G( M06- M06- 6-311( 6-311( 6 6

CTBsec1-2 397 399 398 408 404 406 CTBsec2-2 434 437 435 450 447 449 CTBsec3-2 421 423 420 434 433 434 CTBtert1-2 434 436 435 450 448 446 CTBtert2-2 445 448 446 452 447 446

A comparison of experimentally-estimated [133] and theoretically-estimated values of chain transfer to n-butanol, sec-butanol and tert-butanol rate constants in MA polymerization at 80oC, given in Table 6-6, indicates that: (a) the M06-2X-estimated values are closer to the experimentally-estimated ones; (b) the M06-2X-estimated values of chain transfer to n-butanol and sec-butanol rate constants are very close to the experimentally-estimated ones, and (c) the values of chain transfer to tert-butanol rate constant estimated by M06-2X and B3LYP are, respectively, approximately four and six orders of magnitude smaller than the experimentally-estimated value.

6.3.3. Continuum Solvation Models: PCM and COSMO

The kinetics of CTS reactions are explored using two different solvation models, PCM and COSMO. Two functionals (B3LYP and M06-2X) and two basis sets (6-31G(d,p) and

6-311G(d,p)) are applied to predict the kinetic parameters of the most likely CTS reaction mechanisms (CTB1-2, CTM1-2, and CTX1-2) using the solvation models. n-Butanol,

MEK, and p-xylene are considered as the solvent molecules in PCM to explore CTB1-2, 98

CTM1-2, and CTX1-2, respectively. As shown in Table 6-7, the use of PCM strongly affects the activation energy and rate constant of chain transfer to n-butanol but weakly affects those of chain transfer to MEK and p-xylene. We found that the PCM-calculated activation energy for n-butanol is higher than those obtained via gas phase calculations, but the PCM-calculated rate constant for n-butanol is lower. Table 6-7 also indicates that the relative stability of the reactants to the transition states increases when PCM is used for chain transfer to n-butanol. p-Xylene is nonpolar and applying PCM does not significantly affect the stability of reactants and also transition state in chain transfer to p- xylene reaction. n-Butanol and MEK are both polar solvents, thereby, the impact of using

PCM on the stability of reactants and transition states is more noticeably for chain transfer to n-butanol and MEK in comparison to that for chain transfer to p-xylene. Since oxygen site is much more polarizable than hydrogen site, applying PCM makes CTM1-2 transition state more stabilized than CTB1-2 transition state. That is, the change in the stability of CTM1-2 transition state can compensate for that of reactants in chain transfer to MEK reaction. PCM calculations are also carried out to study CTB1-2, CTM1-2, and

CTX1-2 mechanisms for EA and n-BA. Although PCM has a strong effect on the activation energies and rate constants of chain transfer to n-butanol reactions, as Table

6-7 shows, its impacts on the kinetic parameters of chain transfer to MEK and p-xylene are negligible. Moreover, Table 6-7 shows that using PCM does not reveal any considerable effect caused by the type of the end substituent group.

The solvation model COSMO is applied to investigate solvent effects on the rates and barriers of CTB1-2, CTM1-2, and CTX1-2 reactions for MA, EA, and n-BA. As Table

6-8 shows, unlike PCM, COSMO does not affect appreciably the relative stability of the 99 reactants to the transition states. These results indicate that COSMO is not able to represent the effects of solvent molecules on the kinetic parameters of CTS reactions.

‡ Table 6-6: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy ‡ -1 of Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1s-1, for CTB1-2, CTBsec1-2, and CTBtert1-2 Mechanisms of MA at 298 K

n-Butanol sec-Butanol tert-Butanol

Ea 45.50 45.30 79.50 ∆H‡

) 40.60 40.30 74.00 ‡

d,p ∆G 95.10 93.90 127.00 logeA 12.65 13.02 13.26 B3LYP

6-31G( k 3.30E-03 5.20E-03 6.63E-09 ߢw 3.27 3.25 3.15 kw 1.07E-02 1.69E-02 2.10E-08 n-Butanol sec-Butanol tert-Butanol

Ea 22.90 24.10 64.30 ∆H‡

) 17.90 19.20 59.40 ‡

d,p ∆G 78.40 77.20 109.70 logeA 10.25 11.27 14.34 M06-2X

6-31G( k 2.78E+00 4.60E+00 9.00E-06 ߢw 3.41 3.37 3.28 kw 9.48E+00 1.55E+01 2.95E-05 n-Butanol sec-Butanol tert-Butanol Experimental k (353 K) 1.10E+01 5.50E+01 1.50E+00 [133] B3LYP k (353 K) 5.76E-02 8.93E-02 9.87E-07 6-31G(d,p) M06-2X k (353 K) 1.16E+01 2.10E+01 5.20E-04 6-31G(d,p)

In COSMO, non-electrostatic solute-solvent interactions, such as dispersion, repulsion and cavitation are considered as well as electrostatic interactions. Since the contribution of non-electrostatic interactions in COSMO are included in a simple form, large errors for 100

‡ ‡ -1 Table 6-7: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG ) in kJ mol ; Frequency Factor (A) and Rate Constant (k) in M-1s-1, for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of MA, EA, and n-BA at 298 K, Using PCM ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p B3LYP B3LYP B3LYP B3LYP B3LYP M06-2X M06-2X M06-2X M06-2X M06-2X M06-2X M06-2X B3LYP 6-311( 6-311( 6-311( 6-311( 6-311( 6-31G( 6-31G( 6-31G( 6-31G( 6-31G( 6-31G( 6-311(

MA CTB1-2 CTM1-2 CTX1-2 Ea 61.20 65.30 38.10 35.20 68.30 74.70 48.40 45.20 69.50 73.00 56.50 57.10 ∆H‡ 56.40 60.10 33.20 29.50 63.00 70.10 43.40 40.10 65.00 67.60 51.50 52.00 ∆G‡ 102.10 105.00 84.70 81.10 111.10 119.20 103.50 101.00 114.30 118.40 105.60 103.10 logeA 16.02 16.62 13.86 13.98 15.42 14.82 10.60 10.00 15.06 14.58 13.02 13.86 k 1.70E-04 5.91E-05 2.20E-01 8.00E-01 5.30E-06 2.20E-07 1.31E-04 2.60E-04 2.28E-06 3.43E-07 5.63E-05 1.02E-04

EA CTB1-2 CTM1-2 CTX1-2 Ea 60.10 72.50 34.40 30.10 66.20 69.30 46.30 45.00 72.20 70.00 59.50 55.30 ∆H‡ 54.50 67.20 28.60 25.00 61.40 64.20 40.70 39.50 66.50 65.10 54.20 50.10 ∆G‡ 100.00 112.00 80.20 78.00 114.40 114.20 98.40 98.10 115.00 113.00 105.60 100.40 logeA 16.62 16.38 14.21 13.37 13.01 14.22 11.57 11.21 15.42 15.30 13.62 14.57 k 4.80E-04 2.54E-06 1.38E+00 3.39E+00 1.11E-06 1.07E-06 8.11E-04 9.60E-04 1.10E-06 2.37E-06 3.10E-05 4.30E-04

BA CTB1-2 CTM1-2 CTX1-2 Ea 59.90 64.50 35.50 30.00 64.40 62.60 45.40 47.00 64.50 70.30 57.20 54.60 ∆H‡ 55.10 59.60 30.40 25.20 59.40 58.30 40.40 42.10 60.40 64.50 51.60 50.00 ∆G‡ 101.00 107.10 82.60 77.10 111.00 113.10 98.30 100.00 110.10 113.70 106.00 102.30 logeA 16.02 15.78 13.26 13.50 13.74 12.29 11.21 11.21 14.46 14.82 13.01 13.62 k 2.90E-04 3.52E-05 3.43E-01 4.02E+00 4.77E-06 2.31E-06 8.13E-04 4.30E-04 9.41E-06 1.30E-06 4.20E-05 2.20E-04 101

‡ ‡ -1 Table 6-8: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG ) in kJ mol ; Frequency Factor (A) and Rate Constant (k) in M-1s-1, for CTB1-2, CTM1-2, and CTX1-2 Mechanisms of MA, EA, and n-BA at 298 K, Using COSMO ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p ) d,p B3LYP B3LYP B3LYP B3LYP B3LYP B3LYP M06-2X M06-2X M06-2X M06-2X M06-2X M06-2X M06-2X 6-311( 6-311( 6-311( 6-311( 6-311( 6-311( 6-31G( 6-31G( 6-31G( 6-31G( 6-31G( 6-31G(

MA CTB1-2 CTM1-2 CTX1-2 Ea 44.10 50.60 23.50 24.00 67.40 72.20 51.00 48.10 64.00 69.10 51.60 56.50 ∆H‡ 39.30 46.10 18.40 19.20 62.20 67.20 46.20 43.20 58.50 64.00 46.60 51.60 ∆G‡ 94.00 100.30 78.50 79.50 105.00 108.50 96.60 94.20 105.40 109.10 96.30 101.00 logeA 12.30 12.77 10.49 10.00 17.22 17.83 14.22 13.98 16.26 16.38 14.82 15.06 k 4.10E-03 4.80E-04 2.73E+00 1.37E+00 4.60E-05 1.22E-05 1.72E-03 4.40E-03 6.97E-05 1.00E-05 2.46E-03 4.30E-04

EA CTB1-2 CTM1-2 CTX1-2 Ea 45.10 51.30 20.30 22.10 69.00 70.40 47.10 45.10 68.00 69.30 51.30 52.00 ∆H‡ 40.00 46.40 15.30 17.30 64.20 65.00 42.20 40.30 63.20 64.20 46.50 46.50 ∆G‡ 92.10 97.60 73.50 76.50 105.10 104.40 97.10 94.50 103.30 102.10 94.10 94.60 logeA 13.62 13.50 11.09 10.25 18.18 18.79 12.41 12.65 18.55 19.15 15.42 15.18 k 1.02E-02 7.40E-04 1.81E+01 3.78E+00 6.32E-05 6.61E-05 1.36E-03 3.90E-03 1.37E-04 1.48E-04 5.10E-03 3.00E-03

BA CTB1-2 CTM1-2 CTX1-2 Ea 44.40 51.00 18.00 20.30 61.10 69.30 39.10 42.30 63.50 68.10 54.20 57.40 ∆H‡ 39.10 46.10 13.50 15.10 56.10 64.20 34.20 37.10 59.20 63.00 49.10 52.30 ∆G‡ 92.60 99.30 75.30 76.20 98.00 104.40 88.40 91.00 100.60 103.10 98.50 99.60 logeA 12.89 13.14 9.53 10.00 17.83 18.43 13.01 12.77 17.83 18.43 14.82 15.42 k 6.50E-03 5.85E-04 9.63E+00 6.09E+00 1.08E-03 7.18E-05 6.26E-02 1.35E-02 4.10E-04 1.17E-04 8.62E-04 4.30E-04 102 molecules with specific interactions, such as hydrogen bonding, can be obtained. The insignificant effect of COSMO on CTS reactions is in agreement with that reported for propagation reactions of acrylonitrile and vinyl chloride previously [45].

6.3.4. Effect of the Type of Initiating Radical on CTS

Two types of mono-radicals generated via monomer self-initiation [10, 11] are shown in

)LJXUH,Q the previous sections, the most likely CTS reaction mechanisms of a 2-

▪ monomer-unit live polymer chain initiated via M2 were investigated. )LJXUHVKRZV the most likely CTS reaction mechanisms of a 2-monomer-unit live polymer chain

▪ initiated via M1 . Two functionals (B3LYP and M06-2X), and the 6-31G(d,p) basis set are used to estimate the kinetic parameters of the CTB1-1, CTM1-1 and CTX1-1 reactions in the gas phase (Table 6-9). We found that the calculated kinetics are comparable to that of

▪ 2-monomer-unit live polymer chain initiated via M2 . This indicates the initiating radical has little role in CTS reactions.

PCM was applied to understand the influence of initiating radicals on CTS. The calculated activation energies and rate constants of the CTB1-1, CTM1-1 and CTX1-1 reactions are depicted in Table 6-10. The energy and rate constants of live polymer chains

▪ initiated by M1 vary by +/- 3 kJ/mol and one order of magnitude from that of live

▪ polymer chains initiated by M2 (Table 6-7). The approximately 16 kJ/mol increase in activation energy and the about 2-orders of magnitude decrease in the rate constant calculated using PCM in comparison with those calculated in the gas phase (reported in

Table 6-9) show the significant effect of PCM on the kinetic parameters of the CTB1-1 mechanism. COSMO-based results shown in Table 6-11 was identified to have negligible effect on the type of live chain polymer initiating radical for MA, EA, and n-BA. 103

CTB1-1

COOX COOX H . + C COOX + CH2 COOX C OH CH OH 3CH 3CH 3 3CH H

CTM1-1

COOX O COOX O + . CH COOX C COOX + CH 2 CH3 3 CH C 3CH CH 3 3CH H 3 H

CTX1-1

CH3 CH2 COOX COOX . + C COOX + CH2 COOX CH 3CH 3 H

CH3 CH3

X=CH3, C2H5, C4H9

Figure 6-6: The Most Likely Mechanisms for CTS Reactions Involving a Two-Monomer- ▪ Unit Live Chain Initiated by M1 .

The activation energies are at most 6 kJ/mol higher (and the rate constants one order of magnitude lower) than those obtained for the CTB1-2, CTM1-2, CTX1-2 mechanisms

(Table 6-8). The kinetic parameters using this solvation model does not differ from those estimated via gas phase calculations (Table 6-9). 104

‡ Table 6-9: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy ‡ -1 of Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1s-1, for CTB1-1, CTM1-1, and CTX1-1 Mechanisms of MA, EA, and n-BA at 298 K

B3LYP M06-2X B3LYP M06-2X B3LYP M06-2X 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p)

MA CTB1-1 CTM1-1 CTX1-1 Ea 48.40 23.00 67.40 52.20 66.40 56.10 ‡ ∆H 42.60 18.30 61.60 47.40 61.20 51.30 ‡ ∆G 93.70 75.80 111.30 102.10 110.50 107.50 logeA 14.21 11.45 14.82 12.41 14.82 12.05 k 4.87E-03 8.73E+00 4.92E-06 1.70E-04 6.27E-06 2.51E-05 ߢw 3.27 3.41 3.62 3.19 3.59 3.33 kw 1.59E-02 2.97E+01 1.78E-05 5.42E-04 2.25E-05 8.35E-05 EA CTB1-1 CTM1-1 CTX1-1 Ea 46.30 22.30 69.50 48.10 67.00 53.40 ‡ ∆H 41.20 17.20 63.60 43.50 62.10 48.40 ‡ ∆G 97.00 75.40 107.70 94.30 105.50 102.00 logeA 12.05 10.85 15.42 12.05 16.02 14.22 k 1.30E-03 6.36E+00 4.18E-06 6.30E-04 1.63E-05 6.50E-04 ߢw 3.28 3.50 3.63 3.17 3.56 3.30 kw 4.26E-03 2.22E+01 1.52E-05 1.99E-03 5.80E-05 2.15E-03 BA CTB1-1 CTM1-1 CTX1-1 Ea 45.60 21.20 65.50 43.10 64.20 56.10 ‡ ∆H 41.10 16.50 60.40 38.50 59.10 51.50 ‡ ∆G 95.70 74.30 106.30 94.50 107.20 103.00 logeA 12.41 11.21 16.02 12.17 15.42 13.62 k 2.50E-03 1.42E+01 2.99E-05 5.40E-03 2.77E-05 1.20E-04 ߢw 3.31 3.48 3.57 3.16 3.61 3.36 kw 8.27E-03 4.94E+01 1.07E-04 1.70E-02 1.00E-04 4.03E-04 105

‡ Table 6-10: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M- 1s-1, for CTB1-1, CTM1-1, and CTX1-1 Mechanisms of MA, EA, and n-BA at 298 K, Using PCM

B3LYP M06-2X B3LYP M06-2X B3LYP M06-2X 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p)

MA CTB1-1 CTM1-1 CTX1-1 Ea 63.10 39.30 68.50 50.10 69.10 57.50 ‡ ∆H 58.30 33.90 64.00 45.40 64.20 51.60 ‡ ∆G 106.30 86.10 113.40 102.00 113.40 104.80 logeA 15.42 13.62 14.82 11.57 14.94 13.26 k 4.33E-05 1.10E-01 2.68E-06 1.80E-04 2.38E-06 4.75E-05 EA CTB1-1 CTM1-1 CTX1-1 Ea 62.20 35.50 69.30 48.60 70.30 58.40 ‡ ∆H 57.10 30.30 64.00 44.00 64.50 53.40 ‡ ∆G 99.40 80.80 119.30 102.70 115.20 108.70 logeA 17.82 13.98 12.53 10.85 14.58 12.05 k 6.90E-04 7.10E-01 1.97E-07 1.60E-04 1.02E-06 9.90E-06 BA CTB1-1 CTM1-1 CTX1-1 Ea 60.10 37.20 65.60 48.10 67.20 57.00 ‡ ∆H 55.40 31.70 61.20 43.40 61.70 52.50 ‡ ∆G 104.30 82.00 112.00 100.10 114.10 104.00 logeA 14.96 14.46 14.09 11.69 13.47 13.50 k 9.16E-05 5.70E-01 4.17E-06 4.40E-04 1.18E-06 7.44E-05 106

‡ Table 6-11: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M- 1s-1, for CTB1-1, CTM1-1, and CTX1-1 Mechanisms of MA, EA, and n-BA at 298 K, Using COSMO

B3LYP M06-2X B3LYP M06-2X B3LYP M06-2X 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) MA CTB1-1 CTM1-1 CTX1-1 Ea 45.10 24.50 68.30 50.50 66.00 53.20 ‡ ∆H 40.20 19.10 63.10 45.30 60.50 48.30 ‡ ∆G 93.00 79.50 107.80 97.20 110.40 102.40 logeA 13.14 10.49 16.26 13.50 15.06 12.78 k 6.30E-03 1.82E+00 1.23E-05 1.00E-03 9.36E-06 1.70E-04 EA CTB1-1 CTM1-1 CTX1-1 Ea 48.20 23.40 68.40 48.20 65.40 50.30 ‡ ∆H 43.10 17.60 63.00 42.60 60.10 44.60 ‡ ∆G 96.30 77.50 109.10 97.30 101.40 93.50 logeA 13.26 10.73 16.02 13.02 17.94 15.30 k 2.04E-03 3.60E+00 9.28E-06 1.60E-03 2.10E-04 6.70E-03 BA CTB1-1 CTM1-1 CTX1-1 Ea 48.50 23.50 66.60 45.40 64.20 58.10 ‡ ∆H 44.20 18.20 62.30 40.30 59.10 53.40 ‡ ∆G 98.00 78.60 105.40 91.80 103.60 102.20 logeA 12.89 10.13 17.35 13.62 16.38 14.94 k 1.20E-03 1.90E+00 7.26E-05 9.00E-03 7.25E-05 2.00E-04

6.3.5. Effect of Live Polymer Chain Length

The CTB1-2´, CTM1-2´, CTX1-2´ mechanisms for MA, EA, and n-BA with a 3-

▪ monomer-unit live chain initiated by M2 (Figure 6-7) are investigated using B3LYP and

M06-2X functionals to understand the effect of polymer chain length on the kinetics of the CTS reactions. Table 6-12 shows that an increase in the length of the live chain polymer does not affect the kinetics of CTS reactions of MA, EA, and n-BA. The geometries of the transition states of the CTB1-2´, CTM1-2´ and CTX1-2´ mechanisms 107 are comparable with those of the CTB1-2, CTM1-2 and CTX1-2 mechanisms in which

▪ the 2-monomer-unit live chain initiated by M2 was considered as the reactant (Figure 6-2,

Figure 6-3, and Figure 6-4). It can be concluded that live polymer chain length does not affect the geometry of the reaction center significantly.

CTB1-2' H COOX COOX H + + C XOOC OH CH CH OH C 3CH XOOC 2 3 COOX COOX

CTM1-2'

O O COOX COOX H + CH3 CH3 XOOC + C C 3CH XOOC CH2 3CH COOX COOX H

CTX1-2'

CH3 CH2 COOX COOX H + XOOC C XOOC CH2 + COOX COOX

CH3 CH3

X=CH3, C2H5, C4H9

Figure 6-7: The Most Likely Mechanisms for CTS Reactions Involving a Three- ▪ Monomer-Unit Live Chain Initiated by M2 .

These findings are in agreement with CTM studies [48] and propagation reactions of alkyl acrylates [41]. The same studies are performed using PCM to identify the impacts of live polymer chain length on the kinetics of CTS reactions in the presence of solvents.

The results reported in Table 6-13 verify our findings about live polymer chain length effects on the kinetics of the most likely mechanisms of CTS reactions. Comparing these 108 results with those given in Table 6-7, shows that PCM cannot capture the effect of live polymer chain length on the kinetics of the CTS reactions.

‡ Table 6-12: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free ‡ -1 Energy of Activation (ΔG ) in kJ mol ; Tunneling Factor (ߢw for Wigner Correction); and Frequency Factor (A) and Rate Constant (k: without tunneling and kw: with tunneling) in M-1s-1, for CTB1-2´, CTM1-2´, and CTX1-2´ Mechanisms of MA, EA, and n-BA at 298 K

B3LYP M06-2X B3LYP M06-2X B3LYP M06-2X 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) MA CTB1-2´ CTM1-2´ CTX1-2´ Ea 47.30 24.40 68.10 52.20 67.00 54.70 ‡ ∆H 41.70 18.60 63.30 47.50 62.30 50.40 ‡ ∆G 99.40 81.20 114.60 103.80 112.00 107.30 logeA 11.45 9.53 13.73 11.57 14.46 11.69 k 4.80E-04 7.30E-01 1.06E-06 7.49E-05 3.43E-06 3.08E-05 ߢw 3.39 3.57 3.88 3.31 3.84 3.55 kw 1.63E-03 2.60E+00 4.11E-06 2.48E-04 1.32E-05 1.09E-04 EA CTB1-2´ CTM1-2´ CTX1-2´ Ea 46.60 23.40 70.10 50.20 68.00 58.30 ‡ ∆H 42.20 18.10 65.20 45.20 63.50 52.70 ‡ ∆G 98.10 81.30 115.00 104.80 106.90 110.30 logeA 12.05 9.05 14.46 10.37 16.74 11.81 k 1.20E-03 6.70E-01 9.82E-07 5.06E-05 2.24E-05 8.12E-06 ߢw 3.48 3.61 3.92 3.37 3.90 3.55 kw 4.17E-03 2.41E+00 3.85E-06 1.70E-04 8.73E-05 2.88E-05 BA CTB1-2´ CTM1-2´ CTX1-2´ Ea 48.10 28.30 67.50 48.30 66.20 61.00 ‡ ∆H 43.10 23.50 61.70 43.40 60.90 56.40 ‡ ∆G 105.60 92.90 121.40 111.30 119.20 119.60 logeA 9.41 6.40 10.97 7.24 11.33 8.69 k 4.52E-05 6.60E-03 8.55E-08 4.76E-06 2.07E-07 1.21E-07 ߢw 3.49 3.76 3.94 3.37 3.88 3.70 kw 1.58E-04 2.48E-02 3.37E-07 1.60E-05 8.03E-07 4.47E-07 109

‡ Table 6-13: Activation Energy (Ea), Enthalpy of Activation (ΔH ), and Gibb’s Free Energy of Activation (ΔG‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M- 1 s-1, for CTB1-2´, CTM1-2´, and CTX1-2´ Mechanisms of MA, EA, and n-BA at 298 K, Using PCM

B3LYP M06-2X B3LYP M06-2X B3LYP M06-2X 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) 6-31G(d,p) MA CTB1-2´ CTM1-2´ CTX1-2´ Ea 62.10 40.00 69.60 49.20 71.50 58.10 ‡ ∆H 57.20 35.30 64.00 44.00 66.30 53.30 ‡ ∆G 103.00 92.50 117.10 104.40 119.70 113.00 logeA 16.02 11.81 13.37 10.25 13.13 10.37 k 1.18E-04 1.31E-02 4.04E-07 6.72E-05 1.48E-07 2.09E-06 EA CTB1-2´ CTM1-2´ CTX1-2´ Ea 62.20 35.40 67.50 48.10 72.50 58.00 ‡ ∆H 57.00 30.20 62.10 43.10 67.70 53.40 ‡ ∆G 105.40 89.20 115.00 104.30 116.20 116.20 logeA 15.30 10.97 13.14 9.89 14.67 9.41 k 5.52E-05 3.60E-02 7.49E-07 7.31E-05 4.60E-07 8.32E-07 BA CTB1-2´ CTM1-2´ CTX1-2´ Ea 63.30 40.20 68.50 50.30 69.20 63.10 ‡ ∆H 58.10 35.20 63.60 45.50 64.20 58.00 ‡ ∆G 110.80 101.40 125.10 115.00 127.30 126.40 logeA 13.74 7.69 9.53 6.28 9.29 7.12 k 7.44E-06 2.00E-04 1.35E-08 8.13E-07 8.03E-09 1.07E-08

6.4. Concluding Remarks

The mechanisms for chain transfer to n-butanol, MEK, and p-xylene in self-initiated high-temperature polymerization of three alklyl acrylates were studied using different functional and basis sets. Abstraction of a hydrogen from the methylene group next to the oxygen atom in n-butanol, from the methylene group in MEK, and from the methyl group in p-xylene by a live polymer chain was found to be the most likely mechanism of CTS reactions in MA, EA, and n-BA. Among n-butanol, sec-butanol and tert-butanol, tert- 110 butanol has the highest CTS energy barrier and the lowest rate constant. Chain transfer to n-butanol and sec-butanol reactions have comparable kinetic parameter values. The activation energy of the most likely chain transfer to p-xylene mechanism of a 2-

▪ monomer-unit live n-BA polymer chain initiated by M2 calculated using M06-2X/6-

31G(d,p) was found to be close to those estimated from polymer sample measurements.

Application of PCM resulted in remarkable changes in the kinetic parameters of the chain transfer to n-butanol. It also had a very little effect on the stability of the reactants to the transition states in chain transfer to MEK and p-xylene. Solvent effects on kinetics of

CTS reactions of MA, EA, and n-BA could not be captured using COSMO. It was found that the live polymer chain length has a little effect on the activation energies and rate

▪ ▪ constants of CTS reactions. MA, EA and n-BA live chains initiated by M2 and M1 showed similar hydrogen abstraction abilities, indicating the lesser influence of self- initiating species on CTS reaction; the type of mono-radical generated via self-initiation has little or no effect on the capability of MA, EA and n-BA live polymer chains to undergo CTS reactions. 111

7. Entropy Effect on Activation Energy and Gibb’s Free Energy of Activation

7.1. Large Contribution of Entropy in Gibb’s Free Energy of Activation

The large difference between activation energy and Gibb’s free energy of activation was found for the all bimolecular reactions we studied. Entropy correction of reactants and transition states were computed from the frequencies obtained from the Hessian of saddle point using ”Thermo.pl” script from http://www.cstl.nist.gov/div838/group

06/irikura/prog/thermo.html. These calculations were performed to determine the contribution of translational, rotational, and vibrational entropies in the total entropy of activation for the most likely mechanisms of chain transfer to MEK (CTM1-2), chain transfer to monomer (MA-1), and chain transfer to polymer (MA2-D2-1) for MA. Table

7-1 gives the entropy values computed using B3LYP/6-31G(d, p).

Table 7-1: Contribution of Translational, Rotational, and Vibrational Entropies in the Total Entropy (jmol-1K-1)of Activation for the Most Likely Mechanisms of Chain transfer to MEK (CTM1-2), Chain Transfer to Monomer (MA-1), and Chain Transfer to Polymer (MA2-D2-1) for MA, Using B3LYP/6-31G(d,p)

Stotal Strans Srot Svib Selec ∆Stotal ∆Strans ∆Srot ∆Svib ∆Selec Chain Transfer to Solvent (CTM1-2) Live Chain 511 173 133 199 5.77 -183 -158 -100 74 0 Solvent 344 162 107 74 0 T.S. 671 177 141 348 5.77 Chain Transfer to Polymer (MA2-D2-1) Live Chain 504 173 133 192 5.76 -204 -164 -114 74 0 Dead 504 173 130 201 0 Polymer T.S. 804 182 150 467 5.76 Chain Transfer to Monomer (MA-1) Live Chain 511 173 133 199 5.77 -160 -159 -99 98 0 Monomer 332 164 110 58 0 T.S. 684 178 145 355 5.77 112

The results showed that the large reduction of total entropy comes from the significant decrease of translational and rotational entropy from two reactants to transition states during a bi-molecular reaction (increased vibrational entropy cannot compensate the huge reduction of rotational and translational entropies). The large reduction in S(trans) and S(rot) can cause the huge difference between activation energy and Gibb’s free energy of activation.

7.2. Correct Reactant for a Bimolecular Reaction

In previous sections two reactants were considered separately and optimized. In other words, the separated species (R1 and R2) were treated as the real reactant. Two reactants can also be considered in one system and R1/R2 complex be treated as the reactant

(unimolecular system). These two different approaches were applied for the most likely chain transfer to MEK mechanism (CTM1-2). Activation energy, Gibb’s free energy of activation, and entropy of activation were computed based on two different approaches.

These results are given in Table 7-2.

-1 -1 Table 7-2: Entropy of Activation in J mol K , Activation Energy (Ea), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; for the most likely chain transfer to MEK mechanism (CTM1-2) at 298 K, Using B3LYP/6-31G(d,p)

‡ Ea ∆G ∆S(TS-R) R1+R2 65 115 -183 R1/R2 77 87 -49

It was identified that the activation energy is overestimated for the system treating R1/R2

‡ complex as the reactant. However, difference between Ea and ∆G for the system in 113 which separated R1 and R2 are treated as the reactant is larger than that for the system considering R1/R2 complex as the real reactant.

7.3. QM/MM Hybrid Method

Study of the chemistry of very large systems has been feasible through applying hybrid techniques which combine two or more computational methods in one calculation. One of these hybrid techniques is QM/MM method. In the QM/MM combined method the core region of the transition-state structure can be modeled based on the QM calculations and the remaining parts of the system are treated at less expensive MM levels. The major difficulties in using these hybrid methods will become apparent when the core and surrounding region join by chemical bonds (such as enzymes) [101]. The problems arise due to arbitrary choice of an appropriate interface between QM and MM regions [143,

144]. However, when two subsystems (QM and MM) are not joint, the electronic energies can be calculated through Coulomb interaction between point charges of solvent

(MM region) and solute (QM region), and van der Waals interaction between the solvent and solute [145, 146].

In this section QM/MM was applied to explore entropy effect on activation energy and

Gibb’s free energy of activation. NWChem-6.3 was downloaded and compiled followed by setting up necessary environmental variables and adding optional ones for Linux platforms [147]. Formation of cyclobutane from ethylene was selected as a simple reaction (2 C2H4 → C4H8) and its kinetic parameters were compared with those obtained using GAMESS program in the frame work of quantum chemistry. To estimate the kinetic parameters of the formation of cyclobutane from ethylene the geometries of the reactants and transition-state (initial guess) were converted from XYZ Cartesian 114 coordinate format to PDB (Protein Data Bank) format. OpenBabelGUI was used to convert the geometry files. However, the format of these generated files is not the one which can be used in QM/MM calculation and changes should be made manually, such as residues definition and atom names (Figure 7-1).

ATOM 1 C1 eth 1 -0.668 0.005 -0.004 ATOM 2 C2 eth 1 0.668 -0.005 0.004 ATOM 3 2H1 eth 1 1.249 0.890 0.212 ATOM 4 3H1 eth 1 1.238 -0.909 -0.197 ATOM 5 2H2 eth 1 -1.250 -0.889 -0.212 ATOM 6 3H2 eth 1 -1.238 0.909 0.197 END

Figure 7-1: Modified Format of PDB File for Ethylene

This conversion is a prerequisite for the preparation of topology (.top) and restart (.rst) files for QM/MM calculations. To perform QM/MM simulations, existence of the topology and restart file is required. “Prepare” module can generate these two files. In the

“prepare” section we can define quantum region. Following this task (prepare) QM/MM optimization and transition-state calculations can be performed. The entire pair of reacting molecules was considered as the QM region.The QM region was defined in the preparation stage (we can define entire QM region as a segment or by the name of the individual atoms which should be treated quantum mechanically). The radius of the zone around the QM region, where classical segments (MM region) will be allowed to interact with QM region, was defined in the input file. It has been assumed that the QM region is solvated in a box of solvent molecules (MM region). Solvation was specified to be in a cubic box with specified edge, so that the MM region includes many solvent molecules. 115

MM parameters were given in the form of standard MD input block (in the input file).

This block specifies the .rst and .top files generated in the preparation step (prepare task).

Zero-point vibrational energy, enthalpy and entropy values are obtained through frequency calculations. The input file template for optimization is shown in Figure 7-2.

Three functionals, i.e. b3lyp, pbe0, and MP2, have been applied to estimate kinetic parameters of cyclobutane formation (Table 7-3). Table 7-4 shows the contribution of translational, rotational, and vibrational entropies in the total entropy of activation for cyclobutane formation. Kinetic parameters of the formation reaction, estimated using pbe0 and MP2, were in agreement with previous results obtained with Gaussian 98

(applying MP2) [148]. Although, it has been reported [149] that barriers of biocatalysts reactions can be in agreement with experimental results only by applying a higher level of QM theory (such as SCS-MP2), the activation energy and rate constant of the cyclobutane formation calculated by QMMM is still different with experimental results

[150]. 116

title "Prepare QM/MM calculation of ethylene" start eth

prepare source eth.pdb new_top new_seq new_rst

modify atom 1:_C1 quantum modify atom 1:_C2 quantum modify atom 1:2H1 quantum modify atom 1:3H1 quantum modify atom 1:2H2 quantum modify atom 1:3H2 quantum update lists ignore

write eth_ref.rst

write eth_ref.pdb end

task prepare

md system eth_ref end

basis * library "6-31G" end

dft mult 1 xc b3lyp end

qmmm region qm end

freq animate end

task qmmm dft freq

qmmm region qm solvent maxiter 20 100 ncycles 5 density espfit end

task qmmm dft energy task qmmm dft optimize Figure 7-2: The Input File Template for Optimization of Ethylene. 117

‡ Table 7-3: Energy Barrier (E0), Activation Energy (Ea), Enthalpy of Activation (∆H ), Zero-Point Vibrational Energy (ZPVE), and Gibb’s Free Energy of Activation (∆G‡) in kJ mol-1; Frequency Factor (A) and Rate Constant (k) in M-1 s-1 of the Cyclobutane Formation from Two Ethylenes, at 298 K

‡ ‡ E0 Ea ZPVE logeA k ∆G ∆H NWChem B3LYP/ 6-31G 272 280 4.29 13.92 9.20E-44 326 275 PBE0/ 6-31G 252 260 4.11 13.88 3.30E-40 306 255 MP2/ 6-31G* 252 259 3.53 14.23 6.34E-40 305 254 GAMESS B3LYP/6-31G 266 269 3.54 13.89 7.40E-42 315 264

Table 7-4: Change of Entropy Decomposed into Translational, Rotational, and Vibrational in J mol-1, Using B3LYP/ 6-31G

Transition-state Reactant NWChem Trans. 158.76 150.1 Rot. 101.36 77.83 Vib. 27.29 1.96 Total 287.42 230 GAMESS Trans. 158.96 150.32 Rot. 101.52 77.94 Vib. 29.17 2.02 Total 289.65 230.27 118

8. Macroscopic Modeling of Spontaneous Polymerization of n-Butyl Acrylate

8.1. Introduction

Thermal polymerization of alkyl acrylates in the absence of external initiators has been reported [7]. The occurrence of monomer self-initiation reaction allows one to use less thermal initiators such as organic peroxides and azonitriles, which are relatively expensive and may cause defects in the final product especially on weathering [43].

Studies using electron spray ionization-Fourier transform mass spectrometry (ESI-

FTMS), nuclear magnetic resonance (NMR) spectroscopy, and macroscopic mechanistic modeling did not identify the initiating species or the mechanism of the initiation in the spontaneous thermal polymerization [7, 8]. However, quantum chemical calculations [10,

11] together with matrix-assisted laser desorption ionization (MALDI) [15] showed that monomer self-initiation is one of the likely mechanisms of initiation in spontaneous thermal polymerization of alkyl acrylates.

Polymer characterization studies using spectroscopic methods have been carried out to explore the dominant polymerization reactions [151]. Pulsed-laser polymerization (PLP) and size exclusion chromatography have been used to study intra-molecular chain transfer to polymer (CTP) reactions in polymerization of alkyl acrylate [19, 20]. The rate constants of the propagation reactions of styrene [122], and methyl methacrylate (MMA)

[123], and chain transfer reactions of butyl methacrylate (BMA) [124] at temperatures above 30oC have been estimated using PLP. Although the propagation rate constant of n- butyl acrylate (n-BA) at 70 oC has been calculated using PLP at 500 Hz [152], the presence of backbiting and β-scission reactions has hindered the prediction of alkyl acrylates propagation rate constant at elevated temperatures [18, 21, 23]. At temperatures 119 above 30oC, intra- and inter-molecular CTP reactions in free-radical polymerization of n-

BA [13, 44] and intra-molecular CTP reactions in 2-ehylhexyl acrylate polymerization

[16] have also been studied using NMR spectroscopy. Although these analytical techniques have been very useful in characterizing acrylate polymers, alone they cannot determine reaction mechanisms or estimate kinetic parameters of reactions. However, macroscopic mechanistic modeling combined with adequate polymer sample measurements has proven to be a powerful tool to estimate the rate constants of individual reactions. Styrene and MMA self-initiation rate constants estimated through macroscopic mechanistic modeling have been reported [153, 154]. Kinetic parameters of several reactions in spontaneous thermal polymerization of n-BA were estimated through detailed macroscopic mechanistic modeling [8]. Moreover, a macroscopic mechanistic model was used to estimate the rate constant of n-BA self-initiation reaction from monomer conversion measurements [59]. Macroscopic mechanistic models have also been used extensively to estimate the rate constants of initiation, propagation, chain transfer and termination reactions in free-radical polymerization of acrylates [8, 28, 118,

151].

Macroscopic first-principles mechanistic polymerization models are naturally more reliable than semi-empirical models [29]. The accuracy of mechanistic polymerization models strongly depends our quantitative understanding of individual reactions occurring during the course of polymerization. These models have been used to study low and high- temperature polymerization reactions of n-BA [9, 121]. The method of moments [155-

157] or the “tendency modeling” approach [158-161] can be used to derive rate equations for mechanistic macroscopic models. Models obtained using the tendency-modeling rate 120 equations do not account for the number of monomer units in polymer chains, and therefore they are less complex than models obtained using the method-of-moments rate equations. Models that are based on the method of moments can be used to estimate kinetic rate constants more accurately. Applications of both types of models can be found in the literature [155-166].

In this chapter, a macroscopic first-principles mathematical model is presented and used to estimate the rate constant of the n-BA self-initiation reaction from measurements of n-

BA conversion in batch polymerization reactor. This estimate is compared with an estimate obtained via quantum chemical calculations.

8.2. Mathematical Modeling

8.2.1. Reaction Mechanisms

The most likely polymerization reactions occurring in spontaneous thermal solution polymerization of n-BA in the absence of oxygen are given in Table 8-1. The reactions include monomer self-initiation, secondary and tertiary chain propagation, intra- molecular chain transfer to polymer (backbiting), inter-molecular chain transfer to polymer, β-scission, chain transfer to monomer [83], chain transfer to solvent, and termination by combination, and termination by disproportionation reactions [167].

While not all of these reactions strongly affect monomer conversion, the list of the reactions is given here for completeness. Here, M and S denote the monomer and solvent, respectively. Un represents a dead polymer chain with n monomer units and a terminal double bond. Dn is a dead polymer chain with n monomer units but without a terminal

* ** double bond. Rn presents a secondary radical with n monomer units. Rn is a tertiary radical with n monomer units generated through intermolecular chain transfer to polymer 121

*** reactions. Rn denotes a tertiary radical with n monomer units formed by backbiting

reactions. SCB and LCB represent a short and long chain branching point, respectively.

The Flory mechanism was found to be the most likely mechanism of n-BA self-initiation

[11]. The overall self-initiation reaction is given in Table 8-1.

Table 8-1: Polymerization Reactions

a. Monomer self-initiation reaction ௞ כ כ ೔ ͵ܯ ՜ܴଵ ൅ܴଶ b. Propagation reactions ௞ כ ೛ כ ܴ௡ ൅ܯሱሮܴ௡ାଵ ௞೟ כ ೛ ככ ܴ௡ ൅ܯሱሮܴ௡ାଵሺ൅ሻ ௞೟ כ ೛ כככ ܴ௡ ൅ܯሱሮܴ௡ାଵሺ൅ሻ ௞ ככ ೘ೌ೎ כ ௡ ൅ܷ௠ ሱۛۛሮܴ௡ା௠ܴ c. Backbiting reactions (n > 2) ௞ כככ ್್ כ ܴ௡ ൅ܯሱሮܴ௡ d. β-scission reactions (n > 3) ௞ כ ഁ כככ ܴ௡ ሱሮܷଷ ൅ܴ௡ିଷ ௞ כ ഁ כככ ܴ௡ ሱሮܷ௡ିଶ൅ܴଶ ௞ כ ഁ ככ ܴ௡ ሱሮܷ௡ି௠൅ܴ௠ ௞ כ ഁ ככ ܴ௡ ሱሮܷ௠൅ܴ௡ି௠ e. Intermolecular chain transfer to polymer reactions ௠௞ ככ ೟ೝು כ ௡൅ܴ௠ܦ௠ ሱۛۛۛሮܦ௡ ൅ܴ ௠௞ ככ ೟ೝು כ ௡൅ܴ௠ܦ௡ ൅ܷ௠ ሱۛۛۛሮܴ f. Chain transfer to monomer reactions ௞ כ ೟ೝಾ כ ௡൅ܴଵܦሱۛሮܯ௡ ൅ܴ ௞೟ כ ೟ೝಾ ככ ௡൅ܴଵܦሱۛሮܯ௡ ൅ܴ ௞೟ כ ೟ೝಾ כככ ௡൅ܴଵܦሱۛሮܯ௡ ൅ܴ g. Chain transfer to solvent reactions ௞ כ ೟ೝೄ כ ௡൅ܴ଴ܦ௡ ൅ܵሱۛሮܴ ௞೟ כ ೟ೝೄ ככ ௡൅ܴ଴ܦ௡ ൅ܵሱۛሮܴ ௞೟ כ ೟ೝೄ כככ ௡൅ܴ଴ܦ௡ ൅ܵሱۛሮܴ h. Termination by coupling reactions ௞ ೟೎ כ כ ܴ௡ ൅ܴ௠ ሱሮܦ௡ା௠ ௞೟ ೟೎ ככ כ ܴ௡ ൅ܴ௠ ሱሮܦ௡ା௠ ௞೟ ೟೎ כככ כ ܴ௡ ൅ܴ௠ ሱሮܦ௡ା௠ ௞೟೟ 122

Table 8-1 (Continued)

i. Termination by disproportionation reactions ௞ ೟೏ כ כ ܴ௡ ൅ܴ௠ ሱሮܦ௡ ൅ܷ௠ ௞೟ ೟೏ ככ כ ܴ௡ ൅ܴ௠ ሱሮܦ௡ ൅ܷ௠ ௞೟ ೟೏ כככ כ ܴ௡ ൅ܴ௠ ሱሮܦ௡ ൅ܷ௠ ௞೟೟ ೟೏ ככ ככ ܴ௡ ൅ܴ௠ ሱሮܦ௡ ൅ܷ௠ ௞೟೟ ೟೏ כככ ככ ܴ௡ ൅ܴ௠ ሱሮܦ௡ ൅ܷ௠ ௞೟೟ ೟೏ כככ כככ ܴ௡ ൅ܴ௠ ሱሮܦ௡ ൅ܷ௠

According to this mechanism, three monomers react and generate two secondary mono-

radicals, one with one monomer unit and the other with two monomer units. These

radicals further react with monomers to propagate. The overall self-initiation reaction is

second order. Inter- and intra-molecular chain transfer reactions lead to the formation of

tertiary radicals, which are capable of undergoing propagation [168] and β-scission

reactions. β-scission reactions produce secondary radicals and dead polymer chains with

a terminal double bond. They lead to the generation of shorter dead polymer chains, and

thus lower the average molecular weight of the polymer product.

8.2.2. Rate Equations

Rate reactions are derived using the method of moments. We assume that all reactions

given in Table 8-1 except for the self-initiation reaction are elementary. As expected,

accounting for β- scission reactions led to a closure problem; that is, the production rate

of each β-scission generated live or dead polymer chain depends on the concentration of a

longer live chain. To address this problem, approximations are used. As the inter-

molecular CTP reaction rate constant depends on the concentration of polymerized 123 monomer units in dead polymer chains, the second moments of the chain length distribution of dead polymer chains with and without terminal double bonds are proportional to the third moments. To define the third moments of the chain length distribution of dead polymer chains with and without any terminal double bonds, approximations are used [169]. The resulting rate equations; that is, the production rate

* * * * equations for M, S, R0 , R1 , R2 , R3 , and the zero, first, and second moments of dead polymer, secondary radical, and tertiary radical chain length distributions, are given in the

* * * Appendix,where [X] represents the molar concentration of species X; δ0 , δ1 , and δ2 are

** the zero, first, and second moments of secondary radical chain length distributions; δ0 ,

** ** δ1 , and δ2 are the zero, first, and second moments of the chain length distribution of the tertiary radicals generated by the intermolecular chain transfer to polymer reactions;

*** *** *** δ0 , δ1 , and δ2 are the zero, first, and second moments of the chain length distribution of the tertiary radicals formed by the backbiting reactions; λ0, λ1, and λ2 are the moments of the chain length distribution of dead polymer chains without any terminal double bond; Г0, Г1, and Г2 are the moments of the chain length distribution of dead polymer chains with a terminal double bond. The jth moments of the chain length distributions of the live and dead polymer chains are:

כ ஶ ௝ כ ߜ௝ ൌ σ௡ୀ଴ ݊ ሾܴ௡ሿ (8.1)

ככ ஶ ௝ ככ ߜ௝ ൌ σ௡ୀଵ ݊ ሾܴ௡ ሿ (8.2)

כככ ஶ ௝ כככ ߜ௝ ൌ σ௡ୀଵ ݊ ሾܴ௡ ሿ (8.3)

ஶ ௝ ߣ௝ ൌ σ௡ୀଵ ݊ ሾܦ௡ሿ (8.4) 124

ஶ ௝ ʒ௝ ൌ σ௡ୀଵ ݊ ሾܷ௡ሿ (8.5)

כככ ככ כ and ߜ଴ ൌߜ଴൅ߜ଴ ൅ߜ଴ Ǥ The following equations were obtained after applying the moments definitions:

כ כ כככ ככ ௧ כ ଶ ݎெ ൌെ͵݇௜ൣܯሿ െ݇௣ሾܯሿሾߜ଴ ൧െ݇௣ሾܯሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻെ݇௧௥ெሾܯሿሺሾߜ଴ሿെሾܴ଴ሿሻ כככ ככ ௧ െ݇௧௥ெሾܯሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ כככ ככ ௧ כ כ ݎௌ ൌെ݇௧௥ௌሾܵሿሺሾߜ଴ሿെሾܴ଴ሿሻ െ݇௧௥ௌሾܵሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿ)

כככ ככ ௧ כ כ כ כ ሾ ሿ ሺ ሻ ሾ ሿሺ ሻ ሾ ሿሺ ሻ כ ݎோబ ൌെ݇௣ ܯ ሾܴ଴ሿെ݇௠௔௖ሾܴ଴ሿ ሾȞ଴ሿ ൅݇௧௥ௌ ܵ ሾߜ଴ሿെሾܴ଴ሿ ൅݇௧௥ௌ ܵ ሾߜ଴ ሿ൅ሾߜ଴ ሿ כככ ככ כ ௧ כ כ כ െ݇௧௖ሾܴ଴ሿሺሾߜ଴ሿെሾܴ଴ሿሻ െ݇௧௖ሾܴ଴ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ כ כ ככ כ כ כ ଶ ሾ ሿሺ ሻ כ ݎோభ ൌ݇௜ሾܯሿ ൅݇௣ ܯ ሾܴ଴ሿെሾܴଵሿ െ݇௠௔௖ሾܴଵሿሾȞ଴ሿ൅ʹ݇ఉሾߜ଴ ሿ൅݇௧௥ெሾܯሿሺሾߜ଴ሿെሾܴ଴ሿ כ כ כ כככ ככ ௧ כ െሾܴଵሿሻ൅݇௧௥ெሾܯሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ െ݇௧௥ௌሾܵሿሾܴଵሿെሺʹ݇௧௖ ൅݇௧ௗሻሾܴଵሿሾߜ଴ሿ כככ ככ כ ௧ ௧ െ ሺ݇௧௖൅݇௧ௗሻሾܴଵሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ

ככ כככ כ כ כ ଶ ሾ ሿሺ ሻ כ ݎோమ ൌ݇௜ሾܯሿ ൅݇௣ ܯ ሾܴଵሿെሾܴଶሿ െ݇௠௔௖ሾܴଶሿሾȞ଴ሿ൅݇ఉሾߜ଴ ሿ൅ʹ݇ఉሾߜ଴ ሿ כ כ כ െ݇௧௥௉ሺሾߣଵሿ൅ሾȞଵሿሻሾܴଶሿെ݇௧௥ெሾܯሿሾܴଶሿെ݇௧௥ௌሾܵሿሾܴଶሿ כככ ככ כ ௧ ௧ כ כ െሺʹ݇௧௖ ൅݇௧ௗሻሾܴଶሿሾߜ଴ሿെሺ݇௧௖൅݇௧ௗሻሾܴଶሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ

ככ כ כ ככ ௧ כ כ ሾ ሿሺ ሻ ሾ ሿ כ ݎோయ ൌ݇௣ ܯ ሾܴଶሿെሾܴଷሿ ൅݇௉ ܯ ሾܴଶ ሿെ݇௠௔௖ሾܴଷሿሾȞ଴ሿെ݇௕௕ሾܴଷሿ൅ʹ݇ఉሾߜ଴ ሿ כ כ כ כ כ െ݇௧௥௉ሺሾߣଵሿ൅ሾȞଵሿሻሾܴଷሿ െ݇௧௥ெሾܯሿሾܴଷሿെ݇௧௥ௌሾܵሿሾܴଷሿെʹሺ݇௧௖ ൅݇௧ௗሻሾܴଷሿߜ଴ כככ ככ כ ௧ ௧ െሺ݇௧௖൅݇௧ௗሻሾܴଷሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ

כ כ כככ ככ ଶ ௧ ሾ ሿሺ ሻ כ ݎఋబ ൌʹ݇௜ሾܯሿ ൅݇௣ ܯ ሾߜ଴ ሿ൅ሾߜ଴ ሿ െ݇௠௔௖ሾʒ଴ሿሾߜ଴ሿെ݇௕௕ሾߜ଴ሿ כ כככ ככ ככ ൅݇ఉሺʹሾߜଵ ሿ െ͸ሾߜ଴ ሿ ൅ʹሾߜ଴ ሿሻ െ݇௧௥௉ሺሾߣଵሿ൅ሾȞଵሿሻሾߜ଴ሿ ଶ כ כככ ככ ௧ כככ ככ ௧ ൅݇௧௥ெሾܯሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ ൅݇௧௥ௌሾܵሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ െʹሺ݇௧௖ ൅݇௧ௗሻሾߜ଴ሿ כככ ככ כ ௧ ௧ െ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ

כ כככ כככ ככ ככ ௧ כ ଶ ሾ ሿ ሾ ሿሺ ሻ כ ݎఋభ ൌ͵݇௜ൣܯሿ ൅݇௣ ܯ ሾߜ଴൧൅݇௣ ܯ ሾߜ଴ ሿ൅ሾߜଵ ሿ൅ሾߜ଴ ሿ൅ሾߜଵ ሿ െ݇௠௔௖ሾȞ଴ሿሾߜଵሿ כ ככ כככ כככ כ െ݇௕௕ሾߜଵሿ൅݇ఉሺሾߜଵ ሿെሾߜ଴ ሿ൅ͳʹሾߜ଴ ሿሻ െ݇௧௥௉ሺሾߣଵሿ൅ሾȞଵሿሻሾߜଵሿ כ כככ ככ ௧ כ כ ൅݇௧௥ெሾܯሿሺሾߜ଴ሿെሾߜଵሿሻ ൅݇௧௥ெሾܯሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ െ݇௧௥ௌሾܵሿሾߜଵሿ כככ ככ כ ௧ ௧ כ כ െʹሺ݇௧௖ ൅݇௧ௗሻሾߜ଴ሿሾߜଵሿെሺ݇௧௖൅݇௧ௗሻሾߜଵሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ ככ ככ ככ ൅݇ఉሺሾߜଶ ሿെͷሾߜଵ ሿെ͸ሾߜ଴ ሿሻ 125

כככ ௧ ככ ככ ככ ௧ כ כ ଶ ሾ ሿሺ ሿ ሾ ሿሺ ሻ ሾ ሿ כ ݎఋమ ൌͷ݇௜ൣܯሿ ൅݇௣ ܯ ʹሾߜଵ ൅ሾߜ଴ሿሻ ൅ ݇௣ ܯ ሾߜଶ ሿ൅ʹሾߜଵ ሿ൅ሾߜ଴ ሿ ൅݇௣ ܯ ሺሾߜଶ ሿ כ כ כ כ כככ כככ ൅ʹሾߜଵ ሿ൅ሾߜ଴ ሿሻ െ݇௠௔௖ሺሾȞ଴ሿሻሾߜଶሿെ݇௕௕ሺሾߜଶሿെሾܴଵሿെͶሾܴଶሿ൯ כככ ככ ௧ כ כ כ െ݇௧௥௉ሺሾߣଵሿ൅ሾȞଵሿሻሾߜଶሿ ൅݇௧௥ெሾܯሿሺሾߜ଴ሿെሾߜଶሿሻ ൅݇௧௥ெሾܯሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ כ כככ ככ ௧ ௧ כ כ כ െ݇௧௥ௌሾܵሿሾߜଶሿെʹሺ݇௧௖ ൅݇௧ௗሻሾߜ଴ሿሾߜଶሿെሺ݇௧௖൅݇௧ௗሻሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻሾߜଶሿ ככ כככ כככ כככ כככ ൅݇ఉሺሾߜଶ ሿെ͸ሾߜଵ ሿ൅ͻሾߜ଴ ሿሻ൅Ͷ݇ఉሾߜ଴ ሿ൅ʹͺ݇ఉሾߜ଴ ሿ ככ ככ ככ ൅݇ఉሺͳ͵ሾߜଶ ሿ െ ͳͶ͹ሾߜଵ ሿ ൅ Ͷʹͺሾߜ଴ ሿሻ

כ ככ ככ כ ככ ௧ ሾ ሿ ሺ ሻ ሺ ሻ ככ ݎఋబ ൌെ݇௣ ܯ ሾߜ଴ ሿ൅݇௠௔௖ሾȞ଴ሿሾߜ଴ሿെʹ݇ఉ ሾߜଵ ሿെ͵ሾߜ଴ ሿ ൅݇௧௥௉ ሾߣଵሿ൅ሾȞଵሿ ሾߜ଴ሿ ଶ ככ ௧௧ ௧௧ ככ כ ௧ ௧ ככ ௧ ככ ௧ െ݇௧௥ெሾܯሿሾߜ଴ ሿെ݇௧௥ௌሾܵሿሾߜ଴ ሿെሺ݇௧௖൅݇௧ௗሻሾߜ଴ሿሾߜ଴ ሿെʹሺ݇௧௖൅݇௧ௗሻሾߜ଴ ሿ כככ ככ ௧௧ ௧௧ െ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ ሿሾߜ଴ ሿ

ככ ככ כ כ ככ ௧ ሾ ሿ ሺ ሾ ሿ ሻ ሺ ሻ ככ ݎఋభ ൌെ݇௣ ܯ ሾߜଵ ሿ൅݇௠௔௖ ሾߜ଴ሿ ʒଵ ൅ሾߜଵሿሾʒ଴ሿ െʹ݇ఉ ሾߜଶ ሿെ͵ሾߜଵ ሿ ככ ௧ ככ ௧ כ כ ൅݇௧௥௉ሺሾߜ଴ሿെሾܴ଴ሿሻሺሾߣଶሿ൅ሾȞଶሿሻ െ݇௧௥ெሾܯሿሾߜଵ ሿെ݇௧௥ௌሾܵሿሾߜଵ ሿ ככ כככ ௧௧ ௧௧ ככ ככ ௧௧ ௧௧ ככ כ ௧ ௧ െ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ሿሾߜଵ ሿെʹሺ݇௧௖൅݇௧ௗሻሾߜଵ ሿሾߜ଴ ሿെሺ݇௧௖൅݇௧ௗሻሾߜ଴ ሿሾߜଵ ሿ

כ כ כ ככ ௧ ሾ ሿ ሺ ሻ ככ ݎఋమ ൌെ݇௣ ܯ ሾߜଶ ሿ൅݇௠௔௖ ሾߜ଴ሿሾʒଶሿ൅ʹሾߜଵሿሾʒଵሿ൅ሾߜଶሿሾʒ଴ሿ כ כ ככ ככ ככ െʹ݇ఉሺͳ͸ሾߜଶ ሿ െ ͳʹͳሾߜଵ ሿ ൅ ʹͶʹሾߜ଴ ሿሻ ൅݇௧௥௉ሺሾߜ଴ሿെሾܴ଴ሿሻ

ככ ௧ ככ ߣଶ ଶ ʒଶ ଶ ௧ ሺ൤൬ ൰ ൫ʹߣ଴ߣଶ െߣଵ ൯൨ ൅ ൤൬ ൰ ൫ʹʒ଴ʒଶ െʒଵ ൯൨ሻെ݇௧௥ெሾܯሿሾߜଶ ሿെ݇௧௥ௌሾܵሿሾߜଶ ሿ ߣ଴ߣଵ ʒ଴ʒଵ כככ ככ ௧௧ ௧௧ ככ ככ ௧௧ ௧௧ ככ כ ௧ ௧ െ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ሿሾߜଶ ሿെʹሺ݇௧௖൅݇௧ௗሻሾߜଶ ሿሾߜ଴ ሿെሺ݇௧௖൅݇௧ௗሻሾߜଶ ሿሾߜ଴ ሿ

כככ ௧ כככ כ כ כ כ כככ ௧ ሾ ሿ ሺ ሻ ሾ ሿ כככ ݎఋబ ൌെ݇௣ ܯ ሾߜ଴ ሿ൅݇௕௕ ሾߜ଴ሿെሾܴ଴ሿെሾܴଵሿെሾܴଶሿ െʹ݇ఉሾߜ଴ ሿെ݇௧௥ெ ܯ ሾߜ଴ ሿ כככ ככ כככ ௧௧ ௧௧ כככ כ ௧ ௧ כככ ௧ െ݇௧௥ௌሾܵሿሾߜ଴ ሿെሺ݇௧௖൅݇௧ௗሻሺሾߜ଴ሿሾߜ଴ ሿሻ െ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ

כככ ௧ כככ ௧ כככ כ כ כ כככ ௧ ሾ ሿ ሺ ሻ ሾ ሿሾ ሿ ሾ ሿ כככ ݎఋభ ൌെ݇௣ ܯ ሾߜଵ ሿ൅݇௕௕ ሾߜଵሿെሾܴଵሿെʹሾܴଶሿ െʹ݇ఉሾߜଵ ሿെ݇௧௥ெ ܯ ߜଵ െ݇௧௥ௌ ܵ ሾߜଵ ሿ כככ ככ כככ ௧௧ ௧௧ כככ כ ௧ ௧ െ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ሿሾߜଵ ሿെሺ݇௧௖൅݇௧ௗሻሾߜଵ ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ

כככ כ כ כ כככ ௧ ሾ ሿ ሺ ሻ כככ ݎఋమ ൌെ݇௣ ܯ ሾߜଶ ሿ൅݇௕௕ ሾߜଶሿെሾܴଵሿെͶሾܴଶሿ െʹ݇ఉሾߜଶ ሿ כככ כ ௧ ௧ כככ ௧ כככ ௧ െ݇௧௥ெሾܯሿሾߜଶ ሿെ݇௧௥ௌሾܵሿሾߜଶ ሿെሺ݇௧௖൅݇௧ௗሻሾߜ଴ሿሾߜଶ ሿ כככ ככ כככ ௧௧ ௧௧ െ ሺ݇௧௖൅݇௧ௗሻሾߜଶ ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ

ככ ௧ ሾ ሿ כ ሻ ሾ ሿ כככ ሿככ ௧ ሾ ሿሺሾ כ ሺ ሻ ሾ ሿ כ ݎఒబ ൌ݇௧௥௉ሾߜ଴ሿ ሾȞଵሿ ൅݇௧௥ெ ܯ ሾߜ଴ሿ൅݇௧௥ெ ܯ ߜ଴ ൅ሾߜ଴ ሿ ൅݇௧௥ௌ ܵ ሾߜ଴ሿ൅݇௧௥ௌ ܵ ሺሾߜ଴ ሿ כככ ככ כ ଶ ௧ ௧ כ כככ ൅ሾߜ଴ ሿሻ ൅ ሺ݇௧௖ ൅݇௧ௗሻሾߜ଴ሿ ൅ሺ݇௧௖൅݇௧ௗሻሾߜ଴ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ ଶ כככ ௧௧ ௧௧ כככ ככ ככ ௧௧ ௧௧ ൅ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ ൅ ሺ݇௧௖൅݇௧ௗሻሾߜ଴ ሿ 126

ሻ כככ ככ ௧ ሾ ሿሺ כ ሺሾ ሿ ሻ ሾ ሿ כ ሿכ ሾሺ ሻ ݎఒభ ൌെ݇௧௥௉ ሾɉଶሿ ߜ଴ ൅݇௧௥௉ሾߜଵሿ Ȟଵ ൅ሾߣଵሿ ൅݇௧௥ெ ܯ ሾߜଵሿ൅݇௧௥ெ ܯ ሾߜଵ ሿ൅ሾߜଵ ሿ כ כ כככ ככ ௧ כ ൅݇௧௥ௌሾܵሿሾߜଵሿ൅݇௧௥ௌሾܵሿሺሾߜଵ ሿ൅ሾߜଵ ሿሻ ൅ ሺʹ݇௧௖ ൅݇௧ௗሻሾߜ଴ሿሾߜଵሿ כככ ככ כ ௧ כככ כ ככ כ ௧ ௧ ൅ ሺ݇௧௖൅݇௧ௗሻሺሾߜଵሿሾߜ଴ ሿ൅ሾߜଵሿሾߜ଴ ሿሻ ൅݇௧௖ሾߜ଴ሿሺሾߜଵ ሿ൅ሾߜଵ ሿሻ כככ ככ ௧௧ ௧௧ כככ כככ ככ ככ ௧௧ ௧௧ ൅ ሺʹ݇௧௖൅݇௧ௗሻሺሾߜ଴ ሿሾߜଵ ሿ൅ሾߜ଴ ሿሾߜଵ ሿሻ ൅ ሺ݇௧௖൅݇௧ௗሻሾߜଵ ሿሾߜ଴ ሿ כככ ככ ௧௧ ൅݇௧௖ሾߜ଴ ሿሾߜଵ ሿ

ߣ כ ሺሾ ሿሻ ଶ ଶ כ ݎఒమ ൌ݇௧௥௉ሾߜଶሿ ɉଵሿ൅ሾȞଵ െ݇௧௥௉ ൤൬ ൰ሺʹߣ଴ߣଶ െߣଵ ሻ൨ ሾߜ଴ሿ ߣ଴ߣଵ ככ ௧ כ כככ ככ ௧ כ ൅݇௧௥ெሾܯሿሾߜଶሿ ൅ ݇௧௥ெሾܯሿሺሾߜଶ ሿ൅ሾߜଶ ሿሻ ൅݇௧௥ௌሾܵሿሾߜଶሿ൅݇௧௥ௌሾܵሿሺሾߜଶ ሿ כככ ൅ሾߜଶ ሿሻ ଶ כ כ כ ଶ כ כ כ ൅݇௧௖ሺʹሾߜ଴ሿሾߜଶሿ൅ʹሾߜଵሿ െʹሾߜଵሿሾߜ଴ሿെሾߜ଴ሿ ሻ כככ ככ כ כככ ככ כ כככ ככ כ ௧ ൅݇௧௖ሾሾߜଶሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ ൅ʹሾߜଵሿሺሾߜଵ ሿ൅ሾߜଵ ሿሻ ൅ሾߜ଴ሿሺሾߜଶ ሿ൅ሾߜଶ ሿሻሿ ככ ככ ଶ ככ ככ ככ ௧௧ ൅݇௧௖ሺሾߜଶ ሿሾߜ଴ ሿ൅ʹሾߜଵ ሿ ൅ሾߜ଴ ሿሾߜଶ ሿሻ כככ ככ כככ כככ ככ כככ ௧௧ ൅݇௧௖ሾሾߜ଴ ሿሺሾߜଶ ሿ൅ሾߜଶ ሿሻ ൅ʹሾߜଵ ሿሺሾߜଵ ሿ൅ሾߜଵ ሿሻ כככ ככ כ ௧ כ כ כככ ככ כככ ൅ሾߜଶ ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻሿ ൅݇௧ௗሾߜ଴ሿሾߜଶሿ൅݇௧ௗሾߜଶሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ כככ ככ כככ ௧௧ ככ ככ ௧௧ ൅݇௧ௗሾߜ଴ ሿሾߜଶ ሿ൅݇௧ௗሾߜ଴ ሿሺሾߜଶ ሿ൅ሾߜଶ ሿሻ

כ ሿሻככ ሿ ሾככ ሿ ሾכככ ሺሾ כ ݎʒబ ൌെ݇௠௔௖ሾߜ଴ሿሾʒ଴ሿ൅ʹ݇ఉ ߜ଴ ൅ ߜଵ െ͵ߜ଴ െ݇௧௥௉ሾߜ଴ሿሾʒଵሿ כככ ככ כ ௧ כ כ כ כ ൅݇௧ௗሾߜ଴ሿሺሾߜ଴ሿെሾܴଵሿെሾܴଶሿሻ ൅݇௧ௗሾߜ଴ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ ଶ כככ ௧௧ כככ ככ ככ ௧௧ ൅݇௧ௗሾߜ଴ ሿሺሾߜ଴ ሿ൅ሾߜ଴ ሿሻ ൅݇௧ௗሾߜ଴ ሿ

כ ሻ כככ כככ כככ ሺ כ ݎʒభ ൌെ݇௠௔௖ሾߜ଴ሿሾʒଵሿ൅݇ఉ ͵ሾߜ଴ ሿ൅ሾߜଵ ሿെʹሾߜ଴ ሿ െ݇௧௥௉ሾߜ଴ሿሾʒଶሿ כככ ככ כ ௧ כ כ ככ ככ ככ ൅݇ఉሺሾߜଶ ሿ െ ሾߜଵ ሿ െ͸ሾߜ଴ ሿሻ ൅݇௧ௗሾߜ଴ሿሾߜଵሿ൅݇௧ௗሾߜ଴ሿሺሾߜଵ ሿ൅ሾߜଵ ሿሻ כככ כככ ௧௧ כככ ככ ככ ௧௧ ൅݇௧ௗሾߜ଴ ሿሺሾߜଵ ሿ൅ሾߜଵ ሿሻ ൅݇௧ௗሾߜ଴ ሿሾߜଵ ሿ

ሿሻכככ ሿ ሾכככ ሿ ሾכככ ሾ ሿ ሺሾ כ ݎʒమ ൌെ݇௠௔௖ሾߜ଴ሿ ʒଶ ൅݇ఉ ߜଶ െͶ ߜଵ ൅ͳ͵ߜ଴

ʒଶ ଶ כ ככ ככ ככ ൅݇ఉሺͳͲሾߜଶ ሿ െ ͷͻሾߜଵ ሿ ൅ ͻ͵ሾߜ଴ ሿሻെ݇௧௥௉ሾߜ଴ሿ ൤൬ ൰ ൫ʹʒ଴ʒଶ െʒଵ ൯൨ ʒ଴ʒଵ כככ ככ ככ ௧௧ כככ ככ כ ௧ כ כ ൅݇௧ௗሾߜ଴ሿሾߜଶሿ ൅݇௧ௗሾߜ଴ሿሺሾߜଶ ሿ൅ሾߜଶ ሿሻ ൅݇௧ௗሾߜ଴ ሿሺሾߜଶ ሿ൅ሾߜଶ ሿሻ כככ כככ ௧௧ ൅݇௧ௗሾߜ଴ ሿሾߜଶ ሿ 127

8.2.3. Batch Reactor Model

Mole balances on all species and balances on the moments lead to a batch reactor model that consists of 21 ordinary differential equations:

ௗሾ௃ሿ ൌݎǡሾܬሿሺͲሻ ൌሾܬሿ (8.6) ௗ௧ ௃ ଴ כככ כככ כככ ככ ככ ככ כ כ כ כ כ כ כ where ܬൌܯǡܵǡܴ଴ǡܴଵǡܴଶǡܴଷǡߜ଴ǡߜଵǡߜଶǡߜ଴ ǡߜଵ ǡߜଶ ǡߜ଴ ǡߜଵ ǡߜଶ ǡߣ଴ǡߣଵǡߣଶǡʒ଴ǡʒଵǡʒଶǤ

In this bulk polymerization case, all initial concentrations are zero except for that of monomer, which is nonzero and is denoted by [M]0. The number- and weight-average molecular weights of dead polymer chains are calculated using:

ఒభାʒభ ܯ௡ ൌ ܯ௠ (8.7) ఒబାʒబ

ఒమାʒమ ܯ௪ ൌ ܯ௠ (8.8) ఒభାʒభ and monomer conversion using:

ሾெሿ ܺൌͳെ (8.9) ሾெሿబ where ܯ௠ is the molecular weight of the monomer.

8.3. Experimental and Analytical Procedures

The monomer, 98% n-butyl acrylate stabilized with 50 ppm of 4-methoxyphenol as inhibitor, is from Alfa Aesar. Batch reactors are 4-inch stainless steel Swagelok tubes

(Swagelok Inc., Huntingdon Valley, PA), capped at both ends with Swagelok stainless steel caps. These tubes can withstand pressures up to 3300 psig. Proton NMR is used to determine conversion. The NMR analysis is conducted on a 300 MGz Varian instrument; the NMR free induction decay signal is processed using ACD/NMR analytical software version 12.01.

For each set of batch experiments, 30-50 ml of n-butyl acrylate was dripped through an 128 inhibitor removal column DHR-4, from Scientific Polymer Products of Ontario, New

York, for 150 minutes in order to remove the inhibitor. The inhibitor-free monomer is collected in a 50-ml round bottom flask equipped with a standard taper 24/40 ground glass joint. After one hour of UHP nitrogen bubbling, we remove the needles and wrap the rubber septum tightly with aluminum foil secured with tight rubber bands. The flask with the inhibitor free, oxygen free monomer is then transferred to a glove bag (catalogue number 108D R-37-37 obtained from Glascol Inc. of Terre Haute, Inc.). Oxygen is removed from the glove bag using three ultra-high purity nitrogen gassing/degassing cycles and then refilled with ultra-high purity nitrogen. A small opening is created in the glove bag and ultra-high purity nitrogen at about 2 psi is fed through the glove bag for one hour. The glove bag is then sealed. After the oxygen is removed from the glove bag, the flask containing inhibitor-free, nitrogen purged monomer is opened in the glove bag and let oxygen equilibrate for 30 minutes. 2.5 ml of monomer is transferred into each of the stainless steel reactors (reaction tubes). The reactor tubes are then capped tightly and placed horizontally in a silicone oil bath already heated to 140 oC. GE Silicone oil SF-

96-500 is the heating fluid and is heated by a heater-stirrer (model Ceramag-midi from

IKA Werke) with a fuzzy temperature controller (model ETS-D4, also from IKA). Each heated tube is pulled out at a specified time instant and cooled quickly in a water bath to prevent further reaction from taking place. The tube contents are then emptied into vials, and 0.1 ml polymer samples are then diluted in 0.6 ml deuterated chloroform (CDCl3,

99.8% with 0.03% v/v tetramethylsilane, catalogue number 41389 obtained from Alfa

Aesar) to be taken for proton NMR to determine conversion. 129

8.4. Parameter Estimation

Rate constants of all reactions except the monomer self-initiation reaction are given in

Table 8-2. The only unknown rate constant, ki , is estimated from monomer-conversion measurements. Reaction rate constant definitions and correlations are provided in Table

8-3. Dimensionless kinetic parameter values are also given in Table 8-4. The system of ordinary differential equations (ODEs) is integrated numerically using the MATLAB routine bvp5c. For parameter estimation, the MATLAB function fminsearch, which is based on the Nelder-Mead simplex algorithm, is used to find the value of ki that minimizes the sum of the squared relative differences between measurements and model- predicted values of conversion.

Table 8-2: Reaction Rate Constant Values

Parameter Frequency Factor Activation Ref. Energy kJ.mol-1 -1 -1 ݇௣ 2.21E+07 L.mol .s 17.9 [28] ௧ -1 -1 ݇௣ 1.20E+06 L.mol .s 28.6 [168] -1 ݇௕௕ 7.41E+07 s 32.7 [121] -1 -1 ݇௧௥ெ 2.90E+05 L.mol .s 32.6 [83] -1 -1 ݇௧ 3.89E+09 L.mol .s 8.4 [118] ௧௧ -1 -1 ݇௧ 5.30E+09 L.mol .s 19.6 [9] -1 ݇ఉ 1.49E+09 s 63.9 [9] -1 -1 ݇௧௥௉ 4.01E+03 L.mol .s 29.0 [20] ܥ௧௥ௌ 1.07E+02 35.4 [9]

Using the numerical method, the monomer self-initiation rate constant, ki, is estimated to be 4.39E-15 M-1L-1 at 413 K, which is almost in agreement with the value (1.04E-14 M-

1L-1) obtained through quantum chemical calculations [170]. In numerically solving the 130 minimization (parameter estimation) problem, the self-initiation rate constant value estimated via quantum chemical calculations [170], 1.04E-14 M-1L-1, was used as the initial guess.

Table 8-3: Reaction Rate Constant Definitions and Correlations [9, 167]

݇௧ ൌ݇௧௖ ൅݇௧ௗ ௧ ௧ ௧ ݇௧ ൌ݇௧௖ ൅݇௧ௗ ௧௧ ௧௧ ௧௧ ݇௧ ൌ݇௧௖ ൅݇௧ௗ ݇௧ௗ ൌߜ௦݇௧ ௧௧ ௧௧ ݇௧ௗ ൌߜ௧݇௧ ௧ ௧௧ ݇௧ௗ ൌߜ௦௧ට݇௧݇௧

݇௧௖ ൌ ሺͳെߜ௦ሻ݇௧ ௧௧ ௧௧ ݇௧௖ ൌ ሺͳെߜ௧ሻ݇௧ ௧ ௧௧ ݇௧௖ ൌሺͳെߜ௦௧ሻට݇௧݇௧

݇௧௥ௌ ൌܥ௧௥ௌ݇௣ ௧ ௧ ݇௣ ݇௧௥ெ ൌ ݇௧௥ெ ݇௣ ௧ ௧ ݇௣ ݇௧௥ௌ ൌ ݇௧௥ௌ ݇௣ ݇௠௔௖ ൌߛ݇௣

Table 8-4: Dimensionless Kinetic Parameter Values Needed to Calculate Rate Constants Given in Table 8-3 [9, 167]

Parameter Dimensionless Value ߜ௦ 0.1 ߜ௦௧ 0.7 ߜ௧ 0.9 Γ 0.5

Figure 7-2 compares the model predictions and measurements of conversion at 413 K. It also indicates the sufficient sensitivity of the conversion predictions to the self-initiation 131

0.04 Model Prediction (ki=ki*) 0.035 Model Prediction (ki=1.1ki*) 0.03 Model Prediction (ki=0.9ki*) Measurement 0.025

X 0.02 0.015 0.01 0.005 0 0 50 100 150 200 250 Time (min)

Figure 8-1: Model Prediction and Measurements of Conversion as well as the Sensitivity -1 -1 of Predicted Conversion to 10% Changes in ki (ki*=4.39E-15 L.mol .s ) at 413 K.

0.09 0.08 Model Prediction Measurement 0.07 0.06 0.05 X 0.04 0.03 0.02 0.01 0 0 50 100 150 200 250 Time (min)

-1 -1 Figure 8-2: Model Prediction and Measurements of Conversion (ki=2.09E-14 L.mol .s ) at 433 K. 132 rate constant. Model predictions and measurements of conversion at higher temperatures

(433 K and 453 K) are shown in Figure 8-2 and Figure 8-3. ki is estimated to be 2.09E-14

M-1L-1 and 3.50E-13 M-1L-1 at 433 K and 453 K, respectively. Activation energy and frequency factor are also reported in Table 8-5 and compared with those obtained through quantum chemical calculations. The quantum calculations were based on transition-state

[11] and nonadiabatic transition-state theories [170]. Equation 3.3 is applied to calculate

Ea at 298 K in Table 8-5. We also assumed that activation energy and frequency factor remain constant as the temperature increases.

0.3 Model Prediction 0.25 Measurement

0.2

X 0.15

0.1

0.05

0 0 50 100 150 200 250 Time (min)

-1 -1 Figure 8-3: Model Prediction and Measurements of Conversion (ki=3.50E-13 L.mol .s ) at 433 K.

298[11] 133

-1 Table 8-5: Activation Energy (Ea), Energy Barrier (E0) in kJ mol ; Frequency Factor (A) -1 -1 and Rate Constant (ki) in M s for Monomer Self-Initiation Reaction

TEa E0 ki logeA Quantum 298 172.50 170.8[11] 9.77E-27[11] 9.68[11] Chemical 413 172.50 170.8[11] 2.43E-18 9.68 Calculations [11, 170] 413 115.00 ------1.04E-14[170] 1.38 Macroscopic 413 169.00 4.39E-15 16.04 Model ------

8.5. Concluding Remarks

A macroscopic mechanistic mathematical model of a batch polymerization reactor in

which spontaneous thermal polymerization of n-BA takes place was presented. The

model accounted for monomer self-initiation reaction and was based on the method of

moments. The monomer self-initiation rate constant was estimated from monomer

conversion measurements. The estimated values for rate constant, activation energy, and

frequency factor were in agreement with those obtained through quantum chemical

calculations. The model predicted monomer self-initiation rate constant was pretty close

to the value reported for the rate limiting step of initiation reaction. 134

9. Conclusions and Future Directions

9.1. Conclusions

The mechanisms of chain transfer reactions in self-initiated high-temperature polymerization of three alkyl acrylates were studied using different levels of theory.

Abstraction of a methylene group hydrogen by a live polymer chain in EA and n-BA, and a methyl group hydrogen in MA were found to be the favorable mechanisms for chain transfer to monomer reaction. The kinetic parameters predicted using M06-2X/6-

31G(d,p) are closest to those estimated from polymer sample measurement. Similarity of the predicted NMR spectra of dead polymer chains from the most probable mechanisms of CTM reactions to those from polymer sample analyses confirmed that the products of the chain transfer reactions proposed in this study are formed in polymerization of MA,

EA and n-BA. Increasing the chain length of the live polymer indicated that the polymer chain length had little or no effect on the activation energies and rate constants of the most likely mechanisms of CTM reactions. Two types of mono-radicals formed from the most probable mechanism of self-initiation reaction showed similar hydrogen abstraction abilities. It was found that a tertiary radical has much lower reactivity toward CTM reactions than a secondary radical.

B3LYP, X3LYP and M06-2X functionals were applied to explore different mechanisms of CTP reactions in self-initiated high-temperature polymerization of MA, EA, and n-BA.

Hydrogen atom abstraction from a tertiary carbon atom, which leads to the formation of a stable tertiary radical, was found to be the most favorable mechanism for CTP reaction in alkyl acrylates. Studying different structures of the dead polymers involved in CTP reactions showed that termination through hydrogen abstraction results in the formation 135

of dead polymers having very high reactivity toward CTP reaction. To explore chain

length effects on the kinetic parameters of the CTP reactions monomeric units were

added to the mono-radical involved in the most likely mechanism of CTP reaction. It was

found that the kinetic parameters of the CTP reaction barely changes as the chain length

• • of the live polymer increases. M2 was found to be as reactive as M1 in CTP reactions of

MA. Four different basis sets (6-31G(d), 6-31G(d,p), 6-311G(d) and 6-311G(d,p)) were

applied to validate the transition states and energy barriers for alkyl acrylates. Although

similar transition state geometries for each CTP mechanism for MA, EA, and n-BA were

obtained, the activation energies and rate constants of various CTP mechanisms using

these basis sets varied from each other by at most 10 kJ/mol and 2 orders of magnitude

respectively. It was also found that the end substituent group in the monomers has little

influence on the energy barriers of the chain transfer reactions.

The mechanisms for chain transfer to n-butanol, MEK, and p-xylene in self-initiated

high-temperature polymerization of three alklyl acrylates were studied using different

functional and basis sets. Abstraction of a hydrogen from the methylene group next to the

oxygen atom in n-butanol, from the methylene group in MEK, and from the methyl group

in p-xylene by a live polymer chain was found to be the most likely mechanism of CTS

reactions in MA, EA, and n-BA. Chain transfer to tert-butanol has the highest energy

barrier and the lowest rate constant relative to chain transfer to n-butanol and sec-butanol.

Chain transfer to n-butanol and sec-butanol reactions have comparable kinetic parameter values. The activation energy of the most likely chain transfer to p-xylene mechanism of

▪ a 2-monomer-unit live n-BA polymer chain initiated by M2 calculated using M06-2X/6-

31G(d,p) was found to be in agreement with those estimated from polymer sample 136 measurements. Application of PCM resulted in remarkable changes in the kinetic parameters of the chain transfer to n-butanol. COSMO was also used to explore the solvent effects on the kinetics of CTS reactions of MA, EA, and n-BA. COSMO did not have any significant impact on the kinetic parameters of the reactions. It was found that the live polymer chain length has a little effect on the activation energies and rate

▪ ▪ constants of CTS reactions. MA, EA and n-BA live chains initiated by M2 and M1 showed similar capabilities for undergoing CTS reactions, indicating the lesser influence of self-initiating species on CTS reactions.

A macroscopic mechanistic mathematical model of a batch polymerization reactor in which spontaneous thermal polymerization of n-BA takes place was also presented. The model was based on the method of moments and accounted for monomer self-initiation reaction. The monomer self-initiation rate constant was estimated from monomer conversion measurements. The estimate indicates that the sole contribution of the monomer through self-initiation to the initiation of the polymerization is not significant enough to achieve a very high monomer conversion in a reasonably short period of time.

9.2. Future Directions

Since the conversion measurements from experiments in the presence and absence of oxygen showed the significant contribution of dissolved oxygen to chain initiation in free-radical polymerization of n-BA, a model based on a comprehensive set of the most- likely free-radical polymerization reactions should be developed considering oxygen co- initiation and monomer self-initiation. The rate of initiation reactions can be estimated from conversion measurements made during polymerization batch cycles in the presence and absence of oxygen. Obtained results can be used to gain a better understanding of the 137 role of oxygen and monomer in the thermal polymerization of acrylates.

PPV derivatives are widely used in light emitting devices [171-173] and in spin-coating technique to analyze mechanical properties [174]. Although PPVs are being used because of their great electroluminescent properties [175], oxidation reactions [176, 177], and imbalance of different carrier currents (holes and electrons) [178] are adversely affecting their efficiency (i.e. photons out per charge injected) and performance (i.e. conductivity and mechanical properties).

Oxidation reaction has been shown to be affected by adding side groups, such as alkyl, alkoxy, and nitrile [176, 177]. The electron current in PPVs is usually found to be lower than the hole current (electron mobility is about 3 orders of magnitude lower than hole mobility) [179-181]. PPVs have been shown to be good hole-transport materials while having low hole-injection barriers [182, 183]. The electron traps distribution in the band gap [184], has been reported as a problem in PPVs which causes an imbalance of different carrier currents leading to reduce the device efficiency. It has also been reported that the water molecules appeared within PPV can make complexes with oxygen molecules (hydrated oxygens) [185]. This complex formation leads to an increase in electron affinity which creates electron trap sites [185]. Hydrated oxygen complexes, as the origin of trap sites, have been validated through the evaluation of electron affinity by considering a continuum dielectric (ε=2-10) with respect to the gas phase [185].

Performing the following tasks can great assist to design high performance light emitting diodes (LED) and enhance their efficiency:

a) Exploring different oxidation mechanisms and proposing stable derivatives of

substituted-PPV and PPA against oxidation reaction: 138

The reaction of singlet oxygen with various substituted-PPV and PPA should be

explored. Lower reactivity between triple bond in PPA derivatives and singlet oxygen

makes PPA as a good replacement for PPV derivatives. We will estimate kinetic

parameters of different oxidation reactions to study which side group can reduce the rate

of the oxidation reaction more. Side groups such as nitrile, alkyl, alkoxy, and large side

groups like phenyl groups will be selected.

b) Controlling the rate of β-scission which is followed by another oxidation

reaction:

We will study β-scission reactions for different substituted polymers. We will compare

the rate constant of β-scission with oxidation reaction. Choice of side groups for reducing the rate of β-scission reaction would be useful. For lower-rate β-scission reactions we will try other possible mechanism (such as termination or propagation). Products of oxidation reaction plays significant role in the properties of the final product. Changes in the rate of β-scission reaction (depends on different side groups of polymer derivatives) can control the polydispersity and consequently the performance of LEDs. Reduction in the rate of this reaction can prevent higher degradation of the polymers.

c) Proposing substituted-PPV and PPA derivatives which are capable of

reducing the stability of hydrated oxygen complexes to prevent electron

trapping:

We will study the polarity effects of surrounding environment. We will also explore reaction of solvent-oxygen complex formation and that of polymer with solvent and oxygen molecules (considering different types of solvents having different polarities).

We will compare the mechanisms and stability of hydrated oxygen complexes. Since the 139 hydrated oxygen complexes have been suggested to serve as the origin of electron trapping, formation of unstable complexes can lead to an increase in efficiency.

d) Applying Quantum Mechanics (QM)/Molecular Mechanics (MM) Monte

Carlo method to simulate the mechanisms and explore the solvent effects:

First-principles methods at low computational cost can be applied to model systems with medium-sized molecules. However, for larger systems (i.e. more than 80 atoms) especially in the condensed phase (in solutions) exploring the mechanisms and barriers estimation would be challenging. Computational methods to combine quantum mechanics (QM) and molecular mechanics (MM) have been developed [101, 186-188] during the past decades. The most important problem in this type of simulations is that the QM calculations for the solute should be repeated many times for all configurations of the surrounding (solvent) which are described by MM till gaining satisfactory statistical average [189]. A fast and accurate QM/MM hybrid method has been described to simulate solute in a polar solvent, recently [186]. The response of the solute to the solvent electric field is evaluated by pre-calculated electric moments and polarizabilities

[190]. Therefore, instead of repeating the QM calculations, the solute is described by its pre-calculated molecular energy in an arbitrary-shaped external electrostatic potential of the environment [186].

Instantaneous polarization of the solute by solvent molecules is the main advantage of this approach relative to previous ones. The first-order reduced density matrix, distributed multipoles moments, and generalized multipole polarizability [186] should be calculated for a fixed geometry of solute molecule to be used in evaluation of electrostatic interaction energy between solute and instantaneous configurations of solvent in a Monte 140

Carlo simulation. Neglecting [191, 192] or even approximating [193] the polarization of

QM region by MM region prevents simulating a full recalculation of QM wave function

[145]. 141

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Vita

Name: Nazanin Moghadam Maragheh

Education:

• Ph.D., Chemical Engineering, Drexel University, 2009 to present (Graduation:

March, 2015)

• Master of Engineering, Chemical Engineering, Biotechnology, Sharif University of Technology, Tehran, Iran, 2006

• Bachelor of Engineering, Chemical Engineering, Sharif University of

Technology, Tehran, Iran, 2001

Selected Publication:

Moghadam N., S. Liu, S. Srinivasan, M. C. Grady, M. Soroush, A. M. Rappe,

Computational Study of Chain Transfer to Monomer Reactions in High-Temperature

Polymerization of Alkyl Acrylates, J. Phys. Chem. A (117) 2605-2618, 2013

Moghadam N., S. Srinivasan, M. C. Grady, A. M. Rappe, M. Soroush, Theoretical

Study of Chain Transfer to Solvent Reactions of Alkyl Acrylates, J. Phys. Chem. A (118)

5474-5487, 2014

Moghadam N., S. Liu, S. Srinivasan, M. C. Grady, A. M. Rappe, M. Soroush,

Theoretical Study of Intermolecular Chain Transfer to Polymer Reactions of Alkyl

Acrylates, Ind. Eng. Chem. Res., DOI: 10.1021/ie504110n, 2014