Viscoelastic response of polymers using non-affine lattice dynamics: Bridging the timescale gap

# Presented in "Condensed-Matter-Highlights-2025" at University of Milan (Italy) on 20/02/2025
https://sites.google.com/site/somunimi/

# A similar talk has been presented at DPG-2024 in Berlin
https://www.dpg-verhandlungen.de/year/2024/conference/berlin/part/cpp/session/8/contribution/1

# A similar talk has been presented at DYFP-2024 in Kerkrade (Netherlands)
https://www.dyfp-conferences.org

# Associated publications:
https://pubs.acs.org/doi/10.1021/acs.macromol.4c01360
https://doi.org/10.1063/5.0272171

Transcript

1. Viscoelastic response of polymers using non-affine lattice dynamics: Bridging the timescale gap VINAY VAIBHAV University of Milan Alessio Zaccone, UniMi, Milan Ankit Singh, UniMi, Milan Timothy Sirk, USA Army Lab, Maryland Caterina Czibula, Northwestern University, Evanston
2. Timescale Gap # Highest rate accessible in experiments: ~ 104 s-1 # Lowest rate accessible in simulations: ~ 1010 s-1 Timescale gap of 6 orders of magnitude DMA: oscillatory shear MD Simulation: oscillatory shear Strain, ε Stress, σ Ultimate strength Strain hardening Necking Rise Young's modulus = Slope = Fracture Yield strength Run Run Rise Wikipedia Δx A F l Θ
3. Timescale Gap # Highest rate accessible in experiments: ~ 104 s-1 # Lowest rate accessible in simulations: ~ 1010 s-1 Timescale gap of 6 orders of magnitude DMA: oscillatory shear MD Simulation: oscillatory shear Strain, ε Stress, σ Ultimate strength Strain hardening Necking Rise Young's modulus = Slope = Fracture Yield strength Run Run Rise Wikipedia Δx A F l Θ
4. Timescale Gap # Highest rate accessible in experiments: ~ 104 s-1 # Lowest rate accessible in simulations: ~ 1010 s-1 Timescale gap of 6 orders of magnitude DMA: oscillatory shear MD Simulation: oscillatory shear Strain, ε Stress, σ Ultimate strength Strain hardening Necking Rise Young's modulus = Slope = Fracture Yield strength Run Run Rise Wikipedia Δx A F l Θ
5. Timescale Gap # Highest rate accessible in experiments: ~ 104 s-1 # Lowest rate accessible in simulations: ~ 1010 s-1 Timescale gap of 6 orders of magnitude DMA: oscillatory shear MD Simulation: oscillatory shear Strain, ε Stress, σ Ultimate strength Strain hardening Necking Rise Young's modulus = Slope = Fracture Yield strength Run Run Rise Wikipedia Δx A F l Θ Non-af fi ne Lattice Dynamics: Timescale Bridging ?
6. Af fi ne vs Non-af fi ne deformation # Af fi ne deformation: sum of forces is zeros; homogeneous deformation # Non-af fi ne deformation: non-zero force fi eld; additional displacement M. Born and H. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1954
7. NALD: Non-af fi ne Lattice Dynamics # Derived using particle-bath Caldeira-Leggett model: damped harmonic oscillator — tagged particle in a sea of other atoms — write Hamiltonian considering eigenfrequency dependent interaction with bath atoms Non-af fi ne force Hessian or dynamical matrix vDOS force fi eld correlator Applied strain frequency Born modulus Af fi ne contribution Damping parameter Zwanzig, J. Stat. Phys. (1973) Lemaitre & Maloney, J. Stat. Phys. (2006) Palyulin, Ness, Milkus, Elder, Sirk, Zaccone, Soft Matter (2018) vDOS:
8. NALD: Non-af fi ne Lattice Dynamics # Derived using particle-bath Caldeira-Leggett model: damped harmonic oscillator — tagged particle in a sea of other atoms — write Hamiltonian considering eigenfrequency dependent interaction with bath atoms Non-af fi ne force Hessian or dynamical matrix vDOS force fi eld correlator Applied strain frequency Born modulus Af fi ne contribution Damping parameter Zwanzig, J. Stat. Phys. (1973) Lemaitre & Maloney, J. Stat. Phys. (2006) Palyulin, Ness, Milkus, Elder, Sirk, Zaccone, Soft Matter (2018) Hessian matrix “normal modes” “instantaneous normal modes” λ>0 λ<0 vDOS:
9. Zwanzig, J. Stat. Phys. (1973) Lemaitre & Maloney, J. Stat. Phys. (2006) Palyulin, Ness, Milkus, Elder, Sirk, Zaccone, Soft Matter (2018) # MD simulation Static con fi guration Hessian (3Nx3N symmetric matrix) # Diagonalize the hessian matrix — eigenvalues eigenfrequency vibrational density of state — eigenmodes # Gamma: force fi eld correlator NALD: computational protocol Hessian matrix “normal modes” “instantaneous normal modes” λ>0 λ<0 Storage modulus: Loss modulus: Projection of force- fi eld on eigenmodes
10. Zwanzig, J. Stat. Phys. (1973) Lemaitre & Maloney, J. Stat. Phys. (2006) Palyulin, Ness, Milkus, Elder, Sirk, Zaccone, Soft Matter (2018) # MD simulation Static con fi guration Hessian (3Nx3N symmetric matrix) # Diagonalize the hessian matrix — eigenvalues eigenfrequency vibrational density of state — eigenmodes # Gamma: force fi eld correlator NALD: computational protocol Hessian matrix “normal modes” “instantaneous normal modes” λ>0 λ<0 Storage modulus: Loss modulus: Projection of force- fi eld on eigenmodes Microscopic input: Hessian, force- fi eld Modulus as a function of deformation frequency
11. Coarse-grained polymer: theory vs simulation # Kremer-Grest polymer model (Bead-spring): LJ + Fene; MD simulation (LAMMPS) # 100 monomers; 100 chains # NALD # Non-equilibrium molecular dynamics: oscillatory shear
12. Kremer-Grest polymer: theory vs simulation Storage modulus Loss modulus NALD MD Simulation: oscillatory shear
13. Kremer-Grest polymer: theory vs simulation Storage modulus Loss modulus NALD MD Simulation: oscillatory shear
14. Kremer-Grest polymer: theory vs simulation Storage modulus Loss modulus NALD Strain Stress MD Simulation: oscillatory shear
15. NALD: Silica
16. Cross-linked Epoxy Polymers Widespread application of epoxy Epoxy resins react with cross-linking molecules to form a network
17. Cross-linked Epoxy Polymers Widespread application of epoxy Epoxy resins react with cross-linking molecules to form a network Viscoelstic response? Theory vs Simulation vs Experiment
18. Cross-linked Epoxy Bisphenol A diglycidyl ether (DGEBA) Tg = 445.9 K # Amber force fi eld; Long range interaction # Slow cooling from high temperature to below Tg — Rubbery to glassy state # NPT; LAMMPS # N = 10074 H3C O O O O CH3 O NH2 H2N [ ] n = 3 NH2 R + O R’ N OH R’ R OH R’ 2 Vaibhav, Sirk, and Zaccone, Macromolecules, 57(23) (2024) Poly(oxypropylene) diamine (POP) Commercially available: DGEBA and JEFFAMINE D-230 Huntsman corporation
19. 0 200 400 600 ! (THz) 0.000 0 200 400 600 ! (THz) 0 2 4 6 ˜ °(!) £ 104 (N2kg°1) (b) 0 100 200 300 ! (THz) 0.0 0.1 0.2 0.3 P(!) 0 200 400 600 ! (THz) 0.000 0.004 0.008 0.012 D(!) (a) T = 210 K T = 300 K T = 405 K 0 200 400 600 ! (THz) 0 2 4 6 ˜ °(!) £ 104 (N2kg°1) (b) 0 100 200 300 ! (THz) 0.0 0.1 0.2 0.3 P(!) Cross-linked Epoxy: NALD Vibrational density of states 10°6 10°4 10°2 100 102 104 ≠(THz) 100 101 102 G0(GPa) T = 405 T = 300 T = 210 Af fi ne force fi eld correlator Storage modulus # Response covers a wider spectrum # Plateau storage modulus at small frequency # Non-monotonic response at higher frequency Storage modulus Vaibhav, Sirk, and Zaccone, Macromolecules, 57(23) (2024)
20. Cross-linked Epoxy: NALD vs Simulation 10°3 10°1 101 103 105 ≠ (THz) 101 102 G(GPa) MD NALD °2 0 2 °2 0 2 Stress(GPa) °0.01 0.00 0.01 °2 0 2 °0.01 0.00 0.01 °0.01 0.00 0.01 20.01 50.27 200.10 300.63 400.20 502.65 604.15 700.47 101341.70 Strain Stress-Strain Modulus vs frequency Reasonably good match between theory and MD simulation Diverse stress-strain relationship Vaibhav, Sirk, and Zaccone, Macromolecules, 57(23) (2024)
21. Cross-linked Epoxy: NALD vs Experiment https://www.sandia.gov/polymer-properties/828-d230-shear-modulus-vs-temp/ Plateau storage modulus as a function of temperature Experiment: Dynamic Mechanical Analysis (DMA) data from Sandia Labs 10°6 10°4 10°2 100 102 104 ≠(THz) 100 101 102 G0(GPa) T = 405 T = 300 T = 210 Vaibhav, Sirk, and Zaccone, Macromolecules, 57(23) (2024) °250 °200 °150 °100 °50 0 T ° Tg (K) 10°1 100 101 102 G0(GPa) GA Experiment Theoretical Fit NALD NALD ° Athermal
22. NALD: Thermoplastics Polyetherimide (PEI) Polycarbonate (PC) Polyphenylsulfone (PPSU) Polymethyl methacrylate (PMMA) Work in progress Timothy Sirk US Army Lab Caterina Czibula Northwestern
23. NALD: PMMA 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 -200 -100 0 100 200 300 400 500 600 700 T = 300 K D(ω) (THz) ω Vibrational density of states 250 275 300 325 350 375 400 425 W (THz) 10°2 10°1 100 G0 (GPa) DMA Experiment NALD Theoretical Fit Polymethyl methacrylate (PMMA) Caterina Czibula Northwestern
24. NALD: PMMA 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 -200 -100 0 100 200 300 400 500 600 700 T = 300 K D(ω) (THz) ω Vibrational density of states 250 275 300 325 350 375 400 425 W (THz) 10°2 10°1 100 G0 (GPa) DMA Experiment NALD Theoretical Fit Polymethyl methacrylate (PMMA) Caterina Czibula Northwestern Vibrational Spectra Chemical Structure Mechanical Response Design next generation materials
25. NALD: Viscosity of polymer melt Ankit Singh UniMi Green-Kubo approach NALD approach NEMD approach Coarse-grained polymer
26. Summary and outlook # Bridging the timescale gap: Theory vs Simulation vs Experiment # Mechanical response near Tg: Microscopic description of friction; large systems # Liquid viscosity # Design next generation materials: Vibrational Spectra Chemical Structure Mechanical Response H3 C O O O O CH3 (a) (b) (d) 100 101 102 G0(GPa) GA Submitted Non-af fi ne lattice dynamics approach to compute the viscosity of polymeric systems A. Singh, V. Vaibhav, and A. Zaccone (2025) Ongoing Non-af fi ne softening modes in thermoplastics V. Vaibhav, C. Czibula, T. Sirk, and A. Zaccone (2025) 10°3 10°1 101 103 105 ≠ (THz) 101 102 G(GPa) MD NALD °200 °100 0 T ° Tg 10°1 100 101 102 G0(GPa) GA Experiment Theoretical Fit NALD NALD ° Athermal
27. Summary and outlook # Bridging the timescale gap: Theory vs Simulation vs Experiment # Mechanical response near Tg: Microscopic description of friction; large systems # Liquid viscosity # Design next generation materials: Vibrational Spectra Chemical Structure Mechanical Response H3 C O O O O CH3 (a) (b) (d) 100 101 102 G0(GPa) GA Submitted Non-af fi ne lattice dynamics approach to compute the viscosity of polymeric systems A. Singh, V. Vaibhav, and A. Zaccone (2025) Ongoing Non-af fi ne softening modes in thermoplastics V. Vaibhav, C. Czibula, T. Sirk, and A. Zaccone (2025) 10°3 10°1 101 103 105 ≠ (THz) 101 102 G(GPa) MD NALD °200 °100 0 T ° Tg 10°1 100 101 102 G0(GPa) GA Experiment Theoretical Fit NALD NALD ° Athermal Thank you