Supplementary Information for Atomistic Simulation of Spinodal Phase Separation Preceding Polymer Crystallization

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1 Supplementary Information for Atomistic Simulation of Spinodal Phase Separation Preceding Polymer Crystallization Richard H. Gee * Naida Lacevic and Laurence E. Fried University of California Lawrence Livermore National Laboratory Chemistry and Materials Science Directorate P.O. Box 808, L-268 Livermore, California Computational Details All simulations were performed using Livermore Computing machines. Two different machine architectures were used; the first is a cluster of 2048 Intel Xeon 2.4GHz processors (MCR cluster), the second is a cluster of 4096 Intel Itanium2 Tiger4 1.4GHz processors (Thunder cluster). The peak performance of the MCR and Thunder clusters is TFlop/s and 22.9 TFlop/s, respectively. We used 1024 (Thunder) or 2048 (MCR) processors. The total simulation time of our polymer simulations were ~50 ns, which took ~1x10 6 processor hours for each polymer melt studied. The LAMMPS 1 parallel code was used throughout. Polymer Model The MD simulations of the polymer melts were carried out on bulk amorphous ensemble of both polar and non-polar linear polymers. All polymers were represented as unitedatoms (UA). The simplified UA model affords the investigation of much larger polymer ensembles, thus reducing finite size effects. The polar polymer melt is composed of 8, bead polymer chains (2,073,600 united-atoms; M w = 7,682 g/mol). The non-polar polymer melts are composed of either 20, bead polymer chains (4,915,200 united-atoms; M w = 3,360 g/mol), or 5, bead polymer chains (4,478,976 unitedatoms; M w = 10,752 g/mol). The UA polymer models are representative of either a poly(vinylidene fluoride) (pvdf) polymer analogue (polar linear polymer), or a polyethylene (PE) polymer analogue (non-polar linear polymer). In the UA polymer models, the hydrogen or fluorine atoms are lumped onto the carbon backbone atoms to which they are attached. The force field parameters for pvdf 2, 3 and PE 4 consists of both valence (stretch, bending, and torsion terms) and nonbonded potential terms (van der Waals and Coulomb). All valence degrees of freedom were explicitly treated and unimpeded. The force field parameters for UA pvdf are shown below for completeness E bond = 1 k R R 2 b( o) 2 (2) E angle = 1 k 2 θ θ θ o ( ) 2 (3) * To whom correspondence should be addressed.

2 E torsion = 1 V 2 3( 1 + cos( 3φ ))+ 1 V cos( φ) ( ) (4) 12 r E vdw = D o o r 2 r o r 6 (5) E Coulomb = K Q iq j r ij (6) k b = 700 kcal/mol Å -2, R o = 1.52 Å, k θ = 100 kcal/mol rad -2, θ o = o, V 1 = 3.2, V 3 = 0.8 in kcal/mol 5, D o CF = 0.08 kcal/mol, r o CF = 5.65 Å, D o CH = 0.12 kcal/mol, r o CH = 4.5 Å, Q CF = 0.23, Q CH = -0.23, and the constant K = gives the energies in kcal/mol. The initial starting polymer configurations were generated using a Monte Carlo method. The monomers were allowed to polymerize in a head-to-tail manner with no monomer reversals. The initial atomic positions of the amorphous melt structures were generated using a random distribution of torsional angles, which was generated using a Monte Carlo method that assigns random values to all rotatable torsions in the polymer chain. The resulting amorphous structure was then relaxed by energy minimization. To further characterizatize the model pvdf melt, we have computed the minimum polymer chain-length or critical molecular weight, M c for the formation of stable entanglements using the approach described by Fetters et al. 6 The model polymer is characterized by its monomer density ρ, polymer chain molecular weight M, monomer mass m, square end-to-end distance per monomer ( b 2 <R 2 /N> ), the Kuhn length b k,, the entanglement critical molecular weight, M c = mn c, and the molecular weight between entanglements, M e = mn e, where M c 2M e. N e and N c can be calculated from ρ, b 2, and a fixed length p. In our model pvdf these quantities are calculated using equations 3 and 14 from reference [6]. We obtain M e = 3,349 g/mol and M c = 6,698 g/mol which compares well with the values estimated using the group-contribution method for estimating polymer properties 7 of M e = 2,880 g/mol and M c = 5,759 g/mol. ). The molecular weight of our pvdf polymer chains is 7,680 g/mol, well above the calculated M e. Further; we have calculated these quantities for the polyethylene (768 bead) melts, where the experimental and theoretical M c and M e values for PE are M c = 4,000 g/mol and M e = 2,000 g/mol and M c = 4,200 g/mol and M e = 2,100 g/mol, respectively. 8 Our calculated M c value for PE is 3,986 g/mol (M e = 1,993 g/mol), in good agreement with both the theoretical and experimental values. The molecular weight, M w, of our polymer chains are M w pvdf [240 beads] = 7,680 g/mol; M w PE [240 beads] = 3,360 g/mol; M w PE [768 beads] = 10,752 g/mol, clearly larger than M e. We also calculated the mean-square end-to-end distance, R Å 2, and the meansquare radius of gyrations, R g Å 2 for pvdf at 600 K. The ratio of the mean-

3 square end-to-end distance to the mean-square of the radius of gyration, R g which is consistent with other polymer models. 9 This ratio is ~6.2 for our simulations of PE. Note, that for polymers chains molecules forming random disordered coils in the melt Gaussian chains the ratio of mean-square end-to-end distance to the mean-square radius of gyration, R 2 R g 2 = 6. The ratio reaches a steady state regime after less then 5 ns indicating that the system is equilibrated with respect to chain conformations. We have also measured this ratio for an extensive simulation of PE and observed similar scaling relationships. Due to relatively low molecular weight of our model polymer, the comparison with the experimental values for R 2 2 and R g is difficult because typical experiments are performed at much higher molecular weights. Thus, we provide a characteristic ratio C = R2 Nl = 6 R 2 g. The experimental value for C 2 Nl 2 at T = 463 K is 5.6 ± 0.3, 10 the value for C at T = 300 K estimated using the group-contribution method for estimating polymer properties 7 is 6.9, and the value for C computed by Smith et al. is 6.22 at T = 463 K. Tonelli et al. 11 computed C to be in the range of 3.87 to 14.5 depending on the dielectric. We estimate C at T = 650 K to be Further, we estimate C for polyethylene (768-bead) at a temperature of 600 K and 300 K to be 9.25 and 6.4, respectively, where the experimental values of C at 413 K is Our computed C values are within this experimental range. Molecular Dynamics Method We used three-dimensional cubic periodic boundary conditions. The simulations were generated using constant particle number, pressure, and temperature (NPT) dynamics at a pressure of 0 atm. All computations were carried out using the LAMMPS code. 1 The equations of motion were integrated using the Verlet algorithm 13 with a time step of 2.0 fs for temperatures above the onset temperature of polymer ordering (600 K for pvdf and 450 K for PE) and 4 fs at temperatures at which polymer ordering occurred (600 K for pvdf and 450 K for PE). A Nose-Hoover-type thermostat 14 with a relaxation time of 0.1 ps was used to control the temperature, and the pressure was controlled isotropically. 15 The nonbonded van der Waals interactions were treated by truncating atom pairs with an inter-atomic distance greater than 12 Å. The particle-particle particle-mesh Ewald (PPPM) method 16 was used for the long-range treatment of electrostatic interactions in the polar polymer melts. The polar polymer melts were initially simulated at 750 K, and the non-polar polymer melts were initially simulated at 600 K. The periodic box was allowed to relax under NPT conditions for a minimum of 5 ns. Following this step, the melts were cooled in NPT runs in increments of 50 K and equilibrated for a minimum of 5 ns each at the desired R 2

4 temperature (750 K, 700 K, and 650 K for pvdf melts and 600 K, 550 K, and 500 K for PE melts). At the temperature at which the polymer melts order, the simulations were carried out for more than 30 ns, 37 ns or 20 ns for pvdf, PE (240-bead), and PE (768- bead), respectively. Figure S1; R. Gee et al. Simulation cell dimension versus time from NPT dynamics for our analogue models of pvdf 2, 3 (blue curve) and 240-bead PE 4 (red curve). The simulation cell dimensions approach equilibrium values (are no longer time dependent) within approximately 5 ns or less above the temperature at which polymer ordering occurs (600 K for pvdf and 450 K for PE).

5 Figure S2; R. Gee et al. The structure factor, S(q,t) for model pvdf 2, 3 at 600 K and 650 K at different times during the simulation; the red curves show the lack of growth in the amplitude of S(q,t) as a function of time at 650 K, the blue curve shows a significant increase in the overall amplitude of S(q,t) at long times in the simulation indicative of polymer ordering at 600 K. The inset shows the second-order Bragg peak in S(q,t); the transition from unoriented to oriented dense (smectic) packing is apparent only at a temperature of 600 K. Similar behavior is found for PE.

6 Figure S3; R. Gee et al. The decay of the torsional autocorrelation function, cosφ(t)cosφ(0) cosφ(0) 2 R φ =, for simulated bulk melts of model pvdf 2, 3 at cos 2 φ(0) cosφ(0) K. The relaxation time, τ, was determined by fitting the Kohlrausch-Williams-Watts function 17, 18, γ (t) = exp ( t / τ) β (solid blue curve), to the simulation data (open circles). The relaxation time, τ, is determined to be 7.8 ps and β is The computed relaxation time computed here is comparable to the relaxation times found for other 19, 20 highly flexible linear model polymers.

7 Figure S4: R. Gee et al. Comparison of melt structure factor, S m (q), and single-chain structure factor, S ch (q), (panel a). Structure factor of polymer chains with the Debye functional 21 fits corresponding to the independently measured radius of gyration (panel b). Data shown for simulations of bulk melts of model PE (768-bead) 4 at 600 K. No discernible differences are identifiable in the structural features of the melts over the entire time trajectory. The results shown here compare well with other reported values of these metrics

8 Figure S5: R. Gee et al. The single chain incoherent intermediate scattering function, r ur r ur 1 N sin( qr S inc i(t) rj (0) ) N sin( qr i(0) rj (0) ) ch (q,t) = r ur r ur i, j =1 qr i(t) rj (0), at three different q i, j =1 qr i(0) rj (0) r ur vectors, where r i(t) rj (0) is the distance between (united) atom i at time t and atom j at time 0 (panel a). The dependence of the characteristic ratio on chain length for times of 4.36 (black) and 8.76 (red) ns (panel b). Data shown for simulations of bulk melts of model PE (768-bead) 4 at 600 K. Only slight differences are seen in the characteristic ratio versus time. The results shown here compare well with other reported values of these 25, 26 metrics.

9 Figure S6: R. Gee et al. Cahn-Hilliard plot derived from the early stages of the nucleation process in the polymer crystallization simulations for the 240-bead non-polar polymer model (PE) at 450 K. The solid line is a fit to the data. The linear nature of the plot demonstrates the presence of a spinodal-assisted crystallization process.

10 Supporting References 1. Plimpton, S. J. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comp. Phys. 117, 1-19 (1995). 2. Mayo, S. L., Olafson, B. D. & Goddard, W. A. J.Phys. Chem. 94, 8897 (1990). 3. Gao, G. (California Institute of Technology, 1998). 4. Paul, W., Yoon, D. Y. & Smith, G. D. An optimized united atom model for simulations of polymethylene melts. J. Chem. Phys. 103, (1995). 5. The torsional terms were modified for ease of inclusion into the computer code. 6. Fetters, L. J., Lohse, D. J. & Graessley, W. W. Chain Dimensions and Entanglement Spacings in Dense Macromolecular Systems. J. Polym. Sci.: Part B: Polymer Physics 37, (1999). 7. Bicerano, J. Prediction of Polymer Properties (Marcel Dekker, Inc., New York, 1993). 8. Wool, R. P. Polymer Entanglements. Macromolecules 26, 1564 (1993). 9. Smith, G. D. et al. Molecular dynamics of a 1,4-polybutadiene melt. Comparison of experiment and simulation. Macromolecules 32, (1999). 10. Welch, G. J. Solution Properties And Unperturbed Dimensions Of Poly(Vinylidene Fluoride). Polymer 15, (1974). 11. Tonelli, A. E. Conormational characteristics of Poly(vinylidene fluoride). Macromolecules 9, 547 (1976). 12. Flory, P. J. Statistical Mechanics of Chain Molecules (Interscience, New York, 1969). 13. Verlet, L. Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Phys. Rev. 159, 98 (1967). 14. Nose, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81, 511 (1984), and references therein. 15. Melchionna, S., Ciccotti, G. & Holian, B. L. Hoover's style Molecular Dynamics for systems varying in shape and size. Mol. Phys. 78, 533 (1993). 16. Hockney, R. W. & Eastwood, J. W. Computer Simulation using Particles (McGraw-Hill, New York, 1981). 17. Kohlrausch, R. Ann. Phys. (Lpz.) 12, 393 (1847). 18. Williams, G. & Watts, D. C. Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans. Faraday Soc. 66, 80 (1970). 19. Gee, R. H. & Boyd, R. H. Conformational Dynamics and Relaxation in Bulk Polybutadienes: A Molecular Dynamics Simulation Study. J. Chem. Phys. 101, 8028 (1994). 20. Boyd, R. H., Gee, R. H., Han, J. & Jin, Y. Conformational Dynamics in Bulk Polyethylene: A Molecular Dynamics Simulation Study. J. Chem. Phys. 101, 788 (1994). 21. Doi, M. & Edwards, S. F. The Theory of Polymer Dynamics (Clarendon, Oxford, 1986). 22. Paul, W. & Smith, G. D. Structure and dynamics of amorphous polymers: computer simulations compared to experiment and theory. Reports on Progress in Physics 67, (2004).

11 23. Bennemann, C., Paul, W., Binder, K. & Dunweg, B. Molecular-dynamics simulations of the thermal glass transition in polymer melts: alpha-relaxation behavior. Physical Review E 57, (1998). 24. Paul, W., Binder, K., Heermann, D. W. & Kremer, K. Crossover Scaling In Semidilute Polymer-Solutions - A Monte-Carlo Test. Journal De Physique Ii 1, (1991). 25. Smith, G. D. et al. Local dynamics in long-chain alkane melt from molecular dynamics simulations and neutron scattering experiments. J. Chem. Phys. 107, (1997). 26. Auhl, R., Everaers, R., Grest, G. S., Kremer, K. & Plimpton, S. J. Equilibration of long chain polymer melts in computer simulations. Journal Of Chemical Physics 119, (2003).