74 these states cannot be reliably obtained from experiments. In addition, the barriers between the local minima can also not be obtained reliably fro

Transcription

1 73 Chapter 5 Development of Adiabatic Force Field for Polyvinyl Chloride (PVC) and Chlorinated PVC (CPVC) 5.1 Introduction Chlorinated polyvinyl chloride has become an important specialty polymer due to its high glass transition temperature, high heat distortion temperature, outstanding mechanical, dielectric, and ame and smoke properties, chemical inertness, and low sensitivity tohydrocarbon costs. However, the mechanism through which the various desired and undesired properties are resulted from is not fully understood. Hopefully, simulation at the atomistic level could lead us to a better understanding of those mechanisms. Currently, direct ab initio calculations for polymer systems are not practical. Thus calculations with classical force elds, which are parameterized based on either experimental results or ab initio calculations on smaller model systems, are the method of choice. As a rst step towards the understanding of CPVC, we developed the adiabatic quantum force eld that accurately described the rotational energy surface of the polymer backbone chains. For amorphous polymers, the distribution of backbone conformations and the rates of conformational transitions have a strong eect on their properties, such as moduli, glass temperature, dielectric constant, and diusivity of small molecules. It is critical that the FF leads to the correct relative energies of the minima, e.g., trans versus gauche, and of the barrier heights between them. Thus torsional FF parameters are particularly important for describing amorphous polymers. In many cases, the existence of the molecule in other local minima can be detected, but energies for

2 74 these states cannot be reliably obtained from experiments. In addition, the barriers between the local minima can also not be obtained reliably from the experimental data alone. To circumvent these problems, we use ab initio calculations to provide the torsional potential energy surface. With the 6-31G basis set, the torsional potentials calculated from Hartree-Fock (HF) wavefunctions are adequate. The HF calculations lead to a total torsional potential function E HF (). The classical force eld can be tted to reproduce the quantum energy surface. E HF () ' E FF () (5.1) In determining E FF (), the usual and simplest approach would be to determine the non-adiabatic surface by xing all bonds and angles so that only the torsional angle changes. However, such rigid rotations about backbonds sometimes lead to unfavorable contacts with very short distances between nonbonded atoms. The ab initio wavefunction readjusts the molecular orbitals to minimize repulsion, but the functional forms of nonbond interactions in force eld representation may not accurately describe the inner repulsive wall and often leads to much higher rotational barriers. In order to accurately describe the rotational energy surface, we calculated adiabatic rotational energy surface of molecules with ve backbone carbons. These molecules are used to mimick the corresponding polymer chains. The HF wavefunction was calculated by xing the dihedrals of interest (in increments of 30 o ) and optimizing all other degrees of freedom. These calculations lead to the 2D energy surfaces. Torsional parameters were tted iteratively so that the force eld adiabatic energy surface matchs to ab initio adiabatic energy surface.

3 The Molecular Simulation Force Field (MSFF) The force eld is taken to be of the form E total = E val + E nb : (5.2) The valence part includes bond interactions, angle interactions, and torsion interactions, as E val = E bond + E angle + E torsion : (5.3) The nonbond part has van der Waals interaction and Coulomb interaction. E nb = E vdw + E Q (5.4) The torsion terms involve sums of cosine torsional angles such as E torsion () = The nonbond terms have the form of E Q = X m=0 X i6=j C m cos m: (5.5) q i q j R ij (5.6) for electrostatic, and E vdw = 1 2 X i6=j D e (,12 ij, 2,6 ij ) (5.7) for van der Waals, where ij = R ij =R e. Chain conformation and inter-chain interactions are the dominating factors for amorphous polymers. We'll focus on accurate description of charges and torsional potentials. The forms and parameters of E bond ;E angle, and E vdw are taken directly

4 76 from DREIDING. The charges and torsional parameters are based on ab initio calculations of clusters with ve-backbone carbon atoms Charges In order to determine the proper charges, we did ab initio calculation for various conformations of clusters with ve-backbone carbon atoms: H 3 C, CXY, CHH, CXY, CH 3 (5.8) where X and Y are Cl or H, depending on the form of the actual chloro-polymers. For PVC, X = Cl and Y = H, and for PVDC, X = Cl and Y = Cl. Hartree-Fock (HF) wavefunction with the 6-31G** basis set is used. We considered the following three methods of assigning atomic charges in the chain molecules. Potential derived charge (PDQ). The charge density from the HF wave function is used to calculate the potential energy over a numerical grid surrounding the molecule and a set of point charges on the atoms is optimized to t the potential. We carried out these calculation with PSGVB using a grid of 1000 points outside the van der Waals radii (taken as R C =1:949 (A), R H =1:597 (A), R Cl =1:958 (A), and R F =1:739 (A)). Mulliken charges (Mull). The molecular orbital (MO) coecients are used to estimate a set of atomic charges where overlap terms are assigned equally to each of the two atoms. Charge Equilibration (QEq). The charges of molecules are predicted based on electron anity (EA) and ionization energy (IE). Since the atomic charges vary with the change of molecular conformation, the best description of Coulomb interactions in dynamics simulations should be such, that the

5 77 atomic charges vary with conformation change. However, for amorphous polymer simulations, which require longer chains, assigning charges at every dynamics integration is simply unrealistic. We have to nd a way to best balance various conformations, while compensating the errors made in charge assignment in the torsional force eld. We based the charges on PDQ while considering symmetry property of the clusters Torsional Potential We can write E hf ( 1 ; 2 )=E ff ( 1 ; 2 )+E cor ( 1 ; 2 ) (5.9) where ( 1 ; 2 ) are the two C{C{C{C backbone torsion angles as in Fig (5.1). E hf ( 1 ; 2 ) is ab initio energy, E ff ( 1 ; 2 ) is force eld energy, and E cor ( 1 ; 2 ) is the correction with which the old force eld should be improved. E ff ( 1 ; 2 ) is calculated by minimizing the structures while ( 1 ; 2 ) are constraint, so that the adiabatic 2D ( 1 ; 2 ) rotational energy surface of optimized force eld can match that of ab initio computation well. In order to get accurate adiabatic potentials, we generated D energy surfaces through quantum computation, i.e., constraint the two torsional angles while full Hartree-Fock optimizations were performed. We also used fully optimized structures of all the local minimums. The 2-D torsional potential surfaces were represented by regular grids which are interpolated from the points and all of the local minimum points. The C m in Eq. 5.5 are least-square-tted to minimize E cor ( 1 ; 2 ). Many iterations are performed until the changes of C m are insignicant.

6 78 Cl_2 Cl_1 H_1 Cl_2 H_2 Cl_1 H_1 H_2 C_2 C_2 φ2 H_5 φ2 C_4 C_4 H_10 H_5 H_10 C_1 C_3 C_1 φ1 φ1 C_3 C_5 C_5 H_6 H_7 H_3 H_4 H_8 H_9 H_7 H_4 H_6 H_3 H_8 H_9 Isotactic Syndiotactic Cl_3 Cl_2 Cl_1 H_2 C_2 φ2 C_4 H_5 H_10 C_1 φ1 C_3 C_5 H_7 H_4 H_6 H_3 H_8 H_9 Mixed Cl_3 Cl_4 Cl_2 Cl_1 C_2 φ2 C_4 H_10 H_5 φ1 C_3 C_5 C_1 H_7 H_3 H_4 H_8 H_9 H_6 PVDC Figure 5.1: Chlorinated clusters used in the calculations 5.3 Quantum Mechanical Adiabatic 2D Rotational Energy Surface and Forece Field Parameters For PVC, both syndiotactic and isotactic, we use CH 3 CHClCH 2 CHClCH 3 to represent longer chain polymer, as shown in Fig. 5.1a and Fig. 5.1b. For CPVC of -(CCl 2 CH 2 CHClCH 2 ) n -,we use CH 3 CCl 2 CH 2 CHClCH 3, as in Fig. 5.1c. For PVDC, -(CCl 2 CH 2 )-, molecule of CH 3 CCl 2 CH 2 CCl 2 CH 3 is used as in Fig. 5.1d. Figure 5.2 shows the atomic charges assigned to these clusters. Denote dihedral angle Cl{C 3x{C 3{C 3x as. For C 3x, if >0, we use atomic type (label) C 3R, while if < 0, we use atomic type (label) C 3L. C 32 is used for backbone carbon atoms bonded to two Cl atoms, and C 3 is used for backbone

7 Isotactic Syndiotactic Mixed PVDC Figure 5.2: Atomic charges of the chlorinated clusters carbon atoms that are not bonded to any Cl atoms. By doing so, we can use ipvc torsion parameters for C 3{C 3L{C 3{C 3L, or C 3{C 3R{C 3{C 3R, and spvc torsion parameters for C 3{C 3R{C 3{C 3L. PVDC segments and mixed segments are obvious. All other torsion parameters are default DREIDII parameters (C 0 =1:0000, C 3 =1:0000, and the rest are C m = 0). They are listed in Table 5.1. The optimized torsion parameters are listed in Table 5.2.

8 80 Table 5.1: DREIDING Parameters LJ 12{6 van der Waals Simple Harmonic Bond Atom a R 0 b D 0 Bond c K bond a R 0 H C{C C C{H Cl C{Cl Simple Harmonic Cosine Angle Angle d K angle e 0 Angle d K angle e 0 H{C{H C{C{C H{C{C Cl{C{C H{C{Cl Cl{C{Cl a A; b Kcal=Mol; c Kcal=Mol= A2 ; d Kcal=Mol=Degree 2, e degree. Table 5.2: Optimized DREIDING Torsion Parameters for PVC LLL-CCC-CCC-RRR v 1 v 2 v 3 v 4 v 5 v 6 Isotactic Polyvinyl Chloride (ipvc) C 3-C 31-C 3-C Cl-C 31-C 3-C Syndiotactic Polyvinyl Chloride (spvc) C 3-C 31-C 3-C Cl-C 31-C 3-C Polyvinylidene Chloride (PVDC) C 3-C 32-C 3-C Cl-C 32-C 3-C PVC-PVDC C 3-C 31-C 3-C C 31-C 3-C 32-C Cl-C 31-C 3-C Cl-C 32-C 3-C C 32 is C atom bonded to two Cl atoms, C 31 is C atom bonded to one Cl atom, while C 3 is C atom without Cl atom bonded to. The unit is (kcal/mol).

9 Isotactic Polyvinyl Chloride Figure 5.3: 3D plot of the adiabatic 2D-rotation energy surface. The unit of the two torsion angles is in 10 o, while the unit of energy(z-axis) is in kcal/mol Based on the reection symmetry of the molecule, we can obtain the whole energy surface by reecting half of the 1 { 2 space, as shown in Fig The grid points are generated at interval of 30 o, total of grid points. Considering the reection symmetry and rotational symmetry of two torsion angles, we have 76 independent grid points. For each of the 76 points, as rst step approximation, we optimize the structure by using DREIDII force eld with the two torsion angles xed. This gives us a better starting point for quantum optimization that is quite expensive computationally. Then we perform quantum mechanical structural minimization,

10 82 Figure 5.4: Quantum adiabatic 2D rotational surface for ipvc keeping the two backbone torsions xed. By plotting the whole energy surface, we can extract 6 local minimum grid poins. Starting from those points, we optimized the whole structures, including the backbone torsion angles. Quantum adiabatic energy surface is interpolated based on 76 grid points and the 6 local minimum points. By doing so, we can capture both the rotational barriers and minimum energies of local minimums. Figure 5.3 is the 3D plot of the energy surface, while Fig. 5.4 is 2D contour plot. Based on the procedure outlined in Section 2, we calculated the torsional parameters. Figure 5.5 is the contour map of force eld adiabatic 2D rotational energy surface. They are in good agreement with the ab initio results. The torsion param-

11 83 Figure 5.5: Force eld adiabatic 2D rotational surface for ipvc eters are in Table 5.2. The comparisons of quantum and force eld energy at each gird point are tabulated in Table 5.3, while comparison of local minimums are in Table 5.4. Since ab initio constraint structure minimization are very expensive, most of researchers calculate quantum energy based on generic force eld generated grid points. For 28 grid points(interval of 30 o ), we used DREIDII force eld minimized the structures while keeping the two backbone torsion angles xed. For each of those structures, we did one energy quantum calculation. Figure 5.6 is the 2D contour map based on those grid points. The energies are tabulated in Table 5.5 By focusing on the relative energy dierences of local minimums and the energy barrieres between

12 84 Figure 5.6: One energy quantum calculation of DREIDII adiabatic 2D rotational grids for ipvc those local minimums, we found the adiabatic quantum rotational energy surface is quite dierent from the quantum one energy energy surface and the tted force eld energy surface is a very good approximation of the quantum adiabatic energy surface. In molecular dynamics simulations, by using these force elds, we can generate ensembles with the right distribution of thermodynamic density of states and rates of kinetic conformation. These are the general goals of molecular dynamics simulation of amorphous polymer materials.

13 85 Table 5.3: Adiabatic QM and FF Energy of ipvc(kcal/mol) 1 2 QM FF Error 1 2 QM FF Error

14 86 Table 5.4: QM and FF Local Minimums of ipvc(kcal/mole) 1 2 QM FF Error 1 2 QM FF Error Table 5.5: QM Energy of FF Grids for ipvc(kcal/mol) 1 2 QM 1 2 QM

15 Syndiotactic Polyvinyl Chloride Figure 5.7: 3D plot of adiabatic 2D rotational surface for spvc, z-axis is the energy with unit kcal/mol. x-axis and y-axis correspond to the two backbone dihedral angles with unit 10 o Similar to ipvc, there is also a reection plane on the 2D rotational space for spvc. The reection axis is perpendicular to that of ipvc. 76 grid points are required for an angle increment of 30 o. The 3D plot of quantum adiabatic rotational energy surface is in Fig. 5.7, while the 2D contour map is in Fig Figure 5.9 is the 2D contour map of adiabatic rotational energy surface, base on optimized force eld. The quantum and force eld energies at the grid points are tabulated in Table 5.6, and those of the six local minimums are in Table 5.7. As a comparison, we calculated the quantum one energy on force eld generated grids. The 2D contour map is in

16 88 Figure 5.8: Quantum adiabatic 2D rotational surface for spvc Fig The energies are tabulated in Table [5.8].

17 89 Figure 5.9: Force eld adiabatic 2D rotational surface for spvc

18 90 Table 5.6: Adiabatic QM and FF Energy of spvc(kcal/mol) 1 2 QM FF Error 1 2 QM FF Error

19 91 Figure 5.10: Quantum one energy calculation of DREIDII adiabatic 2D rotational grids for spvc Table 5.7: Local Minimums for spvc(kcal/mole) 1 2 QM FF Error 1 2 QM FF Error

20 92 Table 5.8: QM Energy of FF Grids for spvc(kcal/mole) 1 2 QM 1 2 QM

21 Polyvinylidene Chloride Figure 5.11: 3D plot of adiabatic 2D rotational surface for PVDC (Angle in 10 o ) Polyvinylidene Chloride has higher symmetry than that of ipvc and spvc. For interval of 30 o, we used 50 grid points that covers one quater of the 2D space, plus four local minimums. Figure 5.11 is the 3D plot of quantum adiabatic energy surface, while the 2D contour map is in Fig The results from optimized force eld are plotted in Fig Comparison of energies at each point are tabulated in Table 5.9. We also did quantum one energy calculation on grids generated based on DREIDII force eld. The 2D contour map of rotational energy surface is plotted in Fig. 5.14, and tabulated in Table [5.10].

22 94 Figure 5.12: Quantum adiabatic 2D rotational surface for PVDC

23 95 Figure 5.13: Force eld adiabatic 2D rotational surface for PVDC

24 96 Table 5.9: Adiabatic QM and FF Energy of PVDC (Kcal/Mole) 1 2 QM FF Error 1 2 QM FF Error Local Minimums

25 97 Figure 5.14: Quantum one energy calculation of DREIDII adiabatic 2D rotational grids for PVDC

26 98 Table 5.10: QM Energy of FF Grids for PVDC (Kcal/Mole) 1 2 QM 1 2 QM

27 Mixture of Polyvinyl Chloride and Polyvinylidene Chloride Figure 5.15: 3D plot of adiabatic 2D rotational surface for PVC-PVDC (Angle in 10 o ) In the case of the mixture of Polyvinyl Chloride and Polyvinylidene Chloride, there is no reection symmetry. For an interval of 30 o, 144 grid points plus 9 local minimums are calculated. The 3D quantum adiabatic energy surface is plotted in Fig. 5.15, while the 2D contour map plot is in Fig Figure 5.17 is the 2D contour map based on optimized force eld. The energies at grid points are tabulated in Table 5.11 and Table The 9 local minimums are tabulated in Table Figure 5.18 is the 2D contour map of quantum one energy calculations on DREI-

28 100 Figure 5.16: Quantum adiabatic 2D rotational surface for PVC-PVDC IDII force eld generated grids. The energies are listed in Table Summary The quality of force elds for amorphous polymers are mainly determined by two factors; one is the relative energy dierence between the local minimums of torsional conformationsr, which dictates the equilibrium distributions of torsional states thermodynamically; the other factor is the energy barriers between various local optimum conformations, which determines the rate of conformation transitions, i.e., the rigidity of polymer chains. Compared to the usual one-energy quantum potential based on

29 101 Figure 5.17: Force eld adiabatic 2D rotational surface for PVC-PVDC one torsion angle, adiabatic 2D quantum potential calculations are a big step forward, the new approach included the correlation of adjacent torsion angles. Adiabatic molecular simulation force eld (MSFF) suitable for carrying out molecular dynamics simulations of amorphous polymers (polyvinyl chloride, polyvinylidene chloride) are developed. These force elds can be used in molecular dynamics simulations to study physical properties of amorphous polymers such as glass transition temperature and diusivity ofgas molecules. The force elds are based on adiabatic 2D (two adjacent backbone torsions) rotational energy surfaces generated by using ab initio calculations. Clusters with vebackbone carbon atoms are used to mimmick the polymer chains. These 2D grids

30 102 Figure 5.18: Quantum one energy calculation of DREIDII adiabatic 2D rotational grids for PVC-PVDC are based on rotating two backbone torsions with incement of 30 o. For each conformation (symmetry properties are being used to reduce the number of points), we optimize the whole structure while the two backbone torsions are constrainted. These grid points plus local minimum points (fully optimized) are used to generate the 2-D adiabatic potential energy surfaces. Force elds are tted so that the adiabatic force eld potential energy surfaces match to that of quantum adiabatic potential surface. These force elds will be used in molecular dynamics simulations of glass transition and gas diusions.

31 5.5 References R.G. Parker, G.A., Martello, \Chemical Modications: Chlorinated PVC," Encyclopedia of PVC, 1986, A.K. Rappe, W.A. Goddard III, \Charge equilibration for molecular-dynamics simulations," J. Phys. Chem. 95(8), 1991, S.L. Mayo, B.D. Olafson, W.A. Goddard III, \DREIDING - a generic force-eld for molecular simulations," J. Phys. Chem. 94(26), 1990,

32 104 Table 5.11: Adiabatic QM and FF Energy of PVC-PVDC(I)(Kcal/Mol) 1 2 QM FF Error 1 2 QM FF Error

33 105 Table 5.12: Adiabatic QM and FF Energy of PVC-PVDC(II)(Kcal/Mol) 1 2 QM FF Error 1 2 QM FF Error

34 106 Table 5.13: Local Minimums for PVC-PVDC(Kcal/Mole) Structure 1 2 QM FF Error Table 5.14: QM Energy of FF Grids for PVC-PVDC(Kcal/Mole) 1 2 QM 1 2 QM

35 5.6 Appendix 107 Listed in the tables are the geometry of optimized local minimums for ipvc, spvc, PVDC, and PVC-PVDC.

36 108 Table 5.15: Bonds and Angles of ipvc Local Optimums Structure Bond Distances (A) C 2 C C 2 C C 3 C C 4 C C 2 Cl C 4 Cl C 2 H C 4 H C 3 H C 3 H C 1 H C 1 H C 1 H C 5 H C 5 H C 5 H Angles (degree) C 3 C 2 C C 4 C 3 C C 5 C 4 C Cl 1 C 2 H Cl 1 C 2 C Cl 1 C 2 C Cl 2 C 4 H Cl 2 C 4 C Cl 2 C 4 C H 4 C 3 H Backbone Torsion Angles (degree) C 1{C 2{C 3{C C 2{C 3{C 4{C Minimium Energy (Kcal=Mol) Energy HF (6-31G**) ab initio PS-GVB Constraint Minimizations

37 109 Table 5.16: Bonds and Angles of spvc Local Optimums structure Bond Distances (A) C 2 C C 3 C C 4 C C 4 C C 2 Cl C 4 Cl C 2 H C 4 H C 3 H C 3 H C 1 H C 1 H C 1 H C 5 H C 5 H C 5 H Angles (degree) C 1 C 2 C C 2 C 3 C C 3 C 4 C Cl 1 C 2 H Cl 1 C 2 C Cl 1 C 2 C Cl 2 C 4 H Cl 2 C 4 C Cl 2 C 4 C H 3 C 3 H Backbone Torsion Angles (degree) C 1{C 2{C 3{C C 2{C 3{C 4{C Minimium Energy (Kcal=Mol) Energy HF (6-31G**) ab initio PS-GVB Constraint Minimizations

38 110 Table 5.17: Bonds and Angles of PVDC Local Optimums structure Bond Distances (A) C 2 C C 3 C C 4 C C 4 C C 2 Cl C 4 Cl C 2 Cl C 4 Cl C 3 H C 3 H C 1 H C 1 H C 1 H C 5 H C 5 H C 5 H Angles (degree) C 1 C 2 C C 2 C 3 C C 3 C 4 C Cl 1 C 2 Cl Cl 1 C 2 C Cl 1 C 2 C Cl 2 C 4 Cl Cl 2 C 4 C Cl 2 C 4 C Cl 3 C 2 C Cl 3 C 2 C Cl 4 C 4 C Cl 4 C 4 C H 3 C 3 H Backbone Torsion Angles (degree) C 1{C 2{C 3{C C 2{C 3{C 4{C Minimium Energy (Kcal=Mol) Energy HF (6-31G**) ab initio PS-GVB Constraint Minimizations

39 111 Table 5.18: Bonds and Angles of PVC-PVDC Local Optimums Property opt1 opt2 opt3 opt4 opt5 opt6 opt7 opt8 opt9 Bond Distances (A) C 2 C C 2 C C 3 C C 4 C C 2 Cl C 4 Cl C 2 Cl C 4 H C 3 H C 3 H C 1 H C 1 H C 1 H C 5 H C 5 H C 5 H Angles (degree) C 3 -C 2 -C C 4 -C 3 -C C 5 -C 4 -C Cl 3 -C 2 -Cl Cl 1 -C 2 -C Cl 1 -C 2 -C Cl 2 -C 4 -H Cl 2 -C 4 -C Cl 2 -C 4 -C Cl 3 -C 2 -C Cl 3 -C 2 -C H 4 -C 3 -H Backbone Torsion Angles (degree) C 1 C 2 C 3 C C 2 C 3 C 4 C Minimium Energy (Kcal=Mol) Energy HF (6-31G**) ab initio PS-GVB Constraint Minimizations