parametric studies in multiscale modeling of high ...gmodegar/papers/aiaa-2007-2173.pdfparametric...

Parametric Studies in Multiscale Modeling of High-Performance Polymers

P.K. Valavala*,1, T.C. Clancy†,2, G.M. Odegard‡,1, and T.S. Gates§,3

1Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931 2National Institute of Aerospace, Hampton, VA 23666 3NASA Langley Research Center, Hampton, VA 23681

A computational parametric study has been performed to establish the effect of Representative Volume Element (RVE) size, force field type, and simulation temperature on the predicted mechanical properties of polyimide and polycarbonate materials modeled atomisticially. The results of the simulations indicate no clear effect of RVE size and force field type on the predicted mechanical response of the polyimide and polycarbonate polymer systems. A multiscale modeling technique was utilized to determine the equivalent-continuum Young’s moduli, density, and stress-strain behavior for the set of mentioned modeling parameters. Parametric studies also indicate no clear effect of the simulation temperature on the predicted material densities of LaRC-CP2 when the AMBER force field is used. However, the MM3 force field predicts a steady decrease in the density of LaRC-CP2 as the temperature increases up to and beyond the glass transition temperature. These force fields vary slightly in form and with the associated parameters

I. Introduction

Novel nanostructured materials, i.e. those materials that have definable structure at the nano-scale, have the potential to be structured to provide measurable gains in mechanical properties relative to materials currently used for a wide range of engineering applications. Of particular interest is the development of durable polymer and polymer nanocomposite materials for aerospace structural applications. The nature of atomic interactions and their influence on bulk-level mechanical properties is not fully understood and computational studies are often more efficient and cost effective than experimental tools in understanding and characterizing interactions at this scale. To facilitate the development of polymer-based materials for structural applications, multiscale modeling techniques must be developed that provide reliable structure-property relationships.

Molecular Dynamics (MD) and Molecular Mechanics (MM) have been used in numerical simulations for prediction of the structure and mechanical properties of polymer-based material systems over multiple length scales.1-5 These studies have demonstrated that molecular modeling techniques can be effectively used to determine structure-property relationships of polymer-based material systems. MM is a procedure through which the potential energy of a molecular structure can be determined under static conditions.6 MD can be interpreted as a kinetic MM technique, which involves determination of the time evolution of a set of interacting particles under the influence of forces due to neighboring atoms. The interaction forces are obtained from a MM potential known as a force field.7 MD simulations can be used to obtain the equilibrated molecular structure of a polymeric material and to predict the behavior of the system when subjected to prescribed external force/displacement boundary conditions.

Accurate material constitutive models of polymeric materials require the understanding of the atomic-scale interactions. One limitation of modeling at this scale is that often atomistic simulations have an inherent limitation on the size of the material model. The authors believe that the choice of a representative volume element (RVE) for the molecular model can have a direct impact on the predicted mechanical properties. Linked to this issue is the fact that the number of atoms included in a model is many orders of magnitude less than the number in the bulk material. To date, few studies have addressed the necessary minimum size of a molecular modeling RVE to accurately predict bulk-scale mechanical properties of polymer-based material systems. Lorenz *Graduate Research Assistant, Student Member, AIAA †Staff Scientist ‡Assistant Professor, Senior Member, AIAA §Senior Research Scientist, Associate Fellow, AIAA

American Institute of Aeronautics and Astronautics

1

48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con23-26 April 2007, Honolulu, Hawaii

AIAA 2007-2173

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

and Stevens 8 considered a simple, binary, coarse-grained Lennard-Jones system to study void nucleation in cross- linked epoxy adhesives. Their study reported a small variation in yield strain for models of different RVE sizes. Despite the ground-breaking accomplishments of this study, it did not address the effective-continuum modeling and constitutive behavior of the polymer, and it did not address the fully atomistic response of the material since a simplified model was employed. Therefore, a better understanding of the effect of RVE size on the predicted mechanical behavior of high-performance polymers is necessary for the development of polymer nanocomposite materials.

The objective of the proposed research is to perform atomistic simulations and investigate the influence of a range of temperature, choice of force field, and size of representative volume element (RVE) on the predicted mechanical behavior of high-performance polymer systems (polyimide and polycarbonate).. The predicted mechanical properties of the polymer are compared to those obtained from experiment.

II. Materials In order to provide a comparative basis for the effectiveness of the current simulation procedure employing MM

and MD, three known polymeric systems were studied. These polymers are typically synthesized in the laboratory from step or condensation polymerization. A polycarbonate and a polyimide system were used to study the effects of RVE size and force fields on the predicted properties and a second polyimide system, (LaRC-CP2), was used to study the effects of simulated temperatures on the mechanical behavior.

Polyimides can be categorized into two broad classes based on their structure: linear polyimides and aromatic hetero-cyclic polyimides. Polyimides have been of use in engineered structures due to their inherent thermal stability, In general, a polyimide will have a glass transition temperature in excess of 300°C.. One of the polyimides used in this study (henceforth referred to as simply “polyimide”) is synthesized from 3,3´,4,4´-biphenyltetracarboxylic dianhydride (BPDA) and 1,3-bis(4-aminophenoxy) benzene (APB) monomers. Figure 1a shows the chemical structure of the polyimide.

The second polyimide, referred to as LaRC-CP2, is an amorphous, thermoplastic, colorless polyimide originally developed at NASA Langley Research Center. It is synthesized from 1,3-bis(3-aminophenoxy) benzene (APB) and 2,2-bis(3,4-anhydrodicarboxyphenyl) hexafluoropropane (6FDA).9 The chemical structure of the LaRC-CP2 repeat unit is shown in Figure 1b. This polyimide has been deemed a good candidate for use in inflatable solar concentrators and antennas.

Table I

Summary of input parameters for various cases studied

Material Force Fields No. of Atoms Temperature (in Kelvin)

No. of Chains (No. of Monomers/Chain)

4214 7 (10) 6622 11 (10) Polyimide AMBER, OPLS-

AA, MM3 8428 14 (10) 3972 6 (20) 5958 9 (20) Polycarbonate AMBER, OPLS-

AA, MM3 7944

300

12 (20) 73 173 296 373 423 473

LaRC-CP2 AMBER, MM3 9954

573

9 (16)

A poly-aromatic carbonate was used to study the effects of RVE size and force field type. It is typically

synthesized in the laboratory from bisphenol-A and phosgene monomers in the presence of sodium hydroxide. The chemical structure of a polycarbonate repeat unit is shown in Figure 1c. Polycarbonates have a wide range of applications due to their ease of manufacture and their desirable optical and mechanical properties.


2

Figure 1. Schematic illustration of chemical structures of (a) polyimide, (b) LaRC-CP2, and (c) polycarbonate monomers.

1a – Polyimide monomer

1b – LaRC-CP2 monomer

1c –Polycarbonate monomer

III. Modeling The multiscale modeling approach used in this study utilizes the concepts of effective-continuum constitutive

modeling to homogenize the discrete molecular structure of the polymer material based on energy equivalence3-5. The motivation for this modeling scheme is to transform a discrete atomistic material to an equivalent continuum which is capable of predicting the constitutive behavior of the material. Once the equivalent continuum constitutive model is established, it can be used with existing continuum mechanics theories to predict mechanical behavior. This multiscale method consists of three steps. First, a representative volume element (RVE) of the polymer is established that is capable of statistically representing the molecular structure of the material. The equilibrium structure of the RVE is obtained via a series of MD simulations. The second step involves establishing a constitutive equation that accurately describes the bulk mechanical behavior of the equivalent continuum based on the specific material symmetries that are applicable to the system under consideration. Examples of the RVEs of the three polymer systems are shown in Figure 2. Finally, the energies of deformation are equated for both molecular and continuum models under identical states of discrete deformation. The chemical potential energy for the molecular


3

model is obtained from MM techniques and the strain energy for the continuum model is calculated from the assumed constitutive relation. A series of deformations are chosen such that the equating of energies from the two models resolves equivalent-continuum material parameters. A. Force Fields

Force fields are used in atomistic simulations to describe the interactions between individual atoms and thus relate molecular configuration to the energy of the molecular system. Most force fields are semi-empirical and assume specific degrees of freedom for a given molecular structure. The total energy of the molecular model is obtained as the summation of energies from the individual degrees of freedom. Three widely used force fields, described fully in ref. 5, were utilized for this study, AMBER, OPLS-AA, and MM3. The AMBER and OPLS-AA force fields have a simple functional form and their implementation in the current simulations did not include the electrostatics interactions which are based on the dipole moments of the atoms. OPLS-AA has a functional form very similar to AMBER force field10-20, however, the force constants differ in magnitude. The MM3 force field has a rather complex functional form and includes higher order interactions. As a result, the MM3 force field requires longer computational times when used in an MD or MM simulation. Therefore, the tradeoff between computational time and complexity between these force fields can be established by comparing these three force fields. Further information on the mathematical form and characterization of these force fields can be found elsewhere11,13-17,20.

B. Representative Volume Elements (RVE)

An equilibrated molecular structure was obtained for each of the three polymer systems for different RVE sizes, force field types, and temperatures using the equilibration procedure described in this sub-section. Periodic boundary conditions (PBC) were applied to the simulation boxes of the MM and MD models to minimize finite size effects and to mimic bulk material conditions. Table II summarizes the results obtained from the MD simulations used to obtain equilibrium molecular RVEs for the polyimide and polycarbonate material models.

Table II

Summary of the physical properties from MD simulations

Material Force Field No. of Atoms Density (in g/cc) Energy (Kcal/mole) AMBER 1.00 2248.2

OPLS-AA 1.25 31979 4214 1.19 868.8 MM3

AMBER 1.15 42833.6 OPLS-AA 1.26 39921.8 6622

1.29 * MM3 AMBER 1.13 18687.9

OPLS-AA 1.26 13764.8

Polyimide

8428 1.28 * MM3

AMBER 1.16 480.5 OPLS-AA 1.18 21668.9 3972

1.11 937.3 MM3 AMBER 1.11 22983.9

OPLS-AA 1.17 6021.2 5958 1.14 * MM3

AMBER 0.99 10531.5 OPLS-AA 1.16 6944.6

Polycarbonate

MM3 7944

1.06 *


4

Polycarbonate Polyimide

LaRC-CP2

Figure 2. Representative volume element of Polycarbonate, Polyimide and LaRC-CP2.

Each molecular models of the polyimide was subjected to a series of energy minimizations and MD simulations

to equilibrate and minimize residual stresses due to the initial placement of chains. The TINKER21software package was used for all minimizations and MD simulations described in this paper. The MINIMIZE and NEWTON subroutines, which correspond to a quasi-Newton L-BFGS method and a truncated Newton22 energy minimization methods, respectively, were employed for energy minimizations. The minimizations were performed to RMS gradients of 0.01 and 0.00001 kcal/mole/Å, respectively. Each of the polyimide structures were subjected to the MD simulations in the following order at 300 K: (1) a 50ps simulation with the NVT (constant number of atoms, volume, and temperature) ensemble to prepare the structure for further equilibration, (2) a 100ps simulation with the NPT (constant number of atoms, pressure, and temperature) ensemble at 100atm to evolve the system to higher densities as the initial structure was prepared from a gas phase, (3) a 100ps NPT simulation at 1atm to reduce the effects of high pressure simulations and to let the system evolve to a state of minimal residual stresses, and (4) a 100ps NVT simulation to allow the system to equilibrate at the simulated temperature and density.

The polycarbonate model was prepared initially in the gas phase, as described in more detail elsewhere5. The resulting small, medium, and large RVEs constructed of the polycarbonate system contained 3972, 5958, and 7944 atoms, respectively, using six, nine, and twelve polymer chains, respectively, containing 20 monomer units. As with the polyimide, a total of seven structures were prepared: the small, medium, and large RVEs for both AMBER and OPLS-AA force fields and the small RVE for the MM3 force field as well. All seven of the polycarbonate models were subjected to the same set of MD simulations as the polyimide models for equilibration.

LaRC-CP2 was modeled to predict the effect of simulated temperature on the constitutive behavior. The amorphous RVE of the LaRC-CP2 polymer material was initially constructed in a low-density gas phase. The molecular model has 9954 atoms consisting of nine polymer chains with sixteen repeat units per chain. Initially, the


5

structure was subjected to a series of energy minimizations followed by a short 50ps MD simulation using the NVT ensemble and the MM3 force field to dynamically evolve the system to prepare for further simulations. The resulting RVE from was subsequently subjected to a series of MD simulations to establish separate thermally-equilibrated structures for seven different temperatures (73K, 173K, 296K, 373K, 423K, 473K, and 573K) for both AMBER and MM3 force fields. Each of these molecular structures were prepared by the following series of five simulations at each temperature: (1) a 50 ps NVT simulation to dynamically evolve the system at the desired temperature, (2) a 50 ps dynamic NPT simulation at a pressure of 100atm, (3) a 200 ps NPT simulation at a pressure of 1atm to minimize any residual stresses of the molecular structure, (4) a 100 ps NVT simulation to establish the equilibrated molecular structure at the desired density, and (5) at 100 ps NPT simulation at a pressure of 1atm to check for changes in density. A total of fourteen structures were thus constructed for the LaRC-CP2 system. A summary of twenty eight different structures used in this study are listed in Table I. The RVE for each of the three polymers systems is shown in Figure 2. Different colors in Figure 2 of the polymer RVE indicate the presence of a variety of atomic species in these material models. C. Equivalent-Continuum Constitutive Relation

The molecular structures of the polymers are amorphous, therefore it is expected that the bulk polymer materials exhibit isotropic symmetry. Based on this material symmetry for the equivalent continuum, a hyperelastic continuum constitutive relation was used to depict the deformation characteristics of the discrete molecular model. For generality, it is desired that the constitutive relationship of the equivalent continuum satisfy the following requirements: (1) Formulated in a finite-deformation framework, (2) Established using a thermodynamic potential, and (3) Incorporating isotropic material symmetry. The assumed form of the strain energy is

vol isoc ψ ψΨ = + (1) where

1 1

2 2

vol

iso

cc

ψψ

= Ω= Ω (2)

and

( )21 3

31 2

2 1 3 23 3

1

30

I

I II I

Ω = −

⎛ ⎞Ω = + −⎜ ⎟

⎝ ⎠ (3) where, Ωvol is the volumetric (shape preserving), Ωiso is the isochoric (shape changing) components of the strain-energy density, c1 and c2 are constants and represent material properties, and I1, I2 and I3 are the scalar invariants of the right Cauchy-Green deformation tensor, C. The second Piola-Kirchhoff stress tensor is therefore

( )3 2

11 2 1 2 21 3 3 2 2 21 3 2 1 3 2 2

3 3 3 3 3

2 12 6 1 6 2 3 63

c I I I I Ic I I c c cI I I I I

−⎡ ⎤⎛ ⎞ ⎛ ⎞∂Ψ= = − − + + + −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ⎝ ⎠ ⎝ ⎠⎣ ⎦

S CC

2

I C (4)

where I is the identity tensor. D. Equivalent Continuum Properties

The constitutive relation contains unknown material parameters c1 and c2 which were evaluated from energy equivalence of the molecular model and the equivalent-continuum model under identical deformation fields, as has been performed in previous studies of other nanostructured materials5. The models of the polymer systems were


6

subjected to two different deformation fields, a pure volumetric deformation and an isochoric deformation. For the molecular model, the energy can be computed from the force fields with

( )0

0

1m mV mΨ = Λ −Λ

(5) where Λ

m and Λm are the molecular potential energies before and after application of the deformations, which are directly computed from the force field, and V0 is the volume of the RVE in the reference configuration.

Finite deformations were applied to the molecular RVEs and the equivalent-continuum model in incremental steps. For the volumetric deformations, volumetric strains (E11 = E22 = E33 where E is the Green strain tensor) of 0.1%, 0.2%, 0.3%, 0.4% and 0.5% were applied to the molecular models. For the isochoric deformations, three-dimensional shear strain levels of γ23 = γ13 = γ12 = 0.1%, 0.2%, 0.3%, 0.4% and 0.5% (γij = 2Eij when i ≠ j) were applied to the molecular models. Further details on these deformations can be found elsewhere 5. The Young’s and shear moduli were calculated for the two material systems for all combinations of RVE size and force field using a previously-described method5.

IV. Results and Discussions Figures 3 and 4 are plots of bulk stress vs. volumetric strain for the polyimide and polycarbonate systems,

respectively. From these figures, there does not appear to be a definite trend in the effect of RVE size on the volumetric mechanical response of both polymers for the AMBER and OPLS-AA force fields. For the polyimide, the predicted volumetric stress increases in successive order for the AMBER, MM3, and OPLS-AA force fields, regardless of RVE size. However, no trend is apparent for the polycarbonate with respect to the force field type.

Table III

Experimental and Predicted Elastic Properties of Polyimide

RVE Size Method (No. of Atoms) Young’s Modulus

(in GPa) Shear Modulus

(in GPa)

Experiment23 N.A. 3.9 1.3

4214 7.39 2.55

6622 0.95 0.32 AMBER

8428 6.33 2.19

4214 37.6 13.8

6622 2.8 0.94 OPLS - AA

8428 9.17 3.16

MM3 4214 5.71 1.95

The shear stress versus shear strain response of the polyimide and polycarbonate systems are shown in Figures 5 and 6, respectively, for different RVE sizes and force fields. From these figures, there is no clear trend in the RVE size versus shear response for the AMBER and OPLS-AA force fields in both polymers. For the polycarbonate, the predicted shear stresses increase in successive order for the MM3, AMBER, and OPLS-AA force fields, regardless of RVE size. No such trend is apparent for the polyimide system for the different force fields.

The predicted elastic properties for the polyimde and polycarbonate materials are listed in Tables III and IV, respectively. Experimentally measured results provided in the literature are also given for the polyimide 23 and polycarbonate 24 systems in Tables III and IV, respectively. Similar to the results of Figures 3-6, the data in Tables III and IV do not demonstrate a clear trend between RVE size and predicted elastic moduli. The data does indicate that the elastic moduli predicted with the OPLS-AA force field generally tend to be larger than those predicted by the AMBER and MM3 force fields. Furthermore, comparison of the experimental and computational data indicates that the properties of both polymer systems tend to be over-predicted in comparison to the experimental values. The method utilized in the current study to determine the elastic properties is different than the experimental method


7

utilized for the polyimide system. The experimental method is unknown for the data reported for the polycarbonate system.

0

50

100

150

200

250

300

350

400

450

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009Volumetric Strain

Bulk Stress(MPa)

AMBER

OPLS-AA

MM3

AMBER

OPLS-AA

MM3

Small RVE

Medium RVE

Large RVE

Small RVE

Medium RVE

Large RVE

Figure 3. Bulk stress versus volumetric strain of three different sizes of RVEs of polyimide.

Table IV

Experimental and Predicted Elastic Properties of Polycarbonate

RVE Size Method (No. of Atoms) Young’s Modulus

(in GPa) Shear Modulus

(in GPa) Experiment24 N.A. 2.2 0.8

3972 4.87 1.66

5958 6.6 2.25 AMBER

7944 6.11 2.13

3972 8.55 3.33

5958 8.00 2.74 OPLS - AA

7944 8.67 2.99

MM3 3972 1.01 0.34

The moduli predicted from the molecular models is expected to be higher than those from the experimental values25,26 as these models represent polymer systems with a polydispersity index (PDI) of unity.


8

0

50

100

150

200

250

300

350

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009Volumetric Strain

Bulk Stress(MPa)

AMBER

OPLS-AA

MM3

AMBER

OPLS-AA

MM3

Small RVE

Medium RVE

Large RVE

Small RVE

Medium RVE

Large RVE

Figure 4. Bulk stress versus volumetric strain of three different sizes of RVEs of polycarbonate.

0

5

10

15

20

25

30

35

40

45

0 0.0005 0.001 0.0015 0.002 0.0025 0.003Shear Strain

Shear Stress(MPa)

AMBER

OPLS-AA

MM3

AMBER

OPLS-AA

MM3

Small RVE

Medium RVE

Large RVE

Small RVE

Medium RVE

Large RVE

Figure 5. Shear stress versus shear strain of three different sizes of RVEs of polyimide.


9

0

2

4

6

8

10

12

0 0.0005 0.001 0.0015 0.002 0.0025 0.003Shear Strain

Shear Stress(MPa)

AMBER

OPLS-AA

MM3

AMBER

OPLS-AA

MM3

Small RVE

Medium RVE

Large RVE

Small RVE

Medium RVE

Large RVE

Figure 6. Shear stress versus shear strain of three different sizes of RVEs of polycarbonate.

1

1.2

1.4

1.6

0 100 200 300 400 500 600 700

Temperature (in Kelvin)

Density (in g/cc)

AMBER

MM3

Figure 7. Density versus simulation temperature for the LaRC-CP2 polymer system.


10

Figure 7 shows the trend in predicted density of LaRC-CP2 with simulated temperature. It can be seen that the AMBER force field predicts a constant trend in the density with respect to temperature, close to 1.2 g/cc. The MM3 force field predicts a decreasing trend in the density with increased simulation temperature. The glass transition temperature for LaRC-CP2 is experimentally27 determined to be around 473 K, thus a discontinuity in the density is expected around this temperature. However, there is no definite discontinuity in the data of Figure 7. The absence of any trend in density for AMBER could be a consequence of the relatively simple functional form of the potential that does not necessarily account for the effects of temperature on the van der Waals interactions.

V. Summary A computational parametric study has been performed to establish the effect of RVE size, force field, and

simulation temperature on the predicted mechanical properties of polyimide and polycarbonate materials. A multiscale modeling technique was utilized to determine the equivalent-continuum Young’s moduli, density, and stress-strain behavior for the simulated parameters.

The results of the simulations indicate that for the simulation conditions modeled in this study, there is no clear effect of RVE size and force field on the predicted mechanical response of the polyimide and polycarbonate polymer systems. In addition, there is no clear effect of the simulation temperature on the predicted material densities when the AMBER force field is used. However, the MM3 force field predicts a steady decrease in the polymer density as the temperature increases up to and beyond the glass transition temperature.

VI. Acknowledgements This research was sponsored by NASA Langley Research Center under grant NNL04AA85G.

References

1Frankland, S. J. V., Harik, V. M., Odegard, G. M., Brenner, D. W. and Gates, T. S., "The Stress-Strain Behavior of Polymer-Nanotube Composites from Molecular Dynamics Simulation," Composites Science and Technology, Vol. 63, 2003, pp. 1655-1661. 2Odegard, G. M., Clancy, T. C. and Gates, T. S., "Modeling the Mechanical Properties of Nanoparticle/polymer Composites," Polymer, Vol. 46, 2005, pp. 553-562. 3Odegard, G. M., Clancy, T. C. and Gates, T. S., "Prediction of Mechanical Properties of Polymers with Various Force Fields," 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Austin, TX, 2005. 4Odegard, G. M., Gates, T. S., Nicholson, L. M. and Wise, K. E., "Equivalent-Continuum Modeling of Nano-Structured Materials," Composites Science and Technology, Vol. 62, 2002, pp. 1869-1880. 5Valavala, P. K., Clancy, T. C., Odegard, G. M. and Gates, T. S., "Nonlinear Multiscale Modeling of Polymer Materials," International Journal of Solids and Structures, Vol. 44, 2007, pp. 1161-1179. 6Leach, A. R., Molecular Modelling: Principles and Applications, Prentice Hall, New York, 2001. 7Cornell, W. D., Cieplak, P., Bayly, C. I., Gould, I. R., Merz, K. M., Ferguson, D. M., Spellmeyer, D. C., Fox, T., Caldwell, J. W. and Kollman, P. A., "A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules," Journal of the American Chemical Society, Vol. 117, 1995, pp. 5179-5197. 8Lorenz, C. D. and Stevens, M. J., "Fracture Behavior of Lennard-Jones Glasses," Physical Review E, Vol. 68, 2003, pp. 68. 9St. Clair, A. K., St. Clair, T. L. and Slemp, W. S., "Optically Transparent/Colorless Polyimides," Recent Advances in Polyimide Science and Technology, edited by W. Weber and M. Gupta, Society of Plastics Engineers, Poughkeepsie, NY, 1987, pp. 16-36. 10Allinger, N. L., Geise, H. H., Pyckhout, W., Paquette, L. A. and Gallucci, J. C., "Structures of Norbonane and Dodecahedrane by Molecular Mechanics Calculations (MM3), X-Ray, Crystallography, and Electron Diffraction," Journal of the American Chemical Society, Vol. 111, 1989, pp. 1106-1114. 11Allinger, N. L., Li, F. and Yan, L., "Molecular Mechanics. The MM3 Force Field for Alkenes," Journal of Computational Chemistry, Vol. 11, 1990, pp. 848-867. 12Allinger, N. L., Li, F., Yan, L. and Tai, J. C., "Molecular Mechanics (MM3) Calculations on Conjugated Hydrocarbons," Journal of Computational Chemistry, Vol. 11, 1990, pp. 868-895. 13Allinger, N. L., Yuh, Y. H. and Lii, J. H., "Molecular Mechanics. The MM3 Force Field for Hydrocarbons," Journal of the American Chemical Society, Vol. 111, 1989, pp. 8551-8566.


11

14Jorgensen, W. L., Maxwell, D. S. and Tirado-Rives, J., "Development and Testing of the OPLS All-Atom Force Field on Conformational Energetics and Properties of Organic Liquids," Journal of the American Chemical Society, Vol. 117, 1996, pp. 11225-11236. 15Kaminsky, G. A., Friesner, R. A., Tirado-Rives, J. and Jorgensen, W. L., "Evaluation and Reparametrization of the OPLS-AA Force Field for Proteins via Comparison with Accurate Quantum Chemical Calculations on Peptides," Journal of Physical Chemistry B, Vol. 105, 2001, pp. 6474-6487. 16Lii, J. H. and Allinger, N. L., "Molecular Mechanics. The MM3 Force Field for Hydrocarbons. 2. Vibrational Frequencies and Thermodynamics," Journal of the American Chemical Society, Vol. 111, 1989, pp. 8566-8575. 17Lii, J. H. and Allinger, N. L., "Molecular Mechanics. The MM3 Force Field for Hydrocarbons. 3. The van der Waals' Potentials and Crystal Data for Aliphatic and Aromatic Hydrocarbons," Journal of the American Chemical Society, Vol. 111, 1989, pp. 8576-8582. 18Lii, J. H. and Allinger, N. L., "Directional Hydrogen Bonding in the MM3 Force Field. I," Journal of Physical Organic Chemistry, Vol. 7, 1994, pp. 591-609. 19Lii, J. H. and Allinger, N. L., "Directional Hydrogen Bonding in the MM3 Force Field. II," Journal of Computational Chemistry, Vol. 19, 1998, pp. 1001-1016. 20Weiner, P. K., and Kollman, P. A., "AMBER: Assisted Model Building with Energy Refinement. A General Program for Modeling Molecules and Their Interactions," Journal of Computational Chemistry, Vol. 2, 1981, pp. 287-303. 21TINKER - Software Tools for Molecular Design, Ver. 4.2, Washington University School of Medicine, St. Louis, MO, 2004. 22Ponder, J. W. and Richards, F. M., "An Efficient Newton-Like Method for Molecular Mechanics Energy Minimization of Large Molecules," Journal of Computational Chemistry, Vol. 8, 1987, pp. 1016-1024. 23Hergenrother, P. M., Watson, K. A., Smith, J. G., Connell, J. W. and Yokota, R., "Polyimides from 2,3,3',4'-Biphenyltetracarboxylic Dianhydride and Aromatic Diamines," Polymer, Vol. 43, 2002, pp. 5077-5093. 24Christopher, W. F. and Fox, D. W., Polycarbonates, Reinhold Publishing Corporation, New York, 1962. 25Fan, C. F., Cagin, T., Chen, Z. M. and Smith, K. A., "Molecular Modeling of Polycarbonate. 1. Force Field, Static Structure, and Mechanical Properties," Macromolecules, Vol. 27, 1994, pp. 2383-2391. 26Fan, C. F. and Hsu, S. L., "Application of the Molecular Simulation Technique to Characterize the Structure and Properties of an Aromatic Polysulfone System. 2. Mechanical and Thermal Properties," Macromolecules, Vol. 25, 1992, pp. 266-270. 27Herring, H. M., "Dynamic Mechanical Characterization of Thin Film Polymer Nanocomposites," NASA/CR-2003-212147, 2003.


12

parametric studies in multiscale modeling of high ...gmodegar/papers/aiaa-2007-2173.pdfparametric...

Documents