Multiscale Modeling of Polymer Materials at Cryogenic and Elevated Temperatures
Pavan K. Valavala* and Gregory M. Odegard†
Department of Mechanical Engineering – Engineering Mechanics Michigan Technological University
1400 Townsend Drive, Houghton, MI 49931
A multiscale constitutive modeling approach has used to model a high-performance polyimide system over a series of temperatures ranging from cryogenic to elevated (approaching the glass transition temperature). The multiscale model has employed computational chemistry techniques to determine the molecular structure and to prescribe deformations for establishing mechanical properties. The resulting predicted mechanical properties have been compared to experimentally-obtain properties for the same polyimide system. The predicted and experimentally-obtained moduli show good agreement in terms of magnitude. However, the predicted moduli did not show the expected temperature-dependence. This discrepancy in the modeling is likely due to the use of harmonic potentials in the force field.
I. Introduction
In recent years, nanostructured materials have shown the potential to provide significant gains in mechanical properties relative to current materials used for a wide range of applications. Of particular interest is the development of durable polymer and polymer nanocomposite materials for aerospace structural applications. For this purpose, polymer materials must demonstrate mechanical reliability under temperatures ranging from cryogenic to the glass transition temperature. To facilitate the development of polymer materials for this purpose, multiscale modeling techniques must be developed that provide reliable structure-property relationships.
Molecular Dynamics (MD) and Molecular Mechanics (MM) have been used in numerous studies for prediction of the structure and mechanical properties of polymer-based material systems over multiple length scales.1-14 These studies have demonstrated that molecular modeling techniques can be effectively used to predict structure and properties of polymer-based material systems. MM is a procedure through which the potential energy of a molecular structure can be determined under static conditions.15 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.16-19 MD simulations can be used to obtain the equilibrated molecular structure of a nanostructured material and to predict the behavior of the system when subjected to prescribed deformations. Because the above-cited studies have used MD and MM techniques to predict mechanical properties of polymer-based materials at fixed temperatures, little is known about the effects of simulated temperature on predicted properties.
The objective of the proposed paper is to understand the influence of temperature in molecular simulations on predicted mechanical behavior of high-performance polymer system (LaRC-CP2) using multiscale simulation techniques. The predicted mechanical properties of the polymer are compared to those obtained from experiment for a wide range of temperatures. In this manner, the effectiveness of the multiscale simulations in predicting mechanical properties over a range of temperatures is established. These results will serve to further understand the multiscale modeling of organic, amorphous materials for aerospace applications. After a brief description of the modeled polymer system, the multiscale modeling procedure is described in detail.
* Graduate Research Assistant, Student Member AIAA † Assistant Professor, Senior Member AIAA
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47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Confere1 - 4 May 2006, Newport, Rhode Island
LaRC-CP2 is an amorphous, thermoplastic, colorless polyimide originally developed at NASA Langley Research Center. LaRC-CP2 is synthesized from 1,3-bis(3-aminophenoxy) benzene (APB) and 2,2-bis(3,4-anhydrodicarboxyphenyl) hexafluoropropane (6FDA).20 The chemical structure of the LaRC-CP2 repeat unit is shown in Figure 1. This polyimide can be used for inflatable solar concentrators and antennas due to its superior UV radiation resistance when compared to other polymeric materials. Solar concentrators are envisioned to be widely used in Gossamer (large ultra-light weight) spacecraft in the future, which offer tremendous cost advantage compared to on-orbit constructions.21,22
Figure 1. Schematic illustration of chemical structure of LaRC-CP2 monomer
III. Multiscale Modeling Procedure
The nonlinear-elastic (hyperelastic) properties of the two material systems were determined using the Equivalent-Continuum Modeling method.23,8,24,10-13,25 This modeling technique is ideally suited for large, amorphous atomic structures with a mixture of covalent and secondary chemical bonds, as the Cauchy-Born rule is ignored because of immense computational complexity that would result if it was incorporated under these conditions. This procedure consisted of three steps. First, a representative volume element (RVE) of the polymer was established that described the molecular structure in thermal equilibrium. The second step involved establishing a constitutive equation that accurately describes the bulk mechanical behavior of the equivalent continuum. Finally, the energies of deformation were obtained for both molecular and equivalent-continuum models under identical states of deformation. The energy for the molecular model was obtained from the force field and the energy for the continuum model was calculated from the assumed constitutive equation. A series of deformations were chosen such that equating of energies from the two models resolved the equivalent continuum material parameters. A. Molecular RVE
A molecular model representing the RVE of the LaRC-CP2 polymer was constructed using the repeat unit shown in Figure 1. The amorphous RVE of the polymer material was initially constructed in a very low-density gas phase with 9,954 atoms. The molecular model had nine polymer chains with sixteen repeat units per chain. The gas-phase molecular model so prepared was subjected to an iterative series of energy minimizations coupled with reductions in the size of the RVE. The minimizations were performed using a quasi-Newton L-BFGS method26 as implemented with the MINIMIZE program in the TINKER modeling package,27,28 and were minimized to reach an RMS gradient of convergence of 0.01 kcal/mole/Å. This method resulted in a RVE of increasing density, up to the point where the expected bulk material density was achieved. The resulting RVE of the polymer was subsequently subjected to a series of 100 ps MD simulations with canonical ensemble (NVT) at increasing temperatures (up to room temperature) to further evolve the molecular system to an thermally-equilibrated structure. A 150 ps MD simulation using isobaric-isothermal ensemble (NPT) was performed at room temperature and a pressure of 1 atm. The resulting molecular structure from this portion of the procedure represented a preliminary thermally-equilibrated RVE of the polymer. Periodic boundary conditions were used for all of the simulations.
The RVE from the above procedure was subjected to a series of MD simulations to establish separate thermally-equilibrated structures for four temperatures: 173 K (cryogenic), 296 K (room temperature), 373 K (boiling point of water), and 423 K. Each of these sets of MD simulations consisted of the following series of five simulations at the
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corresponding 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 100 atm, (3) a 200 ps NPT simulation at a pressure of 1 atmospheres 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 1 atm to check for changes in density. Finally, each of the four molecular structures was subjected to the quasi-Newton L-BFGS method described above followed by a truncated-Newton energy minimization29 to a RMS gradient of convergence of 1×10-6 kcal/mole/Å, as implemented with the NEWTON program in the Tinker molecular modeling package. The energy minimizations established the undeformed molecular structure and the corresponding molecular potential energy, at each of the four temperatures. The AMBER force field and periodic boundary conditions were used for these simulations. An example of the resulting RVE of the polymer is shown in Figure 2.
X1
X2
X3
Figure 2. Representative volume element of LaRC-CP2
B. Continuum Model
A hyperelastic approach was used for the constitutive modeling of the equivalent-continuum polymer material subjected to finite deformations. It was assumed that the strain energy function is associated with stress and strain tensors that are thermodynamic work conjugates in the balance of mechanical energy. In particular, the second Piola-Kirchhoff stress tensor is
( )2 c∂Ψ
=∂
CS
C (1)
where C is the right Cauchy-Green deformation tensor and Ψc is the scalar strain energy density function of the equivalent continuum. The deformation tensor is defined as
T=C F F (2) where F is the deformation gradient tensor whose components are given by
iij
j
xFX
∂=
∂ (3)
where X and x are the material (undeformed) and the spatial (deformed) coordinate vectors, respectively. The functional form of the strain-energy density is restricted by considering the invariance properties of the material such that the strain-energy density remains invariant with respect to the coordinate transformations expressed by the material symmetry. For an isotropic material, the reducible invariants of the deformation tensor C are
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( )
( )( ) ( )( )
1
2 22
3
tr1 tr tr2det
I
I
I
=
⎡ ⎤= −⎣ ⎦
=
C
C C
C (4) where tr(C) and det(C) are the trace and determinate of tensor C, respectively. Therefore, Equation (1) can be rewritten as
( )1 2 3, ,2 c I I I∂Ψ
=∂
SC
(5)
It can be shown25 that the equivalent-continuum strain energy density can be decomposed into volumetric
(shape preserving) and isochoric (shape changing) components, ψvol and ψiso, respectively,
vol isoc ψ ψΨ = + (6)
where
1 1
2 2
vol
iso
cc
ψψ
= Ω= Ω
(7)
and
( )2
1 3
31 2
2 1 3 23 3
1
30
I
I II I
Ω = −
⎛Ω = + −⎜ ⎟
⎝ ⎠
⎞ (8)
In Equation (7) the constants c1 and c2 are material parameters. Substitution of Equations (4) to (8) into Equation (1) yields
( )3 2
11 2 1 2 21 3 3 2 2 21 3 2 1 3 2 2
3 3 3 3 3
2 16 1 6 2 3 63
I I I I Ic I I c c cI I I I I
−⎡ ⎤⎛ ⎞ ⎛ ⎞= − − + + + −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦S C
2
I C (9)
It has been shown that this constitutive equation satisfies the required growth conditions for a hyperelastic material.
C. Energy Equivalence
The energy densities of deformation of the equivalent-continuum, Ψc, and molecular models, Ψm, were equated for identical sets of boundary conditions to determine the mechanical properties of LaRC-CP2 for different temperatures. The molecular potential energy, Λm, is given by the AMBER force field
( ) ( )
( )
2 2
mstretch bend
12 6
12 6torsion
1 cos 42
r eq eq
n IIJ
I J IJ IJ
K r r K
V nr r
θ θ θ
σ σφ ζ ε<
Λ = − + −
⎛ ⎞+ + − + −⎡ ⎤ ⎜ ⎟⎣ ⎦
⎝ ⎠
∑ ∑
∑ ∑ J IJ
(10)
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where the summations are taken over all of the corresponding interactions in the molecular model; Kr and Kθ are the bond-stretching and bond-angle bending force constants, respectively; r and req are the bond length and equilibrium bond length, respectively; θ and θ eq are the bond angle and equilibrium bond angle, respectively; 2nV , ζ, n, and φ are the torsion magnitude, phase offset, periodicity of the torsion, and the torsion angle, respectively; and εIJ, rIJ, and σIJ are van der Waals well depth, non-bonded distance between atoms I and J, and the equilibrium distance between atoms I and J, respectively. The molecular energy density is
( 0
0
1m mV
)mΨ = Λ − Λ (11)
where V0 is the volume of the RVE in the undeformed configuration.
Because the strain-energy density of the equivalent continuum, Ψc, is the sum of the volumetric and isochoric deformation components, volumetric and isochoric modes of deformation were applied to the molecular models to determine the material parameters c1 and c2 for each temperature. For each type of deformation, the strain-energy densities in Equations (6) and (11) were equated by adjusting these two material parameters. To relate these deformations to those typically applied to a specimen during laboratory testing, the deformation levels are expressed in terms of the Lagrangian strain tensor (henceforth referred to as the strain tensor)
(12
)= −E C I (12)
Multiple finite-increment steps in deformations were applied to the molecular RVE and the equivalent-
continuum model to avoid a linear interpolation between the undeformed configuration and the deformed configuration with moderate strain levels, and to avoid potential convergence problems with the energy minimizations between deformation steps. For the volumetric deformations, the deformation equations are
( )
( ) ( )
( ) ( )
( ) ( )
11
22
3 23
4 34
α
α
α
α
=
=
=
=
x X
x x
x x
x x
1 (13)
where α1, α2, α3, and α4 are scalar constants that indicate the magnitude of deformation and the superscripts indicate the deformation step. The spatial coordinates x(1), x(2), x(3), and x(4) correspond to volumetric strains (E11 = E22 = E33) of 0.1%, 0.2%, 0.3%, 0.4% and 0.5%, respectively. The relative deformation gradients, F’, which relate the deformation at a given deformation level to those of the previous deformation level, are
( ) ( )( )
( )( )
( ) ( )( )
( ) ( )( )
( )
1(1) (1)
2 3(2) (3) (4)
1 2
iij ij
j
i i iij ij ij
j j
xF FX
4
3j
x x xF F Fx x x
∂′ = =∂
∂ ∂ ∂′ ′ ′= =∂ ∂
x x
x x x =∂
(14)
Therefore, the deformation gradients that relate the coordinate for each deformation level to those of the material coordinate system are
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( )( )
( )( )
( ) (
( )( )
)
( ) ( )
( )( )
( ) (
1(1)
2(2) (2) (1)
3(3) (3) (2)
4(4) (4) (3)
iij
j
iij im mj
j
iij im mj
j
iij im mj
j
xFX
xF F FX
xF F FX
xF F FX
∂=
∂
∂ ′= =∂
∂ ′= =∂
∂ ′= =∂
x
x x
x x
x x )
x
x
x
(15)
Using above equations the constants α1, α2, α3, and α4 where adjusted to achieve the exact desired volumetric strain levels of 0.1%, 0.2%, 0.3%, 0.4% and 0.5%, in Equation (13) These values are shown in Table I.
The deformation equations for the isochoric deformations are
( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
11 1 2 3 1
12 1 1 3 2
13 1 1 2 3
1 11 2 3 1
1 12 1 3 2
1 13 1 2 3
k k kk
k k kk
k k kk
x X X X
x X X X
x X X X
x x x x
x x x x
x x x x
β
β
β
β
β
β
1
1
1
k
k
k
− − −
− − −
− − −
= + +
= + +
= + +
= + +
= + +
= + +
(16)
where β1, β2, β3, and β4 are scalar constants that indicate the magnitude of the deformation and the superscripts 1, 2, 3, and 4 (k = 2, 3, and 4) correspond to 3-dimensional shear strain levels of γ23 = γ13 = γ12 = 0.1%, 0.2%, 0.3%, 0.4% and 0.5%, respectively (γij = 2Eij when i ≠ j). Similar to the case of the volumetric deformation, the constants β1, β2, β3, and β4 were adjusted such that these shear strains were achieved by using above equations, and are shown in Table I.
Table I. Values of deformation parameters
The differences in the potential energies of the molecular models between the deformed and undeformed
configurations, as shown in Equation (11), were determined using MM simulations. Alternatively, MD simulations can be used to establish the difference in potential energies, however, MM simulations were chosen primarily because of the difficulty in modeling the behavior of polymer molecules at laboratory-related time scales (e.g. seconds, minutes, and hours). For the model sizes considered in this study, simulation of such times scales with MD
Deformation parameters Value (unitless) Finite-valued strain components
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would require prohibitively long simulation times. In MM simulations of a polymer, the changes in the molecular structure that occur in laboratory time scales can be easily simulated in a relatively short amount of time. Therefore, predicted properties are expected to be more accurate with MM simulations for relative short simulation times. The improved accuracy with the use of MM simulations has been previously demonstrated.25
Both volumetric and isochoric deformations were applied to the equilibrium molecular structures for each force field by deforming the RVE and all of the atoms in the models according to the applied deformation field. The quasi-Newton and truncated Newton methods, as described above, were used to establish the molecular structure for each load step. During the energy minimization for each load step, the RVE volume was kept constant as the atoms were shifted to minimize the potential energy.
For MM simulations, periodic boundary conditions were applied such that atoms were free to cross the boundary of the deformed and undeformed simulation cells. Atoms that crossed the boundary entered the simulation cells on the opposite side, as described in detail elsewhere.15 Therefore, none of the atoms in the molecular simulations were kinematically over-constrained, as can occur in simulations of RVEs of heterogeneous material systems with kinematic boundary conditions.
IV. Results and Discussion
A. Molecular Model
In the MD method, the displacements of the interacting atoms in the system are continuously scaled to attain the required kinetic energy to achieve the specified simulation temperature. As a result, the simulation temperatures require a finite amount of simulation time at the beginning of the simulation to achieve a steady, fluctuating temperature. Figure 3 shows the temperature of the polymer system as a function of the simulation time for the MD simulation at 423 K. Each portion of the MD simulation described in section III is shown in Figure 3. The error bars indicate the standard deviation of the data. It is clear that temperature fluctuations during the simulation are not significant compared to the range of temperatures considered in this study and that the temperature quickly approaches the point of stable fluctuation early in the simulation.
380
390
400
410
420
430
440
450
0 100 200 300 400 500Simulation time (ps)
Tem
pera
ture
(K)
NPT NPT NPT NVT NPT
Figure 3. MD simulation temperature as a function of simulation time for a target temperature of 423 K
Figure 4 shows the density variations in the molecular model as a function of the simulation time for the
structure simulated at a temperature of 423 K. The error bars in Figure 4 represent the standard deviations of the particular data point and the individual components of the MD simulation described in section III are also shown for reference. It can be seen that the system evolved to an equilibrium state as the change in density became significantly small towards the end of the simulation. Of course, the density remained constants during the NVT
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simulation. However, during NPT simulations, there existed some fluctuations in the densities which were small relative to the magnitude of the polymer density. The fluctuations in the density of the model can be explained on the basis of fluctuations in volume of the model during the MD simulation.
1
1.05
1.1
1.15
1.2
1.25
1.3
0 100 200 300 400 500Simulation Time (ps)
Den
sity
(g/c
c)
NPT NPT NPT NVT NPT
Figure 4. Polymer density as a function of the simulation time at 423 K
-1500
-1000
-500
0
500
1000
0 100 200 300 400 500Simulation Time (ps)
Pres
sure
(atm
)
NPT NPT NPT NVT NPT
Figure 5. Pressure as a function of the simulation time at 423 K
Figure 5 shows the variation in pressure as a function of the simulation time. The pressure of the system is
calculated at every step of MD using Equation (17). The pressure of the system can be calculated through the use of Clausius virial function and equi-partition theorem30
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⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂Λ∂
−= ∑∑>i ij r
ijBij
rr
DTNk
VP 11 (17)
where, P is the pressure of the system, T is temperature and V is volume of the of the simulation box, and kB is boltzmann constant, Λ is the potential energy function, N is the number of atoms, rij is the distance between i and j atoms and D is the dimensionality of the system. The pressure of the system is maintained through small variations in the volume of the simulation box during the simulation.31 It can be seen from Equation (17) that the pressure control is dependent on the simulation temperature, atomic separation, and instantaneous forces on atoms. As shown in Figure 3, the temperature fluctuations are relatively small. Therefore, it can be assumed that the pressure fluctuations are a consequence of changes in the instantaneous atomic separation and the interatomic forces. However, relatively large changes in volume can drive the system away from equilibrium and result in computational difficulties. The objective of the long NPT simulations was to allow the system to freely evolve to its thermodynamic equilibrium density at the prescribed pressure and temperature through gradual changes in volume of the simulation box
Table II. Values of simulated densities of the polyimideTemperature (in Kelvin) Density (g/cc)
173 1.2134 296 1.2438 373 1.2243 423 1.2343
The resulting final densities from the procedures discussed in Section III for the four temperatures are given in
Table II. It can be seen that there is no clear trend in the predicted densities with respect to temperature. The absence of a decrease in densities with an increase in temperature is likely a consequence of the presence of only second order quadratic polynomial functions in AMBER force field as shown in Equation (10). The effect of temperature is obtained in MD simulations through scaled velocities of the atoms which can result in increased atomic separation at equilibrium to mimic thermal expansion. The use of only quadratic terms in the potential might result in atomic motion about the same mean position resulting in no definite trend in interatomic separation as a function of temperature. The inclusion of inharmonic terms in the force field can potentially result in thermal expansion of the model with atoms settling into other local minimum energy positions that are equilibrium structures for higher temperatures due to presence of higher order terms.
Figure 6. Simulated molecular strain energy density for volumetric deformation
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B. Continuum Model and Energy Equivalence
Figures 6 and 7 show the plot of Ψm versus Ω1 and Ω2, respectively. Since the Ψc is a linear function of the volumetric and isochoric deformation terms, as demonstrated by Equation (6), the slope of the curves in Figures 6 and 7 result in the direct determination of parameters c1 and c2. The resulting values for the parameters for each temperature are shown in Table III.
Figure 7. Simulated molecular strain energy density for isochoric deformation
With the parameters c1 and c2 in hand, the resulting constitutive equation was determined for the polymer at each temperature using Equation (6). Using the resulting constitutive equations, the hydrostatic stress is plotted versus the volumetric strain in Figure 8 for each temperature. Similarly, the shear stress (S12) versus shear strain (γ12) response of the material at each temperature is plotted in Figure 9. The slopes of the curves in Figures 8 and 9 are the bulk and shear modulus, respectively, of LaRC-CP2. Because LaRC-CP2 is assumed to be isotropic in bulk, the Young’s moduli can be determined directly from the bulk and shear moduli. The values obtained for various temperatures are listed in Table IV. The table also shows the values of storage modulus obtained from literature32 for the corresponding temperatures. It can be seen from Table IV that the modeling method used herein, which incorporates the AMBER force field, under-predicts the properties for the polymer at 173 K and 373 K, but over predicts the values for 296 K and 423 K. This could be a consequence of the trend in predicted densities (Table II). The densities of the polymer at 173 K and 373 K are lower than those at 296 K and 423 K. From Table IV, the predicted modulus at room temperature is much higher than the experimental value.
Table III. Predicted material parameters of LaRC-CP2 from static molecular simulations (all parameters have units of Pa)
Temperature (in Kelvin) c1 c2
173 1.35×109 2.11×107
296 2.10×109 4.77×107
373 2.21×109 7.06×106
423 2.30×109 1.67×107
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Figure 8. Predicted hydrostatic stress versus volumetric strain behavior for LaRC-CP2
0
20
40
60
80
100
120
140
160
180
0 0.002 0.004 0.006 0.008 0.01Volumetric Strain
Hyd
rost
atic
Str
ess
(MPa
)
173 K296 K373 K423 K
Figure 9. Predicted shear stress versus shear strain behavior for LaRC-CP2
Table IV. Predicted and Experimental Elastic Properties of LaRC-CP2 (All moduli have units of GPa)
Temperature (in Kelvin)
Experimental Storage Modulus
Simulated Young’s Modulus
Simulated Shear Modulus
173 3.17 (± 0.54) 2.90 1.00
296 3.00 (± 0.21) 8.57 3.00
373 2.57 (± 0.19) 1.19 0.4
423 2.24 (± 0.15) 2.66 0.9
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Acknowledgements This research was sponsored by NASA L t #NNL04AA85G.
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