A Molecular Modeling Approach to Predicting Thermo-Mechanical Properties of Thermosetting Polymers

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1 A Molecular Modeling Approach to Predicting Thermo-Mechanical Properties of Thermosetting Polymers Natalia Shenogina, Wright State University Mesfin Tsige, University of Akron Soumya Patnaik, AFRL Sharmila Mukhopadhyay, Wright State University June 5, 2012

2 Thermoset Polymers Applications Aerospace composites Other structural materials Adhesives Encapsulation and boards for electronics Properties that enable these applications Dimensional stability Thermal stability Mechanical Strength Chemical resistance Electrical Properties Polymer

3 Unique Opportunities/Challenges Variability in Structures and Properties All polymers are prone to structural variations and thermosets have additional variable introduced by curing step. However, these are important materials, having annual market demand well over $15B worldwide. Thousands of variations that have been empirically tailored for specific needs. Computational Models can make tailoring easier and more economical by systematically predicting compositionstructure-property relationships.

4 Predictive Models for Macromolecules Models operate at different levels of detail: Length scales Time Scales Degrees of freedom Etc. Practical considerations: Number of constraints Number of unknowns Realistic Conditions Computational time Ref: Florian Muller-Plathe, SOFT MATERIALS, Vol. 1, pp. 1 31, 2003

5 Scales in Modeling Finite-element methods (FEM) are widely used to study large-scale events in polymers. Often limited by lack of the realistic parameters for inputs to constitutive relations. These parameters can be difficult to obtain from experiments. Molecular Dynamics (MD) simulations can provide detailed microscopic information. Can provide parameters for larger systems (finite element analysis or coarse-grained models) The coarse-grained simulations in turn can be used as a guidance for the all atom (AA) simulations

6 MD simulations of Thermosetting Polymers (TSP) Several reported MD simulations have demonstrated that it can be successfully used as a tool in predicting material properties of TSP. Various approaches of simulating bond formation and cross-linking have been developed to bring models closer to experimental conditions. Some Issues remain: Many obtained structures have low degree of curing Some have high internal stresses and distortions Number of atoms that can be included have been small, typically less than 15K atoms These issues often lead to predicted properties that are far from those of real systems

7 Objectives of this study Construct more realistic models Achieve high degree of cure, closer to engineering materials. Calculate some of the important material properties, and compare with experimental trends: Density Glass transition temperature Coefficient of thermal expansion Elastic constants Investigate influence of simulation size, extent of curing, and length of epoxy strands on these quantities.

8 Long-term Goals Use of atomistic MD predicted parameters to incorporate into larger scale models (coarse-grained etc.) Provide parameters for constitutive relations in Finite Element Methods Allow prediction of the strength and toughness as a function of experimental parameters such as: monomer chemistry, cross-link density and temperature In future, better correlations between these Simulation Predictions (Tsige et al) Tensile tests on similar chemistry (Mukhopadhyay et al)

9 Polymer System Investigated Epoxy-amine cured system: reactants Epoxy resin: DGEBA (diglycidyl ether of bisphenol A) Functionality: 2 Aromatic amine hardener: DETDA (diethylene toluene diamine) Functionality: 4

Epoxy-amine cured system: amorphous cell resin hardener atoms 32 16 ~2000 64 32 ~4000 96 48 ~6000 128 64 ~8000

10 Epoxy-amine cured system: amorphous cell resin hardener atoms ~ ~ ~ ~8000 Stochiometric ratio of epoxy resin and hardener (32/16 molecules) in a box ~ ~ ~35000

11 General Approach Used All simulations performed using Materials Studio package from Accelrys. Generate epoxy-amine systems cured to a given extent of reaction Study volume-temperature behavior: Density, Glass transition temperature, Coefficients of thermal expansion Estimate elastic properties: Young s, Shear and Bulk Modulus, Poisson s ratio Details at: Shenogina, N.; Tsige, M.; Patnaik, S.; Mukhopadhyay, S., Molecular Modeling Approach to Prediction of Thermo-Mechanical Behavior of Thermoset Polymer Networks, Macromolecules, accepted for publication, 2012

12 Maximum bond energy, kcal/mol Maximum bond energy, kcal/mol Building of Thermosets Generate many topologically independent structures Extract energetic and structural information at each cross-linking cycle onthe fly. Maximum bond stretching energy considered as a good measure of distortion. Models that show dramatic increase of maximum bond stretching energy at high extent of reaction are rejected (signifies artificially high stresses ) rejected Extent of reaction Extent of reaction 1.6 accepted

13 Specific Volume, cm 3 /g I. Thermal Property Calculations Constant pressure cooling dynamics runs for each degree of cross-linking Specific Volume Vs Temp Plots cooling 50K/100ps cooling 10K/100ps Coefficient of Thermal Expansion (CTE) Temperature, K V G = V G o +α G T ( K) V R = V R o +α R T ( K) a G in the glassy region a R in the rubbery region

14 Analysis of the V vs. T plots (100% curing) structure Dens.(298K) g/cm 3 Dens.(480K) g/cm 3 Tg,K α G 1/K α R 1/K *10-5 * average Experimental values: Density: 1.16 g/cm 3 (Ratna et al, J. Mater. Sci. 38, (2003)) Tg: 443 K (Jansen et al, Polymer, 40, (1999)) 457 K (Ratna et al, Poly. Int. 52, (2003)) 441 K (Shen et al, Macromol. Mat. Eng. 291 (2006) 476 K (Liu et al, Polymer, 47, (2006)) a G : 24*10-5 /K (Liu et al, Polymer, 47, (2006))

15 Volumetric coeff.of therm.expansion*10-5, 1/K Coefficient of Thermal Expansion (CTE) as a function of degree of cross-link atoms 4000 atoms 6000 atoms 8000 atoms atoms atoms atoms T=480K T=298K Extent of reaction,% CTE decreases with increasing degree of curing Size effect is detected, larger structures show clearly linear trend Exp. value

16 Volumetric coeff.of therm.expansion*10-5,1/k CTE for different chain lengths of epoxy monomer 2 monomers 4 monomers T=480K 30 T=298K Extent of reaction,% The dependence of CTE on degree of curing decreases with chain length

17 Glass transition temperature, K Variation of T g with degree of curing atoms 4000 atoms 6000 atoms 8000 atoms atoms atoms atoms Exp. values Extent of reaction,% T g increases with degree of curing, within range of experimental values

18 Density variation with degree of curing atoms 4000 atoms 6000 atoms 8000 atoms atoms atoms atoms Exp. value T=298K Density, g/cm T=480K Extent of reaction,% Density increases with degree of curing This effect is more pronounced at higher temperature

19 Role of chain length of the epoxy strands on density monomer 2 monomers 4 monomers T=298K Density,g/cm T=480K Extent of reaction,% Density increases with chain length

20 Mechanical Properties Any small (e.g. nanometer scale) volume element of an amorphous material can be characterized by unique distribution of matter. Individual elements viewed as regions. A large number of such elements averaged using Voigt-Reuss and Hill- Walpole approaches. Finally, assuming isotropic symmetry, Lame elastic constants obtained from stress and compliance tensors. Leads to coomonly measured coefficents: Young s modulus Shear Modulus Bulk Modulus Poisson s ratio

21 Mechanical Properties Static Approach Static approach (Theodoru and Suter, 1986): Neglect entropic contributions to elastic response Take into account potential energy contribution Small deformations and low temperatures Three tensile and three shear deformations of magnitude ±0.001 are applied to the system Energy minimization of the structures The obtained stress tensor is then used to estimate stiffness coefficients of the material Elastic constants are calculated: Young s, bulk, shear moduli, Poisson s ratio

22 Young's modulus, GPa Young s modulus vs. degree of curing Static Approach atoms 8000 atoms atoms atoms atoms T=298K T=480K Extent of reaction,% Experimental value is 2.71GPa, lower than simulations [Qi et al, Compos. Struct. 75, (2006)]: tensile test at room temperature

23 Young's modulus,gpa Young s modulus vs. degree of curing for different chain lengths (static Approach) monomer 2 monomers 4 monomers T=298K 6 5 T=480K Extent of reaction,% The rate of increase depends on chain length

24 Sreess, GPa Mechanical Properties Dynamic Approach Strain Tensile test simulations at T=298K Constant strain rate 10 8 s -1 Stress-strain curve is stepwise Regions of increase and relaxation of internal stresses First molecular relaxations are observed at low strains No well defined linear region Young s modulus: slope of the initial portion of the curve Experimental study: linear fit over ε =0.015 Poisson s ratio : at uniaxial tension, identify contraction in lateral directions at which lateral stresses are zero. Bulk modulus: 3-axial compression

25 Poisson's Ratio Poisson s ratio vs. degree of curing Dynamic vs. Static Approach T=480K, dynamic T=298K, dynamic Exp. value T=298K, static T=480K, static Degree of Curing, % Poisson s ratio predictions significantly approved by dynamic approach

26 Young's Modulus, GPa Young s modulus vs. degree of curing. Dynamic vs. Static Approach T=298K, static T=480K, static Exp. value T=298K, dynamic T=480K, dynamic Degree of Curing, % Dynamic effects seem to provide much more realistic predictions. This is despite large strain rates and small sizes, inherent to MD

27 Bulk Modulus, GPa Bulk modulus vs. degree of curing Dynamic vs. Static Approach (compression) 7 T=298K, static 6 5 T=480K, static 4 3 T=298K, dynamic 2 T=480K, dynamic Degree of Curing, %

28 Shear Modulus, GPa Shear modulus vs. degree of curing Dynamic vs. Static Approach 5 4 T=298K, static T=480K, static T=298K, dynamic T=480K, dynamic Degree of Curing, %

29 Summary Stress free thermo-set models up to atoms were generated with high degree of cure containing Densities, CTE and Tg were found in good agreement with experimental data Effects of system size and chain length investigated Mechanical Deformation has been simulated using both static and dynamic approaches and their results compared. Significant improvements are seen by treating this as a dynamic event

30 Publications and Presentations to date Papers: Shenogina, N.; Tsige, M.; Patnaik, S.; Mukhopadhyay, S.M, Molecular Modeling Approach to Prediction of Thermo- Mechanical Behavior of Thermoset Polymer Networks, Macromolecules, N. B. Shenogina, M. Tsige, S. M. Mukhopadhyay, S. S. Patnaik, Molecular modeling of thermosetting polymers: effects of degree of curing and chin length on thermo-mechanical properties, Proceedings of 18 th International Conference on Composite Materials, August, Shenogina, N.; Tsige, M.; Patnaik, S.; Mukhopadhyay, S.M. Molecular Modeling to Predict Elastic Properties of Highly Crosslinked Polymer Networks Using Dynamic Deformation Approach, to be submitted to Macromolecules. Book Chapter: S. S. Patnaik and M. Tsige, Modeling and Simulation of Nanoscale Materials Chapter 6 in Nanoscale Multifunctional Materials, Science and Applications, Edited by S. M. Mukhopadhyay, Wiley, Conferences: N.Shenogina, M.Tsige, S.Patnaik, S.M. Mukhopadhyay Molecular Dynamics Simulations of Mechanical Behavior of Thermosetting Polymers Dayton-Cincinnati Aerospace Sciences Symposium, March, 2011 N.Shenogina, M.Tsige, S.M. Mukhopadhyay, S.Patnaik A Molecular Modeling Approach to Predicting Thermo-Mechanical Properties of Thermosetting Polymers, US National Congress on Computational Mechanics-11, July, 2011 N.Shenogina, M.Tsige, S.M.Mukhopadhyay, S.Patnaik Molecular Modeling Simulations of Highly Cross-linked Polymer Networks: Prediction of Thermal and Mechanical Properties, American Physical Society March Meeting, March, 2012 M. Tsige, N.Shenogina, S.Patnaik, S.M. Mukhopadhyay, Molecular Modeling Simulations of Highly Cross-linked Polymer Networks, Invited Presentation, 10 th International Symposium of Polymer Physics, May 2012, Chengdou, China.

31 Acknowledgements Financial support from AFOSR Program Manager: Dr. Joycelyn Harrison Valuable discussions with Dr. Charles Lee Computer Time: DoD Supercomputing Resource Center High Performance Computing.

32 Thank you!

33 Mechanical Properties Simulation Basic Theory The isothermal elastic constants are defined as the elements of the 4th rank tensor of the second derivatives of the Helmholtz energy per unit volume, equivalent to the first derivative of the stress with respect to strain, i.e. For small deformations stress and strain are related by where C denote the stiffness coefficients, and S the compliance coefficients of the material

34 Mechanical Properties Basic Theory (cont d) Since C and S are both symmetric, there can be at most 21 elastic constants For completely isotropic amorphous materials such as the thermosets, the number of independent constants reduces to just two the so called Lamé constants λ and μ. Written in terms of the Lamé constants, the stiffness matrix takes on the form

35 Mechanical PropertiesTheory (cont d) The more familiar elastic constants in terms of Lamé constants can be written as Bulk modulus Shear modulus Young s modulus Poisson s ratio

Static approach for calculating elastic constants Three tensile and three shear deformations of magnitude ±0.

36 Static approach for calculating elastic constants Three tensile and three shear deformations of magnitude ± are applied to the system. The obtained stress tensor is then used to estimate stiffness coefficients of the material.

37 Mechanical Properties Theory Bounds Calculation Any small (e.g. nanometer scale) volume element of an amorphous material can be characterized by unique distribution of matter within it. Individual elements can be viewed as regions of a nanoscopically heterogeneous composite material. Distribution of the material properties in the macroscopic sample. Upper and lower bounds of the elastic constants of such material can be estimated by considering averages of the individual stiffness and compliance matrices of the elements. Thus we obtain the so-called Voigt and Reuss bounds

38 Mechanical Properties Theory Bounds Calculation(cont d) For typical simulated systems, the difference between Voigt and Reuss bounds can be 10-20% of the mean. To calculate mechanical properties of the thermosets with similar characteristics (e.g. hardeners or extents of reaction) we need improved (narrower) bounds estimates. Hill and Walpole approach can be used to obtain the improved bounds of the elastic constants

39 Mechanical Properties Theory Hill-Walpole Bounds Calculation For the stiffness matrix, the so-called Hill-Walpole average is given by Where n denotes the number of samples (cells) and C* is the stiffness matrix of a reference material

40 Mechanical Properties Theory Hill-Walpole Bounds Calculation(cont d) Two moduli, say B and G, are calculated for the stiffness matrix obtained for each model The extremes of these values over the full set of models are then used to define four comparison matrices by combining (Bmax,Gmax), (Bmin,Gmin), (Bmax,Gmin), (Bmin,Gmax) The 4 comparison matrices are used to generate 4 Hill-Walpole averages <C> HW and the associated sets of moduli {B, G, E, ν} Extreme values of the 4 sets of moduli are taken to define the Hill-Walpole bounds

MOLECULAR MODELING OF THERMOSETTING POLYMERS: EFFECTS OF DEGREE OF CURING AND CHAIN LENGTH ON THERMO-MECHANICAL PROPERTIES

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS MOLECULAR MODELING OF THERMOSETTING POLYMERS: EFFECTS OF DEGREE OF CURING AND CHAIN LENGTH ON THERMO-MECHANICAL PROPERTIES N. B. Shenogina 1, M. Tsige