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Why Nanoreinforced Polymers?: Mechanics Issues. Cate Brinson Frank Fisher Roger Bradshaw T. Ramanathan. Qian, Dickey, et al 2000. Outline. Motivation Why nano reinforced polymers? What are nanotubes anyway? Modeling Top-down: micromechanics Geometry effects Experiments

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Why nanoreinforced polymers mechanics issues

Why Nanoreinforced Polymers?: Mechanics Issues

Cate Brinson

Frank Fisher

Roger Bradshaw

T. Ramanathan

Northwestern University


Qian, Dickey, et al 2000


  • Motivation

    • Why nano reinforced polymers?

    • What are nanotubes anyway?

  • Modeling

    • Top-down: micromechanics

    • Geometry effects

  • Experiments

    • Bulk time dependent behavior (DMA)

    • TTSP  relaxation spectra

    • Probes non-bulk polymer behavior

  • Summary and Future Directions

Northwestern University

Motivation why nanocomposites
Motivation: why nanocomposites?

  • Why nanotubes?

    • 1TPa modulus

    • High tensile strains (5% experimental)

  • Chemical interactions

    • Small volume fraction  large non-bulk polymer phase

    • Functionality of NT-matrix tailorable

  • Geometrical constraints

    • High surface to volume ratio

    • Interparticle distance decreases

    • Entanglement leads to strength?

Northwestern University

Motivation why the interphase

Overlapping interphases

10 mm fiber

Nano tube

  • Overlap interphases 10% vf (not 60%+)

Motivation: Why the interphase?

Surface Area/Volume

  • Nanotube Reinforced

    • Surface area/volume 103 to 104 higher than micron sized fibers




Northwestern University

Motivation goals






Motivation: goals

Expected behavior

  • The holy grail: high stiffness, high strength, high toughness, low weight

    • Understand mobility changes in polymer due to NTs

    • Understand effects of NT geometry

    • Bridging of mechanics models at several length scales

Northwestern University

What are carbon nanotubes

(scale bar = 5 nm)

What are Carbon Nanotubes

  • Hexagonal sheet of carbon atoms rolled into 1D cylinder

  • Different forms of nanotubes: SWNTs, MWNTs, and NT ropes or bundles (Harris 1999)

  • Nanotube diameters from 1 to 50+ nm, and lengths on the order of µm (aspect ratios of 1000+)

Northwestern University

Comparison of reinforcement fillers
Comparison of reinforcement fillers

  • Expected mechanical properties of carbon nanotubes compare quite favorably with other types of structural reinforcement

  • NT fracture strains of 15% (numerical) and 5%+ (experimental)

Northwestern University

Motivation benefits and obstacles

Exploit extraordinary mechanical properties for high stiffness, high strength

Multifunctionality: electrical percolation at <0.1%, tune conductive to semi-conductive with chirality

Increase temperature range of polymer

Possible to use standard polymer processing methods

High cost for high purity SWNT ($100/g)

Poorly understood NT-polymer interface effects

Difficult to achieve uniform dispersion of NTs

Lack of control of NT geometry within composite

Existing contradictory data

Motivation: Benefits and Obstacles

Journet, et. al., 1997

SWNT bundles formed viaarc discharge method.

Northwestern University

Factors influencing nrp effective properties
Factors influencing NRP effective properties stiffness, high strength

  • Properties of NTs

    • Method of fabrication, SWNT vs MWNT vs bundle

  • Matrix-nanotube bonding / load transfer

  • NT dispersion

  • NT geometry

    • Alignment

    • Curvature/waviness

  • Influence of NTs on viscoelastic behavior of NRP

Northwestern University

Modeling nanotube geometric effects
Modeling: Nanotube geometric effects stiffness, high strength

  • Current: top-down approach

    • Use micromechanics tools at nanomechanics level

    • Account for NT geometry and moduli

    • Predict elastic response

  • Future: bottom-up approach

    • Use MD simulations

    • Calculate NT impact on polymer locally

    • Bring key response parameters upscale

Modeling Mini-outline

  • Mori-Tanaka (MT) method

  • Moduli predictions - alignment

  • NT waviness - hybrid FE/MT approach

Northwestern University

Modeling micromechanics
Modeling: Micromechanics stiffness, high strength

  • Mori-Tanaka method

    • Uses Eshelby’s classic inclusion analysis

    • Random to aligned inclusions

    • Quick analytic technique

    • Extendable for viscoelastic behavior

  • Average Fields

  • Strain Concentration Matrix A2 for dilute soln

Strain in


Strain farfield e0

Northwestern University

Modeling micromechanics1
Modeling: Micromechanics stiffness, high strength

Strain on


  • Mori-Tanaka theory

    • Each inclusion “sees” boundarystrain equal to the averagestrain in the matrixe1

    • Particle interaction

    • Stiffness C* in terms of the dilute concentration matrix A2

  • Eshelby dilute solution for A2

Strain in


Strain farfield e0

Eshelby Tensor (inclusion shape, moduli)

Northwestern University

Effective modulus of nrps
Effective modulus of NRPs stiffness, high strength

  • Qian et al, 2000

    • 1% (wt) MWNT’s in polystyrene

    • Nanotube diameter of ~30 nm

    • ~35% increase in modulus

    • ~25% increase in ultimate stress

  • Schadler et al, 1998

    • 5 wt% MWNTs in an epoxy matrix

    • NTs were poorly distributed, but well dispersed (individual tubes)

    • “NTs remained curved and interwoven in the epoxy”

    • 20-25% increase in modulus

Northwestern University

Micromechanical predictions of effective moduli
Micromechanical Predictions of Effective Moduli stiffness, high strength

  • Start with simple micromechanics to estimate the NRP effective modulus

    • ENT = 450 GPa (CVD MWNTs, Rodney Andrews, U. Kentucky)

    • Mori-Tanaka method, 2D and 3D random orientation of inclusions

    • Perfect bonding between the phases

    • Cylindrical inclusion (defines S)

  • Significant modulus increase, but less than simple micromechanics predictions

Schadler, et. al. (1998)

5 wt% MWNTs in epoxy

Northwestern University

Micromechanics prediction for nrp
Micromechanics Prediction for NRP stiffness, high strength

  • Simple MT results overpredicting significantly even with low ENT

  • Important considerations for Nanoreinforced Polymers:

    • NT alignment

    • Accurate NT modulus

    • SWNT vs MWNT vs bundles

    • Matrix-nanotube bonding / load transfer

    • NT dispersion

    • Non-bulk polymer behavior

    • NT Curvature/Waviness

Northwestern University

Micromechanics modeling waviness

Strain on stiffness, high strength


Strain in


Micromechanics: Modeling waviness

  • How account for NT curvature?

  • Use hybrid Finite Element - Analytic approach

    • FE unit cell with wavy NT

    • Fiber shape – infinitely long sinusoid

    • Numerically determine A2

    • Use A2 in MT

y = a cos (2 π z / L)


Northwestern University

Micromechanics modeling waviness1
Micromechanics: Modeling waviness stiffness, high strength

  • Wavy Inclusion Analysis Method

    • Volumetrically averaged fiber strain

    • Applied farfield strains

    • Calculate resulting average fiber strain

      • Element strain eij is at element centroid

    • Calculate A2:

  • Problem reduced to 3 variables: ENT / Em, a / L, L / d

Northwestern University

Micromechanics modeling waviness2

Wavelength L stiffness, high strength

Amplitude a

Micromechanics: modeling waviness

  • Various assumptions

    • Treating NT as continuum

    • Solid cross-section for NT

    • Single shape for NT

  • Effective reinforcing modulus concept

    • EERM

Effective modulus of wavy NT if it were straight

a / L

Northwestern University

Multiphase composite analysis
Multiphase composite analysis stiffness, high strength

  • Variable waviness within the NRP: waviness distribution (as discrete phases)

  • Each phase has a characteristic A2 based on the waviness of the phase

  • Proceed with an appropriate multiphase composite analysis for the effective properties

Northwestern University

Effective modulus predictions for nrp
Effective modulus predictions for NRP stiffness, high strength

  • Schadler, et. al., 1998

  • 5 wt% MWNTs in epoxy

  • Waviness distribution 2 (from table)

  • Andrews, et. al., 2002

  • MWNTs in polystyrene

Fisher, Bradshaw, Brinson: Applied Physics Letters, 2002; Composites Science & Tech., in press

Northwestern University

From elastic modeling to ve experiments
From Elastic Modeling to VE Experiments stiffness, high strength

  • Elastic, micromechanics modeling indicates geometry of NT is a reinforcement limiting mechanism

  • Non-straight geometry may be important for strength, however…. (future work)

  • Beyond elasticity: intriguing impact on polymer time dependent response

    • Experiments for bulk Viscoelastic (VE) response

    • VE response ideal to probe non-bulk polymer behavior

    • Evidence of significant reduced polymer mobility with low volume fractions

Northwestern University

Viscoelasticity of nrps
Viscoelasticity of NRPs stiffness, high strength

Gong et al (2000)

Shaffer and Windle (1999)

  • NTs may drastically alter the viscoelastic behavior of the polymer

  • Tg shift of 35 C with NTs and surfactant as a processing aid

  • Broadening of the high temperature end of the tan d peak

  • Suggest that the NTs impact the mobility of the polymer chains



Northwestern University

Impact of molecular mobility on ve behavior
Impact of molecular mobility on VE behavior stiffness, high strength

Odegard, et. al., 2002

Lordi and Yao, 2000

  • Polymer chemistry

    • long sequences of atoms linked via primary (covalent) bonds

    • Polymer chains are highly entangled, networked, have side chains

  • Viscoelastic response - initial elastic response, followed by long-range coordination and chain rearrangement

  • Mobility results in time- and temperature-dependent properties, which can be investigated via

    • Measurement of the Tg

    • Frequency response

    • Time dependent response

    • Physical aging

Northwestern University

Dynamic mechanical analysis
Dynamic Mechanical Analysis stiffness, high strength

  • TA Instruments DMA 2980

    • -150 to 600 C

    • 0.001 to 200 Hz

    • Film tension clamp (t < 2 mm)

  • Polycarbonate-based NRPs (blank, 1 wt%, 2 wt% MWNTs)

    • Tg measurements (T sweep at constant w)

    • Frequency response (scan w at multiple T, time-temperature superposition)

    • Physical Aging creep testing (time domain)

  • Storage modulus (E’) - measure of the elastic (in-phase) response

  • Loss modulus (E’’) - measure of the viscous (out-of-phase) response

  • Loss tangent (tan d) - ratio of storage to loss modulus

Northwestern University

Pc mwnt samples
PC-MWNT Samples stiffness, high strength

  • Solution based processing

  • Evidence of good dispersion

  • Evidence of interphase on NT

Northwestern University

T g measurement
T stiffness, high strengthg measurement

  • Temperature sweep

    • w = 1 Hz

    • DT = 2 °C/min

    • amplitude = 3 µm

  • Storage modulus

    • Higher glassy storage modulus

    • Much higher rubbery storage modulus

  • Loss Modulus

    • Slight shift in Tg to higher temperatures

    • Broadening of E’’ peak

Northwestern University

Frequency response of nrp
Frequency response of NRP stiffness, high strength

2% MWNT in PC (RPI), Tref =150 C

test range



Experimental data

  • Time-temperature superposition to evaluate over extended w range

  • Fit frequency response to a Prony series model of VE behavior

Storage modulus

Loss modulus

Northwestern University

Relaxation spectrum
Relaxation spectrum stiffness, high strength

  • Given the Prony series, we have the time domain response

  • From E(t), we can find the relaxation spectrum H(t)

  • Alfrey’s approximation

  • Greater width of relaxation spectrum indicative of more modes of relaxation

  • Greater contribution of longer relaxation times - consistent with reduced mobility

Northwestern University

To determine interphase mobility and ve properties

Provide indications of required mobility changes due to NTs stiffness, high strength

To determine interphase mobility and VE properties

  • Frequency domain response - can be modeled using micromechanics

  • Ideally

    • Molecular level simulations

    • Atomic scale experimental characterization

  • As a first approximation

    • Assume properties for the interphase behavior

    • Use micromechanical models to predict the NRP properties

    • Compare predictions with experimental data for NRPs, infer interphase volume fraction and properties

Northwestern University

Prony series to model ve behavior
Prony series to model VE behavior stiffness, high strength




  • Model the interphase as a simple shift in the relaxation times of the polymer (characterized by mobility parametera)

  • Neglect vertical shifting of the modulus response

  • First approximation: choose interphase relaxation times to match loss modulus experimental data for the NRP

Northwestern University

Micromechanical modeling
Micromechanical Modeling stiffness, high strength

  • Mori-Tanaka 3D random alignment using Correspondence Principle

    • RPI 2% MWNTs in PC

    • Matrix moduli from DMA

    • ENT = 200 GPa - to match elastic (high w) response

    • Interphase volume fraction = 10%

    • Infer shift in relaxation times

  • Loss moduli qualitatively agree

  • No contribution from elastic nanotube to loss modulus

a = 100

a = 1000

Northwestern University

Physical aging
Physical Aging stiffness, high strength



  • Need to predict the long-term time-dependent properties

  • Physical Aging:

    • material in a non-equilibrium state below Tg

    • interpreted in terms of free volume

    • Material slowly evolves towards equilibrium (physical aging)

  • Standard physical aging test sequence

    • Rejuvenation

    • Isothermal quench

    • Aging time

Northwestern University

Preliminary aging results

pure PC stiffness, high strength

Preliminary Aging Results

  • Rejuvenated 165°C for 15 min; aging temperature 140°C

  • Description of physical aging

    • Shift factor: shift of compliance curves in log space

    • Shift rate: slope of shift factor vs aging time

  • Shift rates decrease with addition of NTs

  • Consistent with reduced-mobility interphase

    • nanotubes “lock out” free volume

Northwestern University

Summary of experimental results
Summary of experimental results stiffness, high strength

Standard VE test

Most sensitive to NTs?

Micromechanical analysis

  • NRPs have different viscoelastic behavior than bulk polymer

  • Attributed to the influence of NTs on molecular mobility of the polymer chains

  • Experimental data consistently interpreted by the presence of a reduced mobility, non-bulk polymer interphase region

    • Slight increase in effective Tg of the material

    • Broadening of the relaxation spectra

    • Decrease in the physical aging shift rates

Northwestern University

Ongoing work nt functionalization
Ongoing Work: NT Functionalization stiffness, high strength

  • Control interactions with polymer matrix

    • Design stiff/flexible interactions

  • Easier dispersion in solvents & polymer

Northwestern University

Nanotube functionalization
Nanotube functionalization stiffness, high strength

Base functionalization


More ductile composite?


Flexible bond


More Brittle composite?

Stiff bond

Northwestern University

Where we are headed
Where we are headed: stiffness, high strength

  • Strength: not addressed yet

    • Geometric and chemical impacts on strength

  • Bottom up approach to modeling, from MD side

  • Real multiscale modeling :

    • MD  interface strength, nonbulk props  mesoscale models (FE and MT) to address strength

  • Real multiscale experiments:

    • Nanoindentation near NTs  local behavior

    • Nanotube pullout?  strength criterion

    • Couple with modeling

  • Make extremely, stiff, strong, lightweight composites

Northwestern University

Future nanotube pullout
Future: Nanotube Pullout stiffness, high strength

In collaboration with: R. Ruoff group at NU, L. Schadler @ RPI

Northwestern University

Research programs
Research Programs stiffness, high strength

  • Nanoreinforced Polymers

  • Shape Memory Alloys

  • Aging of Polymers & Composites

  • Porous Ti - Bone Implants

Northwestern University

Acknowledgments stiffness, high strength

NASA Langley Research Center

Computational Materials: Nanotechnology Modeling and Simulation program

The NASA URETI BIMat Center grant is also gratefully acknowledged

Northwestern University

Northwestern University stiffness, high strength