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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?

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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




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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)

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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

  • 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

  • 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

  • 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

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Modeling micromechanics1

Modeling: Micromechanics

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

  • 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

  • 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

  • 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


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)


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Micromechanics modeling waviness1

Micromechanics: Modeling waviness

  • 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

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

  • 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

  • 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

  • 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

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



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Impact of molecular mobility on ve behavior

Impact of molecular mobility on VE behavior

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

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Dynamic mechanical analysis

Dynamic Mechanical Analysis

  • 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

  • Solution based processing

  • Evidence of good dispersion

  • Evidence of interphase on NT

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T g measurement

Tg 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

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

  • 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

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




  • 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

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Micromechanical modeling

Micromechanical Modeling

  • 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

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Physical aging

Physical Aging



  • 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

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Preliminary aging results

pure PC

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

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

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Ongoing work nt functionalization

Ongoing Work: NT Functionalization

  • Control interactions with polymer matrix

    • Design stiff/flexible interactions

  • Easier dispersion in solvents & polymer

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Nanotube functionalization

Nanotube functionalization

Base functionalization


More ductile composite?


Flexible bond


More Brittle composite?

Stiff bond

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Where we are headed

Where we are headed:

  • 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

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

Northwestern University

Research programs

Research Programs

  • Nanoreinforced Polymers

  • Shape Memory Alloys

  • Aging of Polymers & Composites

  • Porous Ti - Bone Implants

Northwestern University



NASA Langley Research Center

Computational Materials: Nanotechnology Modeling and Simulation program

The NASA URETI BIMat Center grant is also gratefully acknowledged

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Why nanoreinforced polymers mechanics issues

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