Origins of Mechanical and Rheological Properties of ...N F D N C N ∝v P −v N v P N v C •...

Origins of Mechanical and RheologicalProperties of Polymer Nanocomposites

Venkat Ganesan$$$: NSF DMR, Welch Foundation

Megha Surve, Victor Pryamitsyn

Department of Chemical Engineering

University of Texas@Austin

Polymer Nanocomposites

⊕

Polymers (Blends,Block copolymers)

Nano-FillersNanocomposites

Single- and Multi-WalledCarbon Nanotubes

Fullerenes/Buckyballs Montmorillonite Clays

Polymer Nanocomposites

⊕

Polymers (Blends,Block copolymers)

Nano-FillersNanocomposites

• Processing nanocomposites requires understanding their flow behavior.

• Flow fields provide a versatile approach for controlling dispersions of nanocomposites.

Interplay of hydrodynamics and friction in a viscoelastic medium

Flow-inducedorientation

(SchmidtVermant)

Effect of molecular interactions

Flow-inducedJamming

Flow-inducedgelation

(Schmidt) (Galgali)

• Objective: To model the flow dynamics and structure of nanoparticle-polymer mixtures.

Challenges

The Approach

Explicit Solvent Method

• Captures hydrodynamics and other interactions.

• Due to size asymmetry, is computationally expensive.

(Molecular dynamics)Continuum Methods

• Captures hydrodynamics and is computationally tractable.

• Can’t include interactions withthe solvent.

• Not developed for Non-Newtonianflows.

(Stokesian dynamics, Lattice-Boltzmann)

The Approach

Explicit Solvent MethodCoarse-Grained


Collection of microscopic solvent units

( )ppU r

( )pcU r• Particle and solvent units interact by coarse-grained potentials.

• are derivable from more atomistic representations.

( ), ( )pc ppU r U r

The Approach



Cv

PvN

DF NC

NP vv −∝

NPv

NCv

• Particles interact by momentum conservingthermostat (preserves hydrodynamics).

• Involves (central) dissipative forces dependent upon the normal component of the velocity differences.

• Similar to Dissipative Particle Dynamics.

The Approach



• Does not capture local hydrodynamics.

• Requires tangential (not central) velocity-dependent forces.

Cv

PvTPv

TCv T

DF TC

TP vv −∝

No-Slip

The Approach



• Does not capture local hydrodynamics.

• Requires tangential (not central) velocity-dependent forces.

PvN

DF

NDF

TDF

Composite Particles

The Approach



• Directly incorporate tangential velocity-dependent forces.

Cv

PvTPv

TCv

TDF T

CTP vv −∝

Our proposal

Hydrodynamic Friction Forces

Cv

PvDF

• Conserves linear and angular momentum• Preserves hydrodynamical phenomena

Cv

PvN

DF NC

NP vv −∝

NPv

NCv

• Computationally tractable (for size asymmetric systems)

• Includes tangential friction• Brownian dynamics + Dissipative forces

Cv

PvTPv

TCv

TDF T

CTP vv −∝

⊕

(Espanol, 1998; Pryamitsyn and Ganesan, JCP, 2005)

MomentumCv Pv

Cω Pω CPDF ωω −∝Angular

TDF

RF Random forces

ijU ConservativeDissipative

Coarse-Grained Colloidal Suspension

CCU

CPUPPU

≡

CCU CPU PPU

Repulsive LJ Repulsive LJ

CCσ CPσ

PPσSoft Repulsion

• Mixture of colloid and solvent• Colloid:Solvent Radius = 5:1.

(Pryamitsyn and Ganesan, JCP, 2005)

Our methodm

φφ

Φ =Brownian Dynamics

Experiments

Stokesian Dynamics

1

10

100

1000

0 0.2 0.4 0.6 0.8 1

Hydrodynamic Interactions: Zero-Shear Viscosity

• Coarse-Grained solvent method captures hydrodynamicalinteractions

rη

Φ

Shear Rheology

CvCv

CvCv Rheology is sensitive to lubrication forces

Provides a sensitive test of explicit solventmodel.

0.250.31

0.39

0.48φ =

Stokesian Dynamics(Brady)

0.25

0.31

0.39

0.48

0.32

0.420.47

0.48

Agrees to within 20% of Stokesian dynamics results

shear thickening

Summary So Far..

• Outlined a coarse-grained explicit solvent method to simulate hydrodynamical phenomena involving particles in complex fluids.

• Provided evidence that both hydrodynamical and other interactions can be faithfully captured.

• Results on hard sphere suspensions provided new insightsinto the interplay between glass transition, hydrodynamics and rheology.

⊕

Polymers (Solutions, Blends)

Nano-fillers (Clay, Nanotubes, Fullerenes) Nanocomposites

Silica + PEO#

Reinforcement even at η ~ 2%

# Zhang and Archer, Langmuir, 2002

Addition of small particles – Significant property enhancement!!

Elastic modulus doubles

Viscosity increases by an order of magnitude

Microsized particles vol ~ 30% Nanoparticles vol ~ 1 -5 %

mod

ulus

0.5 %0 %

1 %

2 %

Why Polymer Nanocomposites ?

Issues and Questions

• How do nanoparticles modify the mechanical properties of polymer matrices ?

• What are the mechanisms underlying the above effects ?

• What are the parameters governing the mechanisms ?

Rheology of PNCs: Linear Viscoelasticity

Polycaprolactone + Montmorillonite Clays* PC + Nanotubes**

*:Krishnamoorti and Giannelis; **: Fornes and Paul

• Significant enhancements in elasticity at extremely low loadings

• Change of viscoelastic response to “solid-like” behavior.

Rheology: Explanations

Particle Jamming/PercolationR2

d dR >>2

• Jamming/percolationoccurs at low φ

13 ≈Rφ

• Leads to solid behaviorand the enhancementsin modulus.

(Krishnamoorti)

Rheology: Explanations

Polymer Network Mechanism

Immobilized Monomers

• Elasticity due to transient network formation.

• Plateau modulus due to bridges.

(Kumar and Douglas)

Rheology of PNCs: Model System

)1(2

OR

CC

Pg ≈

σ

h )1(ORh

g

≈Slow

• Mixture of spherical nanoparticles in polymer matrices

• Advantage: A lot is known about spherical colloidal dispersions• Disadvantanges: Orientational effects are absent.

Need much higher loadings.

Model of Polymer Nanocomposites

CCU CPU

Repulsive LJ Repulsive LJ

CCσ CPσ

• Mixture of colloid and polymeric meltCPU

CCU≡

PPF

0.5=PP

CC

σσ

PPb

0.3=PP

PPbσ

FENE Springs

PNCC

PgR

σ2

4

48

45.0

6.1

(VP & VG: Macromolecules, 2006; J. of Rheology)

0.0001

0.001

0.01

0.1

1

10

0.0001 0.001 0.01 0.1 1


)(' ωG

ω0.01

0.1

1

10

0.0001 0.001 0.01 0.1 1

)(' ωG

ω

5.0

33.011.0

01.0

01.0

11.0

33.0

5.0

18.0ω

53.0ω6.1ω

3.0ω

• Lower Loadings: Enhancement in modulus but no apparent change in relaxation behavior.

• Higher Loadings: Significant enhancement in modulus and a solid-like behavior.

24=PN 96=PN


Polycaprolactone + Montmorillonite Clays* PC + Nanotubes**

*:Krishnamoorti and Giannelis; **: Fornes and Paul

• Significant enhancements in elasticity at extremely low loadings

• Change of viscoelastic response to “solid-like” behavior.

0.0001

0.001

0.01

0.1

1

10

0.0001 0.001 0.01 0.1 1


)(' ωG

ω0.01

0.1

1

10

0.0001 0.001 0.01 0.1 1

)(' ωG

ω

5.0

33.011.0

01.0

01.0

11.0

33.0

5.0

18.0ω

53.0ω6.1ω

3.0ω

• Enhancement in modulus but no apparent change in relaxationbehavior at lower loadings: Why ?

• Significant enhancement in modulus and a solid-like behaviorat higher loadings: Why ? (Not in this talk)

24=PN 96=PN

Impact on Polymer Dynamics• Significant impact upon glass transition temperature and polymer dynamics on adding nanoparticles.#

How does the polymer dynamics change due to addition of nanoparticles ?

#: Giannelis, Adv. Pol. Sci, 138, 107

Simulation FeaturesNo glass transition

Coarse-grained model

)(tR Normalmodes )(tX m

)0()( mm XtX

For unentangled polymers,)/exp()0()( mmm tXtX τ−≈

2−∝ mmτ ξτ ∝−21 pN Related to viscosity of media

Effect of Particles on Polymer Dynamics

)/exp()0()( ppp tXtX τ−≈

0

1

2

3

4

5

6

7

8

9

10

0 0.25 0.5 0.75 1

Rp

p

ττ

)1/()1( −− PNp

01.0=φ11.0=φ

33.0=φ

8=PN24=PN48=PN96=PN

BareMatrix

Overall Slowing of Polymer Relaxations

5.0=φ

Effect of Particles on Polymer Dynamics

)/exp()0()( ppp tXtX τ−≈

Slowed Monomers

• Simulations of Grant Smith: Attractions lead to only a weak slowing down in melts.

• Different monomers of a chain access the slower regions:Overall slowing down of the polymers

Effect of Particles on Polymer Viscoelasticity

)/exp()0()( ppp tXtX τ−≈ ∑=

−∝⇒PN

ppttG

1

)/2exp()( τ

1.00E-06

1.00E-03

1.00E+00

0.0001 0.001 0.01 0.1 1

)('pol ωG

ω

Modified G’pol due to changes in relaxation times



−∝⇒PN

ppttG

1

)/2exp()( τ

1.00E-06

1.00E-03

1.00E+00

0.0001 0.001 0.01 0.1 1

)('pol ωG

ω


0.0001

0.001

0.01

0.1

1

10

0.0001 0.001 0.01 0.1 1

)(' ωG

ω

5.0

33.011.0

01.0

6.1ω

3.0ω

24=PN



−∝⇒PN

ppttG

1

)/2exp()( τ

1.00E-06

1.00E-03

1.00E+00

1.00E+03

0.0001 0.001 0.01 0.1 1

)('pol ωG

ω

)(' ωG01.0=φ

11.0=φ

33.0=φ


)(' ωG

Physical Picture of Polymer Rheology at Low Loadings

• Particle-induced changes in polymer dynamics is responsible.

• For the weakly attractive particles, the above manifests as just a change in relaxation times.

• For strongly attractive particles, the above manifests as the modulus due to polymer-bridged networks.

wt% C60

PMMArep

rep

ττ

Comparisons to Experiments• (Kropka, Green and Ganesan, Macromolecules, In Press) Comparison of the relaxation times in nanocomposites to the bare relaxation times.

PMMA + C60 NC

αTaTω (rad/s)

βbTG

′, β T

b TG″

(Pa)

Comparisons to Experiments• (Kropka, Green and Ganesan, Macromolecules, in Press) Superposition of mechanical modulii after renormalization of relaxation times.

Rheology of PNCs: Issues and Questions

• How do nanoparticles modify the rheology of polymer matrices ?

• What are the mechanisms underlying the above effects ?

• At low particle loadings, polymer-bridging of particlesis the responsible mechanism.

• What are the parameters governing the mechanisms ?

• What are the elastic and structural properties of the gels ?

• Why do nanoparticles lead to prevelant gelation ?

• How does the concentration of particles and polymer affect the gelation, stability characteristics of the mixture ?

≈

Integrate out polymer degrees of freedom

Pz

+)2(

EffU)1(EffU+≡

)4(EffU)3(

EffU + +..

Monte Carlo MovesAnd Equilibration

Cluster Counting

βU

Effective Interactions

Bridging

ProbabilityBRP

Gelation, Elastic Modulus

Uβ rThermodynamics,

Phase Behavior

Outline of Approach

Analogous to density functional theories used to obtaininteratomic potentials in quantum chemistry

MS, VP, VG:JCP 2005; Langmuir ’06;Macromolecules ’06.

Bead-spring model of polymer

Self avoiding random walkCharge effects

N segment chain

Integrating Out the Polymer By Mean-Field Theory

Thermodynamics

( )W r

• Single chain in a potential field W(r).• W(r) determined self-consistentlyto match statistical properties ofpolymers, say, the volume fraction.

Idea behind mean-field theory*

*: Helfand (1975)

Bead-spring model of polymer

Self avoiding random walk

Presence of other chainsSelf consistent potential field w(r)

Mean Field Approach

Configurations of a chain subjected to w(r)

)'()0;',();',()();',();',( 2 rrrrGnrrGriwnrrGs

nrrG−=−∇=

∂∂ δ

r'n

G(r, r’; n)

rBCs :Polymer - particle interactions

Charge effectsN segment chain

Integrating Out the Polymer By Mean-Field Theory

Configurations of a chain subjected to w(r)

)'()0;',();',()();',();',( 2 rrrrGnrrGriwnrrGs

nrrG−=−∇=

∂∂ δ

r'n

G(r, r’; n)

r

Density distributions for polymer

Bispherical coordinates

Numerical solution Curvature of particles accounted exactly!!!

Polymer mediated effective interactions

Bridge

Number and probability distribution

LoopTail

Field-Theory Model For Polymers

Smaller particles – tails dominate

0

1

0 1 2 3 4

φ = 2.58

tails

loops

Tails dominate ! Loops dominate !

Frac

tion

of

segm

ents

in

loop

s &

tails

Radius of the particle, R / Rg

Loop

Tail

Adsorbed Layer: Particle Size Effects

Smaller particles – More number of bridges

Bridge

0

4

8

12

0 2 4 6

R/Rg = 0.5R/Rg = 1.0R/Rg = 2.0

Interparticle distance, r/R

# o

f br

idge

s/ar

ea

R/Rg

Bridging: Particle Size Effects

Effective interactions

Monte Carlo Moves & Equilibration

# & size of clusters, Percolation thresholds

bridge

Cluster formation

Volume fraction at which a space spanning cluster is observed

Bonds generation ~ bridging probability

I II

Cluster Statistics and Gelation

0

0.05

0.1

0 0.5 1 1.5 2 2.5

Polymer meltφ = 5.16

Smaller particles – Gel much earlier!

Particle size ratio, R/Rg

Perc

olat

ion

thre

shol

d

R/Rg = 2.0 R/Rg = 1.0 R/Rg = 0.5

η = 8%

Cluster Statistics and Gelation

Post-gel systems Identification of backbone

R/Rg = 2.0 R/Rg = 1.0 R/Rg = 0.5

Simple Network Theories: Elastic Modulii = Number of Bridges in backbone

Graph theory

f

Elastic properties

f

Smaller particles – Gel much earlier! η = 8%

Determining Elastic Properties

Much Stronger Enhancements of Moduli in Smaller Particles

0

400

800

0 0.05 0.1 0.15

R/Rg = 0.5R/Rg = 1.0R/Rg = 2.0

Particle volume fraction, ηparticle

Elas

tic

mod

ulus

, G’

*Surve, Prymitsyn, Ganesan, Phys. Rev. Lett.

Smaller quantities of nanoparticles are required.

Bridging induced clustering of particles responsible for reinforcement.

Smaller particles -> stronger reinforcement.

Elastic Modulii of Gels

G’

Particle volume fraction, (η-ηc)

G’ ~ (η-ηc)1.799

(η-ηc)

1.E-03

1.E-01

1.E+01

1.E+03

1.E+05

0.001 0.01 0.1 1

~1.84

~3.08

Universal scaling of elastic modulus

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0.0001 0.001 0.01 0.1 1

Simulation results* Experimental data

*Surve, Prymitsyn, Ganesan, Phys. Rev. Lett.

Scaling of Elastic Modulii of Gels

GelationFor smaller particles, gelation occurs at very low volume fractionsSmall particles -> dense networksSmall particles -> Much stronger enhancements in modulii

Conclusions – Part II

Origins of Mechanical and Rheological Properties of ...N F D N C N ∝v P −v N v P N v C •...

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