Department of Chemical Engineering University of Texas@Austin Origins of Mechanical and Rheological Properties of Polymer Nanocomposites Venkat Ganesan $$$: NSF DMR, Welch Foundation Megha Surve, Victor Pryamitsyn
Polymer Nanocomposites Polymers (Blends, Block copolymers) Nano-Fillers Nanocomposites Single- and Multi-Walled Carbon Nanotubes Fullerenes/ Buckyballs Montmorillonite Clays
Polymer Nanocomposites Polymers (Blends, Block copolymers) Nano-Fillers Nanocomposites Processing nanocomposites requires understanding their flow behavior. Flow fields provide a versatile approach for controlling dispersions of nanocomposites.
Challenges Objective: To model the flow dynamics and structure of nanoparticle-polymer mixtures. Interplay of hydrodynamics and friction in a viscoelastic medium Flow-induced orientation (Schmidt Vermant) Effect of molecular interactions Flow-induced gelation Flow-induced Jamming (Schmidt) (Galgali)
The Approach Explicit Solvent Method (Molecular dynamics) Captures hydrodynamics and other interactions. Due to size asymmetry, is computationally expensive. Continuum Methods (Stokesian dynamics, Lattice-Boltzmann) Captures hydrodynamics and is computationally tractable. Can t include interactions with the solvent. Not developed for Non-Newtonian flows.
Explicit Solvent Method The Approach Coarse-Grained Explicit Solvent Method U () pc r U () pp r Collection of microscopic solvent units Particle and solvent units interact by coarse-grained potentials. Upc (), r Upp () r are derivable from more atomistic representations.
Explicit Solvent Method The Approach Coarse-Grained Explicit Solvent Method N v C N v P N F D v N P v C v N C v P Particles interact by momentum conserving thermostat (preserves hydrodynamics). Involves (central) dissipative forces dependent upon the normal component of the velocity differences. Similar to Dissipative Particle Dynamics.
Explicit Solvent Method The Approach Coarse-Grained Explicit Solvent Method T vc T v P v C v P T F D No-Slip v T P v T C Does not capture local hydrodynamics. Requires tangential (not central) velocity-dependent forces.
Explicit Solvent Method The Approach Coarse-Grained Explicit Solvent Method Composite Particles N F D N F D v P T F D Does not capture local hydrodynamics. Requires tangential (not central) velocity-dependent forces.
Explicit Solvent Method The Approach Coarse-Grained Explicit Solvent Method T v C T v P T F D v T P v C v P v T C Our proposal Directly incorporate tangential velocity-dependent forces.
F v D v C P Hydrodynamic Friction Forces N F D N v C N v P Angular Momentum v N P v C v N C vc ωc v P T v C T v P T F D v P ωp FD v T P v C v P P v T C ω ω C Conserves linear and angular momentum Uij Conservative Preserves hydrodynamical phenomena T F D Dissipative Includes tangential friction Brownian dynamics + Dissipative forces FR Random forces Computationally tractable (for size asymmetric systems) (Espanol, 1998; Pryamitsyn and Ganesan, JCP, 2005)
Coarse-Grained Colloidal Suspension Mixture of colloid and solvent Colloid:Solvent Radius = 5:1. U CP U PP U CC UCC UCP U PP σ CC σ CP Repulsive LJ Repulsive LJ (Pryamitsyn and Ganesan, JCP, 2005) σ PP Soft Repulsion
Hydrodynamic Interactions: Zero-Shear Viscosity 1000 η r Our method Brownian Dynamics Φ= φ φ m Experiments 100 Stokesian Dynamics 10 1 Φ 0 0.2 0.4 0.6 0.8 1 Coarse-Grained solvent method captures hydrodynamical interactions
Shear Rheology v C Rheology is sensitive to lubrication forces Provides a sensitive test of explicit solvent model. Stokesian Dynamics shear thickening (Brady) v C φ = 0.48 0.39 0.31 0.25 0.48 0.39 0.31 0.25 0.48 0.47 0.42 0.32 Agrees to within 20% of Stokesian dynamics results
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 insights into the interplay between glass transition, hydrodynamics and rheology.
Why Polymer Nanocomposites? Polymers (Solutions, Blends) Nano-fillers (Clay, Nanotubes, Fullerenes) Nanocomposites Reinforcement even at η ~ 2% modulus 2 % 1 % 0.5 % 0 % Elastic modulus doubles Viscosity increases by an order of magnitude Microsized particles vol ~ 30% Nanoparticles vol ~ 1-5 % Silica + PEO # # Zhang and Archer, Langmuir, 2002 Addition of small particles Significant property enhancement!!
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** Significant enhancements in elasticity at extremely low loadings Change of viscoelastic response to solid-like behavior. *:Krishnamoorti and Giannelis; **: Fornes and Paul
Rheology: Explanations Particle Jamming/Percolation 2R d 2R >> (Krishnamoorti) d Jamming/percolation occurs at low φ φr 3 1 Leads to solid behavior and the enhancements in modulus.
Rheology: Explanations Polymer Network Mechanism (Kumar and Douglas) Immobilized Monomers Elasticity due to transient network formation. Plateau modulus due to bridges.
Rheology of PNCs: Model System Mixture of spherical nanoparticles in polymer matrices 2R σ P g CC O(1) Slow h h O(1) R g Advantage: A lot is known about spherical colloidal dispersions Disadvantanges: Orientational effects are absent. Need much higher loadings.
Model of Polymer Nanocomposites Mixture of colloid and polymeric melt U CP N P 4 2 σ P R g CC 0.45 U CC 48 1.6 UCC UCP F PP σ CC σ σ CC PP = 5.0 σ CP b PP σ PP = 3.0 Repulsive LJ Repulsive LJ (VP & VG: Macromolecules, 2006; J. of Rheology) b PP FENE Springs
Rheology of PNCs: Linear Viscoelasticity 10 1 G' ( ω) 0.1 0.01 0.001 0.0001 0.5 0.11 0.33 0.3 ω 1.6 ω 0.01 ω 0.01 0.0001 0.001 0.01 0.1 1 10 1 0.1 G' ( ω) 0.5 0.33 0.11 0.18 ω ω 0.53 0.01 N = 24 = 96 P ω 0.0001 0.001 0.01 0.1 1 Lower Loadings:Enhancement in modulus but no apparent change in relaxation behavior. Higher Loadings: Significant enhancement in modulus and a solid-like behavior. N P
Rheology of PNCs: Linear Viscoelasticity Polycaprolactone + Montmorillonite Clays* PC + Nanotubes** Significant enhancements in elasticity at extremely low loadings Change of viscoelastic response to solid-like behavior. *:Krishnamoorti and Giannelis; **: Fornes and Paul
Rheology of PNCs: Linear Viscoelasticity 10 1 G' ( ω) 0.1 0.01 0.001 0.0001 0.5 0.11 0.33 0.3 ω 1.6 ω 0.01 ω 0.01 0.0001 0.001 0.01 0.1 1 10 1 0.1 G' ( ω) 0.5 0.33 0.11 0.18 ω ω 0.53 0.01 N = 24 = 96 P ω 0.0001 0.001 0.01 0.1 1 Enhancement in modulus but no apparent change in relaxation behavior at lower loadings: Why? Significant enhancement in modulus and a solid-like behavior at higher loadings: Why? (Not in this talk) N P
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? For unentangled polymers, τ m X m 2 m R(t) ( t) X m(0) exp( t / τ m) τ N 2 1 p ξ Normal modes X m X m (t) ( t) X m(0) Simulation Features No glass transition Coarse-grained model Related to viscosity of media # : Giannelis, Adv. Pol. Sci, 138, 107
Effect of Particles on Polymer Dynamics 10 X p ( t) X p (0) exp( t / τ p ) τ τ p R p 9 8 7 6 5 4 3 2 φ = 0.5 φ = 0.33 φ = 0.11 φ = 0.01 N N N N P P P P = = = = 8 24 48 96 Bare Matrix 1 0 0 0.25 0.5 0.75 1 ( p 1) /( N 1) P Overall Slowing of Polymer Relaxations
Effect of Particles on Polymer Dynamics X p ( t) X p (0) exp( t / τ p ) 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 X p ( t) X p (0) exp( t / τ p ) 1.00E+00 ' G pol ( ω) N P G( t) exp( 2t / τ ) p= 1 p 1.00E-03 Modified G pol due to changes in relaxation times 1.00E-06 0.0001 0.001 0.01 0.1 1 ω
X p Effect of Particles on Polymer Viscoelasticity ( t) X p (0) exp( t / τ p ) 1.00E+00 ' G pol ( ω) 1.00E-03 N P G( t) exp( 2t / τ ) p= 1 0.0001 Modified G pol due to changes in relaxation times 10 1 G' ( ω) 0.1 0.01 0.001 0.5 0.11 0.33 0.3 ω 1.6 ω 0.0001 0.001 0.01 0.1 1 p 0.01 N P = 24 ω 1.00E-06 0.0001 0.001 0.01 0.1 1 ω
Effect of Particles on Polymer Viscoelasticity X p ( t) X p (0) exp( t / τ p ) 1.00E+03 N P G( t) exp( 2t / τ ) p= 1 p ' G pol ( ω) G' ( ω) 1.00E+00 φ = 0.01 φ = 0.11 φ = 0.33 G' ( ω) 1.00E-03 Modified G pol due to changes in relaxation times 1.00E-06 ω 0.0001 0.001 0.01 0.1 1
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.
Comparisons to Experiments (Kropka, Green and Ganesan, Macromolecules, In Press) Comparison of the relaxation times in nanocomposites to the bare relaxation times. τ τ rep PMMA rep PMMA + C60 NC wt% C 60
Comparisons to Experiments (Kropka, Green and Ganesan, Macromolecules, in Press) Superposition of mechanical modulii after renormalization of relaxation times. βb T G, β T b T G (Pa) α T a T ω (rad/s)
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 particles is 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 Outline of Approach (1) U Eff + (2) U Eff + βu (3) U Eff Effective Interactions (4) U Eff + +.. z P Monte Carlo Moves And Equilibration Cluster Counting βu P BR r Thermodynamics, Phase Behavior Bridging Probability MS, VP, VG: JCP 2005; Langmuir 06; Macromolecules 06. Gelation, Elastic Modulus Analogous to density functional theories used to obtain interatomic potentials in quantum chemistry
Integrating Out the Polymer By Mean-Field Theory N segment chain Bead-spring model of polymer Self avoiding random walk Charge effects Idea behind mean-field theory* W() r Thermodynamics Single chain in a potential field W(r). W(r) determined self-consistently to match statistical properties of polymers, say, the volume fraction. *: Helfand (1975)
Integrating Out the Polymer By Mean-Field Theory N segment chain Bead-spring model of polymer Self avoiding random walk Charge effects Mean Field Approach Presence of other chains Self consistent potential field w(r) Configurations of a chain subjected to w(r) G( r, r'; n) 2 = G( r, r'; n) iw( r) G( r, r'; n) s G( r, r';0) = δ ( r r') BCs : Polymer - particle interactions r n r' G(r, r ; n)
Field-Theory Model For Polymers Density distributions for polymer Polymer mediated effective interactions Numerical solution Bispherical coordinates Curvature of particles accounted exactly!!! Configurations of a chain subjected to w(r) G( r, r'; n) 2 = G( r, r'; n) iw( r) G( r, r'; n) s G( r, r';0) = δ ( r r') r n r' G(r, r ; n) Tail Loop Bridge Number and probability distribution
Adsorbed Layer: Particle Size Effects Loop Tail 1 Fraction of segments in loops & tails 0 φ = 2.58 tails loops 0 1 2 3 4 Radius of the particle, R / R g Tails dominate! Loops dominate! Smaller particles tails dominate
Bridging: Particle Size Effects Bridge # of bridges/area 12 8 4 R/Rg R/Rg = 0.5 R/Rg = 1.0 R/Rg = 2.0 0 0 2 4 6 Interparticle distance, r/r Smaller particles More number of bridges
Cluster Statistics and Gelation bridge Effective interactions I Bonds generation ~ bridging probability II Monte Carlo Moves & Equilibration Cluster formation # & size of clusters, Percolation thresholds Volume fraction at which a space spanning cluster is observed
Cluster Statistics and Gelation R/Rg = 2.0 R/Rg = 1.0 R/Rg = 0.5 Smaller particles Gel much earlier! Percolation threshold 0.1 0.05 0 Polymer melt φ = 5.16 0 0.5 1 1.5 2 Particle size ratio, R/R g 2.5 η= 8%
Determining Elastic Properties R/Rg = 2.0 R/Rg = 1.0 R/Rg = 0.5 Smaller particles Gel much earlier! η= 8% Post-gel systems Identification of backbone f f Graph theory Elastic properties Simple Network Theories: Elastic Modulii = Number of Bridges in backbone
Elastic Modulii of Gels Elastic modulus, G 800 400 0 R/Rg = 0.5 R/Rg = 1.0 R/Rg = 2.0 0 0.05 0.1 0.15 Particle volume fraction, η particle Smaller particles -> stronger reinforcement. Smaller quantities of nanoparticles are required. Bridging induced clustering of particles responsible for reinforcement. *Surve, Prymitsyn, Ganesan, Phys. Rev. Lett. Much Stronger Enhancements of Moduli in Smaller Particles
G 1.E+03 1.E+02 1.E+01 1.E+00 Scaling of Elastic Modulii of Gels Simulation results * G ~ (η-η c ) 1.799 1.E+05 1.E+03 1.E+01 Experimental data ~3.08 ~1.84 1.E-01 1.E-01 1.E-02 1.E-03 0.0001 0.001 0.01 0.1 1 0.001 0.01 0.1 1 Particle volume fraction, (η-η c ) (η-η c ) *Surve, Prymitsyn, Ganesan, Phys. Rev. Lett. Universal scaling of elastic modulus
Conclusions Part II Gelation For smaller particles, gelation occurs at very low volume fractions Small particles -> dense networks Small particles -> Much stronger enhancements in modulii