Shear Banding in EntangledCarbon Nanotube Networks
Erik K. HobbieNIST Polymers Division
Euromech
2007 –
Shear Banding in Entangled Systems
Generalized Definition of Shear BandingShear-induced separation into two or more macroscopic regions of different strain rate
γ0.
γ1.
= 0
γ2.
>>>
>>>
γ2.γ1
.
Sheared Carbon Nanotube Suspensions
z
x
y
γησ &=xyxv ˆyγ&= ,
Simple Shear Flow
Nanotubes
are in closed periodic Jeffery orbits
cL3
≤
1, cL2d < 1 (dilute)
cL3
>> 1, cL2d < 1 (semi-dilute)
cL3
>> 1, cL2d > 1 (concentrated)
Orbit period scales inversely with shear rate
Sheared Carbon Nanotube Suspensions
Width of image = 200 micrometers
Nanotubes Movie(nanotubes.avi)
Small-Angle Light Scattering from MWNT Suspensions
Good MWNT dispersion
SWNTs
and MWNTsare optically active, so ‘depolarized’
light scattering is needed …… polarization and absorption are bothanisotropic along and normal to the symmetry axis of the tube
E. K. Hobbie
et al, RSI 74, 1244 (2003)
3.5 μm-1
10 μm
h
vh
v
s
p
h
vh
v
s
p
xy
z
μm4 μ
1 μm-1
S = 0.23
Shear-Induced Orientation in Suspensions -Scaling of SALS in the Semi-dilute Regime
Scaling of depolarized light scattering data corroborates the
scaling observed in the dichroismand the birefringence, and a zeroth-order fitting scheme describes all of the data with only one adjustable parameter, a small back ground in a Gaussian ODF, p(θ). Fitting trends consistent with modest to minimal tube deformation.
D. Fry, et. al. Phys. Rev. Lett. 95, 0383304 (2005)
1 μm-1
S = 0.23
Orientation in Sheared Suspensions -Scaling of Birefringence and Dichroism
0.06 nm-15 μm
...)( ||21 +′−′≈′Δ ⊥ Sn
sn ααφ
...)( ||21 +′′−′′≈′′Δ ⊥ Sn
sn ααφ
)1cos3( 221 −= θS
(rotational Peclet
number)
(dilute )3
]8.0)/[ln(3L
dLTkDD B
o πη−
==
Dγ&
=Pe
(semi-dilute - Doi and Edwards)2−∝ φoDD
3/116.0Pe φ∝∝S
D. Fry, et. al. Phys. Rev. Lett. 95, 0383304 (2005)
0.17 wt. % MWNTnL3 ≈
54nL2d ≈
0.23
MWNTsd ≈
50 nmL ≈
10 μmdispersed in PIB
(Mn = 800)η = 10 Pa-s at 25 oC
Flow-Induced Aggregation - Banding in Weak Shear?
S. Lin-Gibson et al, Phys. Rev. Lett. 92, 048302 (2004)
Schmid & Klingenberg, Phys. Rev. Lett. 84, 290-293 (2000)
Generic pattern at early timeexaggerated by confinement effectsat late time …
ψ = area fraction
[width of FFT = 1.2 μm-1]
)0()()( ψψ rr =c
Shear-Induced Aggregation and Coarsening
S. Lin-Gibson et al, Phys. Rev. Lett. 92, 048302 (2004)
Many Other Systems …Clay gels under simple shear flow
Gel of 0.56 % XLGin water (25 oC)
Pignon, Magnin, & Piau, Phys. Rev. Lett. 79, 4689 (1997)
Macroscopic Voriticty Rolls (0.1 s-1 – sped up x10)
Width of image = 1 mm
Banding Movie
(vorticityrolls.avi)
A Simple Cartoon of a Flow Induced InstabilityA number of vastly different systemsexhibit the same shear-induced pattern because they each represent elastic droplets suspended in a viscous fluid
.
HA
S
Generic phase diagram for shear-inducedaggregation in semi-dilute non-Brownian
MWNT suspensions
z
yh
x
‘phase diagram’
.
Bird, Armstrong, & HassagerMontesi, Pena, & Pasquali
50 μm
Unusual Rheological Signal
Negative normal stressLarge intrinsic viscosityDecreasing φConfinement effectsVorticity
elongation
Re ≈10-5, Wi ≈
10
Growth Cycle
Flow orientationPositive normal stressMulti-step processRe ≈10-3, Wi < 1
Dissolution Cycle
N1 = σxx - σyy N = σxx - σzz[J. M. Dealy, J. Rheol. 39, 253 (1995)]
S. Lin-Gibson et al, Phys. Rev. Lett. 92, 048302 (2004)
430 μm
MWNTs
in 500 PIBCone-and-plate geometry
Controlled-Stress
Rheology of Semi-Dilute to Concentrated Suspensions
t (s)
Controlled-Strain
Controlled-Strain
Scaling of Linear Viscoelasticity with Concentration
αφκ ∝
E. K. Hobbie
and D. J. Fry, Phys. Rev. Lett. 97, 036101 (2006)
1.7=α
αφκ ∝
Network Yield Stress
Controlled Strain
E. K. Hobbie
and D. J. Fry, J. Chem. Phys. 126, 124907 (2007)
Relating Shear Modulus and Yield Stress to Network Morphology
E. K. Hobbie
and D. J. Fry, Phys. Rev. Lett. 97, 036101 (2006)E. K. Hobbie
and D. J. Fry, J. Chem. Phys. 126, 124907 (2007)
1.7=ααφκ ∝
5.3=ββφσ ∝0
)3()3(
f
b
dd
−+
=α)3(
2
fd−=β
45.2≈fd1≈bd
W.-H. Shih et al., Phys. Rev. A 42, 4772 (1990).
Processing Phase DiagramMorphology as a function of concentration, confinement, and strain ratefor MWNTs
in 500 PIB -
controlled strain measurements:
E. K. Hobbie
and D. J. Fry, Phys. Rev. Lett. 97, 036101 (2006)
Universal Phase Diagram – Critical Parameters
00 / RR ∝δφδ
1
2
/
)(cos−∝∝
=
γτγδδ
θ
&&S
PS
[ ]00 /)(exp/ SSSac −=σσ
xθ0
R0 ~φ -1/3
ε
θ
3/1
0
φε
∝∝R
SE. K. Hobbie
and D. J. Fry, Phys. Rev. Lett. 97, 036101 (2006)
D. Fry (NIST –
NRC now at JPL)H. Wang (NIST –
now at SUNY)
B. Langhorst
(SURF)S. Lin-Gibson (NIST), J. Pathak
(NIST, now at NRL)
H. Kim (Kyunghee
U)E. Grulke
(U Kentucky)
S. Hudson (NIST)
Thanks!!