For Review. Confidential - ACS Homogeneous shear, wall slip and shear banding of entangled polymeric liquids in simple-shear rheometry: a roadmap of nonlinear rheology Journal: Macromolecules Manuscript ID: ma-2010-01223q.R1 Manuscript Type: Perspective Date Submitted by the Author: n/a Complete List of Authors: Wang, Shi-Qing; University of Akron, Dept. of Polymer Science Ravindranath, Sham; University of Akron Boukany, Pouyan; The Ohio State University, NSEC/CANPBD ACS Paragon Plus Environment Submitted to Macromolecules
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For Review. Confidential - ACS
Homogeneous shear, wall slip and shear banding of entangled polymeric liquids in simple-shear rheometry: a
roadmap of nonlinear rheology
Journal: Macromolecules
Manuscript ID: ma-2010-01223q.R1
Manuscript Type: Perspective
Date Submitted by the Author:
n/a
Complete List of Authors: Wang, Shi-Qing; University of Akron, Dept. of Polymer Science Ravindranath, Sham; University of Akron Boukany, Pouyan; The Ohio State University, NSEC/CANPBD
ACS Paragon Plus Environment
Submitted to Macromolecules
For Review. Confidential - ACS
1
Homogeneous shear, wall slip and shear banding of entangled polymeric
liquids in simple-shear rheometry: a roadmap of nonlinear rheology
Shi-Qing Wang, S. Ravindranath and P. E. Boukany
Department of Polymer Science, University of Akron, Akron, Ohio 44325
Abstract
The recent particle-tracking velocimetric (PTV) observations revealed that well-entangled polymer solutions and melts tend to either exhibit wall slip or assume an inhomogeneous state of deformation and flow during nonlinear rheological measurements in simple-shear rheometric setups. Many material parameters and external conditions have been explored since 2006, and a new phenomenological picture has emerged. In this article, we not only point out the challenges to perform reliable rheometric measurements but also discuss the relation between wall slip and internal (bulk) cohesive breakdown and summarize all available findings in terms of a phase diagram. This map specifies the conditions under which shear homogeneity, interfacial slip, and bulk shear inhomogeneity would prevail respectively. The paper is closed by enumerating a number of unresolved questions for future studies.
Figure 1 (a) 3-D view of homogeneous shear at an apparent shear rate appγ =& V/H in absence
of wall slip. (b) Side view: velocity profile in presence of wall slip on two identical shearing
surfaces, when 1app
γ τ >& , with τ being the dominant chain relaxation time. When a slip
velocity Vs is present, the amount of slip can be quantified by the extrapolation length b defined in the figure involving the bulk shear rate γ& . (c) Side view: bulk shear banding along
with a small amount of wall slip is a characteristic response of well entangled linear polymer solutions during startup shear at high rates when wall slip alone can no longer prevent the sample bulk from shearing at rates higher than the dominant relaxation rate 1/τ.
To verify whether this is the case, we have recently developed an in situ particle-
tracking velocimetric (PTV) method as shown in the middle figure of TOC. 6 The
combination of rheometric and PTV measurements revealed a clear violation of each of
the three basic assumptions listed in the first paragraph. In other words, shear
inhomogeneity was found to take place cone-plate shear cell during startup shear7,8,9,10
and large amplitude oscillatory shear. 11 Entangled solutions 12 and melts 13 were not
quiescent upon shear cessation from a large step strain. Thus, the nonlinear responses of
entangled polymers appear far more complicated than previously revealed. In other
words, when using any commercial rheometer for simple shear, we could no longer
assume a priori that the states of deformation and flow are homogeneous.
In this Perspective we indicate some guidance for future rheological experiments
by showing when, how and why the three basic assumptions in polymer rheology may be
invalid. We first briefly review a primary type of shear inhomogeneity, i.e., wall slip, as
schematically illustrated in Figure 1b. We then describe when nonlinear rheological
Wiws-sb when rough shearing surfaces are used to remove wall slip. For example, surfaces
made of sandpaper can effectively reduce wall slip so that shear banding takes its
place.48,50
5.3 The "phase diagram"
Clearly, bmax/H is the key parameter to control the nonlinear rheology of
entangled polymers, dictating when and what type of shear inhomogeneity may occur for
a given apparent Weissenberg number Wiapp. Here we summarize what we know about
the nonlinear rheological responses to startup shear in terms of a phase diagram as shown
in Figure 2a, where for a fixed gap distance H* the X axis of 2bmax/H* is controlled by
bmax. For monodisperse linear-chain melts, Eq A.1 shows how the molecular weight
dictates the magnitude of bmax. For entangled solutions made with polymeric solvents,
besides the molecular weight of the solute the solvent viscosity ηs can also alter bmax
according to Eq A.4. For a given polymeric system, i.e., for a fixed b*max, varying H
would have the consequence shown in Figure 2b.
1
2bmax/H*
Wiapp
1
homogeneous
hom
ogen
eo
us
H/2lent
2a
1
2b*max/H
Wiapp
1
homogeneousho
mogen
eou
s
uniform
disentanglement
H/2lent
shear banding
H=2lent
Widis
2b
Figure 2 Phase diagrams in the parameter space of Wiapp vs. 2bmax/H (double log scales) to map out the different responses in terms of the state of deformation for (a) a fixed sample thickness so that bmax/H
* simply denotes the level of chain entanglement; and (b) a given sample so that b*
max/H increases with lowering H until the upper bound is reached at H = 2lent when only wall slip is possible, moving along the red vertical line.
In both Figures 2a-b, at small values of bmax/H, we have the vertical
"homogeneous" strips to the left of the vertical white lines. The inclined white borderline
between wall slip and shear banding is given by Eq 2. The boundary between "uniform
disentanglement" and "shear banding" is depicted by Eq A.3 or Eq. A.6, which is inclined
in Figure 2a and horizontal in Figure 2b. Actually, Figure 2a is somewhat analogous to
Figure 6 of Ref. 6. Both Figures 2a and 2b serve as a roadmap for the steady-state
behavior of well entangled polymers in startup shear.
1
2bmax/H*
Wiapp
1
homogeneous
hom
ogen
eou
s
Slip
2c
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1
Wi
= 2
0 <
Wi w
s-sb
= 2
6
Wi
= 2
00 >
Wi w
s-sb
= 2
6
y (µ
m)
V/V0
Figure 2 (c) Regions of the phase diagram where PTV studies have been carried out, including the small bmax/H regime inside the rectangle, and six solutions of different 2bmax/H equal to 22, 26, 136, 140, 259, 1470 respectively. The open symbols indicate approximate shear homogeneity.
Figure 3 PTV observations of steady state shear in a gap of H = 50 µm at two different rates of 0.3 and 3.0 s-1 for an entangled solution of 13 wt. % polybutadiene (Mw ~ 103 kg/mol) in a polybutadiene solvent (Mw ~ 10 kg/mol).
We can summarize the available results from the literature in Figure 2c as well as
in Table 1 that lists five PTV-based studies9,10,50,72,73 on eleven entangled solutions. The
first two papers in Table 1 focused on the phase space circled by the (red) rectangle in
Figure 2c in the region of 2bmax/H < 1. We specifically mark the other regions
corresponding to the other six solutions listed at the bottom of Table 1, having
respectively 2bmax/H equal to 22, 26, 136, 142, 259 and 1470. Particularly worth
mentioning is the system with 2bmax/H = 26 represented by the five circles that shows
both slip dominant and shear banding prevailing features as the applied shear rate
increases. Also special about this study73 is the significantly reduced sample thickness
Appendix A Account of Wall Slip and Ultimate Homogeneous Shear
When the apparent shear rate V/H indicated in Figure 1a is below the terminal relaxation rate 1/τ of the entangled polymer, i.e., (V/H)τ < 1, the amount of external shear deformation is lower than unity within the relaxation time τ. The entanglement network does not deform sufficiently to break up unevenly. In other words, there is no chain disentanglement at either the polymer/wall interface or in the bulk. Homogeneous shear prevails as shown in Figure 1a. The viscosity ratio in Eq 1 is on the order of unity and b
is only molecularly small.
When the apparent Weissenberg number Wiapp = (V/H)τ is much higher than unity, the adsorbed chains cannot remain engaged with the bulk chain entanglement network due to definite intermolecular gripping forces during startup simple shear. Such interfacial yielding34 produces an interfacial layer (of thickness a) with a greatly reduced viscosity ηi. Consequently, b can be millions times of a, i.e., macroscopically large according to Eq 1. For a monodisperse entangled linear polymer melt with (weight-average) molecular weight Mw, the maximum value of b can be estimated upon complete interfacial chain disengagement as22
bmax = (η/ηe)lent = (Mw/Me)3.4
lent, for entangled melts, (A.1)
when ηi in Eq 1 turns into ηe, which is that of a melt with entanglement molecular weight Me. Eq A.1 assumes the slip layer thickness to be the entanglement spacing lent, related to Me as83,84 (Me/ρNAp)1/2, where ρ is the polymer mass density, the packing length p and the Avogadro constant NA. Here we adopted the empirical exponent 3.4 to depict the molecular weight dependence of η. The viscosity ratio in Eq. A.1 can readily approach 106, making bmax macroscopically large, for polymer melts, especially polyethylene, polybutadiene and polyisoprene.
For monodisperse melts, the local shear rate in the entanglement-free slip layer upon maximum slip is related to the bulk shear rate γ& as
disγ& = (η/ηe) γ& = ( γ& τ)/τe ~ 1/τe. (A.2)
where η/ηe = τ/τe follows from the tube model,44 with the confinement τe is related to the dominant relaxation (i.e., reptation) time τ as τe = τ(Me/Mw)3.4, and the product γ& τ is only
on the order of unity at the onset of complete slip. The corresponding Weissenberg number Widis is given by
Widis = disγ& τ = τ/τe = bmax/lent, (A.3)
where the last equality follows from Eq A.1.
For an entangled polymer solution with polymer weight fraction φ, complete slip corresponds to having ηi in Eq 1 reduced to the solvent viscosity ηs, where a may also be thought of as the enlarged entanglement spacing49,85 lent(φ) = lentφ
bmax(φ) = [η(φ)/ηs]lent(φ), for entangled polymer solutions. (A.4)
Since ηs can be as low as 10-3 Pa.s (i.e., that of water), entangled solutions could also display significant wall slip. Similar to Eq A.2, the local shear rate in the slip layer upon complete slip is given by
disγ& = [η(φ)/ηs] γ& , (A.5)
which can be rewritten in terms of the corresponding Weissenberg number Widis as
where the last equality follows from Eq A.4, and the product γ& τ is on the order of unity.
Note that Eq A.6 is of the same form as Eq A.4.
Homogeneous shear should occur upon complete chain disentanglement when the imposed rate is as high as the local shear rate disγ& in the slip layer during maximum wall
slip. This critical shear rate is given by Eq A.2 for melts and by Eq A.5 for solutions respectively corresponds to a Weissenberg number Widis that is related to the maximum slip length bmax as Widis = bmax/lent as shown in Eq A.3 and and A.6.
Appendix B Theoretical developments based on the tube model
The original 1979 tube model44 of Doi and Edwards revealed a non-monotonic constitutive relation in its calculation. They and subsequently others related such a feature to a flow instability (known as the stick-slip transition22) observed in capillary rheometry.86,87 Subsequently, it was proposed in a 1993 theoretical study88 that shear banding could occur in entangled polymer solutions under simple shear. The experimental report for shear banding, inspired by this theoretical analysis,88 was made in 1996 for worm-like micellar solutions.89,90 Because the worm-like micelles are capable of adjusting their sizes in shear and may even suffer breakage by shear,99,100 shear banding in wormlike micellar solutions 91 , 92 , 93 , 94 , 95 , 96 , 97 , 98 , 99 , 100 was not taken to imply that conventional non-charged, non-associative polymers would also shear inhomogeneously. In the decade between 1996 and 2005, a couple of studies reported a hint of shear banding in semi-dilute polyacrylamide/water solutions where chain-chain associations occur through hydrogen bonding. 101 , 102 Apart from this special case, the important premise of shear homogeneity had never been observed to break down for entangled polymer liquids. In absence of a perfect stress plateau in the steady state flow curve and any direct evidence of shear inhomogeneity, major modifications 103 ,104 including the introduction of convective constraint release,105 were made to make sure that the tube model does not predict a non-monotonic constitutive relation for entangled polymers.
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