**Matthew E. Helgeson, PhD**, University of California (Santa Barbara)

**Summary
and impact**

Using a new four roll mill (FRM) device, we have performed exploratory
studies of wormlike micelles (WLMs) with two representative behaviors: shear
thinning and shear banding. In both cases, the fluid “resists” a transition
from extensional to shearing flow. For shear thinning fluids, shear rheology
tends to dominate the flow, resulting in significant gradients in deformation
rate near the boundaries. For shear banding fluids, this gradient occupies an
unobservably small region, with concomitant non-monotonic behavior in both the
local deformation rate __and__ flow type, which we believe is related to the
extensional rheology of the fluid. Our results hold impact for both processes and
formulations involving WLMs. When dealing with highly shear thinning or shear banding fluids,
one should anticipate that geometries producing mixed flows with significant
extensional components will have shear that is highly localized near the
boundaries, with the remainder of the flow being dominated by extension.

This research has built new experimental tools in the PI's lab (including FRM and velocimetry) that complement previous expertise with WLMs. Our work has also benefited from collaboration with Gary Leal, whose expertise in modeling polymeric fluids has enabled predictions of WLM flows which can be compared with our experimental results, providing important physical insight into how underlying fluid behavior influences the observed flows. As a result, the PI and collaborators have become, to our knowledge, the only group in the world currently performing detailed studies of viscoelastic micellar fluids in controlled mixtures of shear and extensional flow.

**Detailed
progress**

We have assembled and validated a
temperature-controlled FRM device for video microscopy measurements (Figure 1). The flow is characterized by a
nominal deformation rate, G_{0}, and applied flow type parameter,
Λ_{0}, equal to the ratio of rotation rates of rollers 2 and 4
relative to 1 and 3 (Λ_{0}=1 for 2D extensional flow, 0 for shear
flow).^{1,2} Tracer particles are introduced into the fluid, the flow
is illuminated from the sides, and a digital video camera takes images of the
flow field at up to 60 Hz. Images can either be used to produce time-lapse
photographs (Figure 1, right, greyscale) or analyzed using standard PTV
algorithms to quantify the velocity field (red arrows).

Corn syrup, WLMs comprised 42 mM hexadecyltrimethylammonium
p-toluenesulfonate (CTAT), 80 mM sodium chloride (NaCl) and 0.1 M cetyltrimethylammonium
bromide (CTAB), and WLMs comprised of 200 mM sodium salicylate (NaSal) were used as representative examples
of Newtonian, shear-thinning, and shear-banding fluids, respectively. Linear
viscoelastic (LVE) and flow curve measurements of the non-Newtonian samples were
fitted with the one-mode Giesekus model (Figure 2).^{3}

In the FRM, linear dependence of
the measured deformation rate
near the stagnation point,
G_{obs},
on the roller speed was obtained for the Newtonian fluid for all L_{0}, validating the device design
(Figure 4). For
WLMs, the deformation
rate is characterized by the dimensionless Weissenberg number, *Wi*=G*τ*, where *τ* is the
characteristic relaxation time from LVE measurements. Time lapse images from the
samples at various *Wi* show interesting behavior (Figure 3). Whereas both
Newtonian and shear thinning samples exhibit similar qualitative behavior (top),
the shear banding sample deviates significantly. At *Wi*<0.2 (middle),
the sample behaves similar to a Newtonian fluid. However, for *Wi*>0.2
and L_{0}>0, the observed flow appears
similar to 2D extensional flow (L_{0}), despite rollers rolling at
different speeds.

Figure 4 shows the observed *Wi*
near the stagnation point, *Wi*_{obs}=G_{obs}*τ*, versus the nominal applied *Wi*,
*Wi*_{0}=G_{0}*τ*, for the WLMs (the flow type parameters,
Λ_{0} and Λ_{obs}, are defined in a similar manner).
The Newtonian result, *Wi*_{0}=*Wi*_{obs}, is shown
for comparison. For both fluids, Newtonian behavior was observed for *Wi*_{0}<0.2.
For the shear thinning sample, *Wi*_{obs} deviates below *Wi*_{0}
for *Wi*_{0}>0.2, with a non-zero slope. By contrast, for the shear-banding
fluid, *Wi*_{obs} first shows a maximum near *Wi*_{0}=0.2, followed by a plateau of Wi_{obs}~0.18
until a slight increase for *Wi*_{0}*>*1. Based on previously published
simulations in the FRM, this non-monotonic behavior could be associated with
extensional thickening of the fluid.

We also studied the behavior of
the fluids in mixed flows (0<Λ_{0}*<*1). For the Newtonian fluid, Λ_{obs}*>*Λ_{0} for Λ_{0}<1 (Figure 4), indicative of non-idealities
in the FRM due to unavoidable contributions from the outside walls. Quantitatively
similar behavior was observed with both WLM samples for *Wi*_{0}*<*0.2.
For both fluids, when *Wi*_{0}*>*0.2, Λ_{obs} is higher than that observed in
the Newtonian case. For the shear thinning fluid,
this deviation becomes continuously more severe as *Wi _{0}* is
increased. For the shear banding fluid, a plateau of Λ

To further probe this behavior, we
analyzed the velocity field near individual rollers (Figure 5). Absent a full
2D model, the flow field around the moving roller can be
approximated by 1D shear flow between two concentric cylinders. For the shear thinning
fluid with *Wi*_{0}>0.2, significant yet smooth gradients in
local shear rate were found close to the rollers. This is typical of a shear
thinning fluid in a geometry with a stress gradient, which explains why *Wi*_{obs}<*Wi*_{0}
for *Wi*_{0}*>*0.2. Predictions of the velocity profile
using the VCM model previously developed for WLMs^{4,5} is in
qualitative agreement, although the model significantly overpredicts the shear
rate gradient.

For the shear-banding fluid at *Wi*_{0}=0.1, the
velocity profile is similar to that obtained with a Newtonian fluid (Figure 5).
For Wi_{0}>0.2, however, we surprisingly find that the dimensional
velocity profile (not shown) was quantitatively identical regardless of *Wi*_{0},
and coincides with the plateau of *Wi*_{obs} under 2D extensional flow (Figure
4). The VCM model
predicts qualitatively similar behavior. This is in contrast to thin gap flow
between concentric cylinders.^{5} **We thus hypothesize that, for
sufficiently curved geometries (or those with significant stress gradients),
shear banding is highly localized near the no-slip surface.**

Copyright © 2014 American Chemical Society