# Soft glassy materials with tunable extensibility

Samya Sen  Department of Mechanical Science and Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801, USA  Rubens R. Fernandes  Department of Mechanical Science
and Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801, USA  Randy H. Ewoldt<EMAIL_ADDRESS>Department of
Mechanical Science and Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801, USA

###### Abstract

Extensibility is beyond the paradigm of classical soft glassy materials, and
more broadly, yield-stress fluids. Recently, model yield-stress fluids with
significant extensibility have been designed by adding polymeric phases to
classically viscoplastic dispersions [1, 2, 3]. However, fundamental questions
remain about the design of and coupling between the shear and extensional
rheology of such systems. In this work, we propose a model material, a mixture
of soft glassy microgels and solutions of high molecular weight linear
polymers. We establish systematic criteria for the design and thorough
rheological characterization of such systems, both in shear and in extension.
Using our material, we show that it is possible to dramatically change the
behavior in extension with minimal change in the shear yield stress and
elastic modulus, thus enabling applications that exploit orthogonal modulation
of shear and extensional material properties.

##  1 Introduction

Yield stress fluids typically exhibit a reversible transition between a solid-
like and a fluid-like state around a critical applied stress. This behavior
can originate from several microstructural interactions [4, 5, 6, 7], such as
the repulsive potential between particles in soft glasses (emulsions and
microgel dispersions). In these materials, particles are dispersed in a liquid
continuous phase at relatively high volume fractions, preventing thermal
relaxation of the microstructure, leading to glassy or jammed states [8]. This
leads to solid-like elasticity from soft particle contact forces and soft
glassy materials are therefore often studied within the paradigm of yield-
stress (elastoviscoplastic) fluids.

Designing extensibility into yield-stress fluids is of particular interest in
applications such as rapid prototyping and direct-ink writing, where a
combination of yield stress and extensibility is beneficial for optimized
printing performance [9, 2, 10]. Applications such as droplet and jet impacts
on surfaces, e.g. as motivated by fire suppression spray coatings, may also
utilize fluids with tunable shear and extensional viscosities [11, 10, 12, 13,
14].

Despite the apparent importance of extensible yield-stress fluids, very few
studies have reported on the formulation and testing of model yield-stress
fluids with significant and tunable extensibility. Only recently, materials
based on emulsions have been designed with significant extensional properties
[15, 1, 9, 2]. However, these materials suffer from experimental artifacts
that hinder their characterization in both shear and extension, e.g. edge
fracture in shear rheology. Also, such materials can be difficult to formulate
with consistent properties such as shear yield stress or elastic modulus,
owing to the imprecise control over the size distribution of the dispersed
phase droplets. Additionally, accurate characterization of the yield stress in
shear and extensional flows has been a topic of debate in the recent
literature. Although soft glassy materials such as Carbopol and emulsions are
often assumed to follow the von Mises yield criterion [16, 17], recent
observations indicate that this may not hold for yield stress fluids in
general [18].

In this work, we design, formulate, and characterize a model extensible yield-
stress fluid that overcomes the limitations described above. We use a Carbopol
microgel suspension, a soft glassy material that is widely used in the
literature as a model yield-stress fluid to study microstructural,
rheological, and flow physics [4, 5, 6, 7, 19, 20, 21, 22, 23], and impart
extensibility to it by making physical mixtures with solutions of a high-
molecular weight poly(ethyleneoxide) (PEO) linear chain polymer. We perform a
thorough rheological characterization in both shear and extension to study the
variation in extensional and shear rheological properties. We show that key
properties in shear, such as yield stress and linear viscoelastic storage
modulus, do not change by more than 10-20% upon addition of PEO. On the other
hand, filament stretching extensional tests show that addition of polymers
changes the extensional properties, such as filament failure strain, by
200-300%. These results strongly support the usefulness of this system as a
model yield-stress fluid with tunable extensibility.

##  2 Materials and methods

###  2.1 Materials used

Carbopol is widely employed to study and model the rheological properties and
flow physics of yield-stress fluids. It is a soft, glassy (repulsion
dominated) fluid [24, 25, 4, 16, 1]. Carbopol particles are microgels made of
cross-linked poly(acrylic acid) chains. When dispersed in water and the $p$H
raised to 6.5-7.5 using a base, the microgel particles swell, leading to a
crowded microstructure, forming a transparent soft glassy material [4, 16].
Carbopol powder (Carbopol 980, Lubrizol) was thoroughly mixed in de-ionized
water for 6 h using a magnetic stirrer at 200 r.p.m. After mixing, 1 M aqueous
NaOH solution was added dropwise until the $p{\rm H}$ was stable around $7\pm
0.10$. The mixture was stirred gently at 50 r.p.m. for 12 h to homogeneously
disperse the base within the sample volume. The samples were tested either in
their pure form, or in a mixture with PEO solution. A schematic of the
structure of Carbopol at various length scales is shown in Fig. 1.

Figure 1:  Hierarchical structures at different length scales for Carbopol and
PEO in water. The orders of magnitude separation in length scales is
indicated.

PEO consists of ethylene oxide monomers which are polymerized into linear
chains. The samples were made from PEO that comes in a fine powder form (Sigma
Aldrich, $M_{\rm v}\simeq$ 4,000 kDa), dissolved in a 19:1 de-ionized water-
ethanol co-solvent. Requisite amounts of the powder were dissolved into the
co-solvent by vigorously mixing in a beaker with a magnetic stir rod and stir
plate at 200 r.p.m. for 12 h and a homogeneous viscoelastic solution was
obtained.

Polymer solutions are in the dilute regime when chains do not overlap or
interact significantly. The semi-dilute regime is when overlap between
hypothetical spheres enveloping the chains start interacting directly [26,
27]. The overlap concentration, $c_{\rm PEO}^{*}$, is used to approximate the
onset of this regime, and is defined as $c_{\rm PEO}^{*}=1/[\eta]_{0}$, where
$[\eta]_{0}$ is its intrinsic viscosity [26, 27]. For PEO, it is empirically
related to the average molecular weight $M_{\text{v}}$ of the polymer as
$[\eta]_{0}\simeq 0.0125M_{\text{v}}^{0.78}$ [26, 27]. Using this, the overlap
concentration for the 4,000 kDa PEO used in this work was calculated as
$c_{\rm PEO}^{*}\approx 0.0567~{}\text{wt}\%$. Among the samples used in our
study, only the 0.01 wt% sample is in the dilute regime, whereas the 0.10,
0.50 and 1.00 wt% samples are in the semi-dilute to concentrated regime. As
the PEO concentration increases, we will see in the following sections that
the extensibility of Carbopol+PEO also increases dramatically, while the
steady and viscoelastic shear properties do not change as much. A schematic of
PEO structure at various length scales in shown in Fig. 1.

The swelled Carbopol microgel units form clusters that lead to a glassy state
due to their repulsive interactions, with the diameter of each swollen
particle $\mathcal{O}(1)~{}\mu{\rm m}$ [4] (i.e. the polymeric unit in Fig.
1). PEO molecules, on the other hand, are orders of magnitude smaller than the
microgel particles, with $R_{\rm g}\sim\mathcal{O}(10)$ nm for a single chain
polymeric unit. Consequently, we hypothesize that the PEO molecules may not
influence the physical interactions between the microgel particles
significantly, although they are expected to alter the behavior of the mixture
in other ways, which will be discussed later.

Figure 2: Plausible visual evidence for (i) the presence of a yield-stress for
Carbopol versus an absence in PEO (over the timescale of observation in the
images, $\sim 1$ h), and (ii) the insignificant extensibility of Carbopol
versus prominent extensional behavior of PEO; samples stretched between plates
under strain rates of $\dot{\epsilon}\sim\mathcal{O}(10)~{}{\rm s}^{-1}$. Upon
mixing, the fluid retains the properties of both its constituent components,
having both a yield stress and large extensibility.

The extensible yield-stress fluid (ExYSF) was obtained by mixing the fully
prepared Carbopol and PEO formulations. Depending on the target weight
fractions of Carbopol and PEO in the final ExYSF sample, requisite amounts of
each material were mixed in a stirrer at 50 r.p.m. for 6 h to make the mixture
as homogeneous as possible. Containers were sealed during stirring to minimize
solvent evaporation, and very high shear rates were avoided during mixing to
prevent scission of the chains in either material.

As can be seen from the protorheology [28] photos in Fig. 2 (a-i), Carbopol
samples are transparent and the yield stress is evident from its ability to
support its own weight when the container is tilted [29]. However, pure
Carbopol dispersions are not appreciably extensible, Fig. 2 (a-ii). On the
other hand, PEO solutions in water are clear, and flow at infinitesimal
stresses, showing no yield stress as is evidenced by it taking the shape of
the container (b-i). The samples show significant stretching between two
parallel plates, and thus have very high extensibility (b-ii). When the
Carbopol suspension is mixed with the PEO solution, the resultant material
shows interesting features: it can support its own weight (c-i), indicating a
yield stress behavior, and also shows high extensibility (c-ii). In the
following sections, we support these qualitative observations with
quantitative rheological measures of characterizing these features in shear
and extension, focusing on the effect of added PEO.

###  2.2 Rheometry methods

Shear rheometry was performed on an ARES-G2 strain-controlled, separated
motor-transducer rheometer (TA Instruments), using a 25 mm parallel-plate
geometry, with adhesive-backed 600 grit sandpaper to mitigate wall slip. All
tests were conducted at $25^{\circ}{\rm C}$ using an Advanced Peltier System
to regulate the temperature. This was chosen to match the ambient temperature
of the room, since controlling temperature in extension experiments is not
possible owing to the large aspect ratio (and hence large exposed surface
area) of the samples in filament stretching.

We use filament stretching extensional rheometry (FiSER) to obtain the
extensional properties of our samples [30, 31, 1, 32]. In these tests, a
cylindrical liquid bridge is initially formed between two circular end-plates.
The plates are then moved apart in a prescribed manner such that the fluid
sample is subjected to a predominantly extensional deformation [33, 34, 31,
35]. With time, the accumulated strain in the sample increases; the filament
starts to neck as it is increasingly stretched, then dramatically thins at its
cross-section where necking first occurred. This is finally followed by pinch-
off failure. The pinch-off or filament breakup is primarily driven by the
boundary conditions at the end plates, although capillary-driven thinning does
play a minor role in our test conditions. In our tests, we used 8 mm parallel
plates (radius $R_{0}=4$ mm) to stretch fluid samples on the ARES-G2 rheometer
in ambient conditions, with an initial gap of $L_{0}=4$ mm. This gives an
initial sample aspect ratio of $\Lambda_{0}\equiv L_{0}/R_{0}=1$, consistent
with previous works on extensible yield-stress fluids [1, 9]. This aspect
ratio was chosen to reduce radial flow effects [30]. The upper plate of the
rheometer was moved up to increase the gap $L(t)$ between the plates and
stretch the sample at a constant nominal true strain rate (Hencky strain rate)
of $\dot{\epsilon}_{0}\equiv\dot{L}(t)/L(t)=0.2~{}{\rm s}^{-1}$. This requires
that the upper plate move at an exponentially increasing linear velocity,
which places a limit on the maximum Hencky strain rate that can be applied to
the sample [1]. After loading the sample, the residual normal stress was
allowed to relax to the noise level of the instrument force transducer
($F_{\rm min}\approx 1~{}{\rm mN}$), and the axial force on the upper plate
was measured during the experiment for further calculations.

##  3 Effect of PEO on steady shear properties

The steady shear data for all Carbopol+PEO samples are shown in Fig. 3.

(a)

(b)

Figure 3: Steady shear data for various Carbopol+PEO samples. The fit lines
are for the Herschel-Bulkley model, Eq. (1). (a) All wt% Carbopol with 0 or 1
wt% PEO added. (b) 0.50 wt% Carbopol with 0, 0.01, 0.10, 0.50, and 1.00 wt%
PEO added. The triangles refer to the Carbopol samples with 1 wt%, and circles
to those with 0 PEO. Each color corresponds to a different wt% of Carbopol
(progressively red to blue colors for 0.25, 0.50, 1.00, 1.50, 2.00 wt%
respectively).

From Fig. 3(a), both types of formulations, with and without PEO, have an
apparent yield stress, evident from the plateauing stress at low shear rates.
Secondly, the shear stress is an increasing function of shear rate, with the
material stress gradually transitioning from the plateau yielding to a flowing
region as the rate increases. The same yielding and flow behavior is also
observed in creep tests, data for which is shown in the Supplementary
Information (SI). These signatures are typical of yield-stress fluids, and
adding PEO to Carbopol does not alter these characteristic features. The
behavior at low shear rates is not very different for a given Carbopol
concentration at 0 _versus_ 1 wt% PEO, except for the lowest Carbopol loading
of 0.25 wt%. The high shear rate behavior is also very similar. The effect of
added PEO, while finite, is not large.

This is seen more clearly from Fig. 3(b), where the yield stress and high
shear scaling do not change appreciably for various PEO wt% added to 0.50 wt%
Carbopol. Further insight can be gained by fitting this data to the Herschel-
Bulkley (HB) model [36, 37, 5, 7], given by

$\displaystyle\sigma=\sigma_{\rm y}+K\dot{\gamma}^{n}\equiv\sigma_{\rm
y}\left[1+\left(\frac{\dot{\gamma}}{\dot{\gamma}_{\rm
crit}}\right)^{n}\right],$ (1)

where $\sigma_{\rm y}$ is the dynamic steady shear yield stress, $K$ is the
consistency index, and $n$ is the flow index. An equivalent form of the model,
introduced by Dinkgreve _et al._ [38], is also shown, which involves a
critical shear rate, $\dot{\gamma}_{\rm crit}$, instead of $K$ as a parameter
[38, 15, 1, 39], to avoid the problem that the dimensions of $K$ depend on
$n$. Steady shear data fits to the HB model for the ExYSF are shown in Fig. 3
as solid lines (Carbopol with 1 wt% PEO) and dashed lines (pure Carbopol). The
dynamic yield stress is shown for all samples in Fig. 4 as a function of
Carbopol loading.

Figure 4: Shear yield stress $\sigma_{\rm y}$ obtained from steady flow data
and LAOS metrics ($G^{\prime},G^{\prime\prime}$ crossover, and peak in
$G^{\prime\prime}$), _versus_ Carbopol loading $c$, for 0 (circles) and 1 wt%
(triangles) PEO, all wt% Carbopol. Co-plotted are results for 0.50 wt%
Carbopol with 0, 0.01, 0.10, 0.50, and 1.00 wt% PEO added (polygonal symbols).
Only the steady shear yield stress is used hereafter. (See Fig. 10 for
crossover and peak in $G^{\prime\prime}$ primary data from LAOS.)

Consistent with our hypothesis, while $\sigma_{\rm y}$ increases with Carbopol
concentration $c$, the addition of PEO does not lead to large variations in
$\sigma_{\rm y}$ for $c>0.50$ wt%. The yield stress of the sample with the
smallest Carbopol loading ($0.25$ wt%, red symbols in Fig. 3) was more
affected by the added PEO. This likely results from the comparatively smaller
yield stress displayed by this sample, meaning that crowding effects are not
dominant over the effects of added PEO. This is true for all estimates of
$\sigma_{\rm y}$ obtained in our experiments, whether from steady shear or
oscillatory shear flows. To obtain yield stress from oscillatory rheology, the
stress at the crossover of $G^{\prime},G^{\prime\prime}$ or the stress at the
peak in $G^{\prime\prime}$ in LAOStrain may be used [6, 40], among other
possibilities. In Fig. 4, we co-plot $\sigma_{\rm y}$ obtained from various
metrics using data from LAOS experiments [41, 40], and the effect of added PEO
is not very significant across any measure of $\sigma_{\rm y}$, and they all
follow the same trend with Carbopol loading. Details of oscillatory shear data
are given in Sec. 5.

At low concentrations, addition of PEO to the Carbopol system may change the
effective volume fraction of particles, enhancing interparticle interactions
between microgel particles, thus increasing the yield stress compared to the
pure microgel system. But this effect is overshadowed by the microgel
particles themselves getting heavily jammed and producing a much larger yield
stress at higher concentrations. This explains the reason for variation
between 0 and 1 wt% PEO for low Carbopol loading, yet a consistent yield
stress across this range of PEO concentration at higher Carbopol loadings.

We have also found that the first normal stress difference in shear, $N_{1}$,
is measurable in our Carbopol samples during steady shear tests. The data is
shown in the SI. We observe that $N_{1}$ increases with Carbopol loading for a
given rate, and for each Carbopol loading, $N_{1}$ increases with the addition
of 1 wt% PEO. Upon PEO addition, $N_{1}$ increased by a factor of 1.5-2 in the
range of shear rates where the data was outside the noisy regime.
Interestingly, this increase is also comparable to the increase in
extensibility (2-3 times increase in failure strains) for Carbopol+PEO when
compared with pure Carbopol, as will be shown in the next section. This
increase in $N_{1}$ suggests that it is correlated to the increased
extensibility.

To summarize, we find that the steady shear yield stress is only marginally
changed upon adding PEO, except at the lowest Carbopol loading, while
increasing shear normal stress differences significantly. In the next section,
we will study the effect of PEO on the extensibility of Carbopol systems.

##  4 Effect of PEO on extensional properties

In FiSER tests, the axial force $F_{z}$ on the end plates can be measured
during the filament stretching process, from which tensile stresses and
apparent extensional viscosities can be calculated [33, 31, 34, 35, 42, 43].
While material functions are difficult to measure precisely for FiSER tests
with these materials, owing to the necking, non-ideal deformation history, and
forces near the measurement limit, there are measurable quantities that are
useful for practical applications. One such quantity is the extensional
strain-to-break [1, 9, 14], as also used in solid mechanics [44], which is the
engineering strain at which the filament breaks. It is calculated as

$\displaystyle\varepsilon_{\rm b}\equiv\frac{\Delta L_{\rm
b}}{L_{0}}=\frac{L_{\rm b}-L_{0}}{L_{0}},$ (2)

where $L_{\rm b}$ is the filament breaking length, and $L_{0}$ is the initial
filament length. Similar criteria connected to the filament breaking length
$L_{\rm b}$ have also been used in the literature [45].

Figure 5: Filament stretching tests and strain-to-break observations for 2 wt%
Carbopol with 0 and 1 wt% PEO. These tests were done at a fixed applied
nominal Hencky strain rate, $\dot{\epsilon}_{0}=0.2~{}\text{s}^{-1}$. The
samples with added PEO display larger $L_{\rm b}$ (and thus larger
$\varepsilon_{\rm b}$), as seen from the delayed filament pinch-off.

Filament breakup criteria (i.e., the stresses and strains at filament failure)
are often used to quantify data obtained from extensional tests conducted with
viscoelastic materials. Either necking or failure strain can thus be used as a
metric for gauging how extensible the material is [46, 47, 48]. The
engineering strain-to-break $\varepsilon_{\rm b}$ is a function of both time
$t$ and the nominal strain rate $\dot{\epsilon}_{0}$ applied to the plates.
The failure of highly elastic solids such as metals is governed by the
accumulated strain, meaning that in these cases $\varepsilon_{\rm b}$ depends
weakly on the nominal strain rate imposed on the material. Elastoviscoplastic
fluids such as the ExYSF used in this work, on the other hand, show both
elastic (strain dependent) and viscous (rate dependent) dynamics, so the
extensional strain-to-break is expected to depend not only on the accumulated
strain, but also on the rate at which the strain is accumulated.

Fig. 5 shows the results of filament stretching tests for 2 wt% Carbopol with
0 and 1 wt% PEO added. From the figure, we see that the filament breaking
length $L_{\rm b}$ is greater for the sample with 1 wt% PEO ($L_{\rm b}\approx
7.30L_{0}$) when compared to the pure Carbopol sample ($L_{\rm b}\approx
2.95L_{0}$). These correspond to a filament breaking engineering strain of
$\varepsilon_{\rm b}\approx 630\%$ for Carbopol with 1 wt% PEO _versus_
$\varepsilon_{\rm b}\approx 195\%$ for pure Carbopol. From these images, the
dramatic differences in extensional properties of the two fluids as a result
of the added polymer becomes very clear. Since the same nominal extension rate
was used for both experiments, a larger extensional strain-to-break also
corresponds to a delayed failure time $t_{\rm b}$ of the material $(t_{\rm
b}\sim\ln(1+\varepsilon_{\rm b})/\dot{\epsilon}_{0})$.

Figure 6: Extensional length-to-break, $L_{\rm b}$, and engineering strain-to-
break, $\varepsilon_{\rm b}$, _versus_ Carbopol loading, $c$ wt%. Also shown
is the theoretical breakup limit for a capillary bridge of an inviscid fluid
at the same initial aspect ratio used in experiments [1].

The effect of PEO addition on $\varepsilon_{\rm b}$ is consistent across all
Carbopol concentrations, as plotted in Fig. 6. We see that $\varepsilon_{\rm
b}$ increases with $c$ for both 0 and 1 wt% PEO. Further, for each value of
$c$, adding PEO increases $\varepsilon_{\rm b}$ many fold: $\varepsilon_{\rm
b}\leq 250\%$ for all pure Carbopol samples, while $\varepsilon_{\rm b}\geq
350\%$ for all Carbopol+PEO samples with 1 wt% PEO.

This can be rationalized by considering the effective concentration of PEO
chains in the Carbopol suspension and the effect of the Carbopol concentration
on the yield stress and viscosity. As is evident in Fig. 3, increasing the
Carbopol concentration leads to an increase in both the yield stress and
viscosity of the base material. Failure by pinch-off results from the surface
tension effects overcoming the resistance imposed by the elastoviscoplastic
nature of the material. The dimensionless plastocapillary number,
$\mathcal{Y}\equiv\frac{\sigma_{\rm y}}{\Gamma/\ell}$, is defined as the ratio
of yield stress to capillary pressure based on a characteristic lengthscale
$\ell$ (e.g. the filament diameter in our tests); $\Gamma$ is the surface
tension of the fluid-air interface [49, 50, 23]. The plastocapillary length is
defined at $\mathcal{Y}=1$, such that $\mathscr{L}\equiv\Gamma/\sigma_{\rm
y}$. For a given material, when $\ell>\mathscr{L}$, surface tension stresses
cannot overcome yield stress and deform the filament. Therefore, materials
with a larger yield stress (for similar surface tension values), and thus
smaller $\mathscr{L}$, can sustain a stable filament to much smaller
lengthscales $\ell$, leading to more stable filaments, and delayed pinch-off.
Additionally, since the extensibility of the designed fluid results from the
elastic contributions from the dispersed polymer, addition of PEO further
increases the extensional strain-to-break. This can be verified by fixing the
Carbopol concentration and varying the amount of PEO. We used 0.50 wt%
Carbopol with 0, 0.01, 01, 0.50, and 1.00 wt% PEO for this test. The data for
these has been co-plotted in Fig. 6. We see that $\varepsilon_{\rm b}$
increases monotonically with PEO concentration, which confirms that the
increased extensional behavior stems predominantly from the added polymers.
Note that the change in $\varepsilon_{\rm b}$ is much larger compared to the
change in $\sigma_{\rm y}$, thus there must be effects other than increased
yield stress upon polymer addition that increase extensibility.

The difference in extensibility due to added PEO shows up both in the
kinematics and the dynamics. Using the engineering axial stress during
filament stretching, $\sigma_{\rm e}$, we can study the effect of added PEO on
forces and stresses. This quantity can be calculated using $F_{z}$ (data shown
in Fig. 1, SI) and the initial filament radius, $R_{0}$, as

$\displaystyle\sigma_{\rm e}(t)=\frac{F_{z}(t)}{\pi R_{0}^{2}}.$ (3)

The engineering stress data is plotted against engineering strain in Fig.
7(a). The initial linear increase of the $\sigma_{\rm e}$ _versus_
$\varepsilon$ data corresponds to the elastic response of the material at
smaller strains. Since we are reporting the engineering stress, and not the
true stress [44], a maximum in the engineering stress is reached since the
effective area at the region of necking and subsequent failure decreases with
deformation. We see that for a given $c$, the maximum engineering stress
increases as PEO is added to Carbopol. The ultimate tensile stress, herein
interpreted as a measure of yield stress in extension, is defined as

$\displaystyle\sigma_{\rm U}\equiv\max_{t>0}\sigma_{\rm
e}(t)\equiv\max_{\varepsilon>0}\sigma_{\rm e}(\varepsilon),$ (4)

and is a measure of this maximum stress, which increases with added PEO. This
is a signature of increased resistance to extensional deformation.

(a)

(b)

Figure 7: Engineering axial stress, $\sigma_{\rm e}$, plotted against
engineering strain, $\varepsilon$, in filament stretching tests. (a) Data for
0 and 1 wt% PEO, for all Carbopol concentrations. Carbopol with PEO has a
higher $\sigma_{\rm e}$ compared to pure samples for all $c$; for a given
Carbopol concentration, $\sigma_{\rm e}$ increases from 0 to 1 wt% PEO. (b)
Data for 0.50 wt% Carbopol with 0, 0.01, 0.10, 0.50, and 1.00 wt% PEO added,
and $\sigma_{\rm e}$ increases as PEO content increases. The triangles refer
to the Carbopol samples with 1 wt%, and circles to those with 0 PEO.

Similar to $\varepsilon_{\rm b}$, $\sigma_{\rm U}$ also increases with $c$ for
a fixed PEO concentration, which further supports the reasoning that
increasingly confined PEO chains in more concentrated Carbopol suspensions
lead to greater extensional stresses. In Fig. 7(b), we have shown $\sigma_{\rm
e}$ for 0.50 wt% Carbopol with 0, 0.01, 01, 0.50, and 1.00 wt% PEO. We see
that $\sigma_{\rm U}$ increases with PEO content, which follows from the
reasoning behind similar monotonic trends observed for $\varepsilon_{\rm b}$.

Based on the data for $\varepsilon_{\rm b}$ and $\sigma_{\rm U}$, we can plot
each against $\sigma_{\rm y}$ data from Sec. 3 to get Ashby-style co-plots
[51]. In Fig. 8(a), we show $\varepsilon_{\rm b}$, measured in extension,
against $\sigma_{\rm y}$, measured in shear. We see that, for a given $c$,
$\varepsilon_{\rm b}$ increases significantly on adding PEO to Carbopol, while
the yield stress increases marginally (except for the lowest Carbopol loading
of $c=0.25~{}wt\%$). This is the most clear evidence that it is possible to
largely decouple the control of extensibility of this model fluid from its
shear properties. Specifically, the extensibility can be increased
significantly while suppressing changes in shear properties. Spanning the
polymer concentrations between 0 and 1 wt% PEO for 0.50 wt% Carbopol changes
the strain-to-break monotonically, which also shows that extensibility can be
systematically varied whilst superimposing different values of the shear yield
stress and extensional failure strains.

(a)

(b)

Figure 8: Ashby co-plots of (a) extensional strain-to-break, $\varepsilon_{\rm
b}$, and (b) ultimate extensional stress, $\sigma_{\rm U}$, _versus_ shear
yield stress $\sigma_{\rm y}$ (from steady shear data), for 0 and 1 wt% PEO,
all wt% Carbopol. Co-plotted are results for 0.50 wt% Carbopol with 0, 0.01,
0.10, 0.50, and 1.00 wt% PEO added. Co-plotted as a dashed line in (b) is the
von Mises criterion for yield stresses in shear and extension. The triangles
refer to the Carbopol samples with 1 wt%, and circles to those with 0 PEO.
Colors indicate the Carbopol concentration and follow the notation established
in Fig. 6.

In Fig. 8(b), we plot $\sigma_{\rm U}$, a measure of yield stress in
extension, against $\sqrt{3}\sigma_{\rm y}$, a measure of effective yield
stress in extension based on shear stresses according to the von Mises
criterion [52, 53, 44]. The dashed line is therefore for materials where the
von Mises criterion holds. Pure Carbopol samples fall on or below this line,
which means that the ultimate stress is smaller than predicted for ideally
elastic materials. Adding PEO to these samples pushes the data above this
boundary, which hints at enhanced extensibility, and a fundamentally different
microstructural consequence of adding PEO to otherwise largely inextensible
yield-stress fluids. Note that the data for 0.25 wt% Carbopol is an outlier
here as well. This might result from the lower value of the yield stress of
the material, such that the values of $\sigma_{\rm U}$ measured under
extension in this case are more significantly affected by capillary forces,
and by experimental noise from the force transducer. The agreement between the
yield stresses in shear and extension with the von Mises criterion for the
higher concentrations of Carbopol are in line with the observations from the
literature [18], where 0.5 wt% Carbopol suspensions were adequately described
by the von Mises yield criterion.

This section has elucidated the significant effect of adding polymers to
Carbopol suspensions: steady shear properties, of which the shear yield stress
is arguably the most important, were relatively unchanged upon addition of
polymer. Additionally, upon PEO addition, shear normal stress increased by a
factor of 1.5-2, which is also comparable to the increase in extensibility
(2-3 times increase in failure strains) for Carbopol samples with PEO when
compared with pure Carbopol. In the following section, we look at the effect
of added PEO on viscoelastic properties of Carbopol, and see if the physics of
largely decoupled of shear and extension properties is retained.

##  5 Effect of PEO on shear viscoelastic properties

In this section, we characterize the oscillatory shear rheology of the ExYSF
to study the effect of added PEO on the viscoelastic behavior of Carbopol.
Each of these tests is aimed at testing if shear viscoelastic properties are
significantly affected by addition of the polymer compared to the change in
extensional behavior. We show frequency sweeps done in the linear regime
(small amplitude oscillatory shear or SAOS) and oscillatory strain amplitude
sweeps spanning linear to nonlinear regimes (large amplitude oscillatory shear
strain or LAOStrain) .

###  5.1 Frequency sweeps in SAOS

Frequency sweeps were done at a constant strain amplitude of $\gamma_{0}=1\%$,
which was in the linear regime as determined from the strain amplitude sweeps
(shown next). The frequency was varied from $\omega=0.1$-$100~{}{\rm
rad~{}s}^{-1}$, and the results are shown in Fig. 9.

(a)

(b)

Figure 9: Comparison of viscoelastic moduli in oscillatory shear frequency
sweeps plotted as a function of frequency for different Carbopol+PEO samples;
$\gamma_{0}=1\%$ and $\omega=$ 0.1-100 rad s-1. (a) All wt% Carbopol with 0 or
1 wt% PEO added. (b) 0.50 wt% Carbopol with 0, 0.01, 0.10, 0.50, and 1.00 wt%
PEO added. The triangles refer to the Carbopol samples with 1 wt%, and circles
to those with 0 PEO.

Two features of the plots stand out: firstly, the values of $G^{\prime}$ are
greater than $G^{\prime\prime}$; $\tan\delta=G^{\prime\prime}/G^{\prime}<0.1$
for all pure Carbopol samples. This indicates that the material has a
predominantly elastic response, which is characteristic of viscoelastic solids
such as gels, arrested systems, crowded and jammed suspensions, and soft
glassy solids [5, 7, 16]. Being a soft glassy system, Carbopol is expected to
display this behavior below the yield stress [4, 54, 16]. This primary
signature is not changed by the addition of PEO, although now we have
$\tan\delta<0.2$, indicating increased dissipative effects due to added PEO
(see Fig. 11).

Secondly, $G^{\prime}$ is only weakly dependent on frequency (average log-log
slope $\ln G^{\prime}/\ln\omega<0.1$). This has also been reported previously
for Carbopol and other microgels [4, 54, 41]. This is characteristic of yield
stress fluids and condensed matter in general, whether gels or soft glasses
[7], and signifies extremely slow thermal relaxation of the microstructure
(and thus a largely constant elastic modulus over a wide range of frequencies)
due to energetically arrested particulate networks or glassy dynamics. This
characteristic is also unchanged upon adding PEO to Carbopol. We also see that
$G^{\prime\prime}$ increases with frequency, for both pure Carbopol and
Carbopol+PEO, suggesting that over shorter timescales the material displays a
more pronounced energy dissipation.

More detailed viscoelastic data is shown by creep tests in the SI, which show
these materials are predominantly solid-like, for stresses below the yield
stress, for timescales up to at least 600 s. This duration is longer than 10 s
(corresponding to $\omega=0.1~{}{\rm rad~{}s}^{-1}$), which is the longest
accessible timescale from the SAOS data shown here. The addition of PEO does
not significantly change the shapes of the viscoelastic creep compliance
curves.

###  5.2 Strain sweeps in LAOStrain

In strain sweeps, the oscillatory strain amplitude was varied from
$\gamma_{0}=0.01$-$1000\%$. All tests were conducted at a fixed frequency of
$\omega=10~{}{\rm rad~{}s}^{-1}$. The results of the strain sweeps are shown
in Fig. 10, with the viscoelastic moduli plotted as a function of the
oscillatory strain amplitude, $\gamma_{0}$. Here, we report the first harmonic
storage $\left(G^{\prime}_{1}\right)$ and loss
$\left(G^{\prime\prime}_{1}\right)$ moduli [55].

(a)

(b)

Figure 10: Comparison of first harmonic storage and loss viscoelastic moduli
in oscillatory shear strain sweeps, plotted as a function of the strain
amplitude during the cycle, for different Carbopol+PEO samples; $\omega=10$
rad s-1 and $\gamma_{0}=$ 0.01-1000%. (a) All wt% Carbopol with 0 or 1 wt% PEO
added. (b) 0.50 wt% Carbopol with 0, 0.01, 0.10, 0.50, and 1.00 wt% PEO added.
The dashed line in each plot shows the low-torque limit of the instrument
[56], below which the data cannot be trusted. The triangles refer to the
Carbopol samples with 1 wt% PEO, and circles to those with 0 PEO.

In the nonlinear viscoelastic regime, higher-order harmonics distort the
periodic stress response [57]. For both pure Carbopol and Carbopol+PEO, we see
that $G^{\prime}_{1}$ is independent of $\gamma_{0}$ at low amplitudes,
indicating a response within the linear viscoelastic regime. At higher
strains, the modulus rolls-off and decreases dramatically, displaying a
softening behavior that is common for colloidal gels and soft glasses [4, 5,
24, 54, 25, 16, 7]. The qualitative behavior is independent of PEO
concentration. Similarly, $G^{\prime\prime}_{1}$ also shows a response within
the linear viscoelastic regime at low strain amplitudes. At intermediate
strain amplitudes, an overshoot is observed, which is characteristic of Type
III behavior in LAOS as defined by Hyun _et al._ [58]. This suggests that the
material displays increased dissipative processes, which have been recently
explained by plastic deformations present at moderate strain amplitudes [59]
in glassy systems. $G^{\prime\prime}_{1}$ eventually rolls-off at larger
strain amplitudes. The critical strain at the departure from linear
viscoelasticity does not seem to be significantly affected by addition of PEO,
since the deviation from linearity in $G^{\prime}_{1}$ happens around
$\gamma_{0}\simeq 10\%$ for all samples. The same features are observed upon
addition of PEO.

Fig. 10(b) shows the moduli when different amounts of PEO are added to the
0.50 wt% Carbopol sample. No significant changes are evident on the values of
$G^{\prime}_{1}$, suggesting that the elastic behavior of the sample is
largely agnostic to the presence of the polymer. This surprising result can be
understood as the linear elastic behavior of the material arising primarily
from the jammed microstructure created by the Carbopol microgel particles,
which is not changed when the polymer is added. A slight increase in
$G^{\prime\prime}_{1}$ can be seen (roughly half an order of magnitude) upon
increasing the amount of PEO. This again suggests that addition of the polymer
increases the energy dissipation in the material, but it is not possible to
identify if this increase in energy dissipation is due to plastic or viscous
deformations.

###  5.3 Summarizing the effect of PEO on LVE of Carbopol

The effect of PEO on the linear viscoelastic properties of Carbopol can be
more clearly seen from the linear regime in the LAOStrain data. We can look at
the linear storage modulus at a fixed frequency calculated from LAOS data, as
$G^{\prime}=\lim_{\gamma_{0}\rightarrow 0}G^{\prime}_{1}$, and the ratio of
the loss to storage moduli, $\tan\delta\equiv G^{\prime\prime}/G^{\prime}$,
where $G^{\prime\prime}=\lim_{\gamma_{0}\rightarrow 0}G^{\prime\prime}_{1}$.
In Fig. 11(a), we plot the power law scaling of $G^{\prime}$ with Carbopol
loading $c$. We see that the scaling exponent in $G^{\prime}\propto c^{\nu}$
is $\nu=0.31\pm 0.08$ for pure Carbopol, which is very similar to the exponent
calculated upon addition of PEO, for which $\nu=0.32\pm 0.04$. These values of
$\nu$ are close to those reported in the literature for Carbopol [4, 15]. The
magnitudes of $G^{\prime}$ are also very close for Carbopol samples with and
without PEO.

(a)

(b)

Figure 11: Co-plots of (a) linear storage modulus $G^{\prime}$, and (b)
$\tan\delta\equiv G^{\prime\prime}/G^{\prime}$, from LAOStrain, at
$\omega=10~{}{\rm rad}~{}{\rm s}^{-1}$, _versus_ Carbopol loading, $c$, for 0
and 1 wt% PEO, all wt% Carbopol. Co-plotted are results for 0.50 wt% Carbopol
with 0, 0.01, 0.10, 0.50, and 1.00 wt% PEO added. The fit lines in (a) are
power-laws for $G^{\prime}$ as function of $c$, and the uncertainties on the
scaling exponents are from data and fit uncertainties. The triangles refer to
the Carbopol samples with 1 wt%, and circles to those with 0 PEO.

In Fig. 11(b), we see that $\tan\delta$ in the linear viscoelastic regime is
much larger with PEO than without. This comes from a more significant increase
in the $G^{\prime\prime}$ compared to $G^{\prime}$ with polymer addition, as
can be seen in Fig. 10(b). The higher value of the loss modulus upon addition
of PEO suggests increased dissipation, as expected due to the increased
amounts of polymer in the suspension. So, although the dissipative component
of the stress increases with polymer addition, the storage modulus is largely
unaffected in both its magnitude and scaling with $c$, since the Carbopol
particles of the ExYSF contribute to the bulk of the stiffness in shear. These
results are in agreement with those seen for the yield stress from the steady
shear data, i.e. the increase in Carbopol concentration increases the yield
stress measured in the steady-state flow curves, and a less significant effect
is observed when PEO is added to the samples.

Figure 12: Ashby co-plots of extensional strain-to-break _versus_ linear
storage modulus from LAOS, $G^{\prime}$ at $\omega=10~{}{\rm rad}~{}{\rm
s}^{-1}$, for 0 and 1 wt% PEO, all wt% Carbopol. Co-plotted are results for
0.50 wt% Carbopol with 0, 0.01, 0.10, 0.50, and 1.00 wt% PEO added. The
triangles refer to the Carbopol samples with 1 wt% PEO, and circles to those
with 0 PEO.

Based on the LAOS data, we can compare the metrics in extension to those in
shear viscoelasticity. Fig. 12 plots $\varepsilon_{\rm b}$ against
$G^{\prime}$, and we see very similar trends as with yield stress, Fig. 8(a).
The values of $\varepsilon_{\rm b}$ increase with added PEO for all Carbopol
concentrations, while $G^{\prime}$ is mostly unaltered upon adding PEO. The
increase in extensibility with PEO content while maintaining similar
$G^{\prime}$ is systematically tunable, as seen from the intermediate PEO
concentrations for 0.50 wt% Carbopol data (yellow symbols). We therefore see
that both steady shear properties such as $\sigma_{\rm y}$ as well as linear
viscoelastic properties such as $G^{\prime}$ are marginally changed when PEO
is added, and the extensibility can be increased almost independently from the
change in shear properties for this model fluid.

##  6 Discussion

The data presented in this paper lies in a rather unique regime of rheological
behavior. In Fig. 13, we propose a map to visualize the amounts of shear
_versus_ extensional nonlinearities in a material. Different metrics could be
used to represent the key rheological features in shear and in extension, so
the choice of which metrics to use in this diagram is open to debate. For
extensional nonlinearity, here we have used $\varepsilon_{\rm b}$ as a metric,
while $1-n_{\rm min}$ is a surrogate for nonlinear behavior in shear, obtained
from the steady shear data. The value of $n_{\rm min}$ is defined as the
smallest gradient of the steady-state flow curve (see Fig. 13 for a
schematic), as

$\displaystyle n_{\rm
min}\equiv\left.\frac{\partial\ln\sigma}{\partial\ln\dot{\gamma}}\right|_{\rm
min},$ (5)

such that smaller values of $n_{\rm min}$ give the most extreme regime of
shear thinning; $1-n_{\rm min}$ correspondingly approaches 1 as shear
nonlinearities become more significant, and approaches zero in the limit of
constant shear viscosity. Materials and models that largely follow a constant
viscosity in shear, such as the Newtonian fluid model and Boger fluids, are
expected to have a value of $n_{\rm min}\rightarrow 1$ (see the bottom-right
inset in Fig. 13). Generalized Newtonian fluid models that describe shear
thinning behavior, such as the power-law model when $n<1$, are expected to
show a variable gradient of the $\sigma$-$\dot{\gamma}$ curve, such that
$n_{\rm min}$ is located at large shear rates. Finally, $n_{\rm min}$ is
expected to be located within the range of stresses where the yielding
transition takes place for yield stress fluids (YSF), since in this region a
dramatic increase of shear rates with a small increase of shear stress. It is
important to highlight that $n_{\rm min}$ therefore does not immediately
distinguish viscoplastic yield-stress fluids from the more general class of
shear-thinning behavior, as it does not take into account the range of shear
rates over which $n_{\rm min}$ occurs, or the value of the yield stress.
However, all yield stress fluids will have a very low value of $n_{\rm min}$.
We see that all our data lie on the right side of the plot, meaning that we
have systematically explored varying ranges of extensional nonlinearity for
materials with significant nonlinear behavior in shear.

Figure 13: Ashby diagram of nonlinearities in shear against extension, with
reference to Boger Fluids [60, 61]. The map helps visualize the different
regimes of shear and extensional nonlinearities shown by materials and
constitutive models. The colored symbols represent the data for all materials
tested in this paper.

The left edge of the map has a name: Boger fluids [36, 37], a class of
rheological response with strong nonlinearity in elastic effects as well as in
extension, and minimal nonlinearity of the shear stress. The right edge of the
map is the limit of maximum shear stress nonlinearity in steady flow, $n_{\rm
min}\rightarrow 0$, the limit of a yield stress fluid, which may also have a
range of extensibility. The model material studied here lies along this edge.

Among the many possible names for this right edge region, we choose the name
“Boger yield-stress fluids” to evoke the connection to Boger fluids that
occupy the opposite edge of behavior. The simplest elastoviscoplastic (EVP)
models, such as the Saramito model [52, 53], also show signatures reminiscent
of Boger yield-stress fluids: yield-stress fluids for which the extensional
behavior can be tuned independently while maintaining the same shear thinning
viscosity exponent.

We have shown that it is possible to formulate samples with various ranges of
yield stress and extensibility, and these two features can be decoupled to a
large extent. However, for certain concentrations of Carbopol, the two effects
are more closely coupled; e.g. for 0.25 wt% Carbopol systems, $\sigma_{\rm y}$
does not follow the general scaling with Carbopol loading $c$, and this hints
at two different microstructural regimes of yielding behavior, which has
already been discussed earlier. Investigating the effect of added polymers to
microgels in this low Carbopol concentration and the extent of decoupling
between shear and extensional properties is an important topic for future
work.

It must be noted that we have not studied the aging and thixotropy of Carbopol
samples with or without PEO. Carbopol is known to show varied degrees of
thixotropy and aging [62, 63, 64, 65, 14], although the degree of structural
breakdown or aging has been shown to not be significant [65]. We have thus not
tested the thixotropic and aging response of our materials.

##  7 Conclusion

In this work, we have formulated a model yield-stress fluid with tunable
extensional properties. The extensibility is largely decoupled from the shear
properties when doping the base Carbopol microgel particle suspensions with
high molecular weight PEO. Adding PEO to Carbopol does not change key
rheological properties measured in shear experiments, namely the shear yield
stress $\sigma_{\rm y}$ and the linear viscoelastic storage modulus
$G^{\prime}$, by more than 10-20%, so the elastic nature of Carbopol is not
changed much, although the dissipation is increased on adding PEO. On the
contrary, properties in extension, such as the extensional strain-to-break
$\varepsilon_{\rm b}$, are increased many fold. These results suggest that the
addition of PEO does not significantly alter the interactions between Carbopol
microgel particles at high loadings, which are responsible for the
elastoplastic shear behavior, while the extensional properties are enhanced by
the PEO chains, which may be largely limited to the solvent phase. This also
exemplifies that it is possible to formulate simple mixtures of complex
fluids, approximately superimposing their individual properties to obtain a
hybrid material with tunable behavior. While there is some variation of shear
properties when varying the extensional behavior of this material system, we
consider it minimal compared to multi-component systems in general where
significant non-trivial coupling may occur. As such, we consider this
Carbopol+PEO system a model fluid which can be used to experimentally study
the isolated effect of changing extensibility in yield stress fluids, the
fundamental understanding of which will benefit applications ranging from
spray coating of fire suppressants [66, 11, 23, 14] to direct-ink writing
additive manufacturing [9, 67], disease control [68, 69], and food texture
[70, 71, 72], among many others.

##  8 Acknowledgments

This work was funded by the National Science Foundation, CAREER Award,
CBET-1351342. The authors thank Dr. C. Saengow and Dr. Y. Wang for helpful
discussions.

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