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Structured Model Pruning of Convolutional Networks on Tensor Processing Units

Anonymous Authors1

††footnotetext:  1Anonymous Institution, Anonymous City, Anonymous Region,
Anonymous Country. Correspondence to: Anonymous Author
<EMAIL_ADDRESS>
Preliminary work. Under review by the International Conference on Machine
Learning (ICML). Do not distribute.

###### Abstract

The deployment of convolutional neural networks is often hindered by high
computational and storage requirements. Structured model pruning is a
promising approach to alleviate these requirements. Using the VGG-16 model as
an example, we measure the accuracy-efficiency trade-off for various
structured model pruning methods and datasets (CIFAR-10 and ImageNet) on
Tensor Processing Units (TPUs). To measure the actual performance of models,
we develop a structured model pruning library for TensorFlow2 to modify models
in place (instead of adding mask layers). We show that structured model
pruning can significantly improve model memory usage and speed on TPUs without
losing accuracy, especially for small datasets (e.g., CIFAR-10).

##  1 Introduction

Convolutional neural networks (CNNs) is the dominant approach for many
computer vision applications, e.g., image classification Krizhevsky et al.
(2012), object detection Girshick et al. (2014), semantic segmentation Long et
al. (2015). The deployment of CNNs in real-world applications, however, is
often constrained by model size, memory usage, and computational time. Model
pruning is one approach to compress model sizes and accelerate inference Liu
et al. (2015); Han et al. (2016); Wen et al. (2016); Zhou et al. (2016); Li et
al. (2017); Scardapane et al. (2017); Anwar et al. (2017); Zhang et al.
(2018); Zhu & Gupta (2018); Deng et al. (2020). Model pruning can be realized
at different levels, e.g., weight-level, channel-level, or layer-level.
Weight-level (unstructured) pruning has the highest flexibility but usually
requires special software or hardware for fast inference Han et al. (2016).
Layer-level structured pruning does not require special packages for inference
acceleration but is only effective when the depth is sufficiently large Wen et
al. (2016). Channel-level structured pruning (shown in Fig. 1) balances
flexibility and ease of implementation on general accelerators Liu et al.
(2017); Li et al. (2017).

Tensor Processing Units (TPUs) are the dominant approach deep learning
accelerators for Google Jouppi et al. (2018; 2020). TPUs can be an order
faster and more energy-efficient than contemporary GPUs or CPUs Jouppi et al.
(2017). At Google, an auto-tuner for the Accelerated Linear Algebra (XLA)
compiler was developed to search for the fastest fusion configurations of an
XLA program on TPUs Kaufman et al. (2020). Unstructured model pruning methods
won’t improve efficiency on TPUs because the XLA layout algorithm will pad
zeros to the removed individual weights Kaufman et al. (2020). Whether
structured model pruning can lead to efficiency gain on Google TPUs with XLA
auto-tuners, however, remains unclear.

In this work, we measure the accuracy-efficiency trade-off for various
channel-level structured model pruning methods and datasets on Google TPUs
with XLA auto-tuners. Performance (accuracy, memory usage and step time) of
pruned models are measured on Google Borg Verma et al. (2015).

Figure 1:  Illustration of the channel-based structured model pruning. Two
channels from a Conv2D layer are pruned.

##  2 Methods

We use a TensorFlow 2 Abadi et al. (2016) implementation of VGG-16 Simonyan &
Zisserman (2015) trained on ImageNet Russakovsky et al. (2015) and CIFAR-10
Krizhevsky & Hinton (2009) as an example to study the accuracy-efficiency
tradeoff of structured model pruning on TPUs. We train the original model,
prune a certain ratio of channels, train the pruned models, and then measure
model performance (accuracy, memory usage and step time) during the last epoch
of training on Google Borg Verma et al. (2015).

To find channels to prune, we use scaling factor-based Liu et al. (2017) and
L1 norm-based pruning algorithms Li et al. (2017). These algorithms estimate
the ”importance” of a channel based on weights after training, e.g., $\gamma$
coefficient in the following batch-normalization layer (scaling factor Liu et
al. (2017)) or L1 norm of channel weights (L1 norm Li et al. (2017)). The
least ”important” channels will be pruned. We also try whether reloading
weights from the original model helps training of pruned models. To create the
pruned models without additional mask layers (so that the actual memory usage
and step time can be measured), we create new models with desired structure
and then reload corresponding weights from original models if desired.

For all trainings (including original models and pruned models), we use the
same hyper-parameters (batch size $128$, max epochs $100$) and optimizers (SGD
Bottou (2010) with momentum $0.9$ and learning rate $0.01$). This is because
we aim at the accuracy-efficiency tradeoff of structured model pruning on TPUs
rather than the best accuracy.

##  3 Results

Figure 2:  Accuracy as a function of channel pruned ratio on TPUs for CIFAR-10
and ImageNet for various pruning methods. Black curves use the scaling factor-
based pruning algorithm Liu et al. (2017) and reload weights from original
models for training. Red curves use the same pruning algorithm but don’t
reload weights from original models. Blue curves use L1 norm-based pruning
algorithm Li et al. (2017) and reload weights from original models. Figure 3:
Memory usage and number of parameters as a function of channel pruned ratio on
TPUs for CIFAR-10 and ImageNet. Figure 4:  Training step time as a function of
channel pruned ratio on TPUs for CIFAR-10 and ImageNet.

Figure 2 shows accuracy as a function of the pruned ratio of channels on TPUs
for CIFAR-10 and ImageNet for various pruning methods. As pruned ratio
increases, the accuracy for ImageNet decreases much faster than accuracy for
CIFAR-10. To sustain original accuracy, VGG-16 can be pruned by $30\%$ on
CIFAR-10 while $10\%$ on ImageNet. At $30\%$ pruned ratio, accuracy for
CIFAR-10 is almost the same, while accuracy for ImageNet decreases by $20\%$,
indicating that VGG-16 is over-parameterized for CIFAR-10. CIFAR-10 requires
fewer parameters to fit accurately compared with ImageNet because the optimal
model size scales with the dataset size Hestness et al. (2017). VGG-16 is more
over-parameterized on CIFAR-10 and a larger pruned ratio can be used without
losing accuracy. Accuracy decreases faster and faster when pruned ratio
increases (the absolute value of the slope increases) because the first
parameters pruned are over-parameterizations. For both datasets, reloading
weights from the original model improves pruned accuracy, and scaling factor-
based pruning Liu et al. (2017) leads to better accuracy than L1-norm-based
pruning Li et al. (2017).

Structured model pruning can improve training memory usage (Fig. 3) and step
time (Fig. 4) on TPUs. While the saved memory usage, number of pruned
parameters, saved step time, and pruned ratio of channels are proportional to
each other, they are not strictly linear with each other. This is because the
number of parameters of each channel is different, and TPUs will pad zeros to
weights according to the XLA layout algorithm Kaufman et al. (2020).

##  4 Conclusion

Using the VGG-16 model as an example, we measure the accuracy-efficiency
trade-off for various structured model pruning methods and datasets (CIFAR-10
and ImageNet) on TPUs. In both cases, reloading weights from the original
model improves pruned accuracy, and scaling factor-based pruning Liu et al.
(2017) leads to better accuracy than L1-norm-based pruning Li et al. (2017).
We show that structured model pruning can significantly improve model memory
usage and speed on TPUs without losing accuracy, especially for small datasets
(e.g., CIFAR-10). This is because VGG-16 is over-parameterized for a small
dataset like CIFAR-10. Our results suggest that structured model pruning is a
promising approach to improve the efficiency of CNNs on TPUs.

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# Unsteady rheo-optical measurements of uniaxially extending liquid polymers

M. Muto<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Department of
Mechanical Systems Engineering, Nagoya Institute of Technology, Gokiso Campus
3-502, Gokiso, Showa-ku, Nagoya-shi, Aichi, Japan  Y. Tagawa
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Department of Mechanical
Systems Engineering, Tokyo University of Agriculture and Technology, Koganei
Campus 6-204, 2-24-16 Nakacho, Koganei-city, Tokyo, Japan

###### Abstract

The viscoelastic behavior of liquid polymers changes depending on the
microstructure of the polymer chains. To investigate the unsteady structural
changes induced by polymer chain morphology under stress, we develop a simple
rheo-optical technique combining a high-speed polarization camera and
dripping-onto-substrate capillary breakup extensional rheometry (CaBER-DoS),
which uniaxially extends and aligns the polymer along the stress direction.
Using this technique, the viscoelastic behavior and birefringence field of an
extending liquid polymer are simultaneously measured. Unlike the conditions
that occur under shear-stress loading with a rotational rheometer, this
technique provides a linear relation between birefringence and extensional
stress. The measurements reflect the structural change in a flexible polymer
(a mixed solution of PEO and CNCs) in the temporal evolution of the
birefringence and orientation angle, which is a measure of the orientation
state and coil-stretch transition of a polymer chain. Remarkably, within the
elasto-capillary regime, the measured birefringence remains constant with a
constant orientation state, while the Weissenberg number gradually increases,
which provides experimental evidence of the coil-stretch transition of the
polymer chain under constant extensional stress loading.

††preprint: AIP/123-QED

##  I Introduction

For a liquid polymer, the stress and velocity distributions of the fluid are
complicated owing to the effects of flow resistance and apparent viscosity
induced by the influence of the microstructure of polymer chains. To
investigate the phenomena occurring in liquid polymers, rheo-optical
techniquesNoto  _et al._ (2020); Philippoff (1956); Lodge (1955, 1956);
Shikata, Dahman, and Pearson (1994); Gortemaker  _et al._ (1976); Yevlampieva,
Pavlov, and Rjumtsev (1999); Meyer  _et al._ (1993); Chow and Fuller (1984);
Ito, Yoshitake, and Takahashi (2016) involving simultaneous rheometric and
optical measurements are often used. A variety of rheo-optical techniques
combining a rotational rheometer, which applies deformation to the fluid with
a moving wall, and birefringence measurement have been reported. With these
techniques, changes in microstructure, indicated by the orientational state
and coil-stretch transition (the degree of stretch from the coil state) of
polymer chains, can be inferred from birefringence measurements. By matching
results at different scales measured separately by rheometry and an optical
method, a comprehensive evaluation of the phenomena occurring in liquid
polymers can be performed. While there have been many studies of the
microstructural behavior of liquid polymers under “shear” conditionsIto,
Yoshitake, and Takahashi (2016); Haward  _et al._ (2019); Zhao, Shen, and
Haward (2016); Haward  _et al._ (2012); Iwata  _et al._ (2019), there have
been few reports on changes in the microstructure under extensional
conditions.

To investigate polymer chain morphology under coil-stretch transition, the
Weissenberg number $Wi$, which is a dimensionless number representing the
ratio of elasticity and viscosity, is often used as an indicator. However,
evidence that $Wi$ is in fact an indicator of coil-stretch transition is
limited to the results of numerical calculationsEntov and Hinch (1997);
Mathues  _et al._ (2018), which depend on the type of constitutive model
chosen Sur and Rothstein (2018); McKinley (2005); Prabhakar, Prakash, and
Sridhar (2006); Doyle  _et al._ (1998); Doyle, Shaqfeh, and Gast (1997). On
the other hand, there have been few reports of experimental investigations of
the changes in molecular structure of polymer chains under extensional
conditions. LodgeLodge (1955, 1956) and PhilippoffPhilippoff (1956) proposed
to measure the flow-induced birefringence of highly viscoelastic solutions
reeled in by a draw wheel. Rothstein and McKinleyRothstein and McKinley (2002)
developed a method for combined measurements by a filament stretching
rheometer and a birefringence measurement system for dilute solutions
undergoing extension. These techniques enable investigation of the effect of
extensional stress on the flow-induced birefringence of liquid polymers.
Nevertheless, to demonstrate the validity of the stress-optic lawNoto  _et
al._ (2020); Janeschitz-Kriegl (2012); Ryu, Inoue, and Osaki (1996); Inoue,
Okamoto, and Osaki (1991) for a flowing polymer fluid, it is important to
simultaneously measure the birefringence and the orientational state of the
polymer chain under extensional stress loading. For this purpose, it is
necessary to establish a simple approach to the investigation of the above-
mentioned complex problems through birefringence measurements and develop a
simple rheo-optical measurement method using a high-speed polarization camera.

The purpose of this study is to develop a simple system for examining changes
in microstructure (i.e., the orientational state and coil-stretch transition
of polymer chains) of liquid polymers under extensional stress loading with
the birefringence measurements. For this purpose, we have chosen dripping-
onto-substrate capillary breakup extensional rheometry (CaBER-DoS)Dinic and
Sharma (2020); Dinic, Jimenez, and Sharma (2017); Sur and Rothstein (2018);
McKinley (2005); McKinley and Tripathi (2000); Clasen  _et al._ (2006) as the
means of applying the extensional stress. Unlike a rotational rheometer, which
measures the shear viscosity, CaBER-DoS is a noncontact method that measures
the extensional viscosity from the radius of the liquid filament, which decays
as the fluid is extended. In this paper, we report the results of an
investigation of the chain morphology of a polymer chain in a stretched flow
field, where elasticity becomes dominant. Selection of the elastic-dominant
regime, i.e., the elasto-capillary (EC) regime is particularly important when
calculating viscoelastic properties such as the extensional viscosity, strain,
and relaxation time. Importantly, within the EC regime, a uniaxial extensional
stress is applied to the fluid, allowing the assumption of uniform flow
without shear stress to be madeMcKinley (2005). Therefore, the polymer chains
inside the fluid are assumed to be oriented in the extensional direction and
uniformly aligned, which should be experimentally demonstrable by the
birefringence measurements. Note that, according to the theory of CaBER-DoS,
three regimes, dominated by inertia, elasticity, and viscosity, respectively,
can be defined on the basis of the temporal decay of the radius of the liquid
filamentSur and Rothstein (2018); McKinley (2005); Entov and Hinch (1997);
Wagner, Bourouiba, and McKinley (2015); Prabhakar, Prakash, and Sridhar
(2006). Thus, the most important requirement for our rheo-optical method is to
be able to simultaneously measure the changes of the three regimes in CaBER-
DoS and the birefringence, which reflects the morphology of the polymer chain.
To do this, a high-speed polarization camera, which enables us to obtain
polarization information (i.e., the flow-induced birefringence and the
orientation angle), is used.

##  II Methods

In this section, the mechanism of flow-induced birefringence of a polymer and
the measurement principle of the rheo-optical technique that uses both a high-
speed polarization camera and the CaBER-DoS system are described.

###  II.1 Flow-induced birefringence of polymer

The flow-induced birefringence of polymer chains is a key indicator of chain
morphology. The flow-induced birefringence $\delta_{n}$ that appears due to
stress loading is proportional to the principal stress difference $\sigma$ in
what is known as the stress-optic law Noto  _et al._ (2020); Janeschitz-Kriegl
(2012); Ryu, Inoue, and Osaki (1996); Inoue, Okamoto, and Osaki (1991):

$\delta_{n}=|n_{\parallel}-n_{\perp}|=C|\sigma_{\parallel}-\sigma_{\perp}|,,$
(1)

where $n_{\parallel}$ and $n_{\perp}$ are the refractive indices for
polarizations respectively parallel and perpendicular to the orientation of
the polymer chain and $C$ is the stress-optic coefficient. In our experimental
setup using birefringence measurements, the intensity of the flow-induced
birefringence can be measured from the polarization of the light transmitted
through the liquid polymer.

When there is no stress loading, polymer chains in the liquid have random
orientations in the coil state, and thus flow-induced birefringence does not
occur [see Fig. 1(a)]. When stress is applied to the liquid polymer, the
polymer chains become highly aligned by stretching, resulting in increased
birefringence $\delta_{n}$ [see Fig. 1(b)]. In this paper, the orientation
angle $\varphi$ is defined as the tilt angle of the polymer chain with respect
to the horizontal ($x$) direction [the vertical ($y$) direction is the
extensional direction].

Figure 1:  Flow-induced birefringence induced by a change in the orientational
state of polymer chains under stress loading.

###  II.2 Rheo-optical technique with high-speed polarization camera

We employed a birefringence measurementsBass (2010); Rastogi (2015) to
determine the phase retardation of polarized light passing thorough polymer
chains under stress loading. Figure 2 shows how the polarization state of
light changes when it is transmitted through a birefringent object.

Figure 2:  Rheo-optical technique with polarization used to measure the phase
retardation induced by changes in the polarization state of a birefringent
object.

The incident electromagnetic wave, which is represented by the red curve,
consists of two orthogonal components, which are represented by the blue and
orange curves. When both components are in phase, the retardation is zero, and
thus the incident light is linearly polarized. When light is transmitted
through a birefringent object (the liquid polymer) under stress loading, the
propagation speed of each component wave differs, since the refractive indices
$n_{\parallel}$ and $n_{\perp}$ are different from one another. As a result,
the phases of the component differ, and the values of the retardation change.
The retardation $\delta$ is given by the integral of the birefringence
$\delta_{n}$ over infinitesimal volume elements within the object along the
optical axis (the $z$ direction):

$\delta=\int\delta_{n}\,dz.$ (2)

We propose a simple calibration technique that combines a CaBER-DoS system and
a high-speed polarization camera based on the theory of birefringence
measurement (CRYSTA PI-1P, Photron Ltd.) (see Fig. 3). Our concept can be
realized by using just a simple setup: a target sample (liquid polymer)
between the polarization camera and an LED light source.

Figure 3:  Experimental setup using a high-speed polarization camera and
CaBER-DoS system.

In the CaBER-DoS system, a fluid dispensing device with a syringe pump (Pump
11 Elite, Harvard Apparatus Ltd.) is used to deliver a drop of the polymer
solution at a relatively low flow rate onto a glass substrate placed at a
fixed distance below the nozzle exit. As the droplet slowly drips from the
nozzle, a filament is formed. By measuring the minimum necking radius
$R_{\mathrm{mid}}$ of the filament, which becomes thinner as it extends, we
can calculate the extensional viscosity, strain, relaxation time, etc. When
utilizing the CaBER-DoS system, we can assume that the orientation of all
chain polymers is uniform, because the stress exerted along the optical axis
is uniform. Therefore, the retardation $\delta$ over the optical depth of the
measurement object $d$ in Eq. (2) can be rewritten as follows:

$\delta=d\delta_{n}=d|n_{\parallel}-n_{\perp}|.$ (3)

This equation is valid for our technique, although it is generally invalid for
experiments under shear stress loading. Note that it is valid only when the
incident polarized light passes through the center of the filament, in which
case we can rewrite $d=2R_{\mathrm{mid}}$ in our setup.

The high-speed polarization camera is capable of capturing photoelastic
phenomena with a pixelated polarizer array at a frame rate of up to $1.5\times
10^{6}$ frames/s. The array consists of four adjacent polarizers ($2\times 2$)
arranged in four different orientations. The instantaneous retardation
${\delta}$ and the instantaneous orientation angle $\varphi$ can be calculated
from the light intensities captured by these four polarizers ($I_{1}$ at 0∘,
$I_{2}$ at 45∘, $I_{3}$ at 90∘, and $I_{4}$ at 135∘)Onuma and Otani (2014):

$\displaystyle{\delta}$
$\displaystyle=\frac{\lambda}{2\pi}\sin^{-1}{\frac{\sqrt{{({I}_{3}-{I}_{1})}^{2}+{({I}_{2}{-I}_{4})}^{2}}}{({I}_{1}+{I}_{2}+{I}_{3}+{I}_{4})/2}},$
(4) $\displaystyle\varphi$
$\displaystyle=\frac{1}{2}\tan^{-1}{\frac{{I}_{3}-{I}_{1}}{{I}_{2}-{I}_{4}}},$
(5)

where $\lambda$ is the wavelength of the incident light. In this experiment,
the incident light was emitted from a green LED with a wavelength of 520 nm
and was polarized by a circular polarization sheet consisting of a polarizer
and a quarter-wave plate. Note that the incident light must be directed
straight into the camera.

In this study, a mixed solution of polyethylene oxide [PEO, 1 wt. %
concentration in ultrapure water, molecular weight $M_{w}=(1\text{--}5)\times
10^{6}$, Sigma-Aldrich Co. LLC] and cellulose nanocrystals (CNCs, 0.5 wt. %,
Cellulose Lab Inc.) was used as the liquid polymer. The solvents were stirred
in ultrapure water for 24 h with a hot stirrer, with the preset temperature
and speed of revolution set to 80 ∘C and 1000 rpm, respectively. Note that the
mixed solutions was a non-Newtonian fluid exhibiting shear-thinning behavior
(see Fig. 4).

Figure 4:  Shear-thinning viscosity curve of mixed solution of PEO and CNCs.

##  III Results and discussion

We divide this section into three subsections. The first reports the
visualized phase retardation field of the extending liquid polymer filament,
captured by the high-speed polarization camera. The second reports the
temporal evolution of phase retardation and orientation angle. The third
presents the flow-induced birefringence.

###  III.1 Visualized phase retardation field of extending liquid polymer
filament

The visualization results for the phase retardation field of the extending
liquid polymer filament captured with the high-speed polarization camera are
discussed here. Figure 5 shows the temporal evolution of the filament, using
the raw images captured by the high-speed polarization camera.

Figure 5:  Temporal evolution of the extending liquid polymer filament shown
by raw images and the visualized phase retardation field for (a) and (b) mixed
solutions of CNCs (0.5 wt. %) and PEO (1 wt. %) with molecular weights
$M_{w}=1\times 10^{6}$ and $5\times 10^{6}$, respectively, and (c) PEO
solution ($M_{w}=5\times 10^{6}$, 1 wt. %) without CNCs. The white triangles
indicate the positions of the minimum necking radius, and the white area
($10\times 10$ pixels indicated by $\delta$) shows the measurement area for
the retardation and orientation angle.

The position of the minimum necking radius was also changing continually
during the measurement, and could be tracked using our field measurement
technique. The retardation field was visualized by converting the intensity
value at each pixel position in the raw images to the corresponding
retardation value.

It was confirmed that the retardation appeared differently as the molecular
weight of the PEO was changed [Figs. 5(a) and 5(b)]. This was due to the
difference in the intensity of stress loading, which depends on the molecular
structure and viscoelasticity of a polymer. Moreover, in the case of the PEO
solution without CNCs [Fig. 5(c)], no retardation could be observed. Thus, we
argue that the retardation in mixed solutions of PEO and CNCs is induced by
microstructural changes in the CNCs, since the induction of birefringence by
CNCs is due to their chain structureKádár, Spirk, and Nypelö (2021);
Calabrese, Haward, and Shen (2021); Hasegawa, Horikawa, and Shikata (2020);
Arai, Horikawa, and Shikata (2018); Chowdhury, Peng, and Youngblood (2017);
Cranston and Gray (2008).

From the retardation field, as the stretching continued, the intensity of
phase retardation also changed, i.e., it increased in the early stages and
decreased thereafter. This is because phase retardation is proportional not
only to the principal stress difference but also to the optical thickness,
i.e., the diameter of the liquid polymer filament, as indicated in Eq. (3).
For an extending liquid filament in a CaBER-DoS system, while the principal
stress differences tend to increase, the diameter of the filament decreases.
Therefore, it can be predicted that retardation will decrease owing to the
reduction in optical thickness.

###  III.2 Phase retardation and orientation angle

We now consider the measurement results for the phase retardation and
orientation angle of the extending liquid polymer filament. To investigate
these quantitatively, we introduced a measurement area ($10\times 10$ pixels,
shown by the white box in Fig. 5) on the axis of symmetry of the filament.
This measurement area migrated together with the minimum necking radius. The
values of the retardation and orientation angle were obtained as spatial
averages over the measurement area. Figure 6 shows the results for radius
ratio, retardation, and orientation angle over time for two different
molecular weights of PEO.

Figure 6:  Temporal evolution of phase retardation $\delta$ and orientation
angle $\varphi$ with radius ratio $R_{\mathrm{mid}}/R_{0}$ for mixed solutions
of CNCs and PEO with molecular weights $M_{w}=1\times 10^{6}$ (a) and $5\times
10^{6}$ (b). The error bars on the curves of the radius ratio show the
standard deviation of the five measurement results.

The measurement results for the temporal evolution of the radius ratio (the
black plots in Fig. 6) can be seen to decrease with time. The elapsed time for
the filament to pinch off increases as the molecular weight of PEO increases.
In the elasto-capillary (EC) regime, which is an important indicator when
determining the viscoelastic properties, the radius $R_{\mathrm{mid}}$ decays
exponentially with time $t$ Sur and Rothstein (2018); McKinley (2005); Entov
and Hinch (1997); Wagner, Bourouiba, and McKinley (2015); Prabhakar, Prakash,
and Sridhar (2006):

$\frac{R_{\mathrm{mid}}(t)}{R_{0}}=\left(\frac{GR_{0}}{2\gamma}\right)^{1/3}\exp\\!\left(-\frac{t}{3\lambda_{E}}\right),$
(6)

where $R_{0}$ is the initial radius, and $G$, $\gamma$, and $\lambda_{E}$ are
the elastic modulus, surface tension, and relaxation time of the fluid,
respectively. Moreover, the duration of the EC regime is the same as the
duration of the period when the strain rate $\dot{\epsilon}_{\mathrm{mid}}$ is
constant:

$\dot{\epsilon}_{\mathrm{mid}}=-\frac{2}{R_{\mathrm{mid}}(t)}\frac{dR_{\mathrm{mid}}}{dt}.$
(7)

Knowing this, we can determine the EC regime of the mixed solutions of PEO and
CNCs. Plots of the radius ratio for common viscoelastic fluids have an
inflection point corresponding to the transition from the inertio-capillary
(IC) regime to the EC regime Sur and Rothstein (2018); McKinley (2005); Entov
and Hinch (1997); Wagner, Bourouiba, and McKinley (2015); Prabhakar, Prakash,
and Sridhar (2006). The results for the mixed solutions of PEO and CNCs (Fig.
6) clearly show the characteristics of the EC regime. The plots of the radius
ratio for the mixed solutions both exhibit an inflection point, and the
temporal images [Fig. 5(b)] clearly show that the filament is cylindrical. The
duration of the EC regime for the mixed solutions of PEO and CNCs of different
molecular weights are 46–78 ms for $M_{w}=1\times 10^{6}$ and 750–1730 ms for
$M_{w}=5\times 10^{6}$.

The blue curves in Fig. 6 show the results for the orientation angle of the
polymer chains. Remarkably, these results confirm that the orientation angle
$\varphi$ remains constant at $\pi/2$, which means that the polymer chains
have become highly aligned along the extensional direction of the filament,
except during the early stage (part of the IC regime) and the final stage
[part of the visco-capillary regime (VC) regime]. Note that the values of the
orientation angle oscillate in the VC regime owing to the lack of spatial
resolution. The relationship between the flow-induced birefringence
$\delta_{n}$ and the orientation angle $\varphi$ is expressed by the following
equationOkada, Urakawa, and Inoue (2016):

$\delta_{n}=f\delta_{n}^{0}=\frac{3\cos^{2}(\pi/2-\varphi)-1}{2}\delta_{n}^{0},$
(8)

where $\delta_{n}^{0}$ is the intrinsic birefringence, which is a material
constant for the orientational birefringence of polymers. Importantly, within
the EC regime, the birefringence can be treated as intrinsic birefringence,
because the orientation angle remains constant at $\pi/2$. In such situations,
the liquid polymer filament is assumed to be an axially uniform cylindrical
column of constant radius $R_{\mathrm{mid}}$, with a free-slip boundary
condition, which does not induce a radial shear flow McKinley (2005); McKinley
and Tripathi (2000). Thus, the principal stress difference in Eq. (1) is equal
to the normal stress, i.e., the uniaxially extensional stress $\sigma_{e}$,
and the stress-optic law within the EC regime can be written as follows:

$\delta=d\delta_{n}=Cd\sigma_{e}.$ (9)

As a result, in our experiment, polymers were oriented along the extensional
direction. Thus, it is confirmed that the CaBER-DoS system provides sufficient
extensional stress to orient the polymer chains.

Each of the red plots of retardation in Fig. 6 exhibits a maximum point, which
is at the same locations as the inflection point in the corresponding plot of
the radius ratio. Thus, the positions of these maxima in the retardation plots
indicate the locations where the IC regime switches to the EC regime, since
the results for the retardation are affected by the optical thickness [see Eq.
(9)]. In the next subsection, we describe a more detailed validation oft the
change in birefringence that is not affected by the decay.

Figure 7:  Temporal evolution of birefringence $\delta_{n}$, Weissenberg
number $Wi$, and strain rate $\dot{\epsilon}_{\mathrm{mid}}$ for mixed
solutions of CNCs and PEO with molecular weights $M_{w}=1\times 10^{6}$ (a)
and $5\times 10^{6}$ (a).

###  III.3 Flow-induced birefringence within coil-stretch transition

We now analyze the results for the flow-induced birefringence measured with
the high-speed polarization camera for an extending liquid polymer. When all
the polymer chains are aligned with the optical axis, the flow-induced
birefringence $\delta_{n}$ due to the polymer chain morphology can be derived
from the measured retardation $\delta$ [Eq. (3)]. Figure 7 show plots of the
birefringence $\delta_{n}$ vs normalized time $\hat{t}$ (normalized by the
transition time from the IC to the EC regime). Here, the normalized times
within the EC regime for the PEO with molecular weights $M_{w}=1\times 10^{6}$
and $5\times 10^{6}$ are from 1 to 1.7 and from 1 to 2.3, respectively. These
results confirm that the birefringence increases exponentially within the IC
regime, has a constant value within the EC regime, and increases again within
the VC regime. Moreover, the plot of birefringence has an inflection point at
the transition from the IC to the EC regime. Within the EC regime, the
variations of the birefringence with respect to the overall range are $\pm
1.45\%$ and $\pm 7.98\%$ for PEO with molecular weights $M_{w}=1\times 10^{6}$
and $5\times 10^{6}$, respectively. The fact that the value of birefringence
remains constant within the EC regime is consistent with the theoretical basis
of CaBER-DoS, according to which the uniaxial extensional stress is constant
within the EC regime Sur and Rothstein (2018); McKinley (2005); Entov and
Hinch (1997); Wagner, Bourouiba, and McKinley (2015); Prabhakar, Prakash, and
Sridhar (2006).

We believe that there are two reasons why the temporal evolution of the
birefringence shows a similar trend to that of the extensional stress. The
first is that, within the EC regime, the birefringence is proportional to the
extensional stress. This is evident from the fact that Eq. (9) holds under
conditions where the birefringence can be measured as the intrinsic
birefringence and only a uniaxial extensional stress is exerted on the
filament within the EC regime [as follows from the derivation of Eq. (9) in
Sec. III.2]. The second reason is that, from the IC to the EC regime, the
tendency of the birefringence matches that of the capillary pressure (i.e.,
the extensional stress) calculated through numerical simulation by Mathues _et
al._ Mathues  _et al._ (2018), who argued that the transition from the IC to
the EC regime can be explained by the temporal evolution of the stress
contributions. Within the IC regime, the polymer chains remain in a coiled
state, and the inertial acceleration balances the capillary pressure in the
fluid column, which results in the strong stretching of the chains in the
coiled state. Immediately after this, the elastic stress rises quickly to
balance the capillary pressure, and the inertial contribution drops to a
negligible value. The inflection point of the capillary pressure occurs when
the contributions of inertial and elastic forces are swapped at the transition
from the IC to the EC regime. As a result, upon entry into the EC regime, the
growth of the uniaxial deformation of the chains suddenly slows down, which
results in the formation of an equilibrium state of the capillary pressure.

To verify whether the birefringence corresponds to the polymer chain
morphology, we introduce the Weissenberg number $Wi$ as an indicator. It is
known that for a polymer coil to stretch, $Wi$ must at least exceed a critical
value of $\frac{1}{2}$ Sur and Rothstein (2018); McKinley (2005); Prabhakar,
Prakash, and Sridhar (2006). In this study, we calculate $Wi$ as the product
of the strain rate $\dot{\epsilon}_{\mathrm{mid}}$ and the relaxation time
$\lambda_{E}$. The latter is constant McKinley (2005), and so the curve of
$Wi$ has the same shape as that of the strain rate. Figure 7 show that $Wi$
remains larger than the critical value of $\frac{1}{2}$ until the pinching off
of the liquid polymer filament, while the strain rate is constant within the
EC regime. Moreover, it is clear that each plot of $Wi$ has two turning
points: a maximum followed by a minimum. The normalized times at the first
turning point are 0.86 and 0.57 for the PEOs with molecular weight
$M_{w}=1\times 10^{6}$ and $5\times 10^{6}$, respectively, both of which are
within the IC region. For both PEOs, the second turning point of $Wi$, which
is at a value greater than $\frac{1}{2}$, is located at the transition from
the IC to the EC regime. The existence of two turning points, which cause the
plot of $Wi$ to have a mountainous shape from the IC to the EC regime, has
also been reported for other polymer solutions Sur and Rothstein (2018);
Prabhakar, Prakash, and Sridhar (2006). From a comparison of the experimental
results for the birefringence and $Wi$, we can confirm that the birefringence
becomes constant at the point of transition from the IC to the EC regime,
where the $Wi$ data show that the polymer chain gradually becomes stretched
from the coiled state. Therefore, we argue that the measured birefringence can
be used as an indicator to investigate the coil-stretch transition in our
rheo-optical technique that uses both a high-speed polarization camera and a
CaBER-DoS system.

##  IV Conclusion

To investigate unsteady changes in the microstructure of liquid polymers under
extensional stress loading, we have developed a simple rheo-optical system
based on CaBER-DoS and birefringence measurements. The CaBER-DoS system was
used to uniaxially extend the liquid polymer under extensional stress loading,
and we successfully measured the phase retardation field of a liquid polymer
(a mixed solution of PEO and CNCs) using a high-speed polarization camera.

As the molecular weight of the polymer was changed, the visualized retardation
changed in appearance, owing to differences in the intensity of stress
loading. By performing measurements in a defined area of the retardation
field, we confirmed that the plot of retardation vs time exhibits a maximum
point, which is at the same location as an inflection point in the plot of the
radius ratio corresponding to the transition from the IC to the EC regime.
From this, we argue that the position of the maximum point of the retardation
indicates where the IC regime switches to the EC regime.

According to the theoretical basis of CaBER-DoS, the direction of the
extensional stress and the orientation of the polymer chain are the same
within the EC regime. To verify this, a simultaneous measurement of the
orientation angle of the polymer chains was carried out, and it was confirmed
that the orientation within the EC regime is the same as the extensional
direction. Thus, we have proved that our system enables measurement of flow-
induced birefringence of polymer solutions in the case where the polymer is
oriented in the extensional direction.

The plot of the temporal evolution of the flow-induced birefringence of the
extending liquid polymer also exhibits an inflection point at the transition
from the IC to the EC regime. We confirmed that there is an exponential
increase within the IC regime and a constant value within the EC regime; the
variations in the birefringence within the EC regime with respect to the
overall range are $\pm 1.45\%$ and $\pm 7.98\%$ for PEOs with molecular
weights $M_{w}=1\times 10^{6}$ and $5\times 10^{6}$, respectively. This trend
matches that of the uniaxially extensional stress obtained from numerical
simulations in the literature and is consistent with the theoretical behavior
according to which the extensional stress is constant within the EC regime.
Moreover, at the same point of transition from the IC to the EC regime, the
$Wi$ data show that the polymer chain gradually becomes stretched from its
coiled state. Thus, we argue that the measured birefringence can be used as an
indicator in investigations of the coil-stretch transition.

###### Acknowledgements.

This work was supported by the Japan Society for the Promotion of Science,
KAKENHI Grant Nos. 17H01246, 20H00223, 19K23483, and 20K14646. We thank D.
Yamada and Y. Matsumoto for support with the experiments.

## Author declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

*

##  Appendix A  Xanthan gum solution

We believe that the difficulties in orienting polymer chains of polyethylene
oxide (PEO) arise because PEO is more flexible than a semirigid polymer such
as xanthan gum (XG) Pereira, Andrade, and Soares (2013). Thus, the fact that
the polymer chains of PEO were oriented by our measurement technique could
probe its generalizability. To investigate the ease of orientation of a
semirigid polymer, the same experiment as for PEO was carried out for XG
solutions with various concentrations. In this experiment (see Fig. 8),
solutions of XG (0.3–0.7 wt. % in ultrapure water, molecular weight
$M_{w}=2\times 10^{6}$, Sigma-Aldrich Co. LLC) were used. It should be noted
that XG solutions are non-Newtonian fluids that show shear-thinning behavior
(see Fig. 9) and induce birefringence due to their chain structureYevlampieva,
Pavlov, and Rjumtsev (1999); Meyer  _et al._ (1993); Chow and Fuller (1984).

Figure 8:  Temporal evolution of an extending liquid polymer filament shown by
raw images and the visualized phase retardation field of an XG solution with a
concentration of 0.5 wt. %. The white triangles indicates the position of the
minimum necking radius, and the white area ($10\times 10$ pixels indicated by
$\delta$) shows the measurement area for the retardation and orientation
angle. Figure 9:  Shear-thinning viscosity curve of XG solutions. Figure 10:
Temporal evolution of retardation $\delta$ and orientation angle $\varphi$
with radius ratio $R_{\mathrm{mid}}/R_{0}$ for XG solutions with
concentrations of (a) 0.3 wt. %, (b) 0.4 wt. %, (c) 0.5 wt. %, (d) 0.6 wt. %,
and (e) 0.7 wt. %.

Figure 10 shows the results for radius ratio, retardation, and orientation
angle over time for five different concentrations of XG. Within the EC regime,
it is known that the liquid filaments of common viscoelastic fluids are
cylindrical, while the temporal images of the XG solutions (Fig. 8) clearly
show that the filament decays with a curvature unlike that in the case of PEO
solutions. Thus, it is difficult to determine the presence of the EC regime.
However, since the radius ratio of the XG solutions obeys the scaling law of
the EC regime [Eq. ( 6)], we assume the existence of the EC regime here and
estimate the durations of the EC regime for XG solutions of different
concentrations as follows: 15–30 ms for 0.3 wt. %, 30–60 ms for 0.4 wt. %,
30–60 ms for 0.5 wt. %, 30–60 ms for 0.6 wt. %, and 50–100 ms for 0.7 wt. %.

The results for the orientation angle in Fig. 10 confirm that the orientation
angle $\varphi$ remains constant at $\pi/2$ except during the early stage
(part of the IC regime) and the final stage (part of the VC regime). Since XG
is a semirigid polymer while PEO is a flexible polymer, we initially expected
that it would be more difficult to orient the XG than the PEO. However, in our
experiment, both types of polymers were oriented along the extensional
direction, and thus it is confirmed that CaBER-DoS provides sufficient
extensional stress to orient the polymer chains. For the retardation of XG
solutions (Fig. 10), the curves are clearly different from those of the mixed
solutions of PEO and CNCs (Fig. 6). The curves for the XG solutions increase
in the early stage and then decrease gradually from the EC to the VC regime.

Figure 11:  Temporal evolution of birefringence $\delta_{n}$, Weissenberg
number $Wi$, and strain rate $\dot{\epsilon}_{\mathrm{mid}}$ for XG solutions
with concentrations of (a) 0.3 wt. %, (b) 0.4 wt. %, (c) 0.5 wt. %, (d) 0.6
wt. %, and (e) 0.7 wt. %.

Figure 11 show plots of birefringence $\delta_{n}$ vs normalized time
$\hat{t}$. It is confirmed that the slope of the birefringence for the XG
solutions is slightly different at the start of the EC regime, showing a
slight increase. However, we cannot determine whether the birefringence can be
an indicator for determining the presence of the EC regime, because it is
difficult to extract any significant trend at the transition from the IC to
the EC regime (this is evident from the data on $Wi$). In other words, the
birefringence results reflect the fact that the presence of the EC regime
cannot be determined from the temporal evolution image of the XG solution, as
mentioned above. We believe this this to be because XG is a naturally
extracted polymer, with a nonuniform molecular weight.

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