1
Supplementary Information
2
1. Optical setup
3
4
FIG. S1. Optical system setup for flow quantification.
5
2. Mixing efficiency
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Mixing quality was estimated using the probability density function (PDF) of the 8-bits grayscale images, which
7
were captured by a high-resolution CCD camera (HiSense, MKII). The grayscale intensity of the recorded
8
images is assumed to be proportional to the actual dye concentration at that instant of recording. For example a
9
PDF with two prominent peaks that are far apart (one of the dyed fluid and the other of the non-dyed fluid)
10
means that the two fluids are not mixed. A single peak at the mid-range of concentration-axis (i.e. x-axis of the
11
plot) means good mixing. The concentration axis of raw PDF was later normalized to a range from 0.0 (i.e. an
12
undyed stream which is the original sidestream’s fluorescence intensity) to 1.0 (i.e. dyed stream which is the
13
original mainstream’s fluorescence intensity). The mixing efficiency eff was then calculated based on the
14
following definition1:
1
 Ci 1

  C i  C  P (C i ) 
C 0
  100%
 1  i


C




1
 eff
2
where Ci is the observed concentration on the normalized concentration scale, C is the concentration on a
3
normalized concentration scale for perfect mixing, and P(Ci) is the probability density function. In this
4
investigation, for an equal flow rate of the main and the side streams, C= 0.5. As a result, with this definition,
5
 eff = 0% indicates no mixing and  eff = 100% indicates perfect mixing.
(1)
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3. Rotational Rheometry
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Advanced Rheometric Expansion System (ARES), Rheometrics Scientific
9
The Advanced Rheometric Expansion System (ARES, Rheometrics scientific) is a rotational rheometer which is
10
capable of measuring a variety of rheological properties, such as shear flow curve (steady shear viscosity) and
11
normal stresses, dynamic properties and relaxation properties. This rheometer is equipped with a 100 FRTN1
12
transducer. The torque range is from 0.004 to 100 g-cm and the normal force range from 0.1 to 100 gmf. Several
13
measuring systems are available with ARES (e.g. cone-and-plate, parallel-plate and Couette geometries). The
14
temperature is bath controlled with a Polyscience (digital temperature controller) circulating bath.
15
The parallel plate measuring system (plate diameter = 50 mm and gap = 1 mm) was used in this investigation.
16
The rheological measurements of PEO fluids, in the semi-dilute regime (i.e. 1.0wt%, 0.8wt% and 0.6wt% PEO
17
in 55 wt% glycerol water), in steady and dynamic modes were carried out at constant temperature of 25 ºC with
18
a shear rate range of 0.1 ≤  ≤ 100 s-1 and frequency of 0.1 ≤ w ≤ 100 rad/s respectively. The steady shear
19
viscosities were determined using strain-controlled-steady sweep test and the dynamic shear viscosity was
20
measured using dynamic frequency sweep test. After loading the sample fluid and setting up the measuring
21
recipe, approximately 5 ~10 minutes were taken for the system to reach equilibrium before the start of each test.
22
Details regarding operation, maintenance and performance specification of the rheometer can be found in the
23
instrument manual (902-30026 REV A) and user guide (902-00107).
2
1
Contraves LS 40
2
The low shear Contraves 40 is a rotational concentric cylinder type rheometer which was mainly used for
3
measurements in the low shear rate domains, especially to get zero shear viscosity. The rheometer is equipped
4
with an inverted bob to eliminate end effects and is able to measure steady and dynamic properties using a small
5
sample size. In this investigation, the rheometer was used for zero shear characterization of dilute aqueous PEO
6
sample fluids (i.e. 0.1 wt% and 0.3 wt % PEO in water) at 25 ºC. However, the relaxation time of such
7
diluteness could not be determined with our facilities for these fluids.
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Typical results of rotational shearing flow measurements
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Typical results of shearing flow measurements on η and N1 for various concentrations of PEO in a mixture of
10
55 wt% glycerol and water were presented in FIG. S2. FIG. S2(d) shows that at low shear zero-shear plateaus,
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the viscosity increased with increasing concentration of PEO, and the shear rate at which shear thinning
12
appeared decreased. These are in good agreements with values reported in the literature. In the present
13
investigation, the zero-shear viscosity, ηo, was determined via a nonlinear regression of the three parameter
14
Carreau model. The evaluated values were different from the values reported in our earlier papers 2, 3 which were
15
fitted by a Ellis model. Better regression fit (1% improvement in fitting results) was obtained from the Carreau
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model.
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The Carreau fluid model was first proposed by Pierre Carreau.4 The full Carreau Model is typically expressed in
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terms of four parameters by the following equation
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2
    (o   ) 1     


n 1
2
(2)
20
21
where, ηo is the zero shear (Newtonian) viscosity, η∞ is the limiting viscosity at high shear rates and λ is a time
22
constant. The high shear viscosity is generally associated with a breakdown of the fluid and is frequently taken
23
as zero.5 Therefore, equation (2) reduces to
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2
  o 1     


25
At low shear rate, 
26
behaves as a power-law fluid.6
n 1
2
(3)
1/  , a Carreau fluid behaves as a Newtonian fluid and at high shear rate, 
1/  ,
3
1
2
3
FIG. S2. Typical rotational shear measurements for various PEO sample fluids in semi-dilute
concentration regime, (a)-(c) oscillatory shear and (d) steady shear. F is denoted as fluorescence additives.
4
5
4. Extensional Rheometry
6
The standard rheological measurements (by rotational rheometer) and the theoretical calculations are inadequate
7
or insufficient for providing a reliable measurement of the relaxation time for the present study. In the
8
experiments of multiple streams flow through abrupt contraction geometry, the viscoelastic sample at the core-
9
stream (or center stream) of the microdevice underwent strong extensional stretching exhibiting nearly pure
10
elongational straining. As such extensional strain measurement was desired. Unlike the typical shear flow
11
measurement, the polymer coils are stretched and not sheared during the measurement and the time required for
12
the polymer coils to relax to the equilibrium state can be expected to be very different. The Capillary Break-up
13
Extensional Rheometer (CaBER) provides a tool to investigate the characteristic timescale of the fluid in a
14
string extensional flow as a function of polymer concentrations.
15
4
1
In the present study, extensional rheology tests were performed on HAAKE CaBER-1 (Thermo Haake GmbH,
2
Karlsruhe, Germany) with a system response of 10 ms. Measurable shear viscosity ranges from 10 to 106 mPas.
3
Typical results of extensional strain measurements for various concentrations of PEO in a mixture of 55 wt%
4
glycerol and water were illustrated in FIG. S3. Due to the limitation of the equipment, the extensional
5
rheological properties of sample fluids in the dilute regime, i.e. 0.1% PW and 0.3% PW, could not be
6
characterized. Therefore, its values were taken from the measurements by Rodd et al..7 As the raw material used
7
and measured viscosity (using LS 40) was in good agreement with Rodd et al. [Ref 6], these values were hence
8
taken to be acceptable for later analysis.
9
10
The CaBER-1 rheometer was equipped with two 6 mm (2  Ro) circular parallel plates. To minimize the
11
influence of gravity and shear flow during the early stages of stretch, the plates were set to an initial gap of h o =
12
3 mm, resulting in an initial aspect ratio o = 0.5 (ho/2Ro). Fluid samples were carefully loaded between the
13
plates using a disposable pipette to ensure the absence of trapped air and contamination. The upper plate was
14
raised suddenly to a pre-set height of 10.43 mm (factory pre-set height) to create a filament. Laser beam aiming
15
at the middle point of the filament measured the changes in diameter. With this experimental arrangement, the
16
extensional flow of the filament was under the influence of the surface tension acting on the fluid surface. 8, 9
5
1
2
3
4
FIG. S3. (a) Typical plot of normalized filament diameter versus time -- Semi-logarithmic plot of necking
for three sample fluids, 1wt% PGW, 0.8wt% PGW and 0.6wt% PGW. (b) Typical plot of apparent
extensional viscosity versus strain. F is denoted as fluorescence additives.
6
1
Figure S3
2
3
4
5
6
FIG. S4. Remaining degree of relaxation (rel)out (at 1.5 mm downstream of a contraction) versus De main.
When (rel)out  unity, sample fluids experienced no relaxation and vice versa. Data were calculated
based on flow cases of 1wt% PGW with 0.1wt% PW (Deratio = 5.9) and 0.8wt% PGW with 0.3wt% PW
(Deratio = 2.2).
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Reference
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2.
3.
4.
5.
6.
7.
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R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids 1. (John Wiley &
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L. E. Rodd, T. P. Scott, D. V. Boger, J. J. Cooper-White and G. H. McKinley, Journal of NonNewtonian Fluid Mechanics 129, 1-22 (2005).
A. V. Bazilevsky, V. M. Entov and A. N. Rozhkov, Proceedings of the Golden Jubilee meeting
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