Supplementary Information for
Continuous sheath-free separation of particles based on shape in
viscoelastic fluids
Xinyu Lu,1 Lin Zhu,2 Ri-mao Hua,3 and Xiangchun Xuan1, *
1
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA
2
School of Engineering, Anhui Agricultural University, Hefei, Anhui 230036, China
3
School of Resource & Environment, Anhui Agricultural University, Hefei, Anhui 230036, China
*
To whom correspondence should be addressed. Email: xcxuan@clemson.edu. Fax: 864-656-7299.
1
1. Determination of the fluid properties in Table 1 in the main text
The zero-shear viscosities, 𝜂, of 500 ppm and 1000 ppm PEO solutions were obtained from the
experimental measurements of Rodd et al.1 That of 2000 ppm PEO was calculated using the
viscosity blending equation,2 with the reported experimental values of 1000 ppm and 3000 ppm.1
The overlap concentration, c*, of PEO solutions was calculated from the expression of
Graessley,3 c* = 0.77/[] = 858 ppm, where [] = 0.072Mw0.65 = 897 mL/g is the intrinsic
viscosity given by the Mark-Houwink relation with Mw = 2106 Da being the molecular weight
of PEO polymser.1 The Zimm relaxation time of PEO solutions was calculated from Zimm
theory4
𝜆𝑍𝑖𝑚𝑚 = 𝐹
[𝜂]𝑀𝑤 𝜂𝑠
(S1)
𝑁 𝐴 𝑘𝐵 𝑇
−3𝜈
where 𝐹 = ∑∞
= 0.463 was estimated from Remann Zeta function using a solvent quality
𝑖=1 𝑖
exponent, ν = 0.55,1 𝜂𝑠 = 1 mPas is the solvent viscosity, 𝑁𝐴 is the Avogadro constant, 𝑘𝐵 is the
Boltzmann constant, and T = 293.15 K is the fluid temperature. The effective relaxation times of
PEO solutions were then estimated according to the following forumla5
𝜆𝑒 = 18𝜆𝑍𝑖𝑚𝑚 (𝑐/𝑐 ∗ )0.65
(S2)
2. Control experiment of shape-based particle separation in water
(a2)
(a1)
Re = 1.11
(b2)
(b1)
200 µm
Peanuts
Spheres
900
800
700
600
500
400
300
200
100
0
(c)
Spheres
Peanuts
PDF
0
0.1 0.2 0.3
FIG. S1. Demonstration of shape-based separation of fluorescent spherical (bright) and plain
peanut-shaped (dark) particles in water through a 25 µm-deep microchannel at a flow rate of 150
µl/h: (a1) and (a2) snapshot images at the channel inlet and outlet, respectively; (b1) and (b2)
superimposed images of peanut and spherical particles, respectively, at the channel outlet; (c) the
plot of particle PDF at the channel outlet. The flow direction is from left to right.
3. Derivations of Eq. (1) and Eq. (2) in the main text
Assuming an Oldroyd-B model for the constitutive equation, the elastic lift force, 𝐅𝑒𝐿 ,
experienced by a particle in a viscoelastic fluid is given by,6,7
2
𝐅𝑒𝐿 ~𝑎3 ∇𝐍1 ~𝑎3 𝑊𝑖𝛾̇ 2
(S1)
where 𝐍1 is the first normal stress difference and 𝛾̇ = 2 𝑉 ⁄𝑤 is the fluid shear rate with 𝑉 being
the average fluid velocity. Substituting the definition of Wi in the main text into Eq. (S1) yields
2𝑉 3
2𝑄
3
𝑎 3
𝑄3
𝐅𝑒𝐿 ~𝑎3 𝜆𝑒 𝛾̇ 3 = 𝑎3 𝜆𝑒 ( 𝑤 ) = 𝑎3 𝜆𝑒 (𝑤2 ℎ) = 8 (𝑤) 𝜆𝑒 (𝑤ℎ)3 ~𝜆𝑒 (𝑎⁄𝑤 )3 𝑄 3
(S2)
The total inertial lift force, 𝐅𝑖𝐿 , scales as8
𝑄
2
𝑄2
𝐅𝑖𝐿 ~ 𝜌𝑉 2 𝑎4 ⁄𝑤 2 = 𝜌 (𝑤ℎ) 𝑎4 ⁄𝑤 2 = 𝜌 ℎ2 (𝑎⁄𝑤 )4 ~𝜌(𝑎⁄𝑤 )4 𝑄 2
(S3)
4. Flow rate effects on shape-based particle separation in different PEO solutions
(a)
PDF
Spheres
Peanuts
0.3
0.2
(b)
Peanuts
450
400
300
200
150
100
350
250
150
50
-50
-150
-450
-350
-250
450
350
250
150
50
-50
-150
0
-450
-350
-250
0.1
Spheres
0.4
PDF
0.3
0.2
450
400
300
200
150
100
350
250
150
50
-50
-150
-450
-350
-250
450
350
250
150
50
-50
-150
0
-450
-350
-250
0.1
FIG. S2. PDF plots for the flow effects on shape-based particle separation in 500 ppm (a) and
2000 ppm (b) PEO solutions through a 50 µm wide, 25 µm deep straight rectangular
microchannel.
5. Flow rate effects on shape-based particle separation in microchannels of different AR
3
50 µL/h 100 µL/h 150 µL/h 200 µL/h 300 µL/h
200 µm
Re = 0.13
Wi = 1.89
Re = 0.27
Wi = 3.78
Re = 0.40
Wi = 5.67
Re = 0.54
Wi = 7.56
Re = 0.80
Wi = 11.3
FIG. S3. Cropped superimposed particle images at the channel outlet for the flow rate effect on
shape-based separation of plain peanut (dark) and fluorescent spherical (bright) particles in 1000
ppm PEO solution through a 50 µm wide and 40 µm deep (i.e., AR = 1.25) straight rectangular
microchannel. The scale bar represents 200 µm.
50 µL/h 100 µL/h 150 µL/h 200 µL/h 300 µL/h
Re = 0.19
Re = 0.37
Re = 0.56
Re = 0.74
Re = 1.1
Wi = 5.04
Wi = 10.1
Wi = 15.1 Wi = 20.1
Wi = 30.2
Off-center distance (µm)
450
350
250
150
50
-50
-150
-250
-350
-450
0
0.2 0.4 0
0.2 0.4 0
0.2 0.4 0
0.2 0.4 0
0.2 0.4
FIG. S4. Flow rate effect (in terms of the Reynolds number, Re, and Weissenberg number, Wi)
on shape-based separation of plain peanut (dark) and fluorescent spherical (bright) particles in
1000 ppm PEO solution through a 50 µm wide and 15 µm deep (i.e., AR = 3.3) straight
rectangular microchannel: (top row) cropped superimposed particle images at the channel outlet;
(bottom row) plots of particle PDF at the channel outlet. The scale bar represents 200 µm.
References
4
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L. E. Rodd, T. P. Scott, D. V. Boger, J. J. Cooper-White, and G. H. McKinley, J. NonNewtonian Fluid Mech. 129, 1-22 (2005).
2
R.E. Maples, Petroleum Refinery Process Economics, 1993, Pennwell Books, Tulsa, Okla.
3
W. W. Graessley, Polymer 21, 258-262 (1980).
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5