Supporting Information

Pressure Driven Water Flow Through Hydrophilic Alumina Nanomembranes

Anil Koklu, Jianing Li, Sevinc Sengor and, Ali Beskok*

Southern Methodist University,

Dallas, TX 75205, USA

Section S-1: Measurement of Average Pore Radius and Porosity

Section S-2: Zeta Potential Measurements

The “used alumina membrane” was grinded for 10 min in a Wig-L-Bug (Crescent/Rinn Dental

Mfg.) mechanical shaker (Kagawa et al. 2006; Roden and Ehleringer 2007). The pulverized

alumina particles were used for zeta-potential measurements. The instrument used in this study

is a Zetasizer Nano ZS (Malvern Instruments, UK) with a 633 nm red laser, and it is capable of

both particle size analysis (using dynamic light scattering as the basic principle of operation)

and zeta-potential measurements. The setup was calibrated using Malvern’s zeta-potential

transfer standard (DTS1230) of ∼300 nm polystyrene latex in a pH 9.2 buffer. The size of

grained alumina particles was measured approximately 1 µm, which satisfies the condition for

zeta potential measurements using Zetasizer Nano ZS (10nm - 10µm). The micro-particles

suspended solution was pipetted to a folded capillary cell (DTS1060). A sufficient sample

volume was used to completely cover the electrodes of the cell. The sample was injected slowly

to avoid air bubbles in the cell. The cell was placed into the Zetasizer and equilibrated at 20℃

(room temperature) for 2 min prior to the particle size measurements. The measurement

b)a)

Fig. S1. Binary images of 40nm pore size alumina membrane (a) Schematics of 4

measurements used to obtain an average diameter for each pore.

temperature was set and maintained by the Peltier elements in the sample holder of the

instrument. The corresponding zeta-potential measurements were taken immediately after the

particle size measurements. Malvern Instruments Dispersion Technology software (Version 4.0)

was used to control and analyze all data from the instrument. For the zeta-potential

measurements, the instrument uses laser Doppler electrophoresis to measure the net velocity of

the nanoparticles. The procedure is explained in (Tantra et al. 2010).

The NaCl electrolyte solutions were prepared from high purity reagents. Hydroiodic (HI) acid

solutions were prepared using an ion exchange column. All the solutions were prepared on

freshly De-ionized (DI) water (σ=5.5×10−6S/m). The zeta-potential measurements performed

for four different NaCl concentration10−2 ,10−4 ,6.61×10−5∧10−5mol/L.

Fig. S2. Zeta-potential vs pH relationships for the NaCl/alumina system. Red line with stars

represents the medium used for current nano-flow experiments.

Section 3: De-Aired Water in the Test

The water was de-aired to ensure a bubble free environment and to eliminate dissolving gases

on the surface of alumina membranes, such as CO2. The water inside the tank was vacuumed

for more than 72 hours to remove the dissolved gases. At beginning of the flow test, air

pressure was applied from the top part of the tank to drive the water through PVC the tubes.

Thus, air starts to dissolve into the de-aired water. The rate of dissolved air transport is

controlled by convection and diffusion effects. Due to the large talk area and low flowrates

convection of water though the tank is smaller than 1 mm, which is negligible for the given size

of the tank. Therefore, only mass diffusion is considered.

Length of the output pipe from the tank to the membrane holder input value is 5 m, which is a

standard for flow experiments (Standard Test Methods for Measurement of Hydraulic

Conductivity of Saturated Porous Materials Using a Flexible Wall Permeameter – ASTM D5084-

00). The maximum water mass was measured 5 g, which is about 5 ml of liquid water volume

(40nm alumina membrane under 0.41 MPa). This much mass transport approximately equals to

35 cm length of water inside the PVC tube. Based on the solution of Fick’s equation, the

concentration of a solute at a distance from the source after a certain time period of initial

injection is given by

C ( x ,t )=C0×erfcx

2(D×t)0.5

where, x=6m, the distance from water/air interface in the tank to the membrane holder;

C0=8gCO2

1kg water, theCO2 solubility of at 20℃ under 0.41 MPa, t=5days (432000 seconds), the

longest test period, and D=1.92×10−9m2/s, diffusion coefficient of dissolved CO2 at 0.41

MPa.

Therefore, dissolved CO2 concentration at the membrane holder after 5 days is

CCO2 (5m ,5days )≈0g /kg

Concentration of other gas components (N 2 and O2) are even less than the concentration of

CO2 under the same condition, due to their less solubility in water.

Oxygen concentration of de-aired water was also tested before the experiments by dissolved

oxygen (DO) meter, which was less than 0.5mg /Lat the membrane holder input. It is a direct

proof that the water was properly de-aired.

Section S-4: Extended Bernoulli Analysis

The extended Bernoulli equation was applied between the pressure gauges. It is expressed as

( piρ +βV i2

2+gz i)−( p0ρ +β

V 02

2+g z0)=hl−hp

where pi and p0 are, the pressures in the inlet and outlet; V iand V 0 are the averaged fluid

velocities in the inlet and outlet; β is the kinetic energy coefficient; z i and z0 are the fluid heights

in the inlet and outlet; hl and hp are the head loss along the entire system and pump head (

hp=0), ρ is the fluid density, respectively. The velocities of inlet and outlet are assumed same

(V i=V 0 ) . The fluid height difference between the inlet and outlet is considered as zero

(z i−z0=0). Therefore, the equation is simplified to

Δ pρ

=p iρ−p0ρ

=hl−hp

The head loss along the entire system contains two parts: major head losses arising from the

viscous effects, hmajor, and minor head losses due to the entrance, area change and exit in the

entire system,hminor. The major head loss is written as;

hmajor=64ℜt

LtD t

V t2

2+ 64ℜh

Lh

Dh

V h2

2

where ℜt and ℜh are the Reynolds numbers in the tube and the membrane holder; Lt(100mm)

and Lh(40mm) are the lengths of the tube and the membrane holder; Dt (2.78mm) and

Dh(20mm) are the diameters of the tube and the membrane holder; V tandV h are the fluid

velocities in the tube and the membrane holder. The minor head loss in the entire system is

given as

hminor=(k i+ke+kc +k0 )V t2

2+(khm+kmh)

V m2

2

where, k i=0.2, k e=2, k hm=0.5, kmh=2, k c=0.5 and k 0=0.2 are the loss coefficients of entrance

from the 90o shoulder to the tube, expansion from the tube to the membrane holder, entrance

from the membrane holder to nano-pores, exit from the nano-pores to the membrane holder,

contraction from the membrane holder to the tube, and exit from the tube to the 90o shoulder

tube (Munson et al. 1990) . Due to mass continuity, the flow rate through the system is a

constant, hence V h=D t2V t /D h

2 and V m=Dt2V t /ϕDm

2 with ϕ as the porosity of the membrane and

Dm as the wetted diameter of the membrane.

pmr−pml=Δ p−Δ ploss

∆ ploss=32 μ LtV t

Dt2 +

32 μ Dt2LhV t

Dh4 + (k i+k e+kc+k 0 )

ρV t2

2+(khm+kmh )

ρDt4V t

2

ϕ2Dm4 2

where, pmr and pml are the pressures on the right hand side (RHS) and left hand side (LHS) of

the membrane. The volumetric flow rate induced by the pressure gradient across the membrane

is given;

Q=nπD p

4( pmr−pml)128μL

=ϕπ Dm

2 D p2 ( pmr−pml)128 μL

where n is the number of pores (n=ϕDm2 /Dp

2), D p is the diameter of nanopore andL is the

thickness of the membrane. Pressure losses are given in the Table S1

Table. S1. Pressure loss for 40 nm pore size nano-membrane.

Dp (MPa) Qexp(m3/ s) V t ¿) hmajor (m¿¿2¿¿ s¿¿2)¿¿¿hminor (m

2/s¿¿2)¿ Dploss(MPa)

0.41 5.81×10−9 9.57×10−4 3.62×10−4 1.69×10−6 3.6×10−7

0.38 5.29×10−9 8.72×10−4 3.47×10−4 1.34×10−6 3.4×10−7

0.34 4.64×10−9 7.66×10−4 3.10×10−4 1.06×10−6 3.1×10−7

0.31 4.24×10−9 6.99×10−4 2.79×10−4 0.88×10−6 2.7×10−7

0.28 3.66×10−9 6.04×10−4 2.51×10−4 0.69×10−6 2.5×10−7

Table. S1. Pressure loss for 20 nm pore size nano-membrane.

Dp (MPa) Qexp(m3/ s) V t ¿) hmajor (m¿¿2¿¿ s¿¿2)¿¿¿hminor (m

2/s¿¿2)¿ Dploss(MPa)

0.41 1.94×10−10 3.2×10−5 1.22×10−5 1.68×10−9 1.2×10−8

0.38 1.85×10−10 3.05×10−5 1.16×10−5 1.53×10−9 1.1×10−8

0.34 1.68×10−10 2.77×10−5 1.06×10−5 1.23×10−9 1.0×10−8

0.31 1.51×10−10 2.49×10−5 0.95×10−5 0.99×10−9 0.9×10−8

0.28 1.37×10−10 2.26×10−5 0.86×10−5 0.82×10−9 0.8×10−8

Table. S1. Pressure loss for 10 nm pore size nano-membrane.

Dp (MPa) Qexp(m3/ s) V t ¿) hmajor (m¿¿2¿¿ s¿¿2)¿¿¿hminor (m

2/s¿¿2)¿ Dploss(MPa)

0.41 7.46×10−12 1.23×10−6 4.68×10−7 2.33×10−12 4.7×10−10

0.38 6.82×10−12 1.13×10−6 4.28×10−7 1.83×10−12 4.3×10−10

0.34 6.32×10−12 1.04×10−6 3.97×10−7 1.58×10−12 4.0×10−10

0.31 5.69×10−12 9.38×10−6 3.58×10−7 1.28×10−12 3.6×10−10

0.28 5.02×10−12 8.28×10−6 3.15×10−7 0.99×10−12 3.2×10−10

Section 5: Analytical Model for the Streaming Potential Effects

Bulk conductivity of the distilled water (σ=10−4S /m) was measured using a conductivity meter

(Con11, Oakton) just after the sample was taken from the tube before it passed through the

nano-pores. The ion concentration of distilled water was approximately calculated from (Graf

and Kappl 2006);

σ=A√ iC

where A is a constant and i is the van’t Hoff factor, which are functions of temperature, and C is

the ionic concentration. Using sodium chloride (NaCl) system as a surrogate we estimated the

ion concentration in distilled water. Using A√ i ≈1.23×10−4 [S /(m√mol/L) ]at room temperature

(20℃) (Burdge and Overby 2012), ionic concentration of distilled water C=6.61×10−5mol /L,

which is consistent with the literature (Jow and Wagner 1979; Sata et al. 2000; Yates et al.

1974).

The zeta potential (ζ ) on the alumina surface is found 15mV at pH 7.4. The EDL thickness is

calculated using Debye-Hückel approximation for a symmetric electrolyte solution as,

λ=√ε0 εr kBT /2n∞e2 z2=37.64nm

where ε 0 is the permittivity of vacuum, ε r is the relative permittivity of water, k B is the Boltzmann

constant, T is the temperature, n∞ is the bulk ionic number density, e is the charge of an

electron, z is the valency of ions. The ratio of EDL thickness to the effective radius of

membranes (r1=3.75nm, r2=7.14nm and r3=15.02nm) are λ1=λ1/r1=¿10.04 ¿,

λ2=λ2 /r2=¿5.27 ¿ and λ3=λ3 /r3=¿2.51 ,¿ respectively.

The streaming potential will trigger an electroosmotic transport that will oppose the pressure-

driven transport. Consequently, the flow rate is lower than the flow rate that would be achieved

in case of a pure pressure-driven transport. Following (Das et al. 2013) one can write,

QPDF+SP

QPDF=1−

8 ζ Ex

Eref [ 12− λ I 1(1/ λ)I 0(1/ λ) ]

where,ζ=(zeζ )/kBT , E x is the induced electric field due to the streaming potential, Eref (

Eref=−( a2 ze2 ϵ 0 ϵ r kBT

× dPdx )¿ is the reference electric field used for nondimensionalization, I 0 ,∧I1

are the modified Bessel functions of the first kind of order 0 and 1, respectively. As shown in

Table S2, QPDF+SP

QPDF≅ 1 for each nano-membrane. Therefore, streaming potential effects on

flowrate can be neglected.

Table. S2. Bessel functions of the first kind of order 0, 1 and 2.

Radius of pore I 0 I 1 I 2 QPDF+SP

QPDF

M 1 0.99750 0.04993 0.00124 1

M 2 0.99002 0.09950 0.00498 1

M 3 0.96039 0.19602 0.01973 1

Section S-6: Contact Angle Measurements

b)

a)

Fig.S3. (a) Schematic of the electric potential distribution in the overlapped EDL, where ψ0 is

the electrical potential at the surface, ψr is the potential at the middle plane, 2 r is the

separation distance and λ represents the EDL thickness. (b) Schematic of the single nano-pore

with positively charged wall, and the EDL forming at the interior of the wall. Direction of the

pressure-driven flow and the electroosmotic transport flow are also shown in the figure.

The mechanism of hydrophilicity was at first revealed by Wenzel (Wenzel 1936) and the contact

angle (θc) is used to quantify the hydrophilicity of surfaces. The θcis the angle that a small drop

of liquid makes as it meets the surface or interface of another phase (Fig. S4). θcis measured

using an automated device (Surface Analyst, SA 1001, Brighton Technologies) that places a

droplet of water (100 μl) on membranes at room temperature (20℃). The measurement is

performed for used and unused membranes. For θcreported, 10 readings from different parts of

the film surface are averaged.

References

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Fig. S4. Pictures of water droplets on used and new 20 nm pore radiusγ−alumina

membranes.

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