Supporting Information
Transport of Multicomponent, Multivalent Electrolyte Solutions
across Nanocapillaries
Kaushik K. Rangharajan, Marie Fuest, A.T. Conlisk and Shaurya Prakash*
Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210
USA.
*E-mail: [email protected]
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UV/VIS calibration
As discussed in the main manuscript, UV/VIS measurements were conducted to determine the permeate content for
methylene blue. In order to determine the amount of methylene blue in the permeate solution, a calibration curve
was developed (Bellman 2011) and a brief description of the methods is presented next. The Beer-Lambert law
provides a connection between the measured absorbance (A) and the concentration (c) for fixed molar absorptivity
(ε) and path length (l),
A = εcl
Solutions of known potassium buffer concentration (Cbulk, in mM) were prepared containing methylene blue (MB)
of varying concentration (c in mM) and the absorbance was measured using Thermo-Scientific Evolution 300
UV/VIS at 665 nm. A linear fit was applied to this data for each of the potassium phosphate buffer concentrations to
generate a detailed calibration curve for absorbance vs. concentration as reported by our group previously (Bellman
2011).
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Fig. S1. Centerline (r = 0 nm) and radial (z = 500 nm) concentration distribution of K+, H2PO4–, HPO42–, and MB+
when the buffer concentration (Cbulk) was 10 mM (a, b), 1 mM (c, d) 0.2 mM (e, f). The Debye length is greater than
the pore radius (5 nm) for a buffer concentration of 0.2 mM and 1 mM thereby giving rise to overlapping or
interacting EDLs, resulting in screening of co-ions (K+) from the nanocapillary. Though the concentration of HPO 42–
is smaller compared to H2PO4– in the source and permeate reservoirs, the concentration of HPO42– is almost equal to
H2PO4– inside the nanocapillary for buffer concentration of 1 mM. With decrease in buffer concentration to 0.2 mM,
concentration of HPO42– is almost double that of H2PO4– inside the nanocapillary. Since the nanocapillary is
positively charged, the concentration of the co-ion is lower near the walls (b, d, and f) in comparison to the center of
the channel. Note: Both the source and permeate reservoir are at 0 V or the rest state. The electroneutrality condition
R
inside the nanocapillary states,   zi ci dr  
0
s
F
, where R = 5 nm. As there is a concentration gradient of MB +
inside the nanocapillary, the concentration of the buffer ions also vary along the nanocapillary in order to satisfy
electroneutrality. Similarly, electroneutrality is enforced in both the source (MB + concentration = 0.14 mM) and
w
Rres
permeate reservoir (MB+ concentration is negligible) where
  z c dr   F
i i
with Rres = 100 nm.
0
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Fig. S2. (a) Plot of centerline built-in potential barrier also known as the Donnan potential, arising due to surface
charge discontinuity at nanocapillary-reservoir junction, plotted as function of the buffer concentration (Cbulk) when
the reservoir-nanocapillary system is in rest state. The enrichment of counter-ions and depletion of co-ions inside the
nanocapillary is more predominant at lower buffer concentrations of 0.2 mM and 1mM (Fig. S1), thus leading to a
higher Donnan potential. Rise in the axial potential inside the nanocapillary (Donnan potential) leads to changes in
the electric field at the nanocapillary-reservoir junction which is plotted for (b) buffer concentration = 0.2 mM (c)
buffer concentration = 1 mM (d) buffer concentration = 10 mM at rest state. The value of Ez is –1.7 MV/m at the
source reservoir-nanocapillary junction (negative sign arising because Ez = d  /dz ) for Cbulk = 0.2 mM,
–0.7 MV/m for Cbulk = 1 mM, and –0.12 MV/m when Cbulk = 10 mM. Note, both the source and permeate reservoir
are at 0 V or the rest state.
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Fig. S3. Variation of electric field in the reservoir-nanocapillary system plotted as a function of applied potential for
buffer concentration of 1 mM. As the Donnan potential (Fig. S2a) is added to the applied potential at the source
reservoir/nanocapillary junction (Fig. 3a, main manuscript) a sharp variation in electric field is observed. However,
inside the nanocapillary, the electric field is constant for the applied potential range of 0.1 – 0.75 V and is seen to
exhibit a 1st order scaling with applied potential as shown in the inset which is consistent with previous reports
(Vlassiouk et al. 2008)
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Fig. S4. The figure show variation of the volumetric space charge density ρs (equation 4 in the main manuscript) in
(a) radial and (b) axial direction at the rest state. From Fig. S4a, ρs is negative for all buffer concentrations, a
consequence of positively charged nanocapillary (Jin et al. 2007). However, the magnitude of ρs is less negative
(-0.09 C/cm3) (Fig. S4b) when the buffer concentration was 10 mM due to poor screening of co-ions by the
nanocapillary. Due to depletion of co-ions near the nanocapillary wall, ρs is minimum (–0.19 C/cm3) near the walls
and increases radially inwards as seen in Fig. S4a when Cbulk = 10 mM. The space charge density for Cbulk = 1 mM
and 0.2 mM was ~ –0.13 C/cm3 implying that nanocapillary is operating under surface charged governed regime.
Note, both the source and permeate reservoir are at 0 V or the rest state.
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Fig. S5. (a) Streamline plot at reservoir/nanocapillary interface for a reservoir surface charge of -0.1 mC/m2, plotted
for a bulk concentration of 10 mM. Color of streamlines indicates axial flow velocity in µm/s. (b) Streamlines plot
inside the source reservoir that show flow reversal nearby bottom wall. (c) Streamline plot at reservoir/nanocapillary
interface for a reservoir surface charge of -0.1 mC/m2, plotted for a bulk concentration of 0.2 mM. (d) Streamlines
plot inside the source reservoir that show evolution of vortices with arrow plots showing clear flow reversal. Such
flow reversal were not observed inside the nanocapillary. Data plotted for an applied inlet potential of 0.75 V.
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Fig. S6. Radial variation of axial velocity Uz (µm/s), as a function of inlet potential for (a) 0.2 mM, (b) 1 mM, and
(c) 10 mM buffer concentration. The maximum centerline (r = 0) velocity was found to increase with increase in
inlet potential and was found to be similar and invariant of the buffer concentration. It is important to note that the
bulk fluid moves due to the net induced body force, –ρs  Φ (equation 8 in the main manuscript) where ρs is the
volumetric space charge density and –  Φ is the electric field inside the nanocapillary. The velocity profiles follow
expected trends as reported in other modeling studies (Bhattacharyya et al. 2005, Jin et al. 2011, Conlisk 2013)
providing further validation of the numerical approach used here. Direction of bulk fluid is from permeate (–) to
source (+) as seen in Fig. 1, main manuscript.
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Table S1. Table of Parameters used in Numerical Model
Parameter
Assigned Value
Unit
K+ (+1) (Jin et al. 2007)
1.96 x 10–9
m2/s
MB+ (+1)(Miložič et al. 2014)
0.57 x 10–9
m2/s
HPO42– (–2) (Jin et al. 2007)
0.70 x 10–9
m2/s
H2PO4– (–1) (Jin et al. 2007)
0.87 x 10–9
m2/s
Temperature (T)
293
K
Vacuum permittivity ( o )
8.85 x 10–12
F/m
Water Permittivity ( w )
80
dimensionless
Faraday Constant
96485.3
C/mol
Diffusion Coefficient (charge)
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