Supplementary Information
Concentration cascade of leading electrolyte using bidirectional isotachophoresis
Supreet S. Bahga and Juan G. Santiago
S1. Protocol for visualizing increase in analyte zone lengths in bidirectional ITP
In the experiments, we used a cross-channel chip architecture, but we only used one straight
channel section and two wells for ITP. The figure below summarizes the protocol.
Figure S1: Protocol for visualizing increase in analyte zone lengths due to shock interaction in
bidirectional ITP. For visualization experiments on Caliper NS 12A chips: (a) we injected LE+/LEmixture by applying vacuum on W well. We then emptied the E and W wells and (b) filled the E well
with TE+/LE- mixture, and the W well with a mixture of LE+, TE-, analytes (S1-, S2-) and nonfocusing tracer (NFT). (c) We then electrokinetically injected S1- and S2- ions by applying voltage
between E and W wells. (d) Next, we turned off the voltage, emptied the W well and filled it with a
mixture of LE+, TE- and NFT. This step ensured injection of finite amounts of S1- and S2- ions in
the channel. (e) We then applied voltage between E and W wells and imaged the focused analyte
zones, prior to the shock interaction. (f) We again imaged the focused analyte zones after the
interaction of cationic and anionic shock waves.
S2. Calculation of species concentrations in bidirectional ITP zones using diffusion-free model
In the current paper, we used a steady state solver based on a diffusion-free model of ITP for
quick estimates of species concentrations and increase in analyte zone lengths in bidirectional ITP.
Here we describe the diffusion-free model and the numerical procedure to solve for steady state
concentrations of analytes, before and after the shock interaction. We note in general ITP zones are
non-stationary, except when electromigration is countered by hydrodynamic flow. Here by steady
state we imply the state in which concentrations of analyte zones do not change with time.
Our steady state solver takes into account acid base equilibrium and the effect of ionic
strength on species mobilities and ionic activity. We benchmark our model with detailed onedimensional simulations using SPRESSO [1-3].
S2.1 Diffusion-free model
We follow the notation of Bercovici et al. [1] and solve for total (analytical) concentrations,
ci , of species i. Total concentration of species i is defined as the sum total of concentrations of all its
ionization states, ci , z ,
pi
ci  ci , z , i  1, , N .
(1)
z  ni
Here z denotes the valence, and ni and pi denote the minimum and maximum ionization states of
species i. To derive the diffusion-free model, we begin with electromigration-diffusion transport
equation for species concentrations ci ,
ci
 · i ci     Di ci   , i  1,, N ,
(2)
t
where i and Di denote the effective mobility and molecular diffusivity of species, i, and  is
the electric field. We here define effective mobility as a signed quantity i  ui / E where ui is
species drift velocity and E is local electric field. Note that, the effective mobility of species i is the
average of the mobilities of its ionization states  i , z  , weighted by the degree of ionization
c
i,z
/ ci  ,
pi
i 

c
i,z i,z
z  ni
pi
c
(3)
i,z
z  ni
See Bercovici et al. [1] for more details on calculating ci , z and i using chemical equilibrium and
electroneutrality. Integrating Eq. (2) over the channel cross-section, neglecting diffusion, and using
  j /  we obtain,
N
N pi
ci
 c 
 j  i i   0,    i ci ,  i   zi , z ci , z F / ci
(4)
t
x   
i 1 z  ni
i 1
where  is the electrical conductivity and j is current density through the channel. Integrating
Eq. (4) over a small element ( x, x  dx)  (t , t  dt ) around an ITP shock wave we obtain the
Hugoniot jump conditions [4] across that shock.
t  dt
t  dx
t
x
 
 ci
   c 
dx  
 j  i i  dxdt  0,
(ci  ci ) 

x    
dt
 t
   c   c  
j  i  i  i i  ,
 
 
(5)
where  and  denote the evaluation of a property behind and in front of a shock, respectively.
Solving for these jump conditions across each shock and for each species, we obtain the
concentrations and shock speeds  dx / dt  .
S2.2 Numerical solver for diffusion-free model
Here we consider a bidirectional ITP experiment similar to that shown in Figure S1. In this
case, finite amounts of anionic sample species, S1- and S2-, focus between anionic leading and
trailing electrolyte ions (LE- and TE-, respectively). Simultaneously, a cationic ITP shock forms
between cationic leading electrolyte (LE+) and trailing electrolyte (TE+) ions. The anionic and
cationic shocks approach each other and later interact.
We solve the Hugoniot conditions across ITP shocks (given by Eq. (5)) iteratively to obtain
the analyte zone concentrations before and after the shock interaction. The ratio of initial to final
concentrations of an analyte zone then gives the gain in its zone length due to shock interaction. To
obtain the initial and final analyte zone concentrations, we perform the following steps:
a) Anionic ITP before shock interaction: We first solve for the concentrations of the sample
analytes focused behind the LE-/LE+ zone (see Figure S1).
b) Cationic ITP before shock interaction: We then solve for the concentration of LE- ions in the
cationic TE (TE+/LE-) zone. Note that before shock interaction LE- is the background
counterion for cationic ITP.
c) Anionic ITP after shock interaction: After the shock interaction TE+/LE- mixture replaces
LE+/LE- as the leading electrolyte for anionic ITP. Using the species concentrations in
TE+/LE- zone, computed in step (b), we solve for the readjusted analyte zone concentrations
behind the TE+/LE- zone.
We here describe an iteration scheme to perform Step (a) for calculating species
concentrations in a given anionic analyte zone prior to the shock interaction. Calculation of species
concentrations in other zones follows analogously. For computing the analyte zone composition the
unknowns are: (i) analyte concentration in analyte zone, cS  , a , (ii) LE+ concentration in analyte
zone, cL  ,a , and (iii) pH (or hydronium ion concentration, cH  , a ) of analyte zone. In our notation,
first subscripts S-, L+ and H+ denote sample, cationic LE and hydronium ions, respectively. The
second subscripts l and a denote LE-/LE+ and focused analyte zones, respectively.
Since analyte ions are not present in the LE- zone and LE- ions are not present in the analyte
zone, Eq. (5) for S- and LE- ions yields,
 S  , a  L  ,l

,   i ci .
a
l
i
(6)
which is the usual ITP condition. Next, using the above relation in Hugoniot condition for LE+ ions
gives an explicit relation between the concentrations of LE+ in LE+/LE- and analyte zone,
 

  
c L  , a  1  L  , a   cL  , l  1  L  , l  .
(7)
 

  
S ,a 
L  ,l 


The remaining equation comes from the electroneutrality of analyte zone. Equations (6) and (7) along
with electroneutrality condition form a coupled set of nonlinear algebraic equations in terms of
variables, cS  ,a , cL  ,a , and pH (or hydronium ion concentration, cH  , a ). The nonlinearity occurs due
to strong dependence of effective mobilities on cH  , a .
We used the following iteration scheme to solve the above nonlinear equations:
Step 1: Knowing the concentrations of S- and LE+ in analyte zone at the end of nth iteration
( cSn ,a and cLn  ,a , respectively) we use acid-base equilibrium and electroneutrality to obtain the pH of
analyte zone. (For the first iteration we choose cS0 ,a  cL ,l and cL0 ,a  cL ,l ).
Step2: We then calculate the effective mobilities of species using the pH obtained in Step 1, and
update  a and cL  , a using Eqs. (6) and (7),
  Ln ,l
Sn,a n
n 1
n
 n  l , cL  ,a  cL  ,l 1  n
 
 L  ,l
L  ,l

1
  Ln ,a 
(8)

1
.

  n 
S ,a 

Step 3: We then update the concentration of analyte, cSn1,a , using cLn1,a and cHn  ,a and the expression
n 1
a


for analyte zone conductivity  an 1 .
We repeat the above steps until solution converges. In practice, we found that updating the species
concentrations using cin 1   cin 1  (1   )cin at every iteration improves the convergence rate, where
ci n 1 is the concentration of species i updated using Eqs. (8). Here  is a predefined scalar
parameter. Choosing 0    1can stabilize otherwise unstable iteration, while   1 can accelerate
the convergence of otherwise stable iteration. For our calculations we used   0.9 . We included
ionic strength effects by using the Onsager-Fuoss model [5] and the Debye-Huckel [6] model to
correct species mobilities and dissociation constants, respectively, in Steps 1 and 2. Numerical
implementation of the Onsager-Fuoss model and the Debye-Huckel model is discussed
elsewhere [3].
S2.3 Comparison of diffusion-free model with SPRESSO
We validated our steady state solver for the diffusion-free model with detailed onedimensional simulations using SPRESSO. We simulated a bidirectional ITP system similar to that
shown in Figure 2 of the paper. For our simulations, we used 100 mM Mops as LE-, 20 mM Taurine
as TE-, Imidazole as LE+ (varying between 200-300 mM), 100 mM Bistris as TE+, and Hepes and
Tricine as the model analytes. We varied the ratio of LE+ to LE- concentration by keeping LEconcentration fixed at 100 mM and varying LE+ concentration from 200 mM to 300 mM. The
relative error between the species concentrations predicted by the steady state solver and SPRESSO
was negligible (on the order of 0.001%). This is expected as steady state solver and SPRESSO solve
respectively the integral and the differential forms of same species-transport equations. In Figure S2,
we show the gain in zone length due shock interaction predicted by the diffusion-free model and
SPRESSO. The diffusion-free model and SPRESSO predict similar gains in zone length over wide
range of LE+ to LE- concentration ratios, with and without ionic strength corrections.
S3. Comparison of analytical model with SPRESSO and diffusion-free model
In Section 3.2 of the manuscript, we presented an analytical model to predict the increase in
zone length due to shock interaction in bidirectional ITP. The analytical model is valid for univalent
ions at safe pH conditions (5<pH<9) and neglects the effect of ionic strength on species mobilities. In
Figure S2, we compare the gain in zone length predicted by the analytical model along with
predictions using diffusion-free model and SPRESSO. Predictions using the analytical model agree
well with those from diffusion-free model and SPRESSO when ionic-strength effects are neglected in
both of the latter models. This shows that the assumption of safe pH in our model is valid for the
conditions of our simulations (and experiments discussed in Section 3.5 of the paper). However, the
gains in zone length calculated using the analytical model do not agree with that obtained using
SPRESSO and diffusion-free model with ionic strength corrections for species mobilities. This
supports our conclusion that the error in prediction of gains in zone length by the analytical model (in
Section 3.5 of the paper) is solely due to our assumption of negligible dependence of ionic strength
on species mobilities. Most importantly, the predictions resulting from the diffusion-free model
including ionic strength effects and the SPRESSO predictions including ionic strength effects agree
nearly exactly; successfully benchmarking the diffusion-free model versus SPRESSO.
Figure S2: Comparison of predicted gains in zone length due shock interaction using analytical
model, diffusion-free model and SPRESSO. (a) and (b) show the variation of calculated gain in zone
length  after / before  of two analytes (Hepes and Tricine) versus the ratio of LE+ to LEconcentration in the initial LE+/LE- mixture  cL  ,init / cL  ,init  . Both plots show that the predictions
using diffusion-free model and SPRESSO agree well with each other, with and without ionic strength
corrections for species mobilities. Whereas predictions using analytical model agree well with the
simulations using diffusion-free model and SPRESSO only when ionic strength effects are neglected
in both of the latter models. For these calculations, LE- is Mops, LE+ is Imidazole, TE- is 20 mM
Taurine and TE+ is 100 mM Bistris. To vary the ratio cL ,init / cL ,init we fixed the concentration of
LE- at 100 mM and varied the concentration of LE+ from 200 to 320 mM.
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