Sample Zone Dynamics in Peak Mode Isotachophoresis

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1 upporting Information ample Zone Dynamics in Peak Mode Isotachophoresis arun K. Khurana and Juan G. antiago* Department of Mechanical Engineering, tanford University, tanford A 94305, UA khurana@stanford.edu, juan.santiago@stanford.edu his document contains the following supporting information for the article: Figure -1: Inverd innsity images of IP sample zone at various locations. Figure -2: Electrophoretic mobility of Alexa Fluor 488 as a function of ionic strength. Analytical solution of sample concentration growth a E-E inrface orrections for electrophoretic mobility and activity coefficient caling argument for effective dispersion coefficient of species in IP inrface. Page -1

2 Figure -1: Raw greyscale innsity images of sample zone at five locations downstream of the E well in an IP experiment. Here, the leading and trailing electrolys were 240 mm histidine-hl and 2.5 mm Na-phenylpropionic acid respectively and current was held constant at 30 µamp. he curvature in the sample zone is evidence of dispersion due to residual (non-uniform) EOF. he curvature changes direction afr x = 21 mm (station 4). Page -2

3 Figure -2: Plot of the electrophoretic mobility of Alexa-Fluor 488 as a function of the ionic strength of histidine-hl buffer (ph = 5.3), as obtained from independent, uniform electroly injection experiments. olid line is a power law fit to the data. he electrophoretic mobility by dermined by electrokinetic injection of Alexa-Fluor 488 inside a simple-cross geometry microchip and we monitored electroosomotic flow in the channel using Rhodamine B as the neutral marker. Analytical solution of sample concentration growth a E-E inrface: implified flux-balance analysis For a univalent IP sysm (weak/strong electroly), the concentration of trailing ion in the regulad E zone is given by (from Jovin s and Alberty s relations) le le le μ μ + μ = le μ μ + μ Page -3 (.1)

4 We use Jovin s relation to arrive at the counrion concentration in the trailing zone le le le le le μ μ + μ = + le μ μ + μ (.2) Using these trailing ion and counrion concentrations, we can obtain the conductivity of the regulad E zone as: ( ) σ = F α μ t + αc μcc (.3) where α and α are the degree of dissociation of the trailing and the counrion in the E zone. hese are obtained from their respective acid equilibrium constants ( and K a, ) and hydronium ion concentration (or ph) as: K a, α α = = K K K a, a, a, + H + + H H + + (.4) H + can be obtained by using the electroneutrality condition substituting for α and α from eq (.4). α = α and 2 Ka, K a, + = H + (.5) 2 Ka, ince the concentration of trailing ion and counrion is known apriori, we can obtain the conductivity of the E well ( σ well ) in similar way (equations (.3)-(.5)) Page -4

5 he sample concentration in the regulad E zone is relad to the initial sample concentration in the well through the flux balance condition across the initial stationary boundary as: well well s α μs σ well = α μ σ (.6) well he net influx of sample ions at the inrface between the E and E zone is equal to the accumulation ra of sample ions. he diffusive fluxes are dominant only at the inrface and only govern the width of the inrface as well as the sample zone. Hence, we can also express the same accumulation ra as the net influx of sample ions into the sample zone, migrating at speed IP between the E-E inrface. dn dt ( μ ) E IP = (.7) he speed of the inrface IP is given by: IP le le j α μ = α μ j σ = σ (.8) le We now substitu for IP in eq (.7) with the expression in eq (.8) and also substitu expression for from eq (.6) to arrive at ( α μ α μ ) well well well α μ α μ σ well dn j dt = (.9) ince the IP inrface moves at constant speed over time, i.e. time t with distance x in eq (.9) ( α μ α μ ) well well well α μ α μ σ well IP x = IPt, we replace dn j dx = (.10) Again, substituting for IP from eq (.8), we obtain Page -5

6 dn dx ( ) α μ α μ α μ σ well well le well = le le αμ α μ σ well (.11) As none of the paramers on the RH of eq (.11) change with distance x, the number of sample species (per unit cross-section area) accumulad at the inrface is simply N ( ) α μ α μ α μ σ = x well well le well le le αμ α μ σ well (.12) he approxima length scale of the inrface can be obtained by solving for the distribution of the leading and trailing electroly at the inrface. In the lab frame of reference, the convective diffusion equation for a univalent (weak/strong) electroly is given by: i i = αμ i ie i + Di t x x Here α i is the degree of dissociation of species i. (.13) We apply the following transformation to the convective diffusion equation to transla the frame of reference with the inrface. τ = t ξ = x t IP (.14) In the transformed set of variables, the convective diffusion equation is given by: i i i IP αμ i iie D = + (.15) i τ ξ ξ ξ he E and E species acquire reach a sady sta distribution in short time scale (~ 1-2 sec for inrface width ~100 µm), we can neglect the species evolution rm on the H Page -6

7 of eq (.15) and solve for the sady sta distribution. he simplified equation is then writn as i ( IPi ) = αμ i ii E+ Di ξ ξ ξ or, = αμ E+ D + F ξ i IP i i i i i i (.16) (.17) where F i is a constant dermined by the boundary conditions. For the leading and trailing ion, F i = 0 since the E and E inrfaces move at identical speed and far away from the inrface, = α μ E = α μ E (.18) IP E E herefore, the flux balance equations for leading and trailing ion simplify to = α μ E+ D IP ξ (.19) and = α μ E+ D IP ξ (.20) We elimina E from eq (.19) and (.20) arrive at the following equation IP IP D D + = α μ α μ α μ ξ α μ ξ (.21) Next we assume negligible change in the ph across the inrface such that α and α do not change significantly (i.e. α = α and α le = α ) and we use Nerst-Einsin relationship to rela the diffusion coefficient of a species to its mobility Page -7

8 D D k μ = μ = e (.22) We then ingra eq (.21) obtain the following relationship between and. 1 α 1 α x, = 0 ξ = exp, x= 0 δ (.23) where δ is the length-scale of the inrface width, given by δ = = k e k e α μ α μ ( α μ α μ ) IP α μ ( αμ αμ ) σ le j (.24) he total length scale over which E changes at the inrface is apprx. equal to the sum of lengthscale over which and drop. 2δ 2δ 2k 1 1 αμ σ le w = + = + α α e α α αμ j αμ ( ) (.25) Noting that α = α, le μ = μ and so on, the scale for concentration of sample ions in le the inrface can now be obtained using the accumulad moles sample species and inrface width length-scale obtained in eq (.12) and (.24). N w le le ( )( ) le le well e αα α μ α μ α μ α μ α μ le le le le 2k α + α α μ α μ α μ well j well σ well x (.26) Page -8

9 orrections for Electrophoretic Mobility and Activity oefficient: he fully ionized mobility of background electroly species (leading, trailing and counrion) is correcd to accommoda the influence of ionic strength using the Pitts equation. 40, μiz, = μiz, ( zz + c1 μiz, + z c2) where 0 iz, 1+ c I 3 (.27) I μ is the fully ionized mobility of the species i with valence z, I is the ionic strength and c1, c2 and 3 are constants, c 1 2 c = 0.23, c = 31.4e 9 and c = 1.5 (obtained afr simplification of Pitts equation for symmetric electrolys). imilarly, we correct for the dependence of activity coefficient of the electroly species on the ionic strength using the ruesdell-jones 42 model of the form: 3 logγ iz, 2 Az I = + ci 1+ Ba I where γ iz, is the activity coefficient of species i with valence z, (.28) I is the ionic strength and constan ts A = 0.51, Ba = 1.5 and c = 0.1. Page -9

10 caling Argument for Effective Dispersion oefficient: Figure -3. (a) chematic of the distord IP inrface showing the bulk velocity field (as black arrows in the frame of reference moving with the plug) and electric field (grey arrows) in the vicinity of the plug. (b) chematic of the net velocity (bulk + electrophoretic) of E and E ions in the frame of reference moving with the IP plug). In b), arrows indica net velocity of E ions and dashed arrows indica net velocity of E ions. he non-uniform bulk (EOF) velocity in the E and E zones induces an axially varying pressure gradient. his distorts the IP inrface since the cenrline bulk velocity is higher than the bulk velocity near the wall. In the frame of reference moving with the inrface, the bulk velocity vector at the inrface is shown in dark arrows in Figure -3(a). he inrface distorts due to this induced pressure driven flow thereby creating a radially non-uniform conductivity and electric field. he electric field vectors, in the vicinity of the inrface, are shown in gray arrows in Figure -3(a). In this modified flow-field, the plug acquires a new sady sta shape. he velocity (and net flux) of the E and E ions near the E-E inrface in the stationary frame of reference, Page -10

11 is positive at the cenrline and negative near the walls, as shown in Figure -3(b). his mismatch in the axial flux at the E-E inrface must be balanced by radial flux of the species. he species will diffuse, advect and electromigra in the radial direction and in suppressed EOF conditions, we hypothesize that radial electromigration is the dominant mechanism contributing to the radial mass flux. For radial electromigration to balance axial dispersion, the two phenomena must occur over the same time scale. For the given case, the time scale for axial dispersion is given by: τ axial ~ eof δ eof (.29) where δeof is the stretched inrface width due to EOF imilarly, time scale for radial electromigration: μey τ radial ~ (.30) a where a is the width of the channel and E y is the radial electric field component. herefore, equating eq (.29) and (.30), we arrive at an expression for δ eof δ eof ~ eof a μ E y (.31) Now, we express the total width of IP plug as the sum of original inrface width (only diffusion) and stretched inrface width Δ= w + δ eof (.32) Here D μ w = μ μ μ ( ) E x (.33) Next we express Δ in the same form as w in rms of effective dispersion coefficient as: Page -11

12 Δ Deff, μ μ μ μ ( ) E x (.34) ubstituting expressions for Δ,δ0 and δ eof in eq (.32), and simplifying to obtain D eff, we get: ( ) E μ x μ Deff, D eofa E y μ or, + (.35) ( + β e ) D D P eff, 1 (.36) where, β ~ E E x y ( μ μ ) μ Page -12

Calculation of species concentrations in ITP zones using diffusion-free model. σ σ. denote the concentration and effective mobility of species i, (1)

Calculation of species concentrations in ITP zones using diffusion-free model. σ σ. denote the concentration and effective mobility of species i, (1) Supplementary Information High sensitivity indirect chemical detection using on-chip isotachophoresis with variable cross-section geometry SS Bahga G V Kaigala M Bercovici and JG Santiago Calculation of