Viktor Shkolnikov, Juan G. Santiago*

Transcription

1 Supplementar Information: Coupling Isotachophoresis with Affinit Chromatograph for Rapid and Selective Purification with High Column Utilization. Part I: Theor Viktor Shkolnikov, Juan G. Santiago* Department of Mechanical Engineering, Stanford Universit, Stanford, CA 9435, USA. * Tel: ; Fa: ; juan.santiago@stanford.edu In this supplementar information document, we present additional information on the following topic: SI. Solution to the simplified advection-reaction equations SI. Separation resolution of ITP-AC

2 SI. Solution to the simplified advection-reaction equations We solve the simplified advection-reaction equations derived in Section. of the main tet. We start b finding the zeroth order solution for c and n following Thomas but using the boundar and initial conditions pertinent to our problem. We then follow a similar procedure to find first and second order solutions. Similar procedure can be applied to obtain higher order solutions. However, in the main tet we discuss and eplain the trends given onl b the zeroth order solution (accurate to O( ε ) ), as it is the simplest engineering approimation to this problem. Fortunatel, the results of the zeroth order solution agree well with eperimental measurements of ke ITP-AC parameters at our eperimental conditions (presented in Part II of this two-paper 3 series). At the end of this section we present the solution accurate to O( ε ) for situations where knowledge of spatiotemporal behavior of ITP-AC is needed to greater accurac. While we continue to work with the non-dimensionalized equations (as we have in Section. of the main tet), we here drop the asterisks above the variables for clarit of presentation. SI. Solution of zeroth order equations We begin b transforming the two zeroth order equations into a coordinate moving with the LE- TE interface velocit (for the non-dimensionalized equations this velocit equals to unit). We then convert the result into a potential function form, thereb collapsing the two equations into a single equation. We then solve the resulting equation using Laplace transforms. We write the zeroth order non-dimensionalized equations and their respective boundar and initial conditions here again for convenience: c c n t z t + + =, () n t c n + n =, c n ( z ) ( z ), =,, =, () (3) = ( ) c, t α π ep t Da 3. (4) We begin b transforming () through (4) into new independent variables, = z and = ( t z) : c n + =, (5)

3 n c n + n =, c n ( ) ( ), =,, =, (6) (7) = ( ) c, α π ep Da 3. (8) Net we introduce a potential function f such that n c =, (9) =. The potential function f automaticall satisfies equation (5). Therefore, we substitute (9) into (6) and obtain f + + =. () Equation () is non-linear and we linearize it b substituting in after which we obtain φ f, = + ln,, () φ = φ. () In this wa, we have condensed equations (5) and (6) into a single equation (). Net we transform the initial and the boundar conditions. We note that we use the onl initial condition for c. The initial condition on n was applied much earlier, when we tacitl assumed that there is no bound target initiall in the affinit region. See the main tet when we first introduce the reaction equation in terms of N (eq. () of main tet). The transformed initial and boundar conditions are φ (,) = = c(, ), φ (,) (3) 3

4 φ (, ) = = + c(, ). φ (, ) (4) We solve the differential equations (3) and (4) and obtain φ, ep ', ', (5) ( ) = c( ) d = H φ(, ) = ep + c(, ' ) d ' = G( ). (6) Net we appl the Laplace transform to equation () and use the result (6) to obtain, dφ dh p = φ+ (7) d d φ is where the overbar denotes the Laplace transform of a function with respect to (e.g., (, p) the Laplace transform of φ and p is the Laplace transform variable). Also, ( p) G( p) φ, =. (8) We solve the differential equation (7) b variation of parameters to obtain ( ') ( ') (, ) dh φ p = G p ep ep d'. p + p (9) d p We then take the inverse transform of (9) to obtain where ( ') φ, = G ' F, ' d ' + dh F ', d ', d () L F L F, = ep, p ( ), ep. p p ( ) = () L F, F,. Taking the inverse Laplace transform of () we obtain 4 designates the Laplace transform of

5 F(, ) = I( ) + δ( ), = ( ) F, I. () Here I and I are modified Bessel functions of the first kind of zeroth and first order respectivel. We substitute () into () and obtain ( ') ' d dh φ, = G + G ' I ' d ' + I ' d '. (3) We then simplif (3) b evaluating the right hand side term via integration b parts and noticing that G = ( ') ( ') d d φ dg dh, = I + I ' d ' + I ' d '. (4) This is the general solution to the zeroth order equations in terms of the potential function φ. Net we appl the initial and boundar conditions specific to ITP-AC. We begin b substituting (7) into (5) and obtain and Substituting, this simplifies (4) to H dh ( ' ) d = ep, (5) = ep. (6) dg φ, = I + I ' d ' + ep ' I ' d'. (7) d We net appl the boundar condition b substituting (8) into (6) and obtain G αda Da 3 3 = ep + erf erf, (8) and 5

6 α dg Da 3 αda Da 3 3 = ep + ep + erf erf. d π (9) Therefore ( ') dg φ, = I + I ' d ' + ep ' I ' d ', (3) d where dg d is given b (9). Now c and n can be easil obtained b substituting (3) into () and substituting the result into (9). For convenience, we evaluated solutions numericall using custom MATLAB scripts as we analzed the spatiotemporal behavior of the target in ITP- AC. SI. Solution of first order equations We solve the first order equations following a procedure similar to the zeroth order equations. We again begin b transforming the two first order equations into a coordinate moving with the LE-TE interface velocit. We then convert the result into a potential function form, thereb again collapsing the two equations into a single equation. We again solve the resulting equation using Laplace transforms. We write the first order non-dimensionalized equations and their respective boundar and initial conditions here again for convenience: ( uc % ) c c n t z t z =, n t c n c + n =, ( z ) ( z ), =,, =, (3) (3) (33) c(, t ) =. (34) We then transform (3) through (34) into new independent variables, = z and = ( t z) : ( uc % ) ( uc % ) c n + + =, (35) 6

7 n c + n =. (36) Net we define new variables C = c + uc %, M = n uc %, (37) and recast (35) and (36) in these variables: C M + =, (38) M ( uc % ) ( C uc ) ( M uc ) =. % % (39) We also recast the initial and boundar condition in these variables % % C z, u z, c z, =, M z, + u z, c z, =, (4) C, t u%, t c, t =. (4) Net we introduce a potential function f such that M= (4) C = and substitute this into (38) and (39). The potential function f automaticall satisfies equation (38), and (39) becomes ( uc % ) f ( + ) uc = Now we appl f to the initial and boundar conditions ( ) = M ( ) = n ( ) u( ) c ( ) %. (43),,, %,,, (44) 7

8 We integrate (44) and (45) Noticing that n ( ',) = and ( ) = C ( ) = c ( ) u( ) c ( ),,, %,,. (45) % (46) = f, n ', u ', c ', d ', % (47) = f, c, ' u, ' c, ' d '. c, ' = we simplif the above equations to % (48) = = f, u ', c ', d ' H, % (49) = = f, u, ' c, ' d ' G. Equations for initial and boundar conditions (48) and (49) are to be evaluated using the result from the zeroth order equations. u% is of the form of an error function (see main tet), i.e., u % = erf t z 3Da Da. Net, we take the Laplace transform of (43) to ield f p H + f = + H ( + + p) w + w(, ), p+ p+ (5) where w= uc %. We solve differential equation (5) using variation of parameters and obtain f G p p p+ = ep ( ') H p ' p ep ep H( ') ( p) w w( ',) d '. p+ p+ p+ (5) We epand this to ( ' ) H f = G( p) F + + H( ' ) + w( ', ) F d' F3 wd ', (5) 8

9 where p F = ep, p+ F p ' = ep, p+ p+ ( p) p( ' ) + + F3 = ep. p+ p+ (53) We recommend finding the inverse Laplace transforms of equations in (53) numericall. We then take the inverse Laplace transform of (5) to obtain f = G ' F, ' d ' ( ') H H ( ' ) w( ', ) F ( ', ) d ' ( ) w ', ' F, ' d ' d '. 3 (54) Now the functions c and n can be obtained fairl easil b substituting (54) into (4), and using the appropriate inverse Laplace transforms for (53) and the result from the zeroth order equations (as inputs for G, H, and w). We recommend evaluating this numericall using MATLAB. SI.3 Solution of second order equations We solve the second order equations following a similar procedure to the zeroth and first order equations. We once again begin b transforming the two first order equations into a coordinate moving with the LE-TE interface velocit. We then convert the result into a potential function form, again collapsing the two equations into a single equation. We then solve the resulting equation using Laplace transforms. We write the second order non-dimensionalized equations and their respective boundar and initial conditions here again for convenience: ( uc % ) c c n t z t z =, (55) n t c + n + c n =, (56) 9

10 c n ( z ) ( z ), =,, =, (57) c(, t ) =. (58) We transform (55) through (58) into new independent variables, = z and = ( t z) : ( uc % ) ( uc % ) c n + + =, (59) n c + n + c n =. (6) Net, we define new variables C = c + uc %, M = n uc %, (6) and recast (59) and (6) in these variables C M + =, (6) M ( uc % ) + C uc + M + uc + cn =. % % (63) We also recast the initial and boundar condition in these variables % % C z, u z, c z, =, M z, + u z, c z, =, (64) C, t u%, t c, t =. (65) Net we introduce a potential function f such that M =, (66) C =,

11 and substitute this into (6) and (63). The potential function f automaticall satisfies equation (6), and (63) becomes ( uc % ) f ( + ) uc + cn = Now we appl f to the initial and boundar conditions and obtain We integrate (68) and (69) and obtain Noticing that n ( ',) = and ( ) = M ( ) = n ( ) u( ) c ( ) %. (67),,, %,,, (68) ( ) = C ( ) = c ( ) u( ) c ( ),,, %,,. (69) % (7) = f, n ', u ', c ', d ', % (7) = f, c, ' u, ' c, ' d '. c, ' =, we simplif the equations above to % (7) = = f, u ', c ', d ' H, % (73) = = f, u, ' c, ' d ' G. Equations for initial and boundar conditions (7) and (73) are to be evaluated using the result from the first order equations. Net, we take the Laplace transform of (67) to obtain f p H + f = + H ( + + p) w + w(, ) w, p+ p+ (74) where w = uc %, w = nc. We solve differential equation (74) using variation of parameters and obtain

12 ( ) p f = G p +Θ ep, p+ (75) where ( ') ' p H Θ= ep H ( ') ( p) w w( ',) w d '. p (76) + p+ We epand (75) to where ( ' ) H f = G( p) F + + H( ' ) + w( ', ) F d' F3 wd ' + F4 wd ', (77) p F = ep, p+ F 4 p ' = ep, p+ p+ ( p) p( ' ) + + F3 = ep, p+ p+ F ( ' ) p = ep. p+ p+ (78) We then take the inverse Laplace transform of (78) to obtain f = G ' F, ' d ' ( ') H H ( ') w ( ',) F ( ', ) d ' ( ) w ', ' F, ' d ' d' 3 + w ', ' F, ' d ' d '. 4 (79)

13 Now functions c and n can be obtained b substituting (79) into (66) using the appropriate inverse Laplace transforms for (78) and the result from the first order equations (as inputs for G, H, w, and w ). Again, we recommend evaluating this numericall using MATLAB. SI.4 Third order accurate solutions Finall, a third order accurate solution is obtained b combining results obtained from (3), (54), and (79) into where once again, ε is defined as c= c + εc + ε c + O ( ε ), ( ε ) 3 n= n + εn + ε n + O 3 ε µ µ, (8) a, TE TEion, TE =. (8) µ TEion, TE Higher order equations can be obtained in a manner similar to that for first and second order equations and hence even higher order of accurac solutions can be obtained. 3

14 SI. Separation resolution of ITP-AC As described in the main tet, the ideal case for ITP-AC separation is that cn target (where subscript cn refers to contaminant). That is, target should be captured approimatel irreversibl while contaminant is removed and transported downstream. For this regime, the target is captured in time p t while contaminants migrate through the column. We present an eperimental demonstration of such separation in the main tet of Part II of this two-part series. We consider here a regime where the target is completel captured (e.g., 6 < ) while the contaminant species remains focused in ITP and migrating at the ITP velocit. For this regime, we define a simple criterion for the time required to achieve sufficient separation between captured target and contaminant. We appl a common definition of resolution, R, given b Giddings 3 as follows: L R=, σ + σ (8) where L is the distance between the two peaks and, σ and σ are the corresponding standard deviations of the two peaks. We let σ be the standard deviation width of the captured (immobile) target peak and σ the standard deviation of the contaminant peak. In ITP-AC the target attains zero velocit at time approimatel p t, and thereafter the width of the target distribution scales as p z. After the target attains zero velocit, the distance between the target and the analte simpl scales as ut. Using the scaling for the ITP peak width from MacInnes and Longsworth 4,5, we obtain the scaling for resolution for ITP-AC, as we did in equation (4) of the main tet. We write it here again for convenience: R ITP-AC ut u µ L in LEµ T in TE kbt p z * + kn u µ L in LE µ T in TE e, (83) where again, k B is the Boltzmann's constant, T is the absolute temperature, e is the electron charge, and µ L in LE and µ T in TE are the mobilities of the LE ion in the LE and the TE ion in the TE respectivel. We note that the resolution for ITP-AC scales as t. This is in contrast to the resolution of traditional electrophoresis or of AC which scale as t. 6 The resolution of ITP-AC increases with increasing ITP velocit u and it asmptotes to a value of tk N p z * for large u. We can achieve 95% of this resolution with u µ L in LEµ T in TE kbt kn..5 µ L in LE µ T in TE e pz * (84) 4

15 Operating at ITP velocit lower than u 95 results in loss of resolution. Operating at a velocit much higher than u 95 leads to larger capture lengths with little gain in resolution and therefore loss in column utilization (see Section 3.. of the main tet). For a good compromise between resolution and column utilization, we therefore recommend operating at around u 95. For an ITP velocit of u 95 and a convenient resolution, target and contaminant zone), the time to capture and purif, c, p If α Da is also less than unit, then, R ITP-AC, value of, sa, (for well separated capture t is approimatel c p p k N. t is approimatel * 8 k N. For tpical ITP-AC conditions such as those we emploed in Part II of this two-paper series (µ L in LE -6-9 m V - s -, µ T in LE - -9 m V - s -, k = 3 M - s -, N 3 µm, and p z *.8), u95 is on the order of. mm/s and t c, p is approimatel 5 min. 7 z References () Thomas, H. C. J. Am. Chem. Soc. 944, 66, () Wolfram Alpha. Wolfram Alpha LLC, 3. (3) Giddings, J. C. Sep. Sci. Technol. 969, 4, (4) MacInnes, D. A.; Longsworth, L. G. Chem. Rev. 93, 9, 7-3. (5) Shkolnikov, V.; Bahga, S. S.; Santiago, J. G. PCCP, 4, (6) Landers, J. P. Handbook of capillar electrophoresis; CRC press, 997. (7) Shkolnikov, V.; Santiago, J. G. Anal. Chem. 4, submitted. 5