Convection-Dominated Dispersion Regime in Wide-Bore

Sep 9, 2009 - for Poiseuille flows in cylindrical capillaries the average residence time grows logarithmically with the Peclet number, while the varia...
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Anal. Chem. 2009, 81, 8009–8014

Convection-Dominated Dispersion Regime in Wide-Bore Chromatography: A Transport-Based Approach To Assess the Occurrence of Slip Flows in Microchannels Alessandra Adrover, Stefano Cerbelli, Fabio Garofalo, and Massimiliano Giona* Dipartimento di Ingegneria Chimica, Sapienza Universita` di Roma, via Eudossiana 18, 00184 Roma, Italy This article develops the theoretical analysis of transport and dispersion phenomena in wide-bore chromatography at values of the Peclet number Pe beyond the upper bound of validity of the Taylor-Aris theory. It is shown that for Poiseuille flows in cylindrical capillaries the average residence time grows logarithmically with the Peclet number, while the variance of the outlet chromatogram scales as the power 1/3 of Pe. In the presence of slip boundary conditions, both the mean and the variance of the outlet chromatograms saturate at high Pe, and this phenomenon provides an indirect transport-based way to detect slip flow conditions at the solid walls and, more generally, flow distributions in channel flows. The increasing interest for analytical microfluidic devices (micrototal analysis systems and lab-on-chip) has triggered a deeper understanding of dispersion properties in microchannels and capillary columns and of the influence of channel shape and geometry on the Taylor-Aris dispersion coefficient.1-5 In analytical studies involving separations, chemical reactions, or simply detection of solutes, hydrodynamic dispersion may negatively influence the performance of the microfluidic device, thus reducing the quality of the measurement or of separation efficiency. While the theoretical analysis of dispersion has been mainly focused on very long microchannels, for which the Taylor-Aris dispersion theory applies,6,7 in recent years wide-bore chromatography has been proposed and successfully applied for resolving small molecules or for the determination of particle size in the nanometer regime.8-10 By definition, wide-bore chromatography operates with columns and microchannels with relatively small length-to-radius aspect ratio R ) L/R e 300 and for very high * To whom correspondence should be addressed. E-mail: max@ giona.ing.uniroma1.it. (1) Dutta, D.; Leighton, D., Jr. Anal. Chem. 2003, 75, 3352–3359. (2) Tallarek, U.; Rapp, E.; Scheenen, T.; Bayer, E.; Van As, H. Anal. Chem. 2000, 72, 2292–2301. (3) Blom, M. T.; Chmela, E.; Oosterbroek, E.; Tijssen, R.; van den Berg, A. Anal. Chem. 2003, 75, 6761–6768. (4) Zhao, H.; Bau, H. H. Anal. Chem. 2007, 79, 7792–7798. (5) Ajdari, A.; Bontoux, N.; Stone, H. Anal. Chem. 2006, 78, 387–392. (6) Taylor, G. Proc. Royal Soc. A 1953, 219, 186–203. (7) Aris, R. Proc. Royal Soc. A 1956, 235, 67–77. (8) Fischer, C.-H.; Giersig, M. J. Chromatogr., A 1994, 688, 97–105. (9) Harada, M.; Kido, T.; Masudo, T.; Okada, T. Anal. Sci. 2005, 21, 491– 496. (10) Okada, T.; Harada, M.; Kido, T. Anal. Chem. 2005, 77, 6041–6046. 10.1021/ac901504u CCC: $40.75  2009 American Chemical Society Published on Web 09/09/2009

Peclet number (which is the ratio of the characteristic time scales for diffusion and advection). This is exactly the range of operations for which Taylor-Aris dispersion theory does not apply.11-13 In point of fact, a theory for transport and dispersion in widebore chromatography is still lacking, and the working region of this analytical technique falls in the no-man’s land of dispersion diagrams, where no theoretical results are available. As a consequence the design of these devices rests upon case-by-case numerical studies (see, e.g., Figure 20.5.2 of a classical reference on transport phenomena12 based on the classical work on dispersion).11 When dealing with fluid dynamics in liquid-filled microchannels, a further issue is represented by the possible occurrence of slip flows, giving rise to a nonzero velocity at the channel walls.14-16 Experimental results on aqueous flow in micrometer channels are contradictory, and the use of velocimetric techniques (µPIV) may be not always be adequate, especially when the size of the seed particles (used for particle image velocimetry) is on the order of 200 nm up to 1 µm and the channels are a few micrometers wide (see the review by Neto el.16 and references cited therein). A recent theoretical work on the properties of Brownian particles in Poiseuille flow has shown the conceptual possibility of discriminating between slip and no-slip flows by the analysis of solute transport in microchannels.17 The analysis proposed by Giona et al.17 is purely theoretical but suggests that it should be possible to approach a fully fluid dynamic characterization (the occurrence of slip flow) via an indirect approach based on the analysis of transport of solute particles advected by the flow and diffusing in it. The scope of this article is twofold: (i) to provide a theoretical understanding of dispersion for very high Peclet values in finitelength microcolumns and microchannels, correspoding to the typical operating conditions of wide-bore chromatography, and (11) Ananthakrishnan, V.; Gill, W.; Barduhn, A. AIChE J. 1965, 11, 1063–1072. (12) Bird, R. B.; Stewart, W. N.; Lightfoot, E. N. Transport Phenomena, 2nd ed.; Wiley: Chichester, 2002. (13) Vanderslice, J. T.; Rosenfeld, A. G.; Beecher, G. Anal. Chim. Acta 1986, 179, 119–1129. (14) Tretheway, D. C.; Meinhart, C. D. Phys. Fluids 2002, 14, L9–L12. (15) Huang, P.; Guasto, J. S.; Breuer, K. J. Fluid Mech. 2006, 566, 447–464. (16) Neto, C.; Evans, D.; Bonaccurso, E.; Butt, H.-J.; Craig, V. S. J. Rep. Prog. Phys. 2005, 68, 2859–2897. (17) Giona, M.; Cerbelli, S.; Creta, F. J. Fluid Mech. 2008, 612, 387–406.

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(ii) to propose a chromatographic approach for determining the possible occurrence of slip conditions for channel flows. In point of fact, these two issues are tightly related to each other: wide-bore hydrodynamic chromatography intrinsically operates outside the Taylor-Aris regime, and since it is strongly controlled by the flow structure, it permits quantitative information on the velocity profile to be derived directly from the analysis of the outlet chromatogram of slowly diffusing solutes and particles. Throughout this article we consider mainly cylindrical microchannels (although the results obtained apply to axial channel flows in arbitrary geometries, as described in the Discussion section), in the presence of a laminar Poiseuille flow. We develop a scaling theory for the behavior of the moments of the outlet chromatogram for slowly diffusing particles. For no-slip flows, it is shown that the first order moment (mean residence time) diverges in a logarithmic way for increasing Peclet number, while the variance of the outlet chromatogram grows as Pe1/3. This is one of the main results which differs from the classical TaylorAris theory which predicts the proportionality between the variance and Pe. Conversely, whenever slip conditions characterize the velocity profile, both the mean residence time and the variance saturate with Pe toward a constant value that can be theoretically predicted as a function of the slip length. This article is organized as follows. Preliminarly, we introduce the relevant nomenclature and notation for wide-bore chromatography, highlighting the transition from Taylor-Aris to convectiondominated transport (and dispersion) regimes. Subsequently, we develop the scaling theory of the convection-dominated regime, providing a theoretical justification for the moment scaling. Finally, we consider the influence of slip conditions on the chromatographic outcome, showing that the quantitative analysis of dispersion permits to estimate the possible occurrence of slip flows. WIDE-BORE CHROMATOGRAPHY: NOTATION AND TRANSPORT REGIMES Let us consider a cylindrical capillary of length L and radius R. Laminar conditions within the device imply a Poiseuille structure for the flow, by an axial velocity profile vz(r), solution of the Stokes equation, that depends solely on the radial coordinate r. The space-time evolution of the nondimensional concentration φ(t,ζ,F) of a solute for a radially symmetric inlet profile obeys the dimensionless equation ∂φ ∂φ 1 ∂2φ R2 1 ∂ ∂φ ) -v(F) + + F ∂t ∂ζ Pe ∂ζ2 Pe F ∂F ∂F

( )

(1)

where φ is the dimensionless solute concentration, F ) r/R and ζ ) z/L are the dimensionless radial and axial coordinates (0 e F,ζ e 1), respectively, t is the dimensionless time (corresponding to the physical time rescaled with respect to the flow residence time L/Vref, where Vref is the mean velocity within the capillary), v(F) ) vz(RF)/ Vref is the normalized axial velocity so that the mean velocity equals 1, Pe ) VrefL/D is the Peclet number (D is the solute diffusivity), and R ) L/R . 1 is the aspect ratio. Let us suppose that slip-boundary conditions occur at the walls of the capillary. This phenomenon can be described by means of the Navier’s boundary condition ls 8010

dvz(r) dr

|

) -vz(r) r)R

|

(2) r)R

Analytical Chemistry, Vol. 81, No. 19, October 1, 2009

where ls is the dimensional slip length. The normalized dimensionless velocity profile is thus given by

v(F) )

2 (1 - F2 + γ) 1 + 2γ

(3)

where γ ) 2ls/R. For γ ) 0, one recovers the no-slip Poiseuille flow. The factor “2” entering the definition of the nondimensional slip length in cylindrical capillaries has been added to keep the formal analogy between eq 3 and the corresponding velocity profile in a two-dimensional straight conduit (see eq. (S41) in the Supporting Information of this article). Equation 1 describes the outcome of a wide-bore chromatographic experiment in which the inlet profile does not depend on the angular coordinate. If the solute is impulsively injected into the column at time t ) 0 then φ|ζ)0 ) δ(t) (δ(t) is the Dirac’s delta distribution). In practice, an impulsive injection means that the inlet concentration pulse duration should be much smaller than the characteristic time L/Vref and has been experimentally implemented by Harada et al.9 and Okada et al.10 in their separation studies involving short columns. Equation 1 is also equipped with the initial condition φ|t)0 ) 0 and complemented with the remaining boundary conditions ∂φ/∂F|F)0,1 ) 0. For the outlet boundary condition, one can consider either the infinite-column approximation or the Danckwerts’ outlet boundary condition. The latter condition dictates ∂φ/∂ζ|ζ)1 ) 0. In point of fact, for high Peclet numbers (which is the operating regime considered throughout this article), the role of the outlet boundary condition is immaterial since one can neglect the contribution of axial diffusion (see below). In chromatographic experiments, one measures the average outlet profile φout(t) at the exit section ζ ) 1, where φout(t) ) 2∫10φ(t,ζ ) 1,F)F dF. A compact quantitative description of φout(t) is provided by its moments

(n) mout )

∫

∞ n

0

t φout(t) dt

(4)

(0) (1) For n ) 0, mout ) 1, and for n ) 1, mout corresponds to the dimensionless mean residence time, which generally deviates from the normalized flow residence time tflow ) 1, while the second order moment m(2) out quantifies solute dispersion. Specif(2) (1) 2 ically, the second order central moment σout2 ) mout - (mout ), corresponding to the variance of φout(t), is the typical measure of dispersion in chromatography. Figure 1A shows the shape of the outlet chromatogram for R ) 100 at increasing values of the Peclet number. Numerical details on the computational issues can be found in the Supporting Information of this article. At Pe ) 104, one recovers the classical, almost Gaussian-shaped profile, possessing a mode located close to dimensionless flow residence time, that is, t ) 1. This is the typical chromatogram in the Taylor-Aris regime. As Pe increases further, the modal abscissa tmode (i.e., the peak location) of the chromatogram moves progressively toward the minimum residence time tmin ) 1/2 and attains a highly asymmetric shape. The modal abscissa tmode of the outlet chromatogram provides a direct indication of the regime

Figure 2. tmax(Peeff) for different values of Peeff at R ) 100. Symbol kin (O) corresponds to points of coordinates (tmax(Peeff),φout (tmax(Peeff))). Dashed lines represent the outlet chromatograms at Peeff ) 101, 2 × 101, 5 × 101, 102, respectively. Solid line (e) is the kinematic limit kin φout (t) ) 1/(2t 2).

chromatogram becomes negligible (Supporting Information Figure S1 and related discussion). Thus, eq 1 simplifies as

Figure 1. (A) Normalized outlet chromatogram φout(t) vs t in a noslip Poiseuille channel at R ) 100 for different Peclet values. The arrow indicates increasing values of Pe ) 104, 5 × 104, 105, 2 × 105, 5 × 105. (B) Modal abscissa tmode vs the effective Peclet number Peeff for cylindrical no-slip channels at different aspect ratios. Line (a) and (O) R ) 20, line (b) and (b) R ) 100.

transition occurring in a finite-length capillary whenever the diffusivity of the solute particles decreases, that is, for increasing values of the Peclet number (Figure 1B). The dimensionless parameter characterizing the transition is the effective Peclet number Peeff ) Pe/R2 )(VrefR/D)(R/L), which is the dimensionless group controlling the relative significance of diffusion in the transverse cross-section (i.e., along the radial coordinate). The effective Peclet number can be also defined as the ratio of the characteristic time for transverse diffusion R2/D and the characteristic axial advection time L/Vref. Figure 1B depicts the behavior of tmode vs Peeff for two cylindrical capillaries with aspect ratios R ) 20 and 100. A critical effective Peclet number Pe*eff can be defined as the value of the effective Peclet number marking a sudden transition in the behavior of tmode. In cylindrical capillaries Pe*eff = 5. Close to Pe*eff, the modal abscissa abruptly switches from values close to 1 (below Pe*eff) to values close to tmin (above Pe*eff) and converges to it as Pe*eff f ∞. The sudden transition depicted in Figure 1A,B has been experimentally observed by Harada et al.9 and Okada et al.10 by considering micelles and nanoparticles in short-length columns. As observed by several authors,9,10 wide-bore chromatography provides a separation if one considers two different solute possessing values of the effective Peclet number below 1 and well above 1 (Peeff g 10). Observe that our definition of Peeff is just the reciprocal of the dimensionless parameter τaν introduced by Harada et al.10 Indeed, the critical value Pe*eff marks the transition from the Taylor-Aris regime (Peeff < Pe*eff) to the convection-dominated regime (Peeff . Pe*eff), which characterizes the operating conditions occurring in wide-bore chromatography. It is worth observing that for Peeff > 5 and R . 1, the contribution of axial diffusion on the shape of the outlet

∂φ 1 1 ∂ ∂φ ∂φ ) -v(F) + F ∂t ∂ζ Peeff F ∂F ∂F

( )

(5)

This physically means that transport and dispersion phenomena occurring in the convection-dominated regime are essentially due to the interplay between the axial velocity profile within the capillary and solute cross-sectional (transverse) diffusion. CONVECTION DOMINATED REGIME For any finite-length capillary column (characterized by a large but finite aspect ratio R), whenever the Peclet number is greater that the threshold Pe*eff ) 5, one enters what can be referred to as convection-dominated transport. From the discussion at the end of the previous Section, it follows that convection-dominated transport in capillaries can be viewed as a singular perturbation (due to the contribution of transverse diffusion) of purely convective transport, which corresponds to the solution of eq 5 for ε ) 1/Peeff ) 0. For ε ) 1/Peeff f 0, the outlet chromatogram converges to the purely kin kinematic residence time probability density function φout (t), associated with kinematically advected solute particles in the absence of diffusion. Simple calculations (Supporting Information, eqs S18-S22) kin show that for the no-slip Poiseuille flow φout (t) takes the form

kin φout (t) )

{

0 1/(2t2)

t < 1/2 t g 1/2

(6)

Therefore, as Peeff increases, the outlet chromatogram attains a shape progressively similar to the pure kinematic limit φkin out(t). This phenomenon is depicted in Figure 2. Line (e) in Figure 2 corresponds to the kinematic limit, while dotted lines (from (a) to (d)) refer to the behavior of φout(t) for increasing Peeff. kin As can be observed φout(t) is practically identical to φout (t) at short time scales. More precisely, there is an interval of times (tmode,tmax(Peeff)) in which φout(t) = φkin out(t), while for t > tmax(Peeff) the outlet chromatogram deviates from the kinematic limit due to transverse diffusion. The time abscissa tmax(Peeff) is an increasing function of the effective Peclet number. Analytical Chemistry, Vol. 81, No. 19, October 1, 2009

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SCALING THEORY OF CONVECTIONDOMINATED TRANSPORT There are several ways of deriving the scaling of the moment (n) hierarchy mout in the convection-dominated regime (see the discussion in Supporting Information). The simplest and physically more intuitive approach consists of deriving an expression for tmax(Peeff) and thus approximating the behavior of φout(t) for t > tmax. This is the approach followed below. Diffusion in the direction transverse to the flow controls the value of tmax(Peeff). Specifically, tmax(Peeff) can be determined as the largest value of the kinematic residence time associated with solute particles lying at the distance 1 - F* from the wall, such that the interaction with the wall is still negligible. In symbols, tmax(Peeff) )

1 v(F*)

(7)

Since the motion in the transverse direction is controlled solely by diffusion, 1 - F* equals the diffusive length scale traveled by a Brownian particle in the transverse direction, that is, 1 - F* ) σdif(tmax)

(8)

2 where A ) φkin out(tmax), and the tail variance σtail (Peeff) scales with Peeff as

σtail2(Peeff) = σt,02Pe2/3

where σt,02 is a constant. By collecting together these results, the following approximation for φout(t) is obtained

{

0

φout(t) = 1/(2t2) kin φout (tmax)g(t - tmax, σtail)

(9)

Particles possessing a mean residence time less than tmax experience transverse diffusion as a small perturbation of the mean convective motion controlled by the axial velocity. Collecting together eqs 7-9, it results that tmax can be computed as the root of the following equation tmax )

1 v(1 - σdif(tmax))

(10)

Since for high Peeff solely the velocity near the solid walls influences the value of tmax, one can expand v(1 - σdif) in a Taylor series of σdif retaining the leading order term. For noslip Poiseuille flow, v(1 - σdif) ) 4σdif + O(σdif2) and, therefore, eq 10 within this approximation can be solved in closed form for tmax obtaining (see Supporting Information Figure S3). tmax(Peeff) )

Peeff1/3 2/3 22

(11)

Figure 2 (symbols O) depicts the estimate of tmax(Peeff) obtained from eq 10 (or equivalently from eq 11) for large Peeff, showing a good agreement between simulations and theory. The second issue that should be approached is the behavior of φout(t) for t > tmax, corresponding to the tail of the outlet chromatogram that is controlled by transverse diffusion. For t > tmax, φout(t) behaves approximately in a Gaussian way (Supporting Information Figure S5) φout(t) ) A exp[-(t - tmax)2 /2σtail2(Peeff)] ) Ag(t - tmax, σtail) (12) 8012

Analytical Chemistry, Vol. 81, No. 19, October 1, 2009

t > tmax (14)

(1) mout )

∫

∞

0

tφout(t) dt )

1 log(tmax) + 2

kin (tmax)[σtail2 + √πtmaxσtail] ∼ φout

2tmax Peeff

t < tmode tmode < t < tmax

See Supporting Information, Figure S4, and related discussion. First-Order Moment. The mean residence time of solute particles corresponds to the first order moment m(1) out of the outlet (1) chromatogram. Within the approximation eq 14, mout can be evaluated in closed form for no-slip Poiseuille flow

where σdif2(tmax) )

(13)

1 log(Peeff) 6

(15)

which derives from the scaling of tmax(Peeff) and σtail(Peeff) in eqs 11 and 13 and on the functional form of the kinematic residence kin (1) time distribution φout (t) in eq 6. Equation 15 indicates that mout ∼ Const. log(Peeff) for large Peeff; that is, it diverges logarithmically with the effective Peclet number. Figure 3 depicts the behavior of the first order moment (symbols O and b) as a function of Peeff for no-slip Poiseuille flow. Simulation results agree with the predicted scaling in eq 15. Dispersion. The dispersive behavior of solute particle in the convection-dominated regime is described by the variance σout2 of φout(t) for Peeff . Pe*eff. By enforcing eq 14, after some calculations, one obtains for no-slip channel flow (2) mout )

∫

∞ 2

0

t φout(t) dt )

1 1 t + 2 max 2

(

kin φout (tmax)[√2πσtail3

)

+ 2tmaxσtail2 + √πtmax2σtail]

(16)

kin and therefore, by enforcing the shape of φout (t) and the scaling of tmax and σtail with Peeff, it follows that

(2) (1) 2 - (mout ) ∼ Peeff1/3 σout2 ) mout

(17)

Numerical simulations confirm the theoretical scaling in eq 16 (see Figure 3B, symbols O and b). Equation 17 shows the qualitative difference characterizing dispersion in the convection-dominated regime with respect to the Taylor-Aris scaling (see also Figure 3B). Equations 15 and 17 are the two major results of this article that can be used for estimating and designing the operating conditions in wide-bore chromatography using cylindrical capil(1) laries. Indeed, the divergence of mout and σout2 is not surprising since the first and second order moments of the kinematic

where tmin ) (1 + 2γ)/[2(1 + γ)] and tsup ) (1 + 2γ)/(2γ), it follows straightforwardly that for large effective Peclet number the first and second order moments of the outlet chromagram saturate toward constant values (Supporting Information eqs S27-S28), which depend on the dimensionless slip length γ.

(1) lim mout ∼

Peefff∞

Figure 3. First-order moment and variance of the outlet chromatogram vs the effective Peclet number Peeff for different aspect ratios (1) associated with slip and no-slip flows. (A) mout vs Peeff. Symbols (0) and (9) refer to R ) 100 and γ ) 0.02 and γ ) 0.1, respectively. (1) Line (a) represents the scaling mout (Peeff) ∼ A log(Peeff) with A ) 0.162, and lines (b) and (c) depict the saturation values predicted by the kinematic limit in eq 19. (B) σout2 vs Peeff. Symbols (O) refer to R ) 20, (b) to R ) 100 in the presence of no-slip velocity profile. Symbols (0) and (9) refer to R ) 100 and γ ) 0.02 and γ ) 0.1, 2 respectively. Line (a) represents the Taylor-Aris scaling σ out (Peeff) ) 2 2Peeff/48 and line (b) the scaling σ out (Peeff) ∼ Peeff1/3. Lines (c) and (d) are the saturation values predicted by the kinematic limit in eq 19.

distribution φkin out(t) diverge. The transition from the Taylor-Aris regime toward convection-dominated transport is rather complex, as depicted in Figure 3B. While the Taylor-Aris regime holds for Peeff < Pe*eff = 5, a fully developed convection-dominated dispersive behavior (i.e., the scaling σout2 ∼ Pe1/3) sets in solely ** ** for Peeff > Peeff = 5 × 103. In between Pe*eff and Peeff , a smooth 2 transition occurs where the function σout (Peeff) smoothly relax from the linear scaling (line (a) in Figure 3B), to the asymptotic scaling (line (b) in Figure 3B). CHROMATOGRAPHIC ANALYSIS OF SLIPBOUNDARY FLOWS The shape of φout(t) in wide-bore chromatography within the interval (tmode,tmax) is exclusively controlled by the properties of the axial velocity profile. For this reason, wide-bore chromatography provides the simplest and experimentally feasible transport experiment to assess fluid-dynamic properties within a microchannel. Conversely, the analysis of Taylor-Aris dispersion is inadequate to provide such description (see Supporting Information Figure S2 and related discussion). A first important issue that can be tackled is the possible occurrence of slip flow at the solid walls of the microchannel. Indeed, by enforcing the properties that φout(t) approaches kin kin φout (t) for Peeff f ∞ and by observing that φout (t) for slip flows is equal to kin φout (t) )

{

0 2

(1 + 2γ)/(2t )

t < tmin and t > tsup tmin e t e tsup (18)

log γ , 2

lim σout2 ∼

Peefff∞

1 2γ

(19)

This phenomenon is depicted in Figure 3B. Therefore, within the convection-dominated regime there is a major qualitative difference in the dispersive behavior of solute particles for large Peeff for slip and no-slip channel flows that is exclusively controlled by the local behavior of the velocity profile near the solid walls. For no-slip flows, the outlet variance is a monotonically diverging function of Peeff, while slip flows in microcapillaries are characterized by a saturating behavior of σout2(Peeff) toward a constant value that depends on the slip length γ. It is important to observe that this result applies to any finite length microchannel operating in the convection-dominated regime. In the region of validity of the Taylor-Aris scaling (Peeff < Pe*eff), no significant qualitative differences can be observed for slip and no-slip flows (see Figure 3B). This result can be easily proved theoretically (Supporting Information eq S17 and Figure S2). To sum up, moment analysis of outlet concentration profiles provides a chromatographic approach toward the estimate of possible slip conditions in microcapillaries, in a way that does not make use of direct velocimetric techniques and allows the use of nanometer particles, as in many applications involving wide-bore chromatography. DISCUSSION In this article we have discussed how the use of wide-bore chromatography can be extended to the analysis of velocity fields in microchannels. The operating conditions, typical of wide-bore chromatography (i.e., Peff > 10), and the corresponding occurrence of convection-dominated dispersion regimes can be exploited to discriminate between slip and no-slip flows. This is made possible by the occurrence of a convection-controlled boundary layer (in no-slip conditions) near the stagnation points of the velocity fields (close to the walls), which cause the divergence of the hierachy of moments associated with the outlet concentration profile. We have mainly focused on the detection of slip and no-slip flow conditions. In point of fact, the analysis of simple transport experiments in capillaries and microchannels can be extended to obtain more comprehensive information of the flow structure, as discussed in the Supporting Information (see Figures S6 and S7 and related discussion). This approach can be termed as chromatographic velocimetry, and intrinsically operates in the convection-dominated dispersion regime. The results obtained in Figures 1 and 3 refer to an impulsive inlet loading of solute particles that is uniform throughout the inlet section of the capillary. It is possible to exploit further the properties of the convection boundary layer, by considering localized inlet conditions, in which solute particles are injected close to the walls. A schematic representation of a chromatoraphic Analytical Chemistry, Vol. 81, No. 19, October 1, 2009

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Figure 4. σout2 vs Peeff for a 2D channel. Line (a) and symbols (O): no-slip flow; line (b) and symbols (b): slip flow γ ) 0.02. (A) Uniform inlet. (B) Localized inlet condition with w ) 0.04.

unit operating in this way is depicted in Figure S8 of Supporting Information). The device is just a T-junction structure; in this, the carrier flow is mainly injected through the major arm of the junction, while solute particles are injected from the minor arm close to one wall. Let w be the normalized width of the localized inlet. In a two-dimensional modeling of the channel, where ζ and η represent the nondimensional axial and transverse coordinates, this means that the localized inlet loading can be mathematically expressed as φ(t,ζ,η)|ζ)0 ) δ(t)/w for 0 < η < w and φ(t,ζ,η)|ζ)0 ) 0 for w < η < 1. Figure 4A depicts the behavior of the outlet variance σout2 in a two-dimensional model of the structure for a uniform inlet loading φ(t,ζ,η)|ζ)0 ) 1 in the case of no-slip (symbols O) and slip (symbols b) flows with a nondimensional slip length γ ) 0.02. Figure 4B refers to a localized inlet with w ) 0.04 (twice the slip length). It can be observed that a more significant difference in the outlet variance σout occurs for a localized inlet profile even for moderate values of the effective Peclet number in the range 10 < Peeff < 500, which is the typical range of Peclet values if one considers a microcapillary operating with nanometer particles (e.g., with diameter of order 30-50 nm). In point of fact, the use of a localized inlet not only extends the domain of application of wide-bore chromatography for fluid dynamic investigations (as those discussed throughout this article)

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but also increases the performance of this analytical technique when used for separation purposes. This aspect is beyond the scope of this article and will be discussed in future work. In this article we have discussed mainly the dispersion properties of cylindrical capillaries in the convection-dominated dispersion regime, by deriving the scaling of the first-order moment and of the variance of the outlet chromatogram from the properties of the outlet concentration profile. The occurrence of dispersion-dominated regime and its scaling properties (associated with the scaling of the hierarchy of moments as a function of Peeff) are a new phenomenology that generalizes Taylor-Aris dispersion theory in finite length channels. A first experimental evidence of the transition from Taylor-Aris dispersion to the convection-dominated regime has been presented by Harada et al.9 (Figures 5 and 6 in the cited article) and by Okada et al.10 (Figures 1 and 3 in the cited article) and constitutes the essence of wide-bore hydrodynamic separation. In this article we have shown that this chromatographic technique can be used in nonstandard way, as an analytical technique for quantifying surface effects (occurrence of slip) and flow characterization in microchannels. Indeed, a general boundary-layer theory of dispersion in widebore microcapillaries possessing a generic cross section has been developed for no-slip Stokes flows.18 Depending on the geometry of the cross section, it is possible to obtain an asymptotic scaling of the variance of the form σout2 ∼ Peeffβ, where the exponent β attains values in the interval (1/3,2). The occurrence of this scaling, with an exponent β > 1/3 (as for cylindrical capillaries), does not alter the results of this article as it regards the use of chromatographic equipment for discriminating slip and no-slip flow conditions in microchannels. SUPPORTING INFORMATION AVAILABLE Additional details and figures for mathematical formulation, numerical issues, Taylor-Aris dispersion and slip flows, kinematic analysis, structure and properties of the outlet chromatograms, chromatographic velocimetry, and wall-enhanced wide-bore chromatography. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review May 12, 2009. Accepted August 19, 2009. AC901504U (18) Giona, M.; Adrover, A.; Garofalo, F.; Cerbelli, S. Phys. Fluids 2009, submitted.