WO2006052882A1

WO2006052882A1 - Formation of eddies in constrained fluidic channels and uses thereof

Info

Publication number: WO2006052882A1
Application number: PCT/US2005/040282
Authority: WO
Inventors: Stella Park; Chris Russo; Daniel Branton; Howard Stone
Original assignee: President And Fellows Of Harvard College
Priority date: 2004-11-09
Filing date: 2005-11-07
Publication date: 2006-05-18

Abstract

The invention features devices that enable the formation of eddies in fluidic channels. In general, the devices include a fluidic channel having one or more constrictions (i.e., regions of reduced cross-sectional area), and the application of an EOF induces eddy formation proximal to a constriction. Eddy formation may be used in a variety of applications including fluidic mixing, sensors, and reactors.

Description

FORMATION OF EDDIES IN CONSTRAINED FLUIDIC CHANNELS AND USES THEREOF

BACKGROUND OF THE INVENTION

The invention relates to the field of fluidics.

Many particle characterization techniques involve passing a particle through a constriction or aperture using various driving forces and detection methods. In the first practical and successful instrument for counting and analyzing cells, the Coulter Counter, a single aperture separates two reservoirs of ionic solution where the cells from one reservoir are driven into a second reservoir by hydrostatic pressure or electrical field; the resulting change in electrical conductivity is used to count and analyze the cells (W. H. Coulter. Technical report, U.S. Patent 2,656,508, 1953.). With fabrication technologies used in Micro-Electro-Mechanical- Systems, miniaturization of such instruments into microfluidic devices is possible. Unlike traditional fluidics in which volumes of fluids are usually driven through channels using pressure gradients, in microfluidic (and increasingly nanofluidic) systems a new set of engineering challenges arise for the integration and quantitative understanding of essential components such as mixers, pumps, sensors, or reactors. To address some of these challenges, many have pointed to microfluidic flows driven by electrokinetics, e.g., electroosmosis. Electroosmotic flow (EOF), or the flow of solvent with respect to a charged surface in the presence of an applied electric field, offers many advantages for microfluidic environments and has been the subject of considerable investigation (e.g., see C. L. Rice and R. Whitehead. J. Phys. Chem., 69:4017-4024, 1965; and R. J. Gross and J. F. Osterle. J. Chem. Phys., 49:228-234, 1968). In addition to the classical applications in colloid and interface science (R. J. Hunter. Foundations of Colloid Science, volume 1. Oxford Science, 1986), EOF has been investigated for use as a microchannel mixer (T. J. Johnson, D. Ross, and L. E. Locascio. Anal. Chem., 73:3656-61, 2002; and A. D. Stroock, M. Week, D. T. Chiu, W. T. S. Huck, P. J. A. Kenis, R. F. Ismagilov, and G. M. Whitesides. Phys. Rev. Lett., 84:3314-3317, 2000), in binding enhancement between a biological pore and a biological molecule (L.-Q. Gu, S. Cheley, and H. Bayley. Proc. Natl. Acad. Sci. USA, 100:15498-15503, 2003), as an artificial synaptic transmitter (M. C. Peterman, J. Noolandi, M. S. Blumenkranz, and H. A. Fishman. Anal. Chem., 76:1850- 1856, 2004), and as an electrokinetic battery (J. Yang, F. Z. Lu, L. W. Kostiuk, and D. Y. Kwok. J. Micromechanics Microengineering, 13:963-970, 2003). The revival of the study of this classic electrokinetic phenomenon is directly related to the increasing importance of microfluidic devices through which laminar low-Reynolds-number flows dominate (H. A. Stone, A. D. Stroock, and A. Ajdari. Annu. Rev. Fluid Mech., 36:381-411, 2004). In particular, we note the increasing importance of studying EOF in nanoscale fluidic devices such as a nanopore (J. Li, D. Stein, C. McMullan, D. Branton, M. J. Aziz, and J. A. Golovchenko. Nature, 412:166-169, July 2001).

Generally, EOF exhibits a uniform velocity distribution outside a thin charged layer adjacent to the boundary. However, it is known that surface charge variations can give rise to recirculating flows (A. D. Stroock, M. Week, D. T. Chiu, W. T. S. Huck, P. J. A. Kenis, R. F. Ismagilov, and G. M. Whitesides. Phys. Rev. Lett., 84:3314-3317, 2000; J. L. Anderson and W. K. Idol. Chem. Eng. Commun., 38:93-106, 1985; A. Ajdari. Phys. Rev. Lett., 75:755-758, 1995; D. Long, H. A. Stone, and A. Ajdari. J. Colloid Interface Sci., 212:338-349, 1999; S. Ghosal. J. Fluid Mech., 459:103-128, 2002; C. A. Keely, T. A. A. M. van de Goor, and D. McManigill. Anal. Chem., 66:4236- 4242, 1994; J. G. Santiago. Anal. Chem., 73:2353-2365, 2001; and A. E. Herr, J I. Molho, J. G. Santiago, M. G. Mungal, and T. W. Kenny. Anal. Chem., 72:1053-1057, 2000). Non-uniform charge distributions can be imposed (through, for example, patterned adsorption of differently charged molecules (A. E. Herr, J. I. Molho, J. G. Santiago, M. G. Mungal, and T. W. Kenny. Anal. Chem., 72:1053-1057, 2000)), or induced on polarizable (e.g., metal electrode) surfaces with uniform or non-uniform applied electric fields (see for example M. Z. Bazant and T. M. Squires. Phys. Rev. Lett., 92(6):0661011-4, 2004). It has also been theorized that transverse electroosmotic flows in slab geometries are also possible where an induced pressure gradient along the direction normal to the applied electrical field leads to a transverse recirculation of the fluid (A. Ajdari. Phys. Rev. E, 65:016301, 2001). These cases of eddy formation are the result of a nontrivial surface heterogeneity, which are difficult or costly to fabricate.

Thus, there is a need for new (micro)fluidic systems for producing eddies in channels.

SUMMARY OF THE INVENTION The invention features devices that enable the formation of eddies in fluidic channels. In general, the devices include a fluidic channel having one or more constrictions (i.e., regions of reduced cross-sectional area), and the application of an EOF induces eddy formation proximal to a constriction. Eddy formation may be used in a variety of applications including fluidic mixing, sensors, and reactors.

In one aspect, the invention features a fluidic system including a fluidic channel having first and second ends and either a constriction, or a location capable of being constricted, disposed between the first and second ends, wherein the walls of the fluidic are charged. Such a fluidic system may further include one or more additional constrictions or locations capable of being constricted. In addition, a fluidic system may include a plurality of fluidic channels, e.g., that are fluidically connected, each having one or more constrictions or locations capable of being constricted. Each constriction in a channel will divide the channel into proximal and distal sides relative to that constriction. A fluidic system may further include one or more voltage sources operationally coupled to each fluidic channel and capable of generating an electroosmotic flow (EOF) between the first and second ends of the channel. When a plurality of channels is present, the voltage source may be able to control the application of voltage to at least two of the channels independently. In one embodiment, a fiuidic system further includes one or more electrodes disposed proximally to a constriction, wherein application of a voltage to the one or more electrodes alters the surface charge on the walls of the fluidic channel, e.g., in a manner capable of augmenting or disrupting the formation of one or more eddies proximal to the constriction. A fluidic system may also include a detector for monitoring species in the fluidic channel.

In various embodiments, a fluidic channel has a height or width of less than 1 nun, e.g., less than 1 μm. A fluidic channel may also include a carbon or silicon nanotube. A constriction in a fluidic channel may include a nanopore. A constriction may also be shaped such that one or more eddies form on only the proximal or distal side of the constriction. A constriction may also have a symmetric or asymmetric cross-sectional geometry. In one embodiment, a constriction blocks at least 5% of a channel. Moreover, a fluidic system may be capable of forming at least one central eddy, either in the presence or absence of a perimeter eddy, or at least one perimeter eddy, either in the presence or absence of a central eddy, under EOF. The surface charge of the walls of a fluidic channel may also be modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, salt concentration, absorption of chemical species, or pH. The invention further features a method of trapping a species employing a fluidic system, as described herein. The method includes inducing the formation of one or more eddies proximal to a constriction in a fluidic system by generating an EOF, and allowing a first species to traverse a fluidic channel, wherein, when the first species contacts an eddy, at least a portion of the first species becomes disposed within the eddy. The trapped species may subsequently be released, with or without intervention, e.g., by altering a voltage applied to an electrode proximal to the constriction and as described herein. A property of the first species may also be detected as it traverses a fluidic channel. A method may further include applying energy, e.g., electromagnetic radiation, electrical energy, magnetic energy, or thermal energy, to the first species in contact with the eddy, e.g., wherein the first species undergoes a chemical or physical change after the application of the energy. In addition, the first species may traverse the channel through the effect of bulk solvent flow under EOF or electrophoresis or by other active means, such as optical trapping and as described herein.

A method may also include introducing a second species into the fluidic channel and contacting the first and second species in the eddy, e.g., so that the first and second species chemically or physically interact or mix. Any interaction between the first and second species may also be detected, and the product of the contacting of the first and second species may be released from the eddy. Exemplary species include a polymer, a particle (e.g., a cell or virus), a nucleic acid, amino acid, sugar, lipid, or an ion (e.g., to alter the salt concentration or pH). A first or second species may also include a label, e.g., a fluorophore, a chromophore, a radionuclide, an antibody, or a nucleic acid. When the method employs a system having more than one constriction, either in a single channel or in a plurality of channels, the first species may contact an eddy disposed proximal to each constriction, or a subset thereof, as it traverses the fluidic channel. Species may also contact multiple eddies proximal to the same constriction. In one embodiment, the first species contacts another species, either the same or different from the first species, at each eddy contacted, e.g., wherein the first species chemically or physically interacts with each contacted species.

In various embodiments, only a portion of the first species is disposed within an eddy. In these embodiments, portions of the first species may be trapped in two or more eddies, e.g., both on the proximal or distal side of a constriction or with a portion of the species traversing one or more constrictions. In other embodiments, the method further includes altering the shape or size of the constriction either before or after the first species contacts an eddy, e.g., either to cause the formation or alteration (increase or decrease) in size of an eddy.

The invention further features methods for designing fluidic systems capable of forming eddies in desired locations and for determining the location of eddies in a system by employing the theoretical treatment described herein.

By "central eddy" is meant an eddy disposed proximal to the geometric center of the cross section of the channel.

By "perimeter eddy" is meant an eddy disposed distal to the geometric center of the cross section of the channel. Typically, perimeter eddies will be disposed proximal to the side walls of the channel.

By "cross-section" is meant the cross-section perpendicular to the direction of fluid flow. For example a cylindrical channel has a circular cross- section.

The terms "proximal side" and "distal side" of a constriction refer to sides of the constriction relative to the direction of fluid flow. For example, the proximal side of a constriction may refer to the segment of a fluidic channel divided by the constriction that is closer to the source of fluid flow, and the distal side may then refer to the segment of the channel farther from the source of fluid flow. When a plurality of constrictions is present, a single segment of a channel may be on the proximal side of one constriction and the distal side of another. Other features and advantages will be apparent from the following description and the claims. BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 : (a) Diagram of the device used to observe particles interacting with the eddies. For the particular case illustrated here, the channel has a negative surface charge with pressures P₁ = p₂ and electrical potentials V₂ > V₁, resulting in the net flow indicated by the black arrows, and circulations on both sides of the constricted region, (b) Photograph of the drawn glass capillary used to observe the electroosmotically induced eddies. The scale bar is 1.0 mm, and the region observed in c is outlined by the dashed box. (c) Time lapse image of a particle trapped in an electroosmotically induced eddy. Arrow 1 shows the direction of net electroosmotic flow and highlights the path of a particle which is not trapped in the eddy but moves with the net flow. Arrow 2 indicates a circulating particle that has moved through several circulations of the eddy. The inset traces a single elliptical traverse of the particle.

Figure 2: Schematic of an axisymmetric cylinder with radial shape variation viewed as a cross section along the Z-axis. The "perimeter" eddy paths are mapped out in grey with the direction of the eddies indicated by the curved arrows. The larger arrows indicate the overall direction of EOF. The perturbation wavelength is L, and the unperturbed radius of the tube is aø.

Figure 3: (a): the dimensionless induced pressure gradient ( dtl I dZ ) and (b): the electrical body force φ_o(dχ/dZ ) for δ = 0.4, a= 0.41, and k ≡κa_Q= 10. Dashed lines show the outline of the constricted channel where R is the dimensionless radius, and Z the dimensionless axial distance. Note that R = O corresponds to the center of the channel. The color bars next to the contour plots show the magnitudes of these force densities. Extra contours with the corresponding numerical values are plotted near Z = 0.1 and 0.9 for both a and b to show the central region in which the pressure gradient is larger than the electrical body force. Figure 4: Streamlines for k = 10, and (a): δ = 0.2, a = 0.45 showing "central" eddies; (b): δ = 0.4, a= 0.4 showing "perimeter" eddies; and (c): δ = 0.4, a = 0.41 showing "central" and "perimeter" eddies. Dashed line shows the outline of the constricted channel and the arrows indicate the flow direction. A scan in the thin region along the surface of the channel in a showed the absence of "perimeter" eddies, and a scan in the empty region towards the center in b revealed an absence of "central" eddies.

Figure 5: An eddy "phase" diagram for various amplitudes a and aspect ratios (inverse perturbation wavelength) δ for k = 10 solid lines and k = 30 dashed lines. The circles are the calculated boundary points, and the lines

(solid for k = 10 and dashed for k = 30) represent approximate boundary lines. The diagram shows four regions: with perimeter eddies, with central eddies, both eddy types, and no eddies (unidirectional flow). The arrow indicates the shift in a and δ values needed to observe eddies for the larger k value.

DETAILED DESCRIPTION OF THE INVENTION We have discovered that simple geometric variations, e.g., constrictions, can give rise to eddies under EOF in a fluidic channel. Devices including these constrictions are useful for trapping species in one or more eddies. Such trapping may be useful for the analysis of species, the inducement of chemical or physical change, the mixing of species under laminar flow conditions, and the retention at some position in the fluidic channel of beads, chips, or other solid phase components around which and through which the fluid in the fluidic channel can flow.

Theory

We provide for electroosmotic flow using an axisymmetric, constricted geometry, e.g., a bottleneck or hourglass shape. For the purpose of modeling our theory, we have used a constant surface charge density along the channel. In this system, even in the absence of an applied pressure difference, there is an induced pressure gradient that arises with the application of an external electric field, E. This induced pressure gradient is a combination of resistive and osmotic pressure effects. The resistive pressure gradient is due to the physical constriction opposing the fluid flow. The osmotic component arises because both the equilibrium surface electrostatic potential, and an "excess" electric potential that arises because of EOF, maintain an ion concentration gradient normal to the surface. As a consequence of this induced pressure gradient, we find eddies (referred to as "recirculating cellular flow" in A. D. Stroock, M. Week, D. T. CMu, W. T. S. Huck, P. J. A. Kenis, R. F. Ismagilov, and G. M. Whitesides. Phys. Rev. Lett., 84:3314-3317, 2000) in certain constricted geometries without requiring any variation of surface charge densities. We utilize a perturbation expansion method in the field strength and go beyond the usual lubrication approximation to represent the effects of the geometric variations more fully. A geometric criterion for the existence of eddies in electroosmotic flow and a phase diagram summarizing the phase regimes with eddies are provided.

In most theoretical approaches to EOF, the Debye screening length κ^~ι is assumed to be much smaller than the mean tube radius ao, and there is an effective slip velocity, μ_EE along the surface, where μ_E is the electroosmotic mobility, which is proportional to the surface charge density σ or the zeta (ς) potential, and E is the applied external electric field. If there is no applied pressure, and the solution and surface properties are uniform and constant (e.g., μ_E = constant), then we show that the fluid velocity at any position x is everywhere proportional to the electric field, i.e., u(x) = μ_EE(x). The pressure distribution then remains constant and unaffected by E since both fields (u and E) are incompressible (V ^• u = 0 and V ^• E = 0). Nevertheless, there are situations in which this collinearity of u and E no longer holds true. For example, when the electroosmotic mobility μ_E is no longer constant, as is the case when the surface charge density varies, then we believe that the pressure distribution is affected by the applied field (J. L. Anderson and W. K. Idol. Chem. Eng. Commun., 38:93-106, 1985). Also, the influence of finite permittivity of the channel boundaries, which gives rise to a small normal component of the electric field at the wall, has been argued to produce pressure gradients in EOF that lead to eddies (P. Takhistov, K. Duginova, and H.-C. Chang. J. Colloid Interf. Sd., 263:133-143, 2003).

It is expected then that perturbations such as a radial shape variation and/or a finite Debye layer thickness may disrupt the perfect alignment of u and E. Here we provide an explicit calculation that shows the disruption of the alignment of u and E₅ and predict eddies in the center and perimeter of a channel based on variations of the wavelength and amplitude of channel radius and Debye length.

In the following theoretical treatment, we consider cylindrical geometries and so utilize a cylindrical (r, z) coordinate system. The theory is however applicable to arbitrary channel geometries. We assume a constant surface charge density σ along the channel and a constant applied potential difference V across a distance L representative of the constriction (see Fig. 2). The radius of the channel is denoted a(z) and the unperturbed radius is a₀. The usual discussion of electroosmosis assumes that the equilibrium ion distribution is unperturbed by the applied electric field E and, conversely, that the applied electric field is unperturbed by the ion distribution. In a cylinder with radial shape variations, the changes in the applied field and the ensuing fluid motion perturb the charge distribution. At steady state, we can write the mass conservation equation for the ith ion species 4 as

V • |w'v - ω'(k_BTVn^ι +

0 (1) where ω¹ denotes the ion mobility, ez_; is the charge on an ion, n¹ the number density of the ith ionic species (p_e = ^'∑^.__ιez_in^ι ), and φ is the total electric potential (D. A. Saville. Ann. Rev. Fluid Mech., 9:321-337, 1977). The first term on the left-hand side is due to the ion convection with the fluid velocity v, and the second and third terms are relative motion of the ions due to thermal diffusion and electric forces, respectively.

Because we allow for the coupling between the electric field E and the ion distribution, the ensuing governing equations become prohibitively complicated. To tackle this complication, we will follow the formalism of Saville (D. A. Saville. Ann. Rev. Fluid Mech., 9:321-337, 1977) and Sherwood and Stone (J. D. Sherwood and H. A. Stone. Phys. Fluids, 7:697-705, 1995) and write a perturbation expansion — of the fluid velocity v, electric potential φ , the number density of the ith ion species n¹, and therefore the charge density P_e — in the dimensionless field strength β = eV/k_BT. It is most common to study the low-field limit, i.e., β < 1, so we assume {γ,φ,nⁱ,p_e} = {O,φ_o,n_o ⁱ ,p_o} + β{u,φ_ι,n₁',p₁} +A where we have used the set notation and each variable can be written out independently, e.g., φ = φ_Q + βφ_x +Λ . The subscript 0 refers to an equilibrium quantity and 1 refers to a perturbation (due to the applied field) from that quantity. We recognize at the outset that the applied electric field though often small in uniform sections, can be large in constrictions, and so the approximation of β < 1 cannot be quantitatively accurate in all cases, but we believe that it captures the most essential qualitative features, as shown below.

Using the continuity equation for the fluid motion (V ^• v = 0), and assuming small equilibrium electric potentials and small Peclet numbers (i.e., ion convection negligible compared to thermal motion), Eq. 1 becomes at 0(1)

O = V - (k_BIVn_o ⁱ + ez_in_oVφ_o) (2) This result provides a Boltzmann distribution of ions, n₀ ^l = n_∞' exp(- ez$₀ /k_BT), and via Gauss's law (V- E = pje, where e is the dielectric constant of the solution) leads to the usual Poisson-Boltzmann equation, or the Debye-Hϋckel equation with the approximation of small equilibrium electrostatic potentials (R. J. Hunter. Foundations of Colloid Science, volume 1. Oxford Science, 1986), i.e.,

lk_BT\ «1 so that

V ²φ₀ = κ²φ₀ , where κ² =

/(εk_Bl) . As usual, K^'1 represents the

Debye screening length and p₀ = -εκ²φ_Q . At O(β), the ion-transport equation (Eq. 2) becomes 0 = ez_in₀ ^ι 7²φ_ι + ez_tn[V²φ₀ + k_B1V²n[ (3)

Consistent with the small equilibrium potential approximation, the second term on the right-hand side of Eq. 3 can be neglected relative to the third term. So we multiply Eq. 3 by ez/kβT and sum over i to obtain

VV₁ H- V² ^ = O

K ε or V²χ = 0, where χ = φ₁ + p_ι l\κ²ε) now plays the role of the effective applied electric potential.

Using the above results, the total electrical body force on the fluid p_eΕ = -p_eVφ can be expanded to give

This result can be substituted into the Navier-Stokes equation

( - Vp + 77V² v + pβ, = 0 , where 17 is the fluid viscosity) which describes the momentum transport in the fluid. Writing only the terms of order β, i.e., only the terms that arises due to EOF, we obtain

0 = -Vn + 77V²U + εκ²φ_Q Vχ (5) where the total pressure p = p_o + pTL +Λ and p₀ = -εκ²φ² is the equilibrium osmotic pressure that results in the absence of hydrostatic pressure.

The above equation can be made dimensionless and we specifically consider the case with radius a(z) and the unperturbed channel radius a₀. Thus we define A(Z) = a(z)/a₀, R = r/a₀, Z = z/L, π = π l{κσV), φ₀ = εκφ₀ /σ, χ = χ/V, U_z = u₂ηL /(κσVa% ). The volume flux

Q = 2π \ u_zrdr is made dimensionless according to Q ≡ QηLlψalσV). We next wish to treat explicitly geometric variations in shape and so introduce δ ≡ a_o /L , which represents the magnitude of the assumed slow variations, |da/dz| « 1. The applied electric potential has boundary condition, χ → 0 at Z -»0 and χ -> 1 at Z -» 1, the pressure boundary condition, π -> 0 at Z → 0, 1, and the radial velocity U_R -» 0 at Z ->■ 0, 1. With Eq. 5, the fluid continuity equation V -U = O, and the above boundary conditions, π , U_R, and U_z can be determined using the lubrication (or the long wavelength, i.e., δ « 1) approximation, e.g., (S. Ghosal. J. Fluid Mech., 459:103-128, 2002; and A. Ajdari. Phys. Rev. E, 65:016301, 2001), though here we retain more of the geometric complexity by working beyond the leading-order terms.

We can expand the variables, defined above, perturbatively in δ,

{U_Z,U_R ,Π,X,Φ₀,Q}= ^_z,o,U_Rfi,Ω_Qd_Q,fi\Q_o}+δ²{u_Zt2,U_Rt2,U₂,χ₂,φ^,Q₂}+A

where we have again used the set notation. We first find the electrical components, φ₀ and χ , which can be determined from the Debye- Hϋckel equation and boundary conditions, and then solve for the dimensionless velocity field U. The Debye-Hϋckel equation for φ₀ (R, Z; δ),

^J where k ≡ κn₀ , can also be written as an expansion in δ². The constant surface

charge density boundary condition states that ^dR R=A(Z) = k and ^δR R=A(Z) = 0 for which the solution to O(δ ) is

φ₀(R,Z;δ) = ^Α + δ²[B₁I₀(kR) + B₂RI_ι(kR)]+ O(δ⁴)+- - (7)

I₁(M) where Bi and B2 can be found through Eq. 6 and the boundary conditions. From Eq. 4, i.e., V²χ = 0 , and the condition that the surface is impermeable to the ions, i.e., n • V² χ - 0 , the effective electric field in the Z- direction is then

where A'= dA/dZ, A"= U²AJdZ², and b₀ is a constant.

In order to complete the solution, we find it convenient to work in terms of average quantities. For example, the average potential difference is,

— -~ι— I which, using Eq. 8, defines the integration constant b₀ = f dZ I A² , which

involves the entire shape variations.

We can substitute Eqs. 7 and 8 into the Navier Stokes equation (Eq. 5) and separate the R- and Z-components: an i a 3²U

+ δ² — dX

^~(RU_R) (10) dR dR R dR dZ^{2 Ψo} dR

which can be further separated into terms of the same order in δ².

The leading-order term in Eq. 10 implies an R-independent π ₀ , and therefore the leading-order term in Eq. 11 can be integrated with respect to R (and the integration constant is determined by imposing the no-slip condition) to yield

where A(Z) brings in the shape variation. Eq. 12 is the solution to the usual pressure- and electrically-driven flow in a locally cylindrically geometry (C. L. Rice and R. Whitehead. J. Phys. Chem., 69:4017-4024, 1965), though here it is applicable at leading order in A(Z). With the pressure boundary condition imposed in the form

jV— W = 0 , where (dflldz) = (21 A²) ζ [dill dz) RdR, the volume flux, and

the induced pressure gradient can be calculated

(15)

(14).

The above equations are substituted into Eq. 12 to obtain the axial velocity distribution, which for R = O, reduces to

I₀(M) 1 (I₀(M) -I) U (0 7) - ²®^{0 4b}° πA² k²A² 2I₁(M) M k 2²A _Λ 2 I₁ (M)

(15).

In the next section we will show that the leading-order velocity field (Eq. 12) can change sign even in the small Debye length limit, i.e., k » 1, and that reversal in sign of the velocity at R = 0, Uz_,o(O,Z), is indicative of eddies in the center of the channel.

To proceed to higher order in the shape variations, the O(δ²) term in Eq. 10 can be integrated using the continuity equation i[d(R U_R>0 )/ di?J) = -R{dU_zfi )/ dZ . The Z-derivative of the result can be substituted into Eq. 11 along with the boundary condition U_Z;2(R = A) = 0, and the pressure boundary condition is then used to obtain U_z>2. The details are straightforward but laborious and are not reported here, but are used in the calculations below. We expect the perturbation expansion, which is in powers of δ², to be a good representation of the physical situation since retaining just two terms has errors of O(δ ). To illustrate the basic effect of a geometric constriction in EOF, we take a cylinder that has a sinusoidal perturbation in channel radius (see Fig. 2), i.e.,

A(Z) = αcos(27rZ) + (1 - α),

(16) where a is the amplitude (0 < a < 1/2) and 0 < Z < 1. The geometry is thus characterized by a dimensionless wavenumber (or aspect ratio) δ and amplitude a. The axial velocity U_z, the induced pressure gradient ( δtl I dZ ), and the electrical body force, which in dimensionless terms is φ_o(dχ/dZ) , can be calculated analytically. We expect that as the amplitude a and the wavenumber δ change, both the induced pressure gradient and the effective electrical body force, as well as the velocity distribution, will be affected. The magnitudes of each will depend on the dimensionless Debye parameter k ≡ κn₀. Note that below we consider values of δ as large as 0.4-0.5, which may be considered large for the lubrication approximation, but the expansion is developed in powers of δ² with an error O(δ⁴) which is then expected to be 2.5-6.3%. In this long wavelength approximation, the amplitude a of the constriction can have any value.

There are two effects that arise from the presence of the constriction in the channel. One is that there is an induced pressure gradient due to the geometric resistance to flow at the constriction. This resistance decreases the flow rate and alters the uniform EOF profile to that similar to pressure- and electrically-driven flows in a straight uniform channels, where an applied pressure flow can oppose the electrically-driven flow. This effect can be seen in the leading-order velocity expression (Eq. 12). The second effect is from the redistribution of ions from the static equilibrium ion distribution due to flow through the constricted geometry, and arises at O(δ²) through the χ term.

These effects can be illustrated with plots of the dimensionless induced pressure gradient dfl/dZ and the dimensionless electrical body force φ_odχ/dZ . In Fig. 3 we report these quantities for δ = 0.4, a. = 0.41, and κn₀ ≡ k = 10. Here δ and a are large, and k is small enough (i.e., the Debye length is large enough), so that the dimensionless induced pressure gradient is comparable to the electrical body force.

The result of the competition between these two force densities is illustrated in Fig. 4 where streamlines are plotted for various a and δ values for k = 10, each of which shows eddies either at the center and/or near the boundary of the channel. If the induced pressure gradient is larger than the electrical body force near the center of the channel, then a reversal of flow is expected in this region, similar to combined pressure- and electrically-driven flows in straight channels (C. L. Rice and R. Whitehead. J. Phys. Chem., 69:4017-4024, 1965). For example, for Z < 0.1 and Z > 0.9, the pressure gradient dfl/dZ is larger in magnitude than the electrical body force φ_o(dχ/dZ) at R = 0 (Fig. 3). There exists then a recirculating "central" eddy in this region (Fig. 4a). The direction of the resulting recirculation should be counter to the net flow direction near the center and aligned with the net flow direction near the surfaces. Because the induced pressure gradient must disappear in the unperturbed portion of the channel, the size of this central recirculating eddy must roughly correspond to the size of the perturbation wavelength. On the other hand, as shown in Fig. 3, near the surface R «A, and for

0.25 < Z < 0.4 and 0.6 < Z < 0.75, the pressure gradient is also comparable or larger in magnitude than the electrical body force. Then, the local flow will oppose the net flow direction and there exists a recirculating "perimeter" eddy in this region as can be seen in Fig. 4b. The size of such a perimeter eddy should correspond to the size of the region with the largest induced osmotic pressure distribution. Further, if the pressure gradient is large enough in both regions, then we should see both types of eddies as in Fig. 4c.

A phase diagram summarizing the existence of both central and perimeter eddies as a function of the amplitude (α) and the wavenumber (δ) parameters is plotted in Fig. 5 for k = 10 and 30. As the arrow in Fig. 5 indicates, the parameter regimes containing eddies decrease in size as k increases (i.e., as the Debye layer becomes thinner).

Since the central eddies are predominantly due to the leading-order induced pressure term, we can evaluate the leading-order force densities and obtain a criterion for one aspect of eddy formation. Using Eqs. 13 and 14, the volume flux can be estimated as Q₀ « [2τώ₀ Zk² Jl /2 - I /[M_n^ )], where A_n^_n = (1 - 2ά) is the minimum channel radius for the specific shape given by Eq. 16, which leads to dfl₀ IdZ « (l62>₀ Zk³A⁴XlZA_n^_n - IZA)). This force density should be larger than the maximum electrical body force (i.e., bo/ A²) to obtain an estimate for a "central eddy criterion":

The left-hand term has a maximum for A = (3/2)A_mj_n. The typical minimum constriction radius A_n^_n that meets the central eddy criterion is then

*_ ^<w w which states that as the Debye length decreases, i.e., as k increases, the constriction radius must decrease to obtain a central eddy. Since this is a leading-order estimate, it only involves the constriction amplitude, and not the perturbation wavelength. Hence, only the central eddies are present in Fig. 5 for small values of δ. We note that the prediction Eq. 18 is in good agreement with the results in Fig. 4, e.g., for k = 10, we predict central eddies when a > 0.43 and for k = 30, when a > 0.48.

As the wavenumber increases, the second-order effect due to the O(δ²) induced pressure term becomes more important and perimeter eddies appear. We attribute this pressure effect to the redistribution of ions during EOF; increasing k, or decreasing the Debye length, confines this redistribution effect to a thinner layer around the cylinder surface. Indeed, the phase region containing perimeter eddies in Fig. 5 is smaller for k = 30 than for k = 10 and the amplitude a. and wavenumber δ values required for the perimeter eddies increase with k.

In general, given the qualitative shifts observed theoretically, we predict that eddies should be only observable for large amplitude perturbations as the

Debye length becomes small. Further, we predict that only the central eddies should be observable in the small Debye length.

The Debye length in current microfluidic systems is always considered much smaller than the channel width, i.e., k ≡ κa₀ range from O(10⁵) to O(IO ). With fabrication of even smaller channels, the values of k decrease. For example, in experiments performed with ion-beam sculpted nanopores at IM

KCl (J. Li, D. Stein, C. McMullan, D. Branton, M. J. Aziz, and J. A.

Golovchenko. Nature, 412:166-169, July 2001), k «100 and A_1111n =0.05, which corresponds to a =0.48 if we assume a sinusoidal perturbation. Given such a marked radial variation and moderate value of k, we predict that eddies exist in this system.

Devices

In the simplest embodiment, a device of the invention includes a fluidic channel and a constriction located in the channel. The constriction will divide the channel into two compartments, denoted proximal (i.e., closer to the source of fluid flow) and distal (i.e., farther away from the source of the fluid flow). Although the discussion above focuses on channels having cylindrical cross- sectional symmetry perpendicular to the direction of fluid flow, eddy formation may occur in fluidic channels having other cross-sectional geometries, e.g., arbitrary, oval, square, triangular, or rectangular. The size (e.g., the amount the channel blocked) and shape (e.g., the cross-sectional geometry of the constriction) of a constriction may be fixed, or the size or shape of the constriction may be variable, either reversibly or irreversibly. The surface charge of the channel may be substantially uniform or may vary, e.g., the surface charge proximal to the constriction may differ from the rest of the channel or the surface charge may be asymmetric with respect to the cross- sectional geometry perpendicular to fluid flow. Devices of the invention may also include the presence of multiple constrictions in a single channel or the presence of multiple channels, e.g., that are fluidically connected, having one or more constrictions. Devices may be fabricated for eddy formation in the central portion of the channel or the perimeter of the channel, or both, as described above. The device may also be configured to allow for eddy formation on one, e.g., proximal or distal, or both sides of a constriction. In devices having eddies on only the proximal or distal side of a constriction, the change from the initial diameter to the constriction diameter can be extreme (a large) on either the proximal or distal side of the constriction, and the change very gradual (a small) on the other. In addition, eddy formation may be asymmetric with respect to the cross-sectional geometry of the channel, e.g., in devices that contain constrictions that have asymmetric cross-sectional geometries. In one example, if the cross-sectional geometry of a constriction is arbitrarily divided in half, and one half has a relatively large a, and the other half has an a of zero, then eddies will form on the half with large abut not the half with zero a. Other variations in the cross-sectional distribution of eddies can be formed by appropriate shaping of the constriction. Cross-sectional asymmetry in eddy formation may also be achieved in devices having cross- sectional variation in surface charge proximal to the constriction. The largest dimension of an eddy is, for example, at most 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 66, 60, 65, 70, 75, 80, 85, or even 90% of the width or height of a channel. The smallest dimension of an eddy is, for example, at most 0.1, 0.5, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 66, 60, 65, 70, 75, 80, 85, or even 90% of the width or height of a channel.

In addition to the constriction, a fluidic channel may include one or more inlets and outlets. Inlets and outlets may also be fluidically connected to a fluid reservoir. A fluidic channel may have one or more inlets and outlets on the proximal or distal sides or both of a constriction. Such inlets may be employed to introduce or remove species, e.g., ions or reactants, on one or both sides of a constriction. Devices may further include additional elements as described herein. In one embodiment, the height or width or both of the fluidic channel is less than 1 mm, e.g., less than 750 μm, 500μm, 400 μm, 300 μm, 200 μm, 100 μm, 50 μm, 25 μm, 10 μm, 5 μm, 1 μm, 500 nm, 400 nm, 300 nm, 200 nm, 100 nm, 50 nm, 25 nm, 10 nm, 5 nm, or even 1 nm.

The constriction may block at least 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75 , 80, 85, 90, 95, or even 99% of the cross-section of the channel.

Fabrication. A variety of techniques can be employed to fabricate a device of the invention, and the technique employed will be selected based in part on the material of choice. Exemplary materials for fabricating the devices of the invention include glass, quartz, silicon, steel, nickel, silicon nitride, aluminum oxide, silicon oxide, poly(methylmethacrylate) (PMMA), polyamide, polytetrafluoroethylene, polycarbonate, polystyrene, polyethylene, polyolefms, epoxy resins, poly(ethylene glycol), silicones (e.g., poly(dimethylsiloxane) (PDMS)), carbon nanotubes, silicon dioxide nanotubes (Fan et al., J. Am Chem Soc. 2003 125:5254-5), and combinations thereof. Devices may include elastomeric materials, rigid materials, or both. Other materials are known in the art. In one embodiment, the device is fabricated, at least in part, from a transparent material to allow for visual inspection or optical measurements (e.g., fluorescence or absorbance) or manipulations (e.g., photochemical modification). Devices of the invention may also include thermally or electrically insulating or conducting materials or magnetic materials (e.g., ferrofluids). Methods for fabricating channels and/or constrictions in these materials are also known in the art. These methods include, photolithography (e.g., stereolithography or x-ray photolithography), molding, casting, embossing, silicon micromachining, wet or dry chemical etching (e.g., reactive ion etching or deep reactive ion etching), milling, diamond cutting, Lithographie Galvanoformung and Abformung (LIGA), electroplating, and ion beam sculpting (J. Li, D. Stein, C. McMullan, D. Branton, M. J. Aziz, and J. A. Golovchenko. Nature, 412: 166-169, July 2001). For example, for glass, traditional fabrication techniques of photolithography followed by wet (KOH) or dry etching (reactive ion etching with fluorine or other reactive gas) can be employed. Techniques such as laser micromachining can be adopted for plastic materials with high photon absorption efficiency. Thermoplastic injection molding and compression molding are also suitable for plastic materials. A device may be fabricated in one or more pieces that are then assembled. Layers of a device may be bonded together by clamps, adhesives, heat, anodic bonding, or reactions between surface groups (e.g., wafer bonding). Alternatively, a device with channels in more than one plane may be fabricated as a single piece, e.g., using stereolithography, multi-layer fabrication techniques (e.g., Unger et al. Science, 288:113-116, 2000), or other three-dimensional fabrication techniques.

Constrictions may be fabricated as part of a channel, e.g., through molding or other three dimensional fabrication techniques. Alternatively, a constriction may be added to an already formed channel. In one embodiment, a constriction is created in a glass (or other thermoplastic material) channel by heating and subsequently deforming the channel. In another embodiment, a constriction is created by adding material inside a channel at a designated location, e.g., through deposition or other suitable techniques.

A constriction may be fixed or variable. Variable constrictions may be able to range from completely open (i.e., no constriction in the channel) to completely closed (i.e., no fluid flow), or within an intermediate range. Variable constrictions may also be reversibly or irreversibly variable. For example, in an irreversibly variable system, the constriction may be degraded or augmented over time, e.g., through dissolution, other degradation process (e.g., chemical, thermal, electrical, or light induced etching), or deposition techniques. In a reversibly variable system, the constriction acts like a valve in the channel and can be opened or closed, as desired. In one embodiment, a reversibly variable constriction is fabricated in an elastomeric material, to which externally applied pressure may be applied to deform the material and thus constrict a channel. An exemplary reversibly variable constriction is fabricated in PDMS using multilayer soft lithography (e.g., Unger et al.

Science, 288:113-116, 2000). In another embodiment, the reversible variable constriction employs a mechanical valve to determine the size and shape of the constriction in the channel. An exemplary valve is an iris. Ferrofluids may also be used to form a constriction in which an applied magnetic field is used to control the size and/or shape of the constriction. Other valves which may act as variable constrictions are known in the art.

A constriction in a channel may also be a nanopore. Any nanopore of the appropriate size may be used in the devices of the invention. Nanopores may be biological, e.g., proteinaceous, or solid-state. Suitable nanopores are described, for example, in U.S. Patent Nos. 6,746,594, 6,673,615, 6,627,067, 6,464,842, 6,362,002, 6,267,872, 6,015,714, and 5,795,782 and U.S. Publication Nos. 2004/0121525, 2003/0104428, and 2003/0104428. Solid- state nanopores can be fabricated with arbitrary size apertures. Because of how physically robust they are, solid-state nanopores may be used at extremes of temperature, voltage, and pH conditions that would destroy biological pores. An exemplary method for fabricating solid-state membranes is the ion beam sculpting method described in Li et al. Nature 2001, 412:166. The ion beam sculpting process as described herein allows structures to be fabricated with desired nanometer scale dimensions from solid state materials like silicon nitride. Solid-state encompasses both organic and inorganic materials including, but not limited to, microelectronic materials, insulating materials such as Si₃N₄, Al₂O₃, and SiO, organic and inorganic polymers such as polyamide, plastics such as polytetrafluoroethylene, or elastomers such as two- component addition-cure silicone rubber, and glasses, although there is no specific limitation to the materials that may be used according to the invention.

In addition to channels and constrictions, other components, such as electrodes, heaters, valves, and sensors (e.g., to detect specific conditions or components of the products of the device, such as pH, conductivity, or specific ions), may be fabricated in the device. Techniques are known in the art for the fabrication of such components. For heaters, resistive elements (e.g., metal or ceramic strips) may be molded or inserted into a device or evaporated or otherwise deposited onto the device. When a voltage is applied, the resistive element emits heat. For electrodes, conductive elements (e.g., metals) may be molded or inserted into a device or evaporated or otherwise deposited onto the device (Schasfoort, et al. Science, 286:942-945, 1999). Connections to external fluid sources or receptacles may be made by any appropriate means, e.g., Luer locks, compression fittings, and threaded fittings.

Surface Modification. The surfaces of the channels may be treated in order to modify the surface properties of the channel, e.g., surface charge, wettability, permittivity, shape, and attraction or repulsion of species. Alternatively, the device may be fabricated out of a material that provides the desired properties. Surface coatings whose properties may be changed, e.g., by the application of a pH change, salt concentration change, electromagnetic radiation, thermal energy, an electric field, a magnetic field, specific or non¬ specific absorption or desorption, physical forces (e.g., sheer forces in a fluid), chemical reaction (e.g., oxidation, reduction, covalent or noncovalent reaction, or cleavage such as enzymatic), may also be employed. Examples of such coatings include titanium oxide, polypyrrole, polyvinylpyrrolidone and polybrene (X. Liu, D. Erickson, D. Li, and U. J. Krull, Analytica Chimica Acta, 507:55-62 (2004)), poly(allylamine hydrochloride), poly(diallyldimethylammonium chloride), sulfonated polystyrene (P. T. Hammond and G. M. Whitesides, Macromolecules, 28:7569-7571 (1995)); polylysine, and other polyions (either polycations or polyanions). In addition, a surface may be derivatized with a binding partner, e.g., an antibody or ligand (such as biotin), and the species that binds to the binding partner, e.g., antigen or receptor (such as avidin) can be introduced into the channel to effect a change in surface properties. To reduce non-specific adsorption of cells or compounds introduced, released, or formed during operation of the device onto the channel walls, one or more channel walls may also be chemically modified to be non-adherent or repulsive, for example through use of a thin film coating (e.g., a monolayer) of commercial non-stick reagents, such as those used to form hydrogels. Additional examples of chemical species that may be used to modify the surfaces of a device include oligoethylene glycols, fluorinated polymers, organosilanes, thiols, poly-ethylene glycol, poly- vinyl alcohol, proteins, poly-HEMA, methacrylated PEG, and agarose. Charged polymers may also be employed to repel or attract oppositely charged species or to modify the surface charge of the channel. Pretreatment of the channels with blocking agents such as tRNA and BSA may also be used to reduce non- specific adsorption. Surfaces of the device may also be treated in order to capture materials produced or released in the device, e.g., small molecules, membrane fragments, or proteins. Mixtures of surface coatings may also be employed. The type of chemical species used for surface modification and the method of attachment will depend on the nature of the surfaces and the species being attached. Surface coatings may be covalently or non-covalently attached. Channels may be modified uniformly or asymmetrically with respect to the cross-section. Asymmetric modification may be achieved by streaming multiple solutions laminarly through the channel (P. J. A. Kenis, R. F. Ismagilov, G. M. Whitesides, Science, 285:83-85, (1999)). For example, one of the multiple solutions contains a polycation and the others do not, such that the surface charge density is changed asymmetrically. Other surface modification techniques are well known in the art. The surfaces of the device may be functionalized before or after the device is assembled.

Methods

The devices of the invention may be used for a variety of applications for studying or manipulating species flowing through the channel. Many applications rely on the trapping of a species, or a portion thereof, in the eddy. Such a trapping may serve to localize a species for a period of time or to delay the rate of traversal of the species through a channel. Any suitable method can be employed to direct species to eddies. For example, species may randomly encounter eddies as they traverse a channel. The number of species that encounter an eddy may be controlled in part by the concentration of that species in the fluid flowing through the channel. Alternatively, species may be actively directed to an eddy, e.g., by optical trapping, magnetic or electrical fields, physical means (e.g., via atomic force microscopy), or chemical reaction (e.g., the covalent reaction of a molecule with surface of a channel proximal to an eddy). Typically, the devices of the invention will employ aqueous based solutions, but nonaqueous solutions, or mixtures of water and non-aqueous solvent, may be employed.

Eddy formation in channels may be controlled by the applied electric field, a pressure differential, the surface charge of the walls, the zeta potential, the ionic concentration of the fluid, pH, temperature, and the size and shape of the constriction. Dynamic alteration of any of these parameters during operation of a device can be employed to cause dynamic changes in the eddies in the channel. Such control may be used to alter the presence, absence, size, or location of an eddy, thereby allowing the capture or release of species in a specific eddy. The devices of the invention are advantageous in that small amounts of species, e.g., reactants, analytes, or reagents, are required, thereby potentially reducing the cost compared to macroscale methods, minimizing the consumption of rare species, or minimizing the production or need for hazardous species.

Any species that can traverse a fluidic channel capable of supporting EOF may be employed in the methods of the invention. Exemplary species include molecules (e.g., lipids, amino acids, nucleic acids, sugars, drugs or drug candidates, and ligands or polymers, e.g., proteins, polypeptides, synthetic polymers, polysaccharides, nucleic acids, and other large or small molecules), coordination complexes, particles (e.g., cells (such as bacterial, fungal, protozoan, or mammalian), beads, nanocrystals, or combinations thereof), ions (e.g., to change the salt concentration or pH), and solvents (e.g., ethanol and other organic solvents).

Specific uses of the device of the invention are discussed below. For each of the uses specified, the methods may employ (i) a single fluidic channel with a constriction, (ii) more than one fluidic channel with a constriction in parallel, e.g., for high throughput uses, or (iii) the use of multiple constrictions in a single channel, e.g., to subject a species to a series of alterations (e.g., chemical or physical modifications) or assays. The methods of the invention may employ the trapping of one or a few species in an eddy, the trapping of continuous streams (or a discrete plug) of species, or combinations thereof. In addition, species, e.g., polymers, passing through a small-scale device, such as a nanopore (J. Li, D. Stein, C. McMullan, D. Branton, M. J. Aziz, and J. A. Golovchenko. Nature, 412:166-169, July 2001), will be influenced by the fluid motion near such a device, particularly in the presence of eddies where the velocity field is non-uniform. With non-uniformity in the velocity field, it is possible to stretch, stall, or locally concentrate species in a very small fluid volume for use as sensors or reactors. Devices having more than one eddy, e.g., either located on either the proximal or distal side of a constriction, both sides of a constriction, or proximal to two or more constrictions, may be used for serial or parallel action on a species. For example, portions of one species, e.g., a polymer, may be trapped by two or more eddies, e.g., all either on the proximal or distal side of a constriction or with a portion of the species traversing one or more constrictions. In addition, portions of channels may be subjected to variations in external stimuli, e.g., in a gradient. Such external stimuli include thermal energy (e.g., heat or cold), electric field, magnetic field, and electromagnetic radiation.

Mixers. Eddies, as is well known in inertial flows, can be used as fluid mixers. The devices of the invention may thus be used to mix species, e.g., to mix a small amount, e.g., as few as one, of two or more species, to mix continuous streams or discrete plugs containing species, or combinations thereof. Mixing may be employed in conjunction with other methods described herein, e.g., in a reactor or sensor.

Sensors. The devices of the invention may be employed to trap species, or a portion thereof, in order to assay the trapped species or another species introduced into the channel. The devices may be employed to assay for the presence or absence of a species, the amount of a species, or characteristics of a species (e.g., molecular weight, length, composition, or other property). The assay may detect an intrinsic property of a species, the interaction between two or more species, or a labeled species, e.g., containing a fluorophore, a chromophore, a radionuclide, an antibody, or a nucleic acid. Species may be detected by any suitable method, e.g., optical, absorbance, fluorescence, phosphorescence, amperometric, or radioactivity. In one embodiment, fluorescence resonance energy transfer (FRET) may be employed, e.g., with one fluorophore on a species trapped in an eddy and one fluorophore attached to the surface of the channel or to a different portion of the trapped species. In one desirable embodiment, the device employs a nanopore as the constriction, as described above, and transport properties of a large molecule such as a polymer, or the fluid in which the molecule is disposed, (current, conductance, resistance, capacitance, charge, concentration, optical properties (e.g., fluorescence and Raman scattering), and chemical structure) are detected as the molecule traverses the nanopore, as described, for example, in U.S. Patent Nos. 6,746,594, 6,673,615, 6,627,067, 6,464,842, 6,362,002, 6,267,872, 6,015,714, and 5,795,782 and U.S. Publication Nos. 2004/0121525, 2003/0104428, and 2003/0104428. In these embodiments, the eddies may be used to trap a portion of a large molecules and thereby delay the passage of the molecule, e.g., a polymer, through the nanopore, leading to an increase in the time available to measure the transport properties of the polymer.

Reactors. Devices of the invention may also be employed as reactors in which species, or portions thereof, trapped in an eddy are modified, e.g., by reaction with chemical reagents, physical combination (e.g., by entanglement or other specific or nonspecific non-covalent bonding interaction) or by supplying external energy, .e.g., electromagnetic radiation, thermal energy, or electrical energy. The products of such reactions may be detected or the product may be subjected to further reaction or modification in the device. Reactions of species in a device may occur in series or in parallel. For example, a serial approach may be used to synthesize, sequentially cleave, or modify polymers, e.g., nucleic acids or polypeptides. In another example, a molecule, e.g., a protein or nucleic acid, is serially contacted with a series of binding partners, e.g., to form a biologically active functional complex, e.g., complexes of proteins, cofactors, ligands, and/or nucleic acids. A serial, or parallel, approach may also be used to synthesize random combinations of species, e.g., as in combinatorial chemistry. A single species may react while it is trapped in one eddy and a series of reactants are introduced into the channel. Alternatively, a single species may be modified as it travels through a channel and contacts multiple eddies at which reactants are present. Different portions of a single species may also be modified in parallel by trapping the portions in multiple eddies. A device having multiple channels with constrictions also allows for high throughput reactions of many species simultaneously. The use of devices of the invention as reactors has applications in drug discovery (e.g., by determining which species binding, activate, or deactivate biological molecules or cells), on chip modification of analytes prior to or after detection, synthesis of polymers or other compounds, and sequencing of polymers.

Example Device

An exemplary device of the invention is a 100 μL borosilicate glass capillary (VWR), originally with inner and outer diameters of 1.5 and 1.7 mm, respectively. A cylindrically symmetric constriction was created by heating a small region of the capillary with a thermal coil. As a result, in the center of the tube is a single constriction with a minimum diameter of approximately 67 μm. The length L over which the diameter varies is approximately 8.5 mm (see Fig. Ib). The fluid used was 1.0 M KCl, which has a Debye screening length of 0.3 nm. In the data shown below, the voltage across the capillary was 100 Volts, which was applied via Ag/AgCl electrodes in the solution baths attached to each side of the capillary (Fig. Ia). The electrical current was measured with an ammeter in series, and was 2.14 mA during this experiment. The hydrostatic pressure on each side of the capillary was also controlled, and set such that the velocity of the fluid was zero in the absence of an applied electric field. The solution was seeded with glass spheres, which have diameters between 10 and 60 microns. We expect that the particles have surface charge densities similar to that of the capillary and so the motion of the particles is due to a combination of EOF and electrophoresis. Video data was recorded using a computer-attached CCD camera, which was coupled to an optical microscope (Zeiss) with dark- field trans-illumination. The data was post-processed to remove background noise and track the position of the particles as a function of time. An eddy was observed in this device, and a particle was trapped in the eddy for at least 4 minutes (Fig. Ic). Eddies were observed on both sides of the constriction, though the particle tracks shown are from circulations on the entrance side of the bottleneck. Additional experiments were performed using hydrostatic pressure (with ΔV = 0) to drive flow through the constriction at rates similar to that of the EOF case. For the steady-state pressure-driven flow, no eddies in the flow were observed, and no particle traversing the outlined region crossed the cross-sectional plane more than once. The experimental parameter values, δ =0.1 and a «0.48, suggest the presence of a "central" eddy and we propose that the experimentally observed eddies are the central ones.

Although it is common to consider electroosmotic flow to be uni¬ directional, we have demonstrated that a sinusoidal perturbation of the channel radius of sufficient amplitude can have significant influences on the local velocity profiles. We have also shown that controlling the amplitude and wavenumber of the shape perturbation can lead to eddies in the center and/or the perimeter regions of the channel (Fig. 4). In addition, a criterion for the formation of a central eddy was established. The various parameter values that give rise to eddies are summarized in a phase diagram (Fig. 5) where we have further shown that as k decreases, eddies become more observable for a larger array of a and δ values.

Other Embodiments

Modifications and variations of the described methods of the invention will be apparent to those skilled in the art without departing from the scope and spirit of the invention. Although the invention has been described in connection with specific desirable embodiments, it should be understood that the invention as claimed should not be unduly limited to such specific embodiments. Indeed, various modifications of the described modes for carrying out the invention, which are obvious to those skilled in the art, are intended to be within the scope of the invention. AU publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually to be incorporated by reference. Other embodiments are within the claims.

What is claimed is:

Claims

1. A method of trapping a species, said method comprising the steps of:

(a) providing a device comprising:

(i) a fluidic channel comprising first and second ends and a constriction disposed between said first and second ends, wherein the walls of said fluidic channel comprise a surface charge, and wherein said constriction divides said channel into a proximal and a distal side; and

(ii) a voltage source operationally coupled to said fluidic channel and capable of generating an electroosmotic flow (EOF) in said fluidic channel between said first and second ends;

(b) inducing the formation of one or more eddies proximal to said constriction by generating said EOF; and

(c) allowing a first species to traverse said fluidic channel, wherein when said first species contacts one of said one or more eddies, at least a portion of said first species becomes disposed within said eddy, thereby trapping said first species.

2. The method of claim I₃ further comprising releasing said first species from said eddy.

3. The method of claim I₃ wherein said device further comprises one or more electrodes proximal to said constriction, wherein application of a voltage to said one or more electrodes alters the surface charge on the walls of said fluidic channel

4. The method of claim 3, further comprising releasing said first species from said eddy by changing a voltage applied to said electrode.

5. The method of claim 1, further comprising applying energy to said first species in contact with said eddy.

6. The method of claim 5, wherein said energy comprises electromagnetic radiation, electrical energy, magnetic energy, or thermal energy.

7. The method of claim 5, wherein said first species undergoes a chemical or physical change after the application of said energy.

8. The method of claim 1, further comprising introducing a second species into said fluidic channel and contacting said first and second species in said eddy.

9. The method of claim 8, wherein upon contacting said first and second species chemically or physically interact.

10. The method of claim 8, wherein upon contacting said first and second species mix.

11. The method of claim 8, further comprising detecting the interaction between said first and second species.

12. The method of claim 8, further comprising releasing the product of said contacting of said first and second species from said eddy.

13. The method of claim 1, wherein said first species is a polymer.

14. The method of claim 1, wherein said first species is a particle.

15. The method of claim 14, wherein said particle comprises a cell.

16. The method of claim 1, wherein said first species comprises a nucleic acid, amino acid, sugar, or lipid.

17. The method of claim 1, wherein said first species comprises an ion.

18. The method of claim 1, wherein said second species comprises a nucleic acid, amino acid, sugar, or lipid.

19. The method of claim 1, wherein said second species comprises an ion.

20. The method of claim 1, wherein said first or second species comprises a label.

21. The method of claim 20, wherein said label comprises a fluorophore, a chromophore, a radionuclide, an antibody, or a nucleic acid.

22. The method of claim 1, wherein said constriction is shaped such that said one or more eddies form on only the proximal or distal sid.e of said constriction.

23. The method of claim 1, wherein said fluidic channel comprises a plurality of constrictions.

24. The method of claim 23, wherein said first species traverses said fluidic channel and contacts an eddy disposed proximal to each constriction.

25. The method of claim 24, wherein said first species contacts another species at each of said eddies proximal to each constriction.

26. The method of claim 25, wherein said first species contacts at least one species different from said first species.

27. The method of claim 25, wherein said first species chemically or physically interacts with another species at each of said eddies proximal to each constriction.

28. The method of claim 27, wherein said first species interacts with at least one species different from said first species.

29. The method of claim 1, wherein said species traverses said iϊuidic channel by use of optical trapping.

30. The method of claim 1, wherein the surface charge of the walls of said fluidic channels is modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, salt concentration, absorption of chemical species, or pH.

31. The method of claim 1 , further comprising detecting a property of said first species as it passes through said constriction.

32. The method of claim 1, wherein said constriction comprises a nanopore.

33. The method of claim 1, wherein said first species comprises a first and a second portion, wherein, in step (c), said first portion is disposed within said eddy.

34. The method of claim 33, wherein said second portion traverses said constriction.

35. The method of claim 33, wherein said second portion is disposed in an eddy, and one of said first and second portions is disposed on the proximal or distal side of said constriction, and the other portion is disposed on the opposite side of said constriction.

36. The method of claim 33, wherein said second portion is disposed in an eddy, and said first and second portions are both disposed on the proximal or distal side of said constriction.

37. The method of claim 1, wherein the fluidic channel has a height or width of less than 1 mm.

38. The method of claim 1, wherein the fluidic channel has a height or width of less than 1 μm.

39. The method of claim 1, wherein said fluidic channel comprises a carbon or silicon dioxide nanotube.

40. The method of claim 1, further comprising applying different temperatures, electric fields, or magnetic fields to said first species as it traverses said fluidic channel.

41. The method of claim 1, wherein the surface charge of the walls of said fluidic channel is modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, or pH.

42. The method of claim 1, further comprising altering the shape or size of said constriction either before or after step (c).

43. The method of claim 1, wherein said constriction blocks at least 5% of said channel.

44. The method of claim 1, wherein said constriction has an asymmetric cross-sectional geometry.

45. The method of claim 1, wherein a central eddy is formed in step (b).

46. The method of claim 1, wherein a perimeter eddy is formed in step (b).

47. A method of trapping a species, said method comprising the steps of:

(a) providing a device comprising:

(i) a fluidic channel comprising first and second ends and a location between said first and second ends capable of being constricted, wherein the walls of said fluidic channel comprise a surface charge; and

(b) constricting said channel at said location to form a constriction that divides said channel into a proximal and a distal side; (c) inducing the formation of one or more eddies proximal to said constriction by generating said EOF; and

(d) allowing a first species to traverse said fluidic channel, wherein when said first species contacts one of said one or more eddies, at least a portion of said first species becomes disposed within said eddy, thereby trapping said first species.

48. The method of claim 47, further comprising releasing said first species from said eddy by altering the size or shape of said constriction.

49. The method of claim 47, further comprising releasing said first species from said eddy.

50. The method of claim 47, wherein said device further comprises one or more electrodes proximal to said constriction, wherein application of a voltage to said one or more electrodes alters the surface charge on the walls of said fluidic channel.

51. The method of claim 50, further comprising releasing said first species from said eddy by changing a voltage applied to said electrode.

52. The method of claim 47, further comprising applying energy to said first species in contact with said eddy.

53. The method of claim 52, wherein said energy comprises electromagnetic radiation, electrical energy, magnetic energy, or thermal energy.

54. The method of claim 52, wherein said first species undergoes a chemical or physical change after the application of said energy.

55. The method of claim 47, further comprising introducing a second species into said fluidic channel and contacting said first and second species in said eddy.

56. The method of claim 55, wherein upon contacting said first and second species chemically or physically interact.

57. The method of claim 55, wherein upon contacting said first and second species mix.

58. The method of claim 55, further comprising detecting the interaction between said first and second species.

59. The method of claim 55, further comprising releasing the product of said contacting of said first and second species from said eddy.

60. The method of claim 47, wherein said first species is a polymer.

61. The method of claim 47, wherein said first species is a particle.

62. The method of claim 61, wherein said particle comprises a cell.

63. The method of claim 47, wherein said first species comprises a nucleic acid, amino acid, sugar, or lipid.

64. The method of claim 47, wherein said first species comprises an ion.

65. The method of claim 47, wherein said second species comprises a nucleic acid, amino acid, sugar, or lipid.

66. The method of claim 47, wherein said second species comprises an ion.

67. The method of claim 47, wherein said first or second species comprises a label.

68. The method of claim 67, wherein said label comprises a fmorophore, a chromophore, a radionuclide, an antibody, or a nucleic acid.

69. The method of claim 47, wherein said constriction is shaped such that said one or more eddies form on only the proximal or distal side of said constriction.

70. The method of claim 47, wherein said fluidic channel further comprises a second constriction or location capable of being constricted to form a second constriction.

71. The method of claim 70, wherein said first species traverses said fluidic channel and contacts an eddy disposed proximal to each constriction.

72. The method of claim 71, wherein said first species contacts another species at each of said eddies proximal to each constriction.

73. The method of claim 72, wherein said first species contacts at least one species different from said first species.

74. The method of claim 72, wherein said first species chemically or physically interacts with another species at each of said eddies proximal to each constriction.

75. The method of claim 74, wherein said first species interacts with at least one species different from said first species.

76. The method of claim 47, wherein said species traverses said fluidic channel by use of optical trapping.

77. The method of claim 47, wherein the surface charge of the walls of said fluidic channels is modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, salt concentration, absorption of molecules, or pH.

78. The method of claim 47, further comprising detecting a property of said first species as it passes through said constriction.

79. The method of claim 47, wherein said constriction comprises a nanopore.

80. The method of claim 47, wherein said first species comprises a first and a second portion, wherein, in step (d), said first portion is disposed within said eddy.

81. The method of claim 80, wherein said second portion traverses said constriction.

82. The method of claim 80, wherein said second portion is disposed in an eddy, and one of said first and second portions is disposed on the proximal or distal side of said constriction, and the other portion is disposed on the opposite side of said constriction.

83. The method of claim 80, wherein said second portion is disposed in an eddy, and said first and second portions are both disposed on the proximal or distal side of said constriction.

84. The method of claim 47, wherein the fluidic channel has a height or width of less than 1 mm.

85. The method of claim 47, wherein the fluidic channel has a height or width of less than 1 μm.

86. The method of claim 47, wherein said fluidic channel comprises a carbon or silicon dioxide nanotube.

87. The method of claim 47, further comprising applying different temperatures, electric fields, or magnetic fields to said first species as it traverses said fluidic channel.

88. The method of claim 47, wherein the surface charge of the walls of said fluidic channel is modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, or pH.

89. The method of claim 47, wherein said constriction blocks at least 5% of said channel.

90. The method of claim 47, wherein said constriction has an asymmetric cross-sectional geometry.

91. The method of claim 47, wherein a central eddy is formed in step (c).

92. The method of claim 47, wherein a perimeter eddy is formed in step (c).

93. A fluidic system comprising:

(a) a plurality of fluidic channels, wherein each fluidic channel comprises first and second ends and either (i) a constriction disposed between said first and second ends or (ii) a location between said first and second ends capable of being constricted to form a constriction, wherein the walls of said fluidic channel comprise a surface charge, wherein said plurality of fluidic channels are fluidically connected, and wherein said constriction divides each channel into a proximal and a distal side; and

(b) one or more voltage sources operationally coupled to said plurality of fluidic channels and capable of generating an electroosmotic flow (EOF) in said plurality of fluidic channels between said first and second ends.

94. The fluidic system of claim 93, wherein said one or more voltage sources are capable of applying voltage independently to at least two of said plurality of fluidic channels.

95. The fluidic system of claim 93, further comprising one or more electrodes disposed proximally to one of said constrictions, wherein application of a voltage to said one or more electrodes alters the surface charge on the walls of said fluidic channel.

96. The fluidic system of claim 95, wherein the alteration of said surface charge is capable of disrupting eddy formation proximal to said constriction.

97. The fluidic system of claim 93, wherein at least one of said plurality of fluidic channels has a height or width of less than 1 mm.

98. The fluidic system of claim 93, wherein at least one of said plurality of fluidic channels has a height or width of less than 1 μm.

99. The fluidic system of claim 93, wherein said at least one of said constrictions comprises a nanopore.

100. The fluidic system of claim 93, wherein at least one of said plurality of fluidic channels comprises a carbon or silicon dioxide nanotube.

101. The fluidic system of claim 93, further comprising a detector for monitoring species in one or more of said plurality of fluidic channels.

102. The fluidic system of claim 93, wherein at least one of said constrictions is shaped such that one or more eddies form on only the proximal or distal side of said constriction.

103. The fluidic system of claim 93, wherein the surface charge of the walls of at least one of said fluidic channels is modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, salt concentration, absorption of chemical species, or pH.

104. The fluidic system of claim 93, wherein at least one of the constrictions has an asymmetric cross-sectional geometry.

105. The fluidic system of claim 93, wherein said system is capable of producing at least one central eddy under EOF.

106. The fluidic system of claim 93, wherein said system is capable of producing at least one perimeter eddy under EOF.

107. A fluidic system comprising:

(a) a fluidic channel comprising first and second ends and either (i) a constriction disposed between said first and second ends or (ii) a location between said first and second ends capable of being constricted to form a constriction, wherein the walls of said fluidic channel comprise a surface charge, and wherein said constriction divides each channel into a proximal and a distal side;

(b) a voltage source operationally coupled to said fluidic channel and capable of generating an electroosmotic flow (EOF) in said fluidic channel between said first and second ends; and (c) one or more electrodes disposed proximally to said constriction, wherein application of a voltage to said one or more electrodes alters the surface charge on the walls of said fluidic channel.

108. The fluidic system of claim 107, wherein the alteration of said surface charge is capable of disrupting eddy formation proximal to said constriction.

109. The fluidic system of claim 107, wherein said fluidic channel has a height or width of less than 1 mm.

110. The fluidic system of claim 107, wherein said fluidic channel has a height or width of less than 1 μm.

111. The fluidic system of claim 107, wherein said constriction comprises a nanopore.

112. The fluidic system of claim 107, wherein said fluidic channel comprises a carbon or silicon dioxide nanotube.

113. The fluidic system of claim 107, further comprising a detector for monitoring species in said fluidic channel.

114. The fluidic system of claim 107, wherein said constrictions is shaped such that one or more eddies form on only the proximal or distal side of said constriction.

115. The fluidic system of claim 107, wherein the surface charge of the walls of said fluidic channel is modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, salt concentration, absorption of chemical species, or pH.

116. The fluidic system of claim 107, wherein said constriction has an asymmetric cross-sectional geometry.

117. The fluidic system of claim 107, wherein said system is capable of producing at least one central eddy under EOF.

118. The fluidic system of claim 107, wherein said system is capable of producing at least one perimeter eddy under EOF.

119. A fluidic system comprising:

(a) a fluidic channel comprising first and second ends, a surface charge on the walls of said channel, and a plurality of (i) a constriction disposed between said first and second ends; a plurality of (ii) a location disposed between said first and second ends capable of being constricted to form a constriction; or at least one constriction of (i) and one location of (ii), wherein each constriction divides said fluidic channel into a proximal and a distal side.

120. The fluidic system of claim 119, further comprising:

(b) a voltage sources operationally coupled to said fluidic channels and capable of generating an electroosmotic flow (EOF) in said fluidic channel between said first and second ends.

121. The fluidic system of claim 119, further comprising one or more electrodes disposed proximally to one of said constrictions, wherein application of a voltage to said one or more electrodes alters the surface charge on the walls of said fluidic channel.

122. The fluidic system of claim 119, wherein the alteration of said surface charge is capable of disrupting eddy formation proximal to said constriction.

123. The fluidic system of claim 119, wherein said fluidic channel has a height or width of less than 1 mm.

124. The fluidic system of claim 119, wherein said fluidic channel has a height or width of less than 1 μm.

125. The fluidic system of claim 119, wherein said at least one of said constrictions comprises a nanopore.

126. The fluidic system of claim 119, wherein said fluidic channel comprises a carbon or silicon dioxide nanotube.

127. The fluidic system of claim 119, further comprising a detector for monitoring species in said fluidic channel.

128. The fluidic system of claim 119, wherein at least one of said constrictions is shaped such that one or more eddies form on only the proximal or distal side of said constriction.

129. The fluidic system of claim 119, wherein the surface charge of the walls of said fluidic channel is modifiable by electromagnetic radiation, thermal energy, applied voltage, magnetic field, salt concentration, absorption of chemical species, or pH.

130. The fluidic system of claim 119, wherein at least one of the constrictions has an asymmetric cross-sectional geometry.

131. The fluidic system of claim 119, wherein said system is capable of producing at least one central eddy under EOF.

132. The fluidic system of claim 119, wherein said system is capable of producing at least one perimeter eddy under EOF.