Electrokinetic flow instabilities: Physics, Modeling, and Experiments
Introduction and Background
Electrokinetic (EK) devices are growing in complexity of design and function. Many EK devices aim to control and analyze heterogeneous electrolytes as in the case of field amplified sample stacking , isotachophoretic sample stacking , and multi-dimensional assays. Another example of heterogenous electrolytes is applications where sample chemistry (e.g., conductivity) is unknown or poorly controlled. Electrokinetic flows of such heterogenous chemistries can lead to flow instabilities known as electrokinetic instabilities (EKI).
Electrokinetic flow instabilities are actually electrohydrodynamic instabilities of the type first described in the 1960’s by GI Taylor and JR Melcher. For example, Hoburg and Melcher studied so-called “leaky-dielectric" liquids (e.g., corn oil) in millimeter and centimeter scale devices. They showed these instabilities are due to a coupling between electric fields and conductivity gradients and are characterized by an electroviscous velocity scale. We term use the “electrokinetic instabilities” to describe the EHD instability microchannel regimes where molecular diffusion is a key component; where ion densities are relatively high and length scales relatively small; and where advective instability effects can couple with electroosmotic flow.
Theory
Electrokinetic instabilities are caused by a coupling of electric fields and ionic conductivity gradients. This coupling results in an electric body force (per unit volume), 
, of the form
where e , and 
are the local values of permittivity, electric, and ionic conductivity, respectively. This electric body force is generated wherever applied electric fields interact with conductivity gradients and so can occur in the bulk liquid-i.e., outside of the electric double layer. This electric force is balanced by viscous forces and the flow generates a disturbance flow velocity that scales as the electroviscous force, U ev
Electrokinetic flows become unstable when the advection of conductivity fields by electroviscous advection dominates over the dissipative effects of viscosity and molecular diffusion.
EKI Visualizations
EKIs create random, complex, three-dimensional fluid flow fields. Suppression or avoidance of electrokinetic flow instabilities is directly applicable to sample stacking as conductivity-gradient-induced instabilities limit stacking efficiency. Promotion of electrokinetic instabilities enables rapid mixing rapid mixing at micron scales, which is critical to increasing the throughput of a variety of binding assays.
Experiment  0.0 sec  0.9 sec  1.4 sec  2.4 sec  4.8 sec |
|
|
|
(a)
(b)
(c)
(d)
(e)
(click on figure to begin movie)
Representative instantaneous scalar concentration field images of unstable electrokinetic flow in 50 mm cross-shaped microchannel (20 um deep) for center-to-sheath conductivity ratios g > 1 (high conductivity in center stream). Images correspond to center-to-sheath conductivity ratio of g = 100, applied field ratio of b = 1.13, and nominal fields Ea shown above each image. The dyed, center stream flows from the west (left) channel and buffer sheath streams flow from the north (top) and south (bottom) channels forming two conductivity interfaces. For stable flow (a), the width of the center stream is a function of the b and g as well as electrolyte chemistry. When the applied field exceeds a critical value, a sinuous pattern in the dye develops and grows as it advects downstream (b). At = 381 V/cm (c), disturbances grow rapidly as they convect downstream and roll up into alternating flow structures. At even larger applied fields (d and e), the flow transitions from a deterministic, periodic regime to a fully chaotic regime and the source of disturbances moves upstream.
Other EKI Visualizations
Movie of Electrokinetic Instability submitted to Gallery of Fluid Mechanics 2004:
a) Low Resolution (4MB)
b) High Resolution (20MB)
Convective flow instabilities in a T-junction
References
1. Melcher, J. R., and Taylor, G. I., 1969, Electrohydrodynamics: a review of the role of interfacial stresses, Annu. Rev. Fluid. Mech. 1, 111-146.
2. Hoburg, J.F., and J. R. Melcher, Journal of Fluid Mechanics 73, 333 (1976).
3. Lin, H., Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Physics of Fluids, Vol. 16, No. 6, 2004
4. Oddy, M.H., and J.G. Santiago, “A Multi-Species Model for Electrokinetic Instability,” in press, Physics of Fluids, 2005.
5. Chen, C.-H., Lin, H., Lele, S.K. & Santiago, J.G., "Convective and absolute electrokinetic instability with conductivity gradients", Journal of Fluid Mechanics, 524, pp. 263 – 303, 2005.
6. Storey, B.D., B.S. Tilley, H. Lin, and J.G. Santiago, "Electrokinetic Instabilities in Thin Microchannels," Physics of Fluids, Vol. 16, No. 6, p.1922-1935, 2004.
7. Posner, J.D. and Santiago, J.G., "Convective Instability of Electrokinetic Flows in a Cross-Shaped Microchannel, " submitted to Journal of Fluid Mechanics, 2005.
8. Lin, H., B.D. Storey and J.G. Santiago, “A depth-averaged electrokinetic flow model for thin microchannels,” submitted to Proceedings of the Royal Society A, 2005.
Introduction and Background
Electrokinetic (EK) devices are growing in complexity of design and function. Many EK devices aim to control and analyze heterogeneous electrolytes as in the case of field amplified sample stacking , isotachophoretic sample stacking , and multi-dimensional assays. Another example of heterogenous electrolytes is applications where sample chemistry (e.g., conductivity) is unknown or poorly controlled. Electrokinetic flows of such heterogenous chemistries can lead to flow instabilities known as electrokinetic instabilities (EKI).
Electrokinetic flow instabilities are actually electrohydrodynamic instabilities of the type first described in the 1960’s by GI Taylor and JR Melcher. For example, Hoburg and Melcher studied so-called “leaky-dielectric" liquids (e.g., corn oil) in millimeter and centimeter scale devices. They showed these instabilities are due to a coupling between electric fields and conductivity gradients and are characterized by an electroviscous velocity scale. We term use the “electrokinetic instabilities” to describe the EHD instability microchannel regimes where molecular diffusion is a key component; where ion densities are relatively high and length scales relatively small; and where advective instability effects can couple with electroosmotic flow.
Theory
Electrokinetic instabilities are caused by a coupling of electric fields and ionic conductivity gradients. This coupling results in an electric body force (per unit volume), 
, of the form
where e , and 
are the local values of permittivity, electric, and ionic conductivity, respectively. This electric body force is generated wherever applied electric fields interact with conductivity gradients and so can occur in the bulk liquid-i.e., outside of the electric double layer. This electric force is balanced by viscous forces and the flow generates a disturbance flow velocity that scales as the electroviscous force, U ev
Electrokinetic flows become unstable when the advection of conductivity fields by electroviscous advection dominates over the dissipative effects of viscosity and molecular diffusion.
EKI Visualizations
EKIs create random, complex, three-dimensional fluid flow fields. Suppression or avoidance of electrokinetic flow instabilities is directly applicable to sample stacking as conductivity-gradient-induced instabilities limit stacking efficiency. Promotion of electrokinetic instabilities enables rapid mixing rapid mixing at micron scales, which is critical to increasing the throughput of a variety of binding assays.
Experiment  0.0 sec  0.9 sec  1.4 sec  2.4 sec  4.8 sec |
|
|
|
(a)
(b)
(c)
(d)
(e)
(click on figure to begin movie)
Representative instantaneous scalar concentration field images of unstable electrokinetic flow in 50 mm cross-shaped microchannel (20 um deep) for center-to-sheath conductivity ratios g > 1 (high conductivity in center stream). Images correspond to center-to-sheath conductivity ratio of g = 100, applied field ratio of b = 1.13, and nominal fields Ea shown above each image. The dyed, center stream flows from the west (left) channel and buffer sheath streams flow from the north (top) and south (bottom) channels forming two conductivity interfaces. For stable flow (a), the width of the center stream is a function of the b and g as well as electrolyte chemistry. When the applied field exceeds a critical value, a sinuous pattern in the dye develops and grows as it advects downstream (b). At = 381 V/cm (c), disturbances grow rapidly as they convect downstream and roll up into alternating flow structures. At even larger applied fields (d and e), the flow transitions from a deterministic, periodic regime to a fully chaotic regime and the source of disturbances moves upstream.
Other EKI Visualizations
Movie of Electrokinetic Instability submitted to Gallery of Fluid Mechanics 2004:
a) Low Resolution (4MB)
b) High Resolution (20MB)
Convective flow instabilities in a T-junction
References
1. Melcher, J. R., and Taylor, G. I., 1969, Electrohydrodynamics: a review of the role of interfacial stresses, Annu. Rev. Fluid. Mech. 1, 111-146.
2. Hoburg, J.F., and J. R. Melcher, Journal of Fluid Mechanics 73, 333 (1976).
3. Lin, H., Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Physics of Fluids, Vol. 16, No. 6, 2004
4. Oddy, M.H., and J.G. Santiago, “A Multi-Species Model for Electrokinetic Instability,” in press, Physics of Fluids, 2005.
5. Chen, C.-H., Lin, H., Lele, S.K. & Santiago, J.G., "Convective and absolute electrokinetic instability with conductivity gradients", Journal of Fluid Mechanics, 524, pp. 263 – 303, 2005.
6. Storey, B.D., B.S. Tilley, H. Lin, and J.G. Santiago, "Electrokinetic Instabilities in Thin Microchannels," Physics of Fluids, Vol. 16, No. 6, p.1922-1935, 2004.
7. Posner, J.D. and Santiago, J.G., "Convective Instability of Electrokinetic Flows in a Cross-Shaped Microchannel, " submitted to Journal of Fluid Mechanics, 2005.
8. Lin, H., B.D. Storey and J.G. Santiago, “A depth-averaged electrokinetic flow model for thin microchannels,” submitted to Proceedings of the Royal Society A, 2005.