Electroosmotic Flow Rate Calculator

Estimate electroosmotic flow with a practical first-pass model

Electroosmotic flow, often shortened to EOF, is liquid motion caused by an electric field acting on the charged layer beside a channel wall. Rather than using a moving part to push fluid as a conventional pump does, electroosmosis uses the applied field. That is particularly useful in microfluidic and nanofluidic devices, where channels are tiny, sample volumes are limited, and mechanical pumping hardware may be difficult to integrate. For a lab-on-a-chip experiment, an early channel design, or a reported-flow check, the practical question is how much liquid this geometry and fluid combination should move.

This electroosmotic flow calculator combines applied voltage, channel length, channel cross-section, wall zeta potential, fluid permittivity, and viscosity to produce a single-channel volumetric estimate in microliters per minute. It is not a replacement for a full electrokinetic simulation, but it is a useful first estimate for prototype planning, channel comparisons, throughput checks, and deciding which input is most worth changing.

Read the EOF estimate as a chain of physical effects. Increasing voltage across the same channel length increases electric field, which increases electroosmotic velocity in this model. Increasing width or height increases the available cross-sectional area, so that velocity carries more liquid. Higher permittivity strengthens the electrokinetic response, while higher viscosity resists it. Zeta potential describes the interfacial charge condition: its magnitude affects flow strength and its sign affects the calculated direction.

Electroosmotic flow inputs in plain language

Channel length is the distance over which the electroosmotic driving voltage drops. It matters because electric field is voltage divided by length, not voltage by itself. Holding voltage constant while shortening the channel raises the field. Channel width and channel height set the channel cross-sectional area. They do not directly alter electroosmotic mobility, but they determine how much liquid passes at a given EOF velocity.

Applied voltage is the most direct electrical EOF control in the form. Zeta potential characterizes the channel wall and adjacent electrical double layer. It is often negative in aqueous systems, so a negative entered value can yield a negative signed flow even when the displayed flow-rate magnitude is useful for comparing throughput. Relative permittivity expresses how the fluid stores electric field energy relative to vacuum. Viscosity is the liquid's resistance to motion; increasing it lowers the predicted EOF velocity for the same field and wall condition.

For a consistent electroosmotic calculation, the page converts every entry to SI units before arithmetic. Millimeters become meters, micrometers become meters, millivolts become volts, and centipoise become pascal-seconds. Those conversions are important because a scale error in a channel dimension or zeta potential can change an EOF result dramatically.

How this electroosmotic flow calculation is structured

This EOF calculator first obtains electric field from the applied voltage and channel length:

It then applies the Helmholtz-Smoluchowski relationship used by the page's calculation, with absolute permittivity written as vacuum permittivity times relative permittivity:

Finally, the calculator converts EOF velocity into volumetric flow by multiplying it by the rectangular channel area:

These steps explain why the form needs geometry as well as fluid and surface properties. Channel length controls field; width and height control area; relative permittivity, zeta potential, and viscosity set electroosmotic mobility. The reported flow-rate magnitude is the absolute value of the signed result converted to microliters per minute, while the result text retains the signed-direction interpretation.

How to interpret the electroosmotic flow result

The electroosmotic flow result reports flow-rate magnitude in microliters per minute together with EOF velocity and electric field. A negative signed direction does not indicate a failed calculation. Under the sign convention in the formula, it means the entered zeta potential and field predict motion opposite the positive field direction. Throughput magnitude and flow direction are therefore best considered separately.

Very small EOF flow-rate values are normal for very small channels. A channel may have meaningful electroosmotic velocity while transporting little volume because its cross-sectional area is microscopic. If the estimate is below a desired throughput, consider whether the device uses parallel channels, a shorter channel, a larger area, or a different allowable electric field than the single-channel model entered here.

Worked electroosmotic flow example before you calculate

Use the values prefilled in the form to see the scale of a single-channel EOF estimate: a 20 mm channel with a 50 µm width and 10 µm height, 100 V applied voltage, -50 mV zeta potential, relative permittivity 80, and viscosity 1 cP. The electric field is 100 V divided by 0.02 m, or 5,000 V/m. The Helmholtz-Smoluchowski calculation gives an electroosmotic speed magnitude of about 0.177 mm/s. Because the channel area is extremely small, the corresponding flow-rate magnitude is about 0.0053 µL/min.

This example shows why EOF design involves more than voltage. Tenfold higher throughput can come from a higher field, but may instead require a shorter channel, a larger width or height, lower viscosity, a wall treatment that changes zeta potential, or multiple channels in parallel. Use the calculator to identify which of those levers changes the single-channel estimate most directly before fabricating or testing hardware.

Electroosmotic flow physics, assumptions, and sample values

Electroosmotic transport lets microscale and nanoscale channels move liquid with an electric field instead of a syringe pump, pressure regulator, or moving membrane. Charged channel walls and the electrical double layer make this possible. EOF is consequently useful in capillary electrophoresis, lab-on-a-chip sample preparation, compact chemical sensors, and experimental nanofluidic systems where dead volume, contamination, or mechanical complexity must be minimized.

The electroosmotic flow velocity $v$ used here follows the Helmholtz–Smoluchowski equation: $v=\frac{\epsilon \zeta E}{\eta }$, where $\epsilon$ is absolute permittivity, $\zeta$ is zeta potential, $E$ is electric field, and $\eta$ is viscosity. Absolute permittivity is vacuum permittivity ${\epsilon }_0$ times relative permittivity. Unit conversion is essential: zeta potential is converted from millivolts to volts, length from millimeters to meters, and viscosity from centipoise to pascal-seconds ($1\text{cP}=0.001\text{Pa·s}$).

After finding EOF velocity, the calculator multiplies it by the rectangular cross-sectional area to obtain volumetric flow rate, $Q=vA$. For the entered width and height, $A=wh$. Width and height entered in micrometers are each converted to meters before multiplication, so their area is correctly expressed in square meters. The final result is shown in microliters per minute rather than raw cubic meters per second for easier bench-scale comparison.

For the prefilled electroosmotic channel example—20 mm long, 50 µm wide, 10 µm high, 100 V, -50 mV zeta potential, relative permittivity 80, and viscosity 1 cP—the electric field is 5,000 V/m. The 50 µm by 10 µm cross-section is 500 µm², or $5\times {10}^{-10}$ m². The implemented model gives an EOF speed magnitude of about 0.177 mm/s and a volumetric flow-rate magnitude of about 0.0053 µL/min. This small value reflects the very small channel area rather than a calculation error.

EOF sensitivity follows directly from the model. Doubling voltage doubles electric field and predicted velocity when the other inputs are fixed. Doubling width doubles the area and flow rate; doubling height does the same. Increasing viscosity lowers velocity in inverse proportion. Increasing the magnitude of zeta potential increases the magnitude of electroosmotic mobility, while its sign changes the signed direction. These proportional relationships make the calculator useful for early design comparisons before second-order effects become the focus.

Real electroosmotic devices can depart from this ideal estimate. Temperature affects viscosity and permittivity. Surface coatings can alter zeta potential. High fields can create Joule heating or electrode-related electrochemical effects. Roughness, corners, nonuniform surface charge, non-Newtonian liquids, and pressure-driven backflow may also change measured flow. In especially small channels, the electrical double layer can occupy a substantial share of the channel dimension, making the thin-double-layer approximation behind Helmholtz-Smoluchowski less exact.

Despite those limitations, a transparent EOF estimate remains useful for DNA handling, capillary electrophoresis, cell-free assays, compact sensors, and similar microscale fluidic work. It indicates whether a concept is in a plausible flow-rate range and highlights which parameter may deserve direct measurement. If laboratory results differ, investigate causes such as bubbles, changing wall chemistry, leaks, evaporation, or an opposing pressure gradient.

Electroosmotic flow comparison table

This electroosmotic comparison keeps channel length and fluid properties fixed while changing voltage or cross-sectional area. The rounded values follow the same single-channel Helmholtz-Smoluchowski calculation used on this page.

The final two rows illustrate two different ways to double this modeled EOF flow rate. Doubling voltage doubles velocity through electric field, while doubling width doubles area. A practical microfluidic design choice between those options depends on electrical limits, heating, electrode chemistry, fabrication constraints, and available chip area.

Practical EOF interpretation and record keeping

If the electroosmotic flow result is smaller than an application requires, electroosmosis is not necessarily unsuitable. The design may need more field, a shorter channel, a larger cross-section, a lower-viscosity buffer, different wall chemistry, or several channels in parallel. Conversely, if the calculated throughput is adequate but the experiment underperforms, check for electrode bubbles, trapped gas, pH-driven surface changes, contamination, or a pressure gradient opposing EOF.

For electroosmotic experiments, record the result together with every entered value after each run. Comparing near-neighbor channel designs is only useful when the voltage, dimensions, viscosity, permittivity, and zeta-potential assumptions can be reproduced. A clear record also helps explain a geometry choice to collaborators and identify which physical input should be measured rather than estimated.

Where an electroosmotic flow estimate is most useful

Electroosmotic pumping is attractive when precise movement of small liquid volumes is needed. In capillary electrophoresis, EOF transports sample through the separation path while charged analytes migrate relative to that bulk movement. In integrated microfluidics, it can replace external pumps and simplify packaging. In biosensors, EOF can move reagents through narrow structures that would be awkward to connect to mechanical pumping hardware. Similar charge-controlled transport is also relevant in porous media, membranes, and engineered surfaces.

EOF design combines geometry, surface chemistry, and electrical control. A pressure-driven system may be changed primarily by changing pressure, but electroosmotic performance can depend as much on a wall treatment that alters zeta potential as on the voltage supply. Likewise, a seemingly modest reduction in channel dimensions can strongly affect flow because cross-sectional area decreases quickly. This calculator makes those dependencies visible rather than treating the estimate as a black box.

This electroosmotic flow calculator runs its calculation in your browser. It needs no external data for the basic estimate, making it suitable for classroom demonstrations, offline planning, and quick lab-bench checks. For work requiring temperature dependence, pressure coupling, non-Newtonian behavior, or electrode effects, use this result as a first checkpoint before a more detailed simulation or experiment.

Electroosmotic flow assumptions worth remembering

This electroosmotic flow page uses a first-order model. It assumes uniform channel properties, a Newtonian liquid, and entered values that reasonably characterize the full channel. It also assumes the electrical double layer is thin enough for the Helmholtz-Smoluchowski relation to be useful. Those assumptions are appropriate for a quick estimate, but extremely small channels, polymer- or surfactant-containing liquids, substantial heating, and patterned or unstable wall charge can require a more detailed treatment.

Use the EOF result as an engineering starting point rather than a final device qualification. A prediction that misses a target by orders of magnitude is already valuable because it can prevent an unsuitable design path. A close comparison between alternatives helps identify what to measure next. Agreement with an experimental flow rate supports, but does not prove, that the dominant electroosmotic physics are represented adequately.