Channel Length (mm): Channel Width (µm): Channel Height (µm): Applied Voltage (V): Zeta Potential (mV): Relative Permittivity: Viscosity (cP): Calculate Copy

Manipulating liquids in nanoscale channels has revolutionized areas from DNA analysis to chemical separations. Electroosmosis provides a convenient way to move fluid without mechanical pumps. When an electric field is applied along a narrow channel, charged surfaces attract a thin layer of counter-ions. These ions drag the surrounding liquid as they migrate, producing bulk flow. The rate depends on geometry and fluid properties.

The velocity of electroosmotic flow $v$ is given by the Helmholtz–Smoluchowski equation: $v=\frac{\mathrm{\backslash varepsilon}\mathrm{\backslash zeta}E}{\mathrm{\backslash eta}}$ where $\mathrm{\backslash varepsilon}$ is absolute permittivity, $\mathrm{\backslash zeta}$ is zeta potential, $E$ is the electric field, and $\mathrm{\backslash eta}$ is viscosity. Absolute permittivity equals the vacuum permittivity $\mathrm{\backslash varepsilon\_0}$ times the relative permittivity. Converting units is essential: zeta potential from millivolts to volts, length from millimeters to meters, and viscosity from centipoise to pascal-seconds ($1\backslash mathrm\{cP\}\; =0.001\backslash mathrm\{Pa\backslash ,s\}$).

Once velocity is known, multiplying by cross-sectional area yields volumetric flow rate $Q=vA$. If width and height are in micrometers, area is $w\times h$ in square micrometers; converting to square meters requires multiplying by ${10}^{-12}$. The calculator outputs flow in microliters per minute for convenience.

Imagine a 20 mm long channel that is 50 µm wide and 10 µm high. Applying 100 V across it with a zeta potential of -50 mV in water (relative permittivity about 80, viscosity 1 cP) leads to an electric field of 5,000 V/m. Plugging the numbers into the formula yields a velocity near 3.5 mm/s. The cross-section of 50 µm by 10 µm equals 500 µm², or $5\times {10}^{-10}$ m². The flow rate becomes roughly 1.05×10^-9 m³/s, equivalent to about 63 µL/min.

Small changes can greatly affect throughput. Doubling the voltage doubles the electric field and thus the flow. Increasing channel height or width boosts cross-sectional area linearly. Conversely, raising viscosity or lowering zeta potential reduces speed. Many labs fine-tune surface coatings or buffer composition to optimize these parameters. Because nanoscale channels confine liquids so tightly, even a slight alteration in wall chemistry can modify zeta potential by tens of millivolts.

Electroosmotic pumping plays a key role in microfluidic and nanofluidic devices. It moves tiny volumes precisely, enabling lab-on-a-chip systems that analyze DNA, proteins, or environmental samples with minimal reagents. In capillary electrophoresis, for example, electroosmosis transports samples through a separation column while an electric field sorts molecules by charge. Biomedical researchers use similar concepts to deliver drugs at controlled rates. Commercial sensors rely on these flows to sample minuscule quantities of fluid from complex mixtures.

While the formula seems simple, designing nanofluidic systems requires paying attention to numerous subtleties. Channel corners may accumulate charges differently than flat surfaces. Temperature changes the viscosity and permittivity of water, altering flow. At extremely small scales, slip effects mean the liquid doesn't stick perfectly to the wall, modifying velocity from the classical prediction. Some fluids contain surfactants or polymers that interact with electric fields in non-linear ways. Computational models can capture these details, but this calculator offers a first-order estimate for typical conditions.

Below is a table illustrating how voltage and channel dimensions impact flow for water at room temperature with a -50 mV zeta potential:

Voltage (V)	Width (µm)	Height (µm)	Flow (µL/min)
50	50	10	31
100	50	10	63
200	50	10	126
100	100	10	126

Notice that doubling voltage or channel width doubles the flow rate, assuming other variables stay constant. This proportionality holds because the underlying physics is linear at these scales. Real devices may show slight deviations due to heat generation or electrochemical reactions at the electrodes.

Proper unit conversion matters for accuracy. Vacuum permittivity is approximately 8.854×10^-12 F/m. Multiplying by the relative constant gives absolute permittivity in farads per meter. Viscosity in pascal-seconds is cP divided by 1000. By default the calculator uses micrometers for dimensions because most nanofluidic channels fall in that range, but you can input other units if you convert them beforehand.

Because electroosmotic flow does not rely on mechanical pumps, systems can be built with no moving parts, reducing contamination and mechanical wear. This makes them attractive for portable analyzers or long-term biological experiments. However, electrolysis near the electrodes may create bubbles or alter pH over time, so careful design is necessary for prolonged operation.

Experimentally measuring nanoliter per minute flows can be tricky. Techniques include tracking fluorescent dye plugs, monitoring pressure changes with miniature sensors, or weighing collected liquid on an ultra-sensitive scale. Comparing these experimental results with predictions from the calculator can validate your setup. If measured flows fall short, check for leaks, bubbles, or unexpected surface charges.

The script here performs calculations entirely in your browser. When you click the button, JavaScript converts inputs to SI units, uses the Helmholtz–Smoluchowski equation for velocity, multiplies by cross-sectional area, and converts the result to microliters per minute. Because no data leaves your device, you can run the tool offline or adapt it for custom prototypes. All the math is transparent so you can double-check every step.

Electroosmotic flow remains an active area of research. Scientists explore ways to enhance zeta potential through novel materials, including graphene and various nanoparticles. Others investigate mixing and separation strategies using alternating electric fields or patterned surface charges. Whether you're designing a tiny pump for a biosensor or studying fundamental ion transport, understanding how parameters interact empowers you to achieve more predictable results. This calculator helps you estimate baseline performance so you can focus on refining your experiments.