PDMS Viscometers: Using Microfluidic Technologies for Everyday Applications

Utilizing the Hagen Poiseuille Equation to Determine Sample Viscosity

The Hagen Poiseuille equation relates the velocity of a newtonian fluid to its viscosity. To solve for viscosity the hydraulic diameter, pressure differential (between fluid interface and channel volume), length of fluid travel, and fluid velocity must be known. Hydraulic diameter, velocity and length can be measured during the experiment. Determining the pressure differential is difficult, thus a reference fluid of known viscosity is utilized to eliminate this variable.

Total pressure differential consists of two components: pressure differential caused by the PDMS reabsorbing air and causing a difference of pressure between the inside channel volume and atmosphere, and pressure differential due to capillary force. Measuring the pressure differential inside the small microfluidic channels would be a difficult endeavor that would require a micro scale pressure sensor. Calculating capillary force

requires the experimenter to know the surface tension and contact angle for the fluid in question. Having to first determine these values before calculating viscosity defeats the purpose of having a quick and user friendly approach to measuring fluid viscosity with one device.

Using a reference fluid eliminates the need to find both components of the pressure differential.

  Beginning with the derived Hagen Poiseuille equations:

Since the vacuum pressure differential caused by the PDMS absorption is equal for both reference and sample fluids {2} and {3} can be set equal to each other to eliminate

To eliminate the capillary force variable a series of assumptions needs to be made. First, is that capillary force is constant throughout the experiment, which in turn implies that dynamic contact angle and surface tension is constant. Secondly, capillary driven flow before a vacuum is established is ignored. Experimentally this means that timing of the fluid’s distance traveled does not begin until the air reservoir has been plugged and the PDMS has been given time to absorb the air reservoir and channel gas volumes. PDMS wets many fluids once plasma treated so capillary flow will begin as soon as the fluid reservoirs are filled. This initial flow should be disregarded from the length and velocity calculations.

Solving {4} in terms of velocity and time  yields;

and if the capillary forces are considered constant {5} takes the linear equation line format of y = mx+b so that slope

By recording multiple length and velocities for one run the data set is plotted to determine slope, which in turn leads to the final solution