A stress-controlled microfluidic shear viscometer based on smartphone imaging

Abstract

We report a stress-controlled microfluidic shear viscometer with flow visualization aided by smartphone technology. The method involves driving the fluid into a microchannel at constant pressure and using the smartphone camera to track the fluid front in a glass capillary attached to the microchannel. We find that videos of interface propagation from the smartphone are of sufficient resolution that accurate pressure drop-flow rate relations can be determined to quantify the viscosity curves for complex fluids. We demonstrate that this simple ‘iCapillary’ device measures the shear viscosity of Newtonian and polymeric fluids over a broad range of shear rates (10–10,000 s⁻¹) that is in quantitative agreement with rotational rheometry. We further show that the simplicity of the iCapillary device allows for parallel analysis of viscosity of several samples. We performed multiplexed measurements of concentration dependence of high shear rate viscosity of globular protein solutions, and the results are in good agreement with models of suspension rheology as well as prior experimental data. Our approach is unique, since no on-chip sensing element is required other than the smartphone camera. This sensor-less approach offers the potential to create inexpensive and disposable devices for point-of-care rheology of complex fluids and biological samples.

Appendix

Wall shear stress relation for the microchannel in the iCapillary device

To calculate the viscosity for a Newtonian fluid, we use a resistive network approach similar to electrical circuits, where the relation between pressure drop and flow rate is specified using the analogy of Ohm’s law. A schematic representation of the iCapillary device is shown in Fig. 1, which also depicts the pressures at different sections of the device. The pressure drop of a fluid flowing through a conduit can be determined from the product of the hydrodynamic resistance of the fluid (R) and the volumetric flow rate (Q). The hydrodynamic resistance is a function of fluid viscosity and conduit dimensions.

Fig. 1

A schematic diagram of the iCapillary device. The top right illustration shows the magnified view of the air-fluid interface having a contact angle, θ, with the wettable surface (i.e. glass capillary in our experiment)

For the microchannel, the pressure drop and flow rate relation is given by

$$ {\mathrm{P}}_{\mathrm{c}}\hbox{--} {\mathrm{P}}_1={QR}_{\mathrm{c}} $$

(1)

where P _c , P ₁ and R _c are the inlet pressure, outlet pressure and hydraulic resistance of the microchannel respectively.

Likewise for the glass capillary, the pressure drop and flow rate relation is given by

$$ {\mathrm{P}}_1\hbox{--} {\mathrm{P}}_{int}={\mathrm{P}}_{\mathrm{h}1}+{QR}_{\mathrm{g}} $$

(2)

where P _int , P _h1 and R _g are the internal pressure, hydrostatic head and the hydraulic resistance of the glass capillary respectively.

Finally, the Laplace pressure jump (P _L ) across the air-water interface is given by

$$ {\mathrm{P}}_{\mathrm{atm}}\hbox{--} {\mathrm{P}}_{int}={\mathrm{P}}_{\mathrm{L}} $$

(3)

where P _atm is the atmospheric pressure, taken as zero.

From Eqs. (1) – (3), we get

$$ {\mathrm{P}}_{\mathrm{c}}+{\mathrm{P}}_{\mathrm{L}}-{\mathrm{P}}_{\mathrm{h}1}=\mathrm{Q}\ \left({\mathrm{R}}_{\mathrm{g}}+{\mathrm{R}}_{\mathrm{c}}\right) $$

(4)

Since R _g  < < R _c , we obtain the pressure drop across the microchannel (ΔP) as

$$ \varDelta \mathrm{P}={QR}_{\mathrm{c}}={\mathrm{P}}_{\mathrm{c}}+{\mathrm{P}}_{\mathrm{L}}-{\mathrm{P}}_{\mathrm{h}1} $$

(5)

Taking a control volume inside the microchannel and balancing the forces due to pressure and wall shear stress gives

$$ {\tau}_w\left[2{L}_{ch}\left(w+h\right)\right]=\varDelta P(wh) $$

(6)

In Eq. (6), L _ch , w and h are the length, width and height of the microchannel respectively. Using Eq. (5) in Eq. (6), we thus obtain the wall shear stress experienced by the microchannel in the iCapillary device as

$$ {\tau}_w=\frac{\left({P}_c-{P}_{h1}+{P}_L\right)wh}{2{L}_{ch}\left(w+h\right)} $$

(7)