Concentration Gradients in Microfluidic Devices

Technical Challenge

Microfluidic devices often contain solutions carrying multiple dissolved chemical species with varying concentrations.  In many biochemical processes the challenge is to minimize the length of concentration gradients.  In other cases, such as the synthesis of composite biomaterials or mimicking in vivo chemical gradients, the challenge is to produce long chemical gradients.

Controlling gradient length is often a critical device or process design feature and requires considering both fluid mechanics and mass transport.

Chemical concentration across alternating segments of chemical and buffer solutions
Figure 1. Chemical concentration across alternating segments of chemical and buffer solutions flowing slowly (left) and quickly (right) through a microchannel. Images show half the channel, cut along the middle for visualization. Concentration gradients between segments are initially sharp, but are stretched by the flow as the segments flow through the channel. This stretching effect is more pronounced for faster flows.


Veryst Solution

To relate the concentration gradient length to the flow and chemical parameters, we developed a COMSOL Multiphysics finite element model to simulate the flow of alternating chemical and buffer solution segments in a microchannel.  Figure 1 shows how chemical concentration evolves in a stream of alternating segments of chemical and buffer solutions for slow and fast flows.  The initially sharp chemical concentration gradients become longer and more diffuse over time, and grow more quickly for faster flows.  Concentration gradient growth is characterized by the Peclet number, which is the ratio of transport by the flow (advection) to that by diffusion, Pe = channel height × velocity / chemical molecular diffusivity.

Slow and fast gradient growth corresponds to slow and fast flows, which in turn correspond to small and large Peclet numbers, respectively.

Because the concentrations are dilute, we used single-phase fluid flow and applied one-way coupling between the fluid mechanics and mass transport.  Furthermore, higher flow rates lead to sharp concentration gradients, accounted for in our model.  We also ensured that numerical diffusion is sufficiently small for these high flow rate cases.   

Stretching of chemical concentration profiles
Figure 2. Stretching of chemical concentration profiles shown in a vertical cross-section of a microchannel for different Peclet numbers. The Peclet number is modified here by pumping the same fluid volume (50% of the channel volume) at different speeds through the channel. Dashed magenta lines indicate 5% and 95% vertically averaged concentrations delineating the concentration gradient.

 

Figure 2 shows chemical concentration profiles in a vertical cross-section of a microchannel for different Peclet numbers.  The Peclet number was modified here by pumping the same fluid volume (50% of the channel volume) at different speeds through the channel.  The higher the flow speed (and Peclet number), the more the concentration profiles are stretched.  For these simulations, the microchannel dimensions are 0.1 mm height, 1 mm width (into page), and 2 cm length and the chemical species has diffusivity 10-9 m2/s .

A short time after the flow stops, vertical concentration gradients equilibrate due to molecular diffusion, leaving only horizontal concentration gradients (Figure 3).  The dashed magenta lines in Figures 2 and 3 indicate the 5% and 95% vertically averaged concentrations; the distance between these is the measure of gradient length.

Concentration profiles from figure 2 after molecular diffusion vertically equilibrates the concentration profiles.
Figure 3. Concentration profiles from figure 2 after molecular diffusion vertically equilibrates the concentration profiles. Dashed magenta lines indicate 5% and 95% vertically averaged concentrations delineating the concentration gradient, whose length increases with increasing Peclet number.

 

The vertically averaged concentration gradients are plotted against axial distance in Figure 4, showing how increasing the Peclet number increases the gradient length, for the same volume of pumped fluid.

Concentration profiles from figure 3 vs. horizontal position, showing how concentration gradient length increases with Peclet number.
Figure 4. Concentration profiles from figure 3 vs. horizontal position, showing how concentration gradient length increases with Peclet number.

 

In Figure 5, the gradient length is plotted for a wide range of Peclet numbers.  Each curve corresponds to the same fluid volume (as % of channel volume) pumped into the channel at different speeds.  For small Peclet numbers (low flow speeds), gradient length increases sharply with increasing Peclet number.  As the Peclet number increases (higher flow speeds), its effect on gradient length decreases.

For a given channel geometry, the approach presented here may be used to identify the flow speeds and pumped volumes necessary to achieve a certain gradient length, or the critical flow speed (and Peclet number) beyond which gradients would exceed a desired threshold length.

. Concentration gradient length vs. Peclet number
Figure 5. Concentration gradient length (distance between 5% and 95% concentrations) vs. Peclet number in a straight microchannel. Each curve corresponds to the same fluid volume (as % of channel volume) pumped into the channel at different speeds. For small Peclet numbers (low flow speeds), gradient length increases sharply with increasing Peclet number. As the Peclet number increases (higher flow speeds), its effect on gradient length decreases.

 

Conclusion
The length of concentration gradients in microfluidic devices depends on the flow rate, chemical molecular diffusivity, and channel geometry, and is characterized by the Peclet number, which is the ratio of transport by the flow (advection) to that by diffusion.


Finite element simulations of the fluid flow and mass transport in a microchannel enable prediction and control of concentration gradients, useful across a range of diagnostics, genomics, and pharmaceutical applications.

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