Gradients in microchannels
Current projects leaders: Du [concept, expts, bio applications], Matt [concept, theory]
Status of current papers
1. cycle flow [status: simulations done; theory & experiment agreement ongoing]
cycle experiment [Jose and Du, mostly DONE, mint version of FITC runs TODO]
theoretical prediction [DONE]
flow & concentration [DONE]
modeling diffusion-only 0-20s "down-time" step [DONE], short [5cm] and long [10cm] channels
expt - pure diffusion to measure D of FITC [DONE]
expt - theory agreement [Matt - ONGOING]
need initial profile in channel for accurate simulation - get from data
use D from expt to predict gradient growth; troubleshoot
writeup [IN PROGRESS, theory done, figures proposed, references - attached below]
2. convergent channels [status: matt completing theory, which will motivate further expts]
Collaborators: Jaesool Shim, Mahesh, Jose [expts], Won Gu [channel design/fabrication]
prelim expts in convergent channels [DONE]
theoretical prediction [ONGOING]
theory concepts [DONE]
simulation coding [TODO]
prediction of channel shape that optimizes gradient [ideas complete, implementation next, then expt]
Experiments to verify the linear gradient [planning]
Simulation results
Summary: Faster is better. Most gradient creation happens on initial fwd flow segment. Gradient can actually decrease in subsequent bkwd/fwd flow segments. This effect is mitigated by the downtime required to reset pump between flow segments, in which only diffusion acts and vertically mixes.
This section contains: explanation of current simulation videos and plots, ongoing simulations and questions, followed by list of files [links to my private Google depot site for the simulation results - email me if you want access].
Videos contain:
3D simulation: top view of horizontal midplane, side view of vertical midplane, center concentration. 5cm and 10cm channels. In longer channels, gradient growth on subsequent fwd/bkwd steps. For shorter channel, majority of growth occurs in first pass (high Pe result).
2D simulation: we assume no variation in lateral horizontal direction across channel; allows higher-resolution 2D simulation, close enough to 3D results
various amounts of diffusion-only down-time are added between flow segments. Gradient does not grow during these down-times, but does not decrease in length as much (or at all) in subsequent flow segments. Simulations for Pe = 1600, 3200, 6400, 12800. Exact solution for 2D/3D diffusion in rectangular duct used and implemented using Matlab's dct/dst functions.
Center gradient evolution: color plot indicating centerline concentration along channel (x-axis = centerline; y-axis = time (increasing downward))
Loglog plots of gradient time evolution:
for one Pe value, we plot evolution of scaled distance Delta=Delta'/H from c=0.9 to c=0.1 [good estimate of gradient length] vs. scaled time t=t' U/H. Black solid line is cross-sectionally avg'd 3D simulation; blue solid is cross-sectionally avg'd 2D simulation; red dashed is centerline for 3D simulation, and magenta dash-dot is centerline for 2D simulation. We notice that centerline gradient predictions are virtually the same for 2D/3D simulations. The effects of the spreading due to the sidewalls [neglected in the 2D sims] is evident in the difference between the gradient lengths for the cross-sectionally averaged concentrations. Plots essentially look the same, but as Pe increases we notice the max gradient length gets larger, and is acheived earlier in the fwd/bkwd steps.
for all Pe values, we plot scaled gradient length Delta=Delta'/H for the cross-sectionally averaged 3D simulations over many Pe values, vs. scaled time t/Pe = t' D/H2. We use t/Pe in order to compare all simulations based on the same physical timescale [in this case the timescale for vertical diffusion, H2/D, about 100 seconds for FITC Dextran]. We notice that the largest and fastest gradients are formed for the largest Pe. For the largest Pe, one fwd flow segment in a 5 cm channel is enough to make a 3cm gradient, and subsequent flow segments must be very short to keep the gradient in the channel, so are associated with less growth. For smaller Pe's, subsequent flow cycles do extend gradient.
Questions: It is not yet clear exactly what Pe is associated with experimental data, and how much growth occurs in the tube before entry into channel, and if part of the gradient gets sucked out inlet on bkwd flow, and if so to what extent gradient is distorted. All FITC data will be processed by Mar 30, 2009; hopefully we'll have answer by then.
What Pe corresponds to experiments? Not sure yet as we have not nailed down mol diff D. Probably between 3200 and 6400 for flow rates 0.04 ml/min.
Files [links go to Matt's Private Google Depot site for simulations - email me if you want access]
3D simulation [5 cm channel] with Pe=6400, with no diffusion down-time
3D simulation [5 cm channel] with Pe=6400, with 0.08 units [dimensionless] of diffusion down-time (approx 10 seconds for D=7e-11 m2/s)
3D simulation [5 cm channel] with Pe=1600, with no diffusion down-time
3D simulation [5 cm channel] with Pe=1600, with 0.08 units [dimensionless] of diffusion down-time (approx 10 seconds for D=7e-11 m2/s)
3D simulation [10 cm channel] with Pe=6400, with no diffusion down-time
3D simulation [10 cm channel] with Pe=6400, with 0.08 units [dimensionless] of diffusion down-time (approx 10 seconds for D=7e-11 m2/s)
3D simulation [10 cm channel] with Pe=1600, with no diffusion down-time
3D simulation [10 cm channel] with Pe=1600, with 0.08 units [dimensionless] of diffusion down-time (approx 10 seconds for D=7e-11 m2/s)
gradient length vs. dimensionless time (units of diffusive timescale) [log, regular]. Pe = 1600, 3200, 6400, 12800 (right to left), varying amounts of diffusion down-time (diffusive down-time does not affect gradient length and is not shown explicitly)
Protocol notes
Calibrating fluoresence signal [from Golpeau, Lonetti, Trouchet, Ajdari, Tabelling 2007]
After you fill the channel with DI water, photograph with the special camera. This give the background image B.
Do the expt, and measure the signal images S(t), where t is time.
After the expt, fill the channel with fluid that has the max concentration. Photograph, this gives the max conc image M.
Then for each image S(t) in the experiment, calibrate by image subtraction and scaling (easy in Matlab): concentration(t) = (S(t) - B) / (M - B)