Gravitational and Magnetic Settling of Assay Beads Problem Statement (Part 1) Numerous biochemical assays use solutions of magnetic beads. Typically, a solution of beads is flushed into a system where a chemical reaction takes place at the surface of the beads (perhaps chemical binding due to antib

Question

Gravitational and Magnetic Settling of Assay Beads

Problem Statement (Part 1)

Numerous biochemical assays use solutions of magnetic beads. Typically, a solution of beads is flushed into a system where a chemical reaction takes place at the surface of the beads (perhaps chemical binding due to antibodies). Magnets are used to control whether beads flow freely or are held stationary, as in Fig. E12.5. Two characteristic settling times are important:

The gravitational settling time (i.e., how long can a solution of beads be stored before the beads fall to the bottom of the container?).
The magnetic trapping time (i.e., how quickly will a magnet pull the particles into a wall?).

Given that [latex]\operatorname{Re}<1[/latex] for a microsphere, derive the terminal velocity for a microsphere. Estimate the settling time in hours for 1- [latex]\mu \mathrm{m}[/latex] diameter magnetic microspheres in water in a container of height [latex]L=1 \mathrm{~cm}[/latex] by calculating the time required for particles at the terminal velocity to travel [latex]0.5 \mathrm{~cm}[/latex]. Assume that the magnetic microspheres have a density of [latex]4 \mathrm{~g} / \mathrm{ml}[/latex].

Problem Statement (Part 2)

Assume that a magnet is used to apply a magnetic force [latex]F_{M}[/latex] of approximately [latex]1 \mathrm{pN}[/latex] to these microspheres, which is a typical value for polystyrene beads with [latex]\sim 15 \%[/latex] ferrite and typical rare-earth magnets. How many seconds would it take for these microspheres to move from one end of a [latex]20-\mu \mathrm{m}[/latex] diameter channel to the other? As before, use the terminal velocity to estimate the elapsed time.

Accepted Answer

Solution Problem Statement (Part 1)

For [latex]\operatorname{Re}<1, F_{D}=3 \pi \mu u_{\infty} D[/latex]. Since the drag force must balance the gravitational body force [latex]\left.\left(\pi D^{3} / 6 ight)\left( ho_{s}- ho_{f} ight) g ight)[/latex] :

[latex]3 \pi \mu u_{\infty} D=\frac{\pi D^{3}}{6}\left( ho_{s}- ho_{f} ight) g .[/latex]

First, solve for [latex]u_{\infty}[/latex] :

[latex]u_{\infty}=u_{t}=\frac{D^{2}\left( ho_{s}- ho_{f} ight) g}{18 \mu}.[/latex]

Then evaluate:

[latex]\begin{aligned}u_{\infty} &=u_{t}=\frac{\left(10^{-6} \mathrm{~m} ight)^{2}\left((4-1) imes 10^{3} \mathrm{~kg} / \mathrm{m}^{3} ight)\left(9.8 \mathrm{~m} / \mathrm{s}^{2} ight)}{18 imes\left(10^{-3} ext { Pa s } ight)}=1.6 \mu \mathrm{m} / \mathrm{s}, \t &=\frac{L}{u_{t}}=\frac{0.5 imes 10^{-2} \mathrm{~m}}{0.16 imes 10^{-6} \mathrm{~m} / \mathrm{s}}=3,125 \mathrm{~s}=0.87 ext { hours. }\end{aligned}[/latex]

Solution Problem Statement (Part 2)

[latex]\begin{aligned}3 \pi \mu u_{\infty} D &=F_{M}, \quad u_{\infty}=u_{t}=\frac{F_{M}}{3 \pi \mu D}, \u_{\infty} &=u_{t}=\frac{10^{-12} \mathrm{~N}}{3 \pi\left(10^{-3} \mathrm{~Pa} \mathrm{~s} ight)\left(10^{-6} \mathrm{~m} ight)}=106 \mu \mathrm{m} / \mathrm{s}, \t &=\frac{L}{u_{t}}=\frac{20 \mu \mathrm{m}}{106 \mu \mathrm{m} / \mathrm{s}}=0.19 \mathrm{~s} .\end{aligned}[/latex]

Question 12.7: Gravitational and Magnetic Settling of Assay Beads Problem S...

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