 ## Drag forces exerted upon a sphere in a rotating fluid

Falvin - 15-7-2021 at 20:06

I am currently working on finding the force with which some microscopic polystyrene beads would adhere to a glass surface. We hope to use the techniques to eventually characterize the adhesion of bacteria, and are currently using polystyrene microspheres simply to help develop a good model before doing so.

This is the method we're using at the present moment:

First, a small, diamond-tipped drill bit is used to mark the center of a microscope slide. Then, a rubber spacer with a hole in the middle is affixed to the slide. The space in the middle is then filled with a suspension of these microspheres in water. A cover slip is then affixed to the top, effectively creating a small, hollow cylinder filled with the microsphere suspension. A spin coater is used to rotate this sample about the marked center.
There are three parameters that could be programmed into the spin coater: The target speed in RPM, the time for which a cycle of rotation lasts, and the angular acceleration in RPM/s. The machine would accelerate the sample from rest at the programmed acceleration until the target speed is reached, then maintain this target speed for the rest of the cycle, then decelerate it back to rest.
Now, we want to calculate what force is exerted upon these microspheres at a known distance from the center of rotation. Since the microspheres are located within a fluid, I would expect that the drag force as a result of there being some relative velocity between the water and the glass would be the most significant. Here's what I'm thinking:

To find the drag force upon one such microsphere, we use Stokes's formula:

F=6πr1μV

Where r1 is the radius of a microsphere, μ is the viscocity of the fluid (Water, in this case), and V is the velocity of the water relative to a microsphere.

The relative velocity has two components: one directed radially, and one tangentially. The total relative velocity can be found simply via the Pythagorean theorem once both components are known.

The tangential component t= (mr2α)/(6πr1μ)
where α is the angular acceleration in rad/s2, m is the mass of one of these microspheres, and r2 is the distance from the center of rotation.

After finding the relative velocity, one simply needs to plug it into Stokes's formula to find the max force that the microsphere would be subjected to. However, since the gap between the cover slip and the glass slide is small (less than one mm) and the radius of a microsphere is around 7.5 micrometers, one would have to account for boundary layer effects, so the force is divided by 2.

Here is what I am wondering: Firstly, would this series of calculations give an accurate assessment of the force exerted upon a microsphere in this environment?
Another paper I've read deals with something similar called a Spinning Disk Assay and suggests to use the following formula for shear stress applied to a cell on the surface of a disk spinning within a fluid:

0.8r(ρμω3)1/2

where ρ is the density of the fluid

(Source: https://sci-hub.st/https://pubs.acs.org/doi/10.1021/acsbioma...)

Another question of mine is thus: Which of these formulae is more applicable to our current setup? Are there any deficiencies in using our calculations?
Finally, how would the calculations need to be modified in the case of a bacterium which is not spherical?

JohnnyBuckminster - 15-7-2021 at 22:33

Not exactly my field, but interesting questions. There are some discussions about fluid dynamics on physics.stackexchange.com, you might want to check those posts.

Out of curiosity, how large are the spheres, how do you detect the positions of the spheres in the fluid, I guess you have a microscope setup?

Falvin - 16-7-2021 at 06:09

Each microsphere has a diameter of 15 micrometers with a standard deviation of 1.43 micrometers from that. To detect how many microspheres are attached to the cover slip at a certain distance from the center, we use a dark-field light microscope fitted with a camera.

[Edited on 16-7-2021 by Falvin]

Falvin - 22-7-2021 at 09:47

On closer inspection, I think I found some problems with the calculations I proposed. Firstly, if using the formulae for the components of the velocity of the fluid, the viscosity terms end up cancelling out. In other words, the model as-proposed would predict that viscosity does not have an effect on microsphere detachment. This is clearly not true.
Secondly, we tried this same experiment, but this time mixed in a solution of polyethylene glycol 6000 in order to increase the viscosity to see how it would affect the results. The liquid in which the spheres were suspended was thus around 1.5% PEG 6000 w/v. Based on a table of PEG solution viscosities and this dilution, the viscosity should correspond to around double the viscosity of water. In two such experiments, we found that a smaller fraction of the microspheres detached compared to when distilled water was used. This would seem to contradict the idea that the force exerted on the spheres is proportional to the viscosity of the fluid. How could this be?

My guess would be that, due to the increased viscosity, the boundary layer is thicker, and so the fluid velocity at 0-15 micrometers from the surface of the glass is lower for the PEG solution compared to water. Would this be a reasonable guess as to why the results were what they were? If so, how could the forces that are exerted on the spheres be modelled more accurately?

Twospoons - 22-7-2021 at 17:02

Are you assuming a Newtonian fluid? It may not be ...
Falvin - 22-7-2021 at 19:09

Yes, I'm assuming that both the plain water and the PEG solution behave as Newtonian fluids.

One thing that I think is worth mentioning, and which I forgot to mention in my initial post, is that the cover slides are submerged into a solution of poly-l-lysine for 5 minutes before being taken out and allowed to dry prior to the experiment.

Twospoons - 22-7-2021 at 19:41

https://www.researchgate.net/figure/Viscosity-as-a-function-of-shear-rate-for-PEG-solutions-The-concentration-of-PEG_fig2_257972647

You may need to rethink your assumptions.

Falvin - 27-7-2021 at 19:51

I've tried the experiment again with glycerol. After rotation, there was absolutely no change in the positions of the spheres at the distances observed. I suppose I should have seen that coming.

Next, I tried the experiment with a mixture of glycerol and water (1 part glycerol to 2.5 parts water by volume). There were significantly more spheres observed on the cover slip at all distances measured (But near the center in particular). My guess is that, since polystyrene is marginally denser than water, then, when the experiment is conducted with water, any spheres that detach or are floating between surfaces would tend to migrate downwards, but the glycerol solution is denser than polystyrene, so any loose spheres would migrate upwards instead. Trying to centrifuge the sample with the cover slip at the bottom only resulted in a large air bubble forming in the center.

I'll check if we have a viscometer. If yes, I'll try using a mixture of ethanol and glycerol so that the density is lower. If not, I'll see if adding a nonpolar solvent such as n-hexane (I'm pretty sure polystyrene is insoluble in hexane) to microsphere suspension and then shaking it will result in a decent amount of the spheres being transferred to the hexane, and also how long it will take for them to settle out of the hexane. Then I'll try another experiment where I attach the spacer to the cover slip first, apply the suspension, and then let the water evaporate off to see what the results with the fluid just being air would be like.

Regardless, I'm thinking that this method is a lost cause and that it would be better to use a conventional spinning disk assay as outlined in this paper:
https://sci-hub.st/https://doi.org/10.1016/s0076-6879(07)26001-x

JohnnyBuckminster - 29-7-2021 at 01:27

 Quote: Originally posted by Falvin Each microsphere has a diameter of 15 micrometers with a standard deviation of 1.43 micrometers from that. To detect how many microspheres are attached to the cover slip at a certain distance from the center, we use a dark-field light microscope fitted with a camera. [Edited on 16-7-2021 by Falvin]

Could you share an image captured after the sample has been rotated according to particular program? What is the concentration of spheres in the starting solution, i.e. what you use to prepare the sample?

Falvin - 29-7-2021 at 08:05

Quote: Originally posted by JohnnyBuckminster Quote: Originally posted by Falvin Each microsphere has a diameter of 15 micrometers with a standard deviation of 1.43 micrometers from that. To detect how many microspheres are attached to the cover slip at a certain distance from the center, we use a dark-field light microscope fitted with a camera. [Edited on 16-7-2021 by Falvin]

Could you share an image captured after the sample has been rotated according to particular program? What is the concentration of spheres in the starting solution, i.e. what you use to prepare the sample?

This is an image taken around 360 micrometers away from the center before the rotation cycle: This is taken after the cycle: The parameters entered were as follows:

Initial acceleration: 100 RPM/S

Target speed: 8000 RPM

Duration of cycle: 160s

The microscope magnification was 40x, and both pictures are of the inner surface of the cover slip.

The stock suspension sold by the manufacturer is of the microspheres in water, with the spheres taking up 2.65% of the total weight, but we usually dilute it to various degrees. This particular fluid was prepared by mixing 360 microliters of glycerol, 900 microliters of deionized water. 100 microliters of the stock suspension was then mixed with 600 microliters of the fluid, IIRC, so the total concentration of spheres in the sample should be around 0.35% by mass.

[Edited on 29-7-2021 by Falvin]