We are working on an application requiring rapid stage translation between multiple fields of view (FOVs). Recently, we got an interesting suggestion from a reviewer, asking us to “comment on the tolerable speed in lateral and axial directions” from the perspective of drag resistance between the immersion medium and the coverslip.
I thought maybe some of you have thought about this in more depth and for longer, so I am curious about any insights you might have?
Of course, any information that would contribute to our response to the reviewer would be properly acknowledged!
Briefly, here is what I’ve come up so far:
intuitively I would think that the translation is less affected in the axial compared to the lateral direction
drag should be proportional to the viscosity of the immersion medium, such that lower viscosity results in less drag
if the timescales of the drag and viscous behaviour of the immersion medium are much lower than the stage response and translation timescales, then there is a separation of timescales and we can ignore the effect of drag for practical applications
I’ve definitely experienced issues with stage speed causing problems with oil in incubation scenarios, when the stage is warmed to 37. Here, conventional oils become less viscous, and slowing the stage down can help, this is the opposite of what it sounds like you want (speed!). A silicone objective may be in order…
Part of the answer depends upon the optical set up. If you are using an inverted microscope (like an old Zeiss IM35) and cover-slip bottomed culture dishes - with cell growing directly on the coverslip and the lens underneath, then translation speed is irrelevant. We used such a set up in intracellular injection studies together with 63X and 100X oil immersion lenses.
If you are using a water immersion lens, then I suspect it depend upon working distance - how close lens actually comes to the cells.
On our automated scope, we have a “slosh” delay to allow any well contents movement to stabilize. Turns out it works well to allow the oil to “catch up” with the objective movement. We are moving a few mm/sec but the need to wait a second or two for the oil drop to re position between the objective and sample. Hope that helps, Chris
Thank you for all your comments! I just wanted to share the approach we took to address this question (sorry for the equation formatting, main conclusions are in bold). Any feedback is welcome!
We expect that the lateral translation is more affected by the viscosity and the interaction forces between the objective, immersion medium and the coverslip, compared to the axial translation which is dictated mostly by capillary forces. Axial translation used is also in steps of ~100 nm, three orders of magnitude smaller than the lateral translation of ~100 µm. Therefore we will focus on the lateral translation since this will have the stronger effect.
To model the system during lateral translation, we consider the immersion medium as a viscous, incompressible Newtonian fluid confined between two parallel planes separated by a distance d, representing the coverslip and the objective, one of which is moving at a velocity U.
Starting from the Navier Stokes equation: ∂u/∂t + (u∙∇)u - ν∇^2u = -∇w + g
Where u is the velocity profile of the fluid, t time, ν the kinetic viscosity of the fluid such that ν=μ/ρ with μ the dynamic viscosity and ρ the density, w an internal source and g and external source of the fluid. Since we are working with an incompressible Newtonian fluid, we ignore the convection and source terms and can rewrite the remaining variation and diffusion terms with respect to the forces experienced by the fluid: μ∇^2 u + f = 0
To consider the lateral motion, we can introduce a force parallel to the two planes as f_x, and we write the equation explicitly along the x coordinate: μ((∂^2 u_x)/(∂x^2 ) + (∂^2 u_x)/(∂y^2 ) + (∂^2 u_x)/(∂z^2 )) = -f_x
Since the gradient will only exist along the y direction, we are left with: μ (∂^2 u_x)/(∂y^2 ) = -f_x
Solving this differential equation with the appropriate boundary conditions gives: u(y) = -f_x/2μ y^2
At the boundary between the moving plane and the fluid, we get: u(d) = U = -f_x/2μ d^2
Approximating the distance d as the focal length of the objective d = 3 mm and the viscosity and density of the immersion oil (Olympus Type F immersion oil) as ν = 450 mm^2/s and density of ρ = (28 g)/(30000 mm^3 ) ≈ 10^(-3) g mm^(-3), gives us a dynamic viscosity of μ = νρ = 0.45 kg m^(-1) s^(-1).
If the translation stage applies a constant force to shift position, the velocity of the stage will therefore scale inversely with the viscosity of the fluid and proportionally to the square of the distance from the objective to the coverslip. Therefore, for certain applications it could be beneficial to increase the separation of the objective from the coverslip before stage translation. Similarly, less viscous immersion media would allow faster translation speed.
This solution closely resembles a phenomenon known as the Couette flow (1,2), in which the time required for a fluid to form a velocity gradient is estimated as: t ~ d^2/ν ≈ (d^2 ρ)/μ ≈ 20 ms
In our case, given the maximal velocity of the stage specified by the manufacturer as 2 mm s^(-1)=2 μm ms^(-1) (Mad City Labs, MicroStage), changing a full field of view (FOV) by translating the stage by 100 µm will require approximately 100 ms (also considering rise and fall time). Since this is larger than the timescale required for the immersion medium to form a velocity gradient, we would expect the stage displacement to be sufficiently slow to avoid significant contribution of the drag force.
References:
Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.
Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.
