A Rheologist’s Perspective: The Science Behind the Couette Cell Accessory

Posted by Bernard Costello on Jul 29, 2013 in Linear Dichroism | 0 comments

Bernard Costello

Senior Application Scientist

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Determining the binding mode of small molecule ligands to DNA, and the orientation of membrane proteins are among the applications of linear dichroism (LD), which is the difference in absorbance between light linearly polarised parallel or perpendicular to an orientation axis. For a sample to have a non-zero response it must be anisotropic, by definition: the anisotropy can be inherent, as in liquid crystals, or induced, for example by an electrical field or a shear field. Shearing is the motion of a plane of liquid parallel to an adjacent plane, and is usually represented, in two dimensions, as can be seen in Figure 1 where y is the thickness of the layer of liquid between the two planes, and  Δx is the relative plane displacement. But the representation can be misleading, because the shear rate can vary across a finite layer, and the diagram is only correct in the limit as y → 0.

Figure 1. Representation of shearing

Figure 1. Representation of shearing

Rigid or semi-rigid particles such as DNA molecules will align in a shear field, while flexible particles such as polymer molecules or micelles will deform by stretching in the direction of shear. In both cases the anisotropy required for LD is produced (you need a chromophore as well of course).

We have introduced a high shear Couette cell as an accessory for the Chirascan range of spectrometers. A Couette cell consists of two concentric cylinders, the outer of which is rotated. The sample resides in the gap between the two cylinders, which in our case, are made of quartz for transparency in the far-UV region. The light passes from the spectrometer’s monochromator, through the cylinders and sample to a detector.

Applied Photophysics’ system is carefully designed to overcome a drawback of the Couette cell which is that the shear rate, varies across the gap: it is higher at the inner surface than at the outer surface. The effect is not insignificant; the shear rate is proportional to the reciprocal of the radius squared. What this means in practice is that the induced anisotropy will vary as well. This may not matter for qualitative measurements, but it will for quantitative measurements, or for comparison with the results of rheological or others measurements in which the sample is sheared. In short, it is preferable that all parts of the sample experience the same shear rate, in the same way that it is preferable that they are all at the same temperature.

The demonstration that the shear rate varies across the gap is straightforward, but it is easiest to go back to the definition of the liquid viscosity, η, as the shear stress, σ, divided by the shear rate. The shear stress is the force acting over an area, with units of pascals, Pa, equivalent to N.m-2 (it is not a coincidence that stress has the same units as pressure, which could be – sometimes is – called the isotropic or bulk stress).

To return to the Couette cell, and assuming that the moving cylinder and each point in the fluid are rotating at constant velocity, then the torque, M, must be constant across the gap. But the torque is equal to the radius, r, i.e. the distance from the axis of rotation, at any point multiplied by the force, F, at that point. Since, A, the area over which the force is acting, is the cylindrical surface at that radius, equal to 2πrl, where l is the length of the cylinder, the stress is proportional to the torque divided by the radius squared. In fact, that is as far as the analysis can go unless some assumption is made about the sample viscosity. The simplest assumption is that the sample is Newtonian, i.e. that the viscosity is independentof the shear rate. This is true (or is at least a very good approximation) for most dilute aqueous solutions of biomacromolecules over the usual range of shear rates, which means that for these materials the shear rate will also be proportional to the reciprocal of the radius squared.

The derivation of the shear rate from what we actually know, the angular velocity, Ω, is a bit more complicated, but it turns out that at the inner cylinder wall is 2ΩRo2 / (Ro2 – Ri2) , and at the outer cylinder wall is 2ΩRi2 / (Ro2 – Ri2), where R is the radius and the subscripts refer to outer and inner. It may seem a bit counterintuitive that the shear rate at the inner cylinder is higher than that at the outer cylinder although the radius is lower, but that is how it is.

What this means for the design of a Couette cell is that the ratio of the radius of the outer cylinder to that of the inner cylinder should be as close to 1 as possible. For the APL high shear cell, the radius of the inner radius is 4.525 mm, and that of the outer cylinder 4.775 mm (Figure 2), so the difference in shear rate across the gap is about 11%, low enough to be negligible for practical purposes.

Figure 2 - Dimensions of the Applied Photophysics Couette Cell Accessory

Figure 2 – Dimensions of the Applied Photophysics Couette Cell Accessory

There are other considerations in the design of a Couette cell for linear dischroism, for example ensuring that the flow is laminar and circular, that high shear rates can be achieved, and the practical considerations of low samples size and ease of use, and these have also been taken into account in the Applied Photophysics design.

Download the technical datasheet for more information

Download the CCA Technical Datasheet



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