Thank you.

Hello, good morning.

Like previous speakers, I am not the first time in an AUC meeting.

I am a veteran.

Unfortunately, I was unable to attend the two last ones in New Zealand and Canada.

So after my last time in Glasgow, I am back again.

And then I am collecting a few works that I have done in the seven years from them,

and which I think of interest to the AUC community.

Let me begin with telling you about a review paper that I published a few years ago,

GMB, and that was a kind of tutorial, a mini-review,

trying to communicate the essential concepts of hydrodynamics, including AUC, of course.

And then I want to give a brief overview of hydrodynamic theory,

which will be surely familiar to most of you, but maybe also welcome for beginners.

And the classical introduction to AUC that you will find in textbooks,

the text talks about the time course of sedimentation,

I mean, how the boundary, either it is absolutely sharp without diffusion or soft when diffusion is present.

Anyway, if you follow the movement of the boundary with time,

then the classical treatment, the basic treatment tells you that the position of the boundary

at some time with respect to the meniscus, the logarithm of that ratio is proportional to the elapsed time.

And the quantity, the proportional constant, which will be the slope in this straight line,

is a square of rotor velocity times a quantity which is called sedimentation coefficient.

Well, the sedimentation coefficient is in principle mechanically defined as the ratio of the linear velocity of translation of the particles

over the angular velocity at the point where they are located.

And the essential, the basic treatment of the equilibrium forces in the process

leads to a famous equation relating the sedimentation coefficient to molecular weight and the friction coefficient.

The friction coefficient, of course, for a spherical particle is 6p viscosity and radius a.

And the text would say that this treatment allows us to obtain the radius of such a spherical particle.

Anyway, keep in mind that the sedimentation coefficient is just a preliminary output of the analysis of the sedimentation process.

Now, if you look a bit into the details of the theory, then this equation is not, in principle, can be written in a more simplified form

if you take into account that all this term except f is the buoyant mass of the particle.

Then the mass of the particle, of course, is molecular weight over an Avogadro number,

and the buoyant mass is corrected by the buoyancy factor, b bar, the specific volume of the particle, and rho, the solvent or solution density.

Well, so it is clear that the sedimentation coefficient is certainly a quantity which is characteristic of the particle,

but not only of the particle, but because in the value of the sedimentation coefficient

the viscosity of the solvent, the density of the solvent or solution, and even the temperature in some way.

So my point is that the sedimentation coefficient certainly gives you an information, conformational information,

about the particle, but in a mixed way because the dependence comes from the mass, which here, and also at the same time from the friction coefficient.

Well, about the friction coefficient, according to fluid mechanics, the friction coefficient of any particle in a fluid solvent is first proportional to the solvent viscosity,

and then if we call f star to the proportionality constant, this is a factor then which depends on the size and shape of the particle.

I like to call this reduced frictional coefficient because it's just removing the effect of the solvent viscosity.

So according to the well-known Stokes equation, friction coefficient is 6 pi eta 0 times the radius,

so f star is again something that depends on the shape, 6 pi for the sphere, and the size radius for the sphere.

The spherical radius then can be obtained easily from this Stokes equation.

So if again you remove the effect of the viscosity dividing by this factor, then you will get what is called the hydrodynamic radius or Stokes radius,

which is clearly then related to the sedimentation coefficient according to the previous equation or the diffusion coefficient according to the Einstein equation.

Okay, so for the analysis of the sedimentation velocity experiments, the quantity that is primarily looked for is the sedimentation coefficient,

but again I emphasize that this quantity includes two effects regarding the size and shape of the particle.

The size of course is here in the mass, and the size and shape is again here in the radius of the equivalent sphere, the Stokes radius.

So the idea that I have in mind is that the analysis of the AUC experiments or the computational prediction of experiments could be done starting not as usual,

I mean the prediction could be done from or the analysis could result in not as usual in terms of sedimentation coefficient and the so-called frictional ratio,