From: Dr A.J. Rowe <ajr@leicester.ac.uk>
  To  : rasmb@bbri.harvard.edu
  Date: Fri, 4 Apr 1997 10:24:09 +0100 (BST)

ks for spheres

Hello all rasmbers !

ks for spheres

My comment at the recent Regensburg AUC Meeting concerning the obsolescence of
the Batchelor value for the s-c dependence of spherical particles aroused a
number of comments afterwards - generally from people under the impression that
the Batchelor value was set in tablets of stone. Nice to find the topic of
sedimentation at *real* concentrations of solute actually does have interest !

A little filling in of things that have happened in this area in recent years
may thus be of some interest. Just very briefly - some other detail is given in
my chapter 21 in Harding, Horton & Rowe (AUC in Biochem & Polymer Science,
1992). 

We are talking ks as in s = s0(1 - ks*c), where for generality c is in *volume
fraction* terms, and hence ks is dimensionless. The problem was first addressed
in detail by Burgers (1941,1942) who showed that the solvent Tback flowU effect
accounted for a ks = 4 (solution density s values) or = 5 (solvent density s
values). Fujita in his 1962 monograph considered the former to be correct
usage: I will quote both in this communication, with an asterisk* against the
authorsU usage.

To this estimate of 4/5 for ks, Burgers then added another term^ covering the
Teffective viscosityU of the solution, giving a total ks = 5.875/6.875*.  A
modification of this treatment by Batchelor (1972) gave ks = 5.55/6.55*. This
value was for years quoted as definitive, especially by polymer chemists and
colloid scientists, although the average of (pretty scattered) data for
Tsettling coefficientsU of glass spheres etc never did agree at all well, the
limiting coefficient determined empirically being lower. Some decidedly dodgy
arguments were used to explain the discrepancy (e.g. Ttransient dimerU
formation !).

Of course, anyone who read the beautiful experimental work on PSL spheres by
Cheng & Schachman (1955) was aware that the empirical value for a clean system
was very close indeed to 4*/5 - but curiously polymer/colloid scientists have
ignored this work - and probably fluid mechanics people donUt read polymer
journals.

In 1977 a new mathematical treatment of a very different nature was published
(Rowe), which gave an expression applicable to ALL types and shapes of
particles, and which for compact spheres reduced to ks = 4*/5. Does that value
sound familiar ?!  The theory also gave an immediate theoretical basis to the
Wales-VanHolde ratio (ks/[eta]) = 1.6 for spheres. Widely used in the former
Soviet Union, this theory has been only modestly applied otherwise.

And in the non-biophysical area ?  Well, in 1988 Brady & Durlovsky published a
completely new treatment, accurate they claimed to a very high degree of
numerical approximation, which predicted the sedimentation of spheres over the
whole possible range of volume fraction (i.e. up to 0.64), a treatment which
agreed very well with empirical evidence from various workers. And the limiting
value for ks ? YouUve guessed it:   ks = 4/5*.  They examined the Batchelor
treatment and showed it to be RaphysicalS in its predictions^^. Just why it is
wrong remains a nice point, but the Brady-Durlovsky analysis leaves no doubts
but that wrong it certainly is. 

Since then IUve been following up a nice clue in one of BurgerUs old papers,
and developed a new treatment applicable to ALL c-values. The final equation is
in my chapter quoted above, the proof will be submitted very shortly - but
anyone with a few minutes and a computer to hand can check that the predicted
relationship is virtually identical numerically to the Brady-Durlovsky
treatment. 

However, lets for the moment stick with the limiting coefficient. Can anyone
out there give me good reason for using anything except ks = 4/5 ?

^the logic of adding this extra term has been disputed (Rowe, 1977)
^^i.e. if terms are retained, it actually predicts a *negative* s values for
even quite modest volume fraction of (denser than solvent) solutes.

References:

Batchelor, G.K. (1955) J. Fluid Mechanics 52, 245
Brady, J.F. & Durlovsky, L.J. (1988) Phys. Fluids 31, 717-727
Burgers, J.M. (1941) Proc. Ned. Akad. Wet. Amsterdam 44, 1045-1051 & 1177-1184
Burgers, J.M. (1942) Proc. Ned. Akad. Wet. Amsterdam 45, 9-16 & 126-128
Cheng,P.Y. & Schachman, H.K. (1955) J. Polymer Sci., 16, 19-30
Rowe, A.J. (1977)  Biopolymers16, 2595-2611

All very best to everyone !

Arthur Rowe

***************************************************
Dr Arthur J Rowe
Director
UK National Centre for Macromolecular Hydrodynamics
Leicester Laboratory
Adrian Building
University of Leicester
Leicester LE1 7RH    UK

Tel: +44 (0)116 252 3448
Fax: +44 (0)116 252 5602
ajr@leicester.ac.uk
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