Centrifugal sedimentation and Brownian motion

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14(1976)187 -188 @ Elsevier Sequoia Sk, Lausanne - Printed in the Netherlands

Short

187

Communication

For an estimation of the lower size limit we require the displacement of the smallest particle by Brownian motion to be ten times shorter than its centrifugal path, i.e. Ar2 = lo-* (F -s)*. Correspondingly, eqn. (3b) takes an equality sign and its righthand side is multiplied by lo*. With numerical values representative for pigments and the Joyce, Loebl disc centrifuge 131, namely

Centrifugal Sedimentation and Brownian Motion

K. BRUGGER C&x-Geigy

Ltd..

Basel

(Received September November 10,1975)

(Switzerland)

9, 1975;

in revised form

Ap = 400 It is well known

that size determination by centrifugal sedimentation is limited for small particles by their Brownian motion Cl], and it seems worth while to have a closer look at this limitation_ The terminal speed ; of a particle at a radial position r in the centrifuge and At, the time required to move radially between s and r, are given by the familiar centrifuge equations [l] a5 F = 1/18-Aplrl.D2W2r

(la)

and At = 18.q/Ap-l/0*0*-ln(r/s)

(lb)

where D represents the Stokes diameter of the particle, Ap the density difference between particle and fluid, q the dynamic viscosity of the fluid, and o the angular velocity of the centrifuge. On the other hand, its mean square displacement in radial direction by Brownian motion in the time interval At is [a] Ars = (2kTjp)At

k

= 1.37

T

= 300K

W* = 7

=

r

s =

X

kg/m3 X lo-=5 10” s-*

K-l (3000

RPM)

0.048m 0.043

m

we find the minimum diameter, Dmin, = 0.03 pm, and conclude that the lower sire limit lies at several hundredths of a micrometer. It is also of interest to ask for the time At in which particles undergo the same radial displacement either under the influence of the centrifugal force or by Brownian motion,

i.e. (Ar)*

= (iAt)*

= A?

From eqns. (4),

At=C/D=

(la),

(4) (2a) and (2b)

we find

(W

Vd

with p = 3nqD

G-1

the coefficient

of the Stokes drag term. For a size determination to be meaningful, the displacements of the particles by Brownian motion must be much smaller than their displacementsundertbe centrifugdforce,hence the condition af<

(r--s)2

or, using eqns. (lb),

(3a) (2a) and (2b),

D3 S- 12/s-kT/Apw*-ln(r/s)/(r-s)*

(3b)

D(vm) Fig. 1. At = C/D5_ Particles of size D undergo in time At the same displacement by Brownian motion as under the influence of the centrifugal force (numerical values in the text)_

168

with

C = 216ja-kTql(Apwar)g

(5b)

With the numerical vaIues above, supplemented by q = 1W3 Pa s, the viscosity of water, we obtain C = 1.5 X low6 pm5 s. This inverse fifth-power law is represented in Fig_ 1.

REFERENCES 1 W. Batel, Einfiihrung in die Korngriissenmesstechnik, Springer, Berlin. 2nd edn.. 1964. 2 R. Becker, Theorie der WZrme, Springer, Berlin, 1964. 3 Joyce, Loebi and Co.. Ltd.. Gateshead, England.