FROM CURRENT LITERATURE

On Svedberg’s Ultracentrifuge

P. A. Rebinder.

Colloid chemistry, developing rapidly, has yielded not only extremely valuable theoretical results, but also many new laboratory methods of investigation. Among them, especially noteworthy are the so-called “ultramethods,” i.e., methods that make it possible to detect the presence of colloidal particles, or even particles of high-molecular substances (and sometimes to separate them from the surrounding dispersion medium).

In 1903 Siedentopf and Zsigmondy1 constructed their ultramicroscope, since then considerably improved, which makes it possible to detect the presence of particles down to 5–3 μμ in diameter (under favorable conditions). Following this, in 1906 Bechhold2 discovered a method of ultrafiltration, i.e., filtration under various pressures through gels compacted to different degrees—a method that makes it possible to separate particles as small as 1 μμ and even to determine their mean diameter. Finally, in 1923–24 Svedberg3 constructed his ultracentrifuge, which makes it possible to determine the rate of sedimentation of particles of highly disperse sols, e.g., gold sols with amicroscopic particles down to 2 μμ in diameter, protein substances, etc.

Long before this, centrifuges were widely used in laboratory practice for precipitating particles suspended in a liquid (separating suspensions and emulsions with particles of about 100 μμ and larger). Under quiet settling, the precipitation (sedimentation) of such suspensions requires a long time; in a centrifuge, however, the action of gravity is replaced by centrifugal force, sometimes 10,000 times greater (for which enormous rotational speeds are required), causing particles heavier than the dispersion medium to settle in the direction from the axis of rotation toward the circumference. An ordinary hand-driven laboratory centrifuge gives up to 200 revolutions per minute; with an electric motor, 600–1800 revolutions; Friedenthal4 constructed centrifuges giving from 11,000 to 27,500 revolutions per minute and enabling him to isolate casein from cow’s milk. Svedberg, however, in his ultracentrifuge uses only 5,000–9,000 revolutions per minute.5 The characteristic features of this in-

of the instrument, which makes it possible to observe the sedimentation of colloidal solutions of any degree of dispersity (down to 2 μμ), consists, first, in prolonged rotation, sometimes continuing uninterrupted for 60 hours, and, secondly, in an extremely ingenious device for photographing the contents of the centrifuged vessel at definite intervals of the rotation time, and in studying the photographs obtained, which show the change in the state of the sol over time; thus, the sedimentation process is not disturbed, the vessel with the sol is not removed from the centrifuge, and with sufficiently prolonged continuous rotation a complete picture of the sedimentation in the solution is obtained. In Figs. 1 and 2 a series of successive states of two gold sols is shown; for Fig. 1 the mean radius of the particles is 3.5 μμ, the number of revolutions is about 5500 per minute, the time between two photographic “samples” is 15 min.; for Fig. 2 (a gold sol obtained by Faraday’s method of reduction with phosphorus) the particle radius is 2.5 μμ, the number of revolutions is the same, and the time between two “samples” is 30 min.

Fig. 1. Ultracentrifugation of a gold sol, 3.5 μμ
(time between photographs 15 min.)

The construction of the ultracentrifuge itself, located in Svedberg’s laboratory at Uppsala University (Sweden), is in its main features as follows: the vessel in which the sol under investigation is placed has the shape of a circular sector with an angle of 5° and plane-parallel walls (see the diagram in Fig. 3); its opening is directed toward the center of rotation and is hermetically sealed; the thickness of the vessel walls is 1 cm; for colored sols it is made of glass, and for sols with absorption in the ultraviolet part of the spectrum, of quartz. The vessel is designed for a liquid volume of about 0.5 cm³. It is subjected to uniform rotation (without jolts) for a long time (up to 60 hours) at 5000–9000 revolutions per minute. By means of a prism with total internal reflection (see Fig. 3), a beam of light is directed through the centrifuge, so that the image of the vessel \(AB\) is thrown (when \(AB\) passes through a definite point of the circumference) onto the photographic plate \(P\).

The use of the ultracentrifuge makes it possible to determine, from the rate of settling of the sol particles (from the displacement of the boundary of the dark part in the photographs), the radius of the particles and their “molecular weight”1. Indeed, the centrifugal force acting on 1 mole (1 gram-molecule) of the dissolved substance is:

\[ F_1 = Nv \cdot \Delta \cdot \omega^2 x. \tag{1} \]

Here \(N\) is Avogadro’s number, \(v\) is the volume of a particle \(\left(v=\dfrac{4}{3}\pi r^3\right)\) of the sol, \(\Delta=D_p-D_s\), the excess of the particle density over the density of the surrounding medium, \(\omega\) is the angular velocity of rotation of the centrifuge, and \(x\) is the distance to the axis of rotation. The force \(F_1\) acts on the particles in the direction away from the center of rotation; the motion of the particles is opposed by the resistance of the medium \((F_2)\); it is directed in the opposite direction and may be taken as proportional to the velocity of motion \(\left(\dfrac{dx}{dt}\right)\):

\[ F_2=f\cdot\frac{dx}{dt} \tag{2} \]

Fig. 2. Gold sol, 255 μμ (time between photographs: 30 min.).

According to the well-known Stokes law, assuming that the particles have a spherical shape with radius \(r\), we find:

\[ F_2=6\pi\eta\cdot r\cdot\frac{dx}{dt}\cdot N, \]

whence

\[ f=6\pi\eta rN. \tag{3} \]

Here \(f\) is the coefficient of friction, and \(\eta\) is the viscosity of the surrounding medium. For steady motion \(F_1=F_2\); substituting the corresponding expressions (1) and (3) for \(F_1\) and \(F_2\), we arrive at the equation:

\[ \frac{9}{2}\cdot\frac{\eta}{\Delta\omega^2}\cdot\frac{dx}{x}=r^2\,dt \tag{4} \]

integrating which, we finally obtain an expression for the mean particle radius:

\[ r=\sqrt{\frac{9\eta\ln\dfrac{x_2}{x_1}}{2\Delta\omega^2(t_2-t_1)}} \tag{5} \]

determining the positions of the boundary \((x_1\) and \(x_2)\), corresponding to two times \(t_1\) and \(t_2\), one can find \(r\). For the gold sol whose sedimentation is shown in Fig. 1, the ultramicroscopic counting method gives \(r=3.2\ \mu\mu\); Svedberg, by formula (5) (from the sedimentation velocity), finds \(r=3.5\ \mu\mu\).

For the sol in Fig. 2, the ultramicroscopic method gives \(2.5\ \mu\mu\); Svedberg found \(2.4\ \mu\mu\).

On the basis of the preceding considerations it is easy to find an expression for the weight of a particle \((M)\), using the known relation \(f=\dfrac{RT}{k}\), where \(k\) is the diffusion coefficient, and noting that \(M=Nv\cdot D_p\):

\[ M=\frac{RT}{k\omega^2}\frac{D_p}{\Delta}\frac{\ln\dfrac{x_2}{x_1}}{t_2-t_1}. \tag{6} \]

With sufficiently long centrifugation of the solution, a “sedimentation equilibrium” is established between the centrifugal force and the force of diffusion, which tends to distribute the substance uniformly throughout the entire volume of the solution. In the state of equilibrium, the amount of substance which, in time \(dt\), under the action of the centrifugal force passes through a section perpendicular to \(x\) is

\[ dm = c(\omega^2 x \cdot Nv \Delta)\frac{1}{f}\,dt, \tag{7} \]

Figs. 3 and 4. Diagram of the ultracentrifuge.

and the same amount passes in the opposite direction (toward the center), under the action of diffusion (\(x\) is measured from the circumference toward the center):

\[ dm = -RT \frac{dc}{dx}\frac{1}{f}\,dt \quad \text{(Fick’s law);} \tag{8} \]

equating these expressions for \(dm\), we obtain for the state of sedimentation equilibrium:

\[ M\frac{\Delta}{D_{\eta}} = \frac{RT}{\omega^2} \frac{\ln \frac{c_1}{c_2}} {\left(\frac{x_2+x_1}{2}\right)(x_2-x_1)}. \tag{9} \]

In this formula \(c_1\) and \(c_2\) are the concentrations at the points \(x_1\) and \(x_2\); from it, among other things, it is seen that, for constant \(x_1\) and \(x_2\), the ratio \(\frac{c_1}{c_2}\) increases with increasing angular velocity \((\omega)\) of the centrifuge.

If the sol is inhomogeneous (consists of particles differing considerably in size), then the sharp boundary between the sol and the pure solvent becomes blurred; this phenomenon can serve for studying the distribution of particles. In order, in this case—as in the sedimentation equilibrium described—to determine the concentration of the sol \((c)\) as a function of the distance from the center \(o\): \(c=f(x)\), photographic samples are analyzed with a self-recording microphotometer, i.e., the brightness of the light passing through different places \((x)\) of the photograph is determined. By carrying out a series of similar experiments with photographs of sols of known concentrations, one can pass from the photometric curve to the curve \(c=f(x)\). The same photographic method, with microphotometric analysis of the photographs obtained, was also applied by Svedberg to the study of diffusion of a liquid at rest in a diffusometer. Even in the case of uncolored substances, for example in studying the diffusion of proteins, Svedberg uses this method, applying lateral illumination with ultraviolet light (mercury lamp), under the action of which proteins fluoresce strongly.

In conclusion, let us note that Svedberg’s method leads to results so interesting and significant that, in all probability, it will soon find application in many laboratories.

1. That is, the relative weight of a sol particle, consisting (in the case of metal sols), in all probability, of several thousand molecules. ↩↩

2. H. Bechhold. Koll.-Zeit. 2, issue 1–2 (1907). Zeit. phys. Chem. 60, 257 (1907). ↩

3. The Svedberg, Journ. Amer. Chem. Soc. 45, 2910 (1923), 46, 2677 (1924), and Zsigmondy Festschrift (Ergänz.-Bd. zur Koll.-Z., Bd. 36, 53–65, 1925). ↩

4. H. Friedenthal, Ber. d. D. chem. Ges. 44, 904 (1911). ↩

5. At higher rotational speeds, vortices arise in the liquid being centrifuged, and uniform sedimentation is disturbed. ↩