APPLICATION OF LIGHT SCATTERING TO DETERMINE MOLECULAR WEIGHT

The study of the molecular scattering of light can serve as a convenient method for determining the molecular weight and dimensions of large molecules in solutions. The theoretical foundations of this method were set forth in sufficient detail in P. Debye’s paper, published in 1944.[^1] The character of the scattering depends essentially on whether we are dealing with large or small particles. Here “small” particles are understood to mean linear particles whose dimensions do not exceed \(1/10\)—\(1/20\) of the wavelength; these maximum dimensions correspond to molecules whose weight lies between 10 and 100 million.

In the case of small particles, the scattered light is characterized by the following two features (for a parallel primary beam): a) the intensities of the light scattered forward and backward are the same; b) the intensity of scattering is proportional to \(\lambda^{-4}\) (if the additional small effects caused by pressure are neglected). The simplest case of scattering by small particles is the case of an ideal gas of low density, studied by Rayleigh. If such a gas contains \(n\) molecules per \(\text{cm}^3\) and the polarizability of each molecule is equal to \(\alpha\),

then the “turbidity” of the gas, caused by scattering, was

\[ \tau=\frac{8\pi}{3}\left(\frac{2\pi}{\lambda}\right)^4 n\alpha^2 . \tag{1} \]

Here “turbidity” is understood to mean the attenuation coefficient of a parallel beam as a result of scattering:

\[ -dI=\tau I\,dx . \tag{2} \]

Equation (1) can be used to determine the number of particles \(n\) in \(1\ \mathrm{cm}^3\). For this it is necessary to know the polarizability \(\alpha\), instead of which one may use the refractive index \(\mu\), since \(\alpha\) and \(\mu\) are related by

\[ \mu-1=2\pi n\alpha . \tag{3} \]

From (1) and (3) we obtain:

\[ \tau=\frac{32\pi^3}{3}\frac{(\mu-1)^2}{\lambda^4}\cdot\frac{1}{n}. \tag{4} \]

Thus, in order to determine \(n\), two measurements must be made: a measurement of the “turbidity” and a measurement of the refractive index.

In the case of liquids the matter is considerably more complicated, since the molecules of a liquid are so close to one another that it is no longer possible to neglect their mutual influence. A. Einstein in his work of 1910 approached these difficulties in the following way. He regarded scattering as the result of local thermal fluctuations of the density of the liquid, owing to which the medium becomes optically inhomogeneous. Einstein calculated the magnitude of these fluctuations by comparing the thermal energy \(kT\) with the work that must be expended in order to produce these changes of density by external pressure. In such a treatment there is no need for a detailed molecular theory, and the result obtained by Einstein can be represented in the form

\[ \tau=\frac{32\pi^3}{3}\cdot\frac{1}{\lambda^4}\frac{kT}{\chi}\left(\mu\frac{\partial\mu}{\partial p}\right)^2, \]

where \(p\) is the hydrostatic pressure, and \(\chi\) is the compressibility.

In the case of solutions there is another cause of irregular fluctuations of the refractive index, namely local changes of concentration. Debye constructs the theory of these fluctuations on the model of Einstein’s theory of density fluctuations, introducing instead of the pressure \(p\) the osmotic pressure \(P\), and instead of the volume \(v\) the concentration \(c\). The work which in this case must be compared with the thermal energy \(kT\) is the work that must be expended from outside (for example, by means of an imaginary semipermeable piston) in order to produce a reversible change of concentration. As a result it turns out that the intensity of scattering caused by concentration fluctuations is proportional to

\[ \frac{kT}{\lambda^4}\, \frac{\left(c\frac{\partial\varepsilon}{\partial c}\right)^2} {c\frac{\partial P}{\partial c}}, \]

where \(\varepsilon\) is the dielectric constant, which may be replaced by the square of the refractive index. Further, in the case of solutions the difference of the refractive indices of the solution and the solvent is proportional to the concentration. The final expression for \(\tau\) is therefore:

\[ \tau=\frac{32\pi^3}{3}\cdot \frac{\mu_0^2(\mu-\mu_0)^2}{\lambda^4}\cdot \frac{1}{c\,\dfrac{d}{dc}\left(\dfrac{P}{kT}\right)} . \tag{5} \]

Finally, for very dilute solutions, for which the van ’t Hoff relation holds,

\[ P = nkT, \]

formula (5) takes the simple form:

\[ \tau = \frac{32\pi^{3}}{3}\cdot \frac{\mu_{0}^{2}(\mu-\mu_{0})^{2}}{\lambda^{4}}\cdot \frac{1}{n}. \tag{6} \]

This formula shows that by measuring the “turbidity” \(\tau\) and the difference of the refractive indices \(\mu-\mu_{0}\), one can determine \(n\), and, knowing \(n\) and the concentration, determine the mass of an individual particle, i.e. the molecular weight.

In order for this method to be applicable in practice, it is necessary that the scattering caused by fluctuations of concentration be large in comparison with fluctuations of density. It may be expected that this will occur when the dissolved substance has a sufficiently high molecular weight. Debye gives the following table, illustrating the influence of particle size on the turbidity due to scattering:

The quantity \(D\), placed in the last column, is the thickness of a layer causing an attenuation of the light by a factor of \(e\). It follows from the table that proteins, i.e. substances important for biology, are an especially suitable object for the application of the scattering method. Another group of substances important for technology, to which this method should be purposefully applied, is formed by high polymers (rubber, etc.).

In a paper published in 1946, P. P. Debye (son) describes apparatus with the aid of which the scattering method was applied to determining the particle sizes of high polymers. To determine \(\tau\), in most cases it proved more advantageous to measure scattering at a definite angle than the attenuation of light. The latter, for the concentrations with which one has to deal, is, generally speaking, very small. As for the difference of refractive indices \(\mu-\mu_{0}\), it also requires careful measurement. The order of its magnitude is \(0.001\) for a 1% solution. In measuring scattering, a cooled air-cooled mercury lamp of medium pressure was used (type AH4) and a monochromatic filter. The instrument was a photoelectric photometer (RCA 929 photocells) with a direct-current amplifier. Owing to the use of four photocells, it was possible, in rapid succession, to measure a) scattering at an angle of \(90^\circ\); b) attenuation of light; c) angular distribution of scattering.

The setup for determining \(\mu-\mu_{0}\) is very simple and convenient. A precision slit \(S\) is illuminated (see the figure) by a mercury lamp through a filter \(F\). The image of the slit, by means of lenses \(L_{1}\) and \(L_{2}\), is obtained approximately at a distance of 190 cm; the position of this image is measured with an ocular filament micro-

method. A solvent is poured into vessel \(C\), and a solution into the hollow prism \(P\). If the refractive indices of the solvent and the solution are not the same, the image of the target is displaced. For a not too large value of the difference \(\mu-\mu_0\), the magnitude of the displacement is proportional to this difference. With this simple instrument it was possible to measure \(\mu-\mu_0\) quickly and conveniently with an accuracy up to 0.000003.

In Oster’s work\(^4\) (from the Rockefeller Institute for Medical Research), the scattering method was used to determine the molecular weight of viruses. The author, evidently not acquainted with the theoretical work of P. Debye, also uses Einstein’s theory of scattering, but for the case of mixtures,*) on the basis of which he obtains the following expression for the molecular weight:

\[ M = 1.69 \cdot 10^{22}\,\frac{D\lambda^{4}}{c\left(\dfrac{d\varepsilon}{dc}\right)^{2}}, \]

where \(c\) is the concentration \((g\ cm^{-3})\) and \(D\) is the optical density of the solution due to scattering. Replacing \(\varepsilon\) by \(\mu^2\) and taking

\[ \frac{d\mu^2}{dc} \simeq 2\mu_0\,\frac{\mu-\mu_0}{c}, \]

he reduces the formula to a form convenient for applications. The measurements necessary for computing \(M\) consist in determining \(\mu_0\) and \(\mu-\mu_0\) (this was done by means of an immersion refractometer) and in the spectrophotometric (obviously visual) determination of \(D\) for various \(\lambda\) and \(c\). In this way two viruses were studied: the influenza virus and the virus inhibiting the growth of tomatoes. For these viruses the values of \(M\) obtained were, respectively, 322 million and 6 million. For the influenza virus, particles of different sizes were obtained—97 \(m\mu\), which agrees well with measurements by means of an electron microscope. The molecular weight of the tomato virus, determined from light scattering, is in agreement with determinations by ultracentrifugation and diffusion.

In view of the development that photometry and spectrophotometry have recently undergone, the scattering method will undoubtedly find wide application in the near future.

E. Shpolsky

REFERENCES

1. P. Debye, J. Appl. Physics 15, 338 (1944).
2. A. Einstein, Ann. d. Phys. 33, 1275 (1910).
3. P. P. Debye, J. Appl. Physics 17, 392 (1946).
4. Gerald Öster, Science 103, 306, 1946.

*) In P. Debye’s work it is pointed out that Einstein’s results for the case of mixtures can be transferred to solutions. However, the method of concentration fluctuations used by Debye is more elegant and simpler.