OPTICAL PROPERTIES OF LATEX WITH UNIFORMLY SIZED PARTICLES

Some time ago we already reported to the readers of our journal on the preparation of latex with uniformly sized spherical particles1. At that time we also indicated that, in addition to the authors’ proposed use of latex particles as size-standard spheres for electron microscopy, this opens up interesting possibilities in the field of research on light scattering. Let us recall that until now Mie’s theory has in essence not been subjected to quantitative experimental verification precisely because of the impossibility of obtaining monodisperse colloids. Since then, a number of works have appeared devoted to the study of the optical properties of monodisperse latexes. Of greatest interest among them is the paper reviewed here2, in which the authors subjected latexes whose particle diameters varied from 0.1 to 1 μ to a comprehensive investigation.

Latex is a suspension of polystyrene or polyvinyltoluene in water. When a small quantity of latex is dried on the surface of glass, it forms a film (3–4 mμ thick), from whose surface a replica can be taken and then subjected to electron-microscopic investigation. The latter shows, above all, the exceptional uniformity of the particles in size.

Table I gives the results of measurements of the sizes and of the standard deviation from the mean size for a series of latexes (designated by Latin letters).

Table I

The measurements were made by comparison with a replica of a diffraction grating (30,000 lines per inch). The absolute error in the measurement of the particle diameter, associated with instrumental inaccuracies, is about 5%. The standard deviations are due not only to the technological causes operating in the preparation of the latex, but also to

aberrations of the electron-optical system. Thus, here we are in fact dealing with systems that are exceptional in monodispersity. Further, as electron microphotographs show, upon drying, latex particles form a close-packed crystalline lattice. If the latex film is heated, a sintering process takes place and the “crystalline,” granulated film gradually becomes continuous.

It is interesting to note that the “crystalline” film breaks up into a large number of crystallization domains with differently oriented “crystalline” lattices.

The refractive index of polyvinyl toluene is equal (for the sodium \(D\)-line) to 1.583, i.e., it differs little from that of water. Its density is \(\rho = 1.030\). In view of the small difference in refractive indices, the following formula must hold for the dependence of the refractive index of latex on its concentration \(c\):

\[ \frac{dn}{dc} = (n_{\text{part}} - n_{\text{water}})/\rho_{\text{part}} = 0.243. \]

Measurements give the value 0.241.

The angular dependence of the scattering intensity in this case (for small \(D\)) is determined by the known relation

\[ I \sim f^2(q) = \left[ \frac{3}{q^3}(\sin q - q \cos q) \right]^2, \]

where

\[ q = \frac{2\pi D}{\lambda}\sin \frac{\theta}{2}, \]

\(\theta\) is the scattering angle, \(D\) is the particle diameter, and \(\lambda\) is the wavelength in water.

If

\[ \frac{D}{\lambda} \ll 1, \]

then the dissymmetry of scattering is

\[ \frac{I_\theta}{I_{180^\circ-\theta}} = 1 + \frac{(2\pi)^2}{5}\frac{D^2}{\lambda^2}\cos\theta. \]

The dissymmetry of scattering was measured on highly diluted latexes (dilution was continued until the dependence of the dissymmetry on the latex concentration disappeared). The measurement results for sample \(A\) (see Table I) are given in Table II. The measurements were made in the light of the green mercury line (546 m\(\mu\)).

Table II

Since for water \(\lambda = 410\) m\(\mu\), the mean particle diameter obtained from these measurements is 164 m\(\mu\), in excellent agreement with the data of Table I. For latexes \(B\) and \(C\), the corresponding measurements led to values of 172 and 272 m\(\mu\).

Such films of dried latex exhibit a sharply pronounced directionality of scattering. Table III gives the values of \(\theta\) corresponding to the first scattering maximum for various latexes and wavelengths (in air), and the particle diameters calculated from them.

Table III

The particle sizes were determined by the formula

\[ \sin \theta_{\max} = m \frac{\lambda}{D}, \]

where the value \(m = 1.14\) was found from the data for latex \(N\) using the value of \(D\) given in Table I. The relative error in determining \(D\) is estimated by the authors to be about 1.5%, i.e., the absolute error is about \(15\ \mathrm{m}\mu\). This method is applicable if \(D \gtrsim 490\ \mathrm{m}\mu\).

The authors note the difference between the value of \(m\) and that obtained for diffraction by a slit and by a circular aperture. We also note that the scattering intensity does not become zero at any angles. The authors explain this by the interaction of particles. Obviously, this circumstance requires more detailed study.

Of great interest are the authors’ observations concerning diffraction by the crystalline structure of a dried latex film. The position \(\theta_{\max}\) of the scattering maximum by a layer was measured as a function of the angle of incidence of the beam illuminating the layer \(\left(\frac{\pi}{2} - \delta;\ \delta \text{ is the glancing angle}\right)\). It turned out that, for all latexes for which \(\theta_{\max}\) exists, there is a linear dependence of \(\theta_{\max}\) on \(\delta\). Table IV gives the values of \(\theta_{\max}\), extrapolated to \(\delta = 0\), for latex \(F\) and various \(\lambda\).

Table IV

Table V gives the values of \(\theta_{\max}\), extrapolated to \(\delta = 0\), for various latexes and \(\lambda = 436\ \mathrm{m\mu}\).

Table V

In the same tables are given the values of \(D\) calculated from the measurement of \(\theta_{\max}\). The particle sizes were determined as follows. Suppose that we have a two-dimensional crystalline structure made up of closely packed spheres of diameter \(D\). The distance between neighboring rows of spheres is

\[ d = \frac{\sqrt{3}}{2}D. \]

Then we have a two-dimensional diffraction grating, and the condition for a maximum may be written in the form

\[ \frac{m\lambda}{d} = \cos\delta - \cos(\theta - \delta), \]

where \(m\) is the order of the spectrum. For \(\delta \to 0\), for \(m = 1\), we have:

\[ D = \frac{\lambda}{\sqrt{3}\sin^2 \frac{\theta}{2}} . \]

It is obvious that this method is applicable to particles whose sizes lie in the interval (for \(\lambda = 436\ \mathrm{m\mu}\)) \(254 < D < 508\ \mathrm{m\mu}\).

The authors note that, owing to the great uniformity of the particles in size, it is possible to prepare diffraction gratings in this way. This can be achieved by metallizing a dried latex film or a replica taken from it. They see the advantage of such a grating in the considerable weakening of “ghosts” and in the extension of the first-order spectrum over the entire range of angles (because of the smallness of \(d\)).

At the same time, the authors note that, for reasons not clear to them, the diffraction spectrum proved to correspond to a two-dimensional structure, whereas in reality the film has a three-dimensional crystalline structure and one would have expected fulfillment of the Bragg condition. The authors suppose that this may be explained by the large scattering coefficient of the particles, as a result of which only the first layer proves to be effective. We note that the considerable background observed by the authors may be explained by multiple scattering.

G. Rozenberg

REFERENCES CITED

1. UFN 39, 142 (1949).
2. A. Turner Jr., E. B. Bradford, J. W. Vanderhoff and G. Oster, IOSA 44, No. 8, 603 (1954).
3. See, for example, K. S. Shifrin, Scattering in Turbid Media, Gostekhizdat, 1951.