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RADIOGRAPHIC DETERMINATION OF PARTICLE SIZES1
Cameron and Patterson
Introduction. It has long been known that particle sizes are one of the most important factors determining the physical properties of matter in the colloidal state. Therefore, the measurement of particle sizes is of extraordinarily great importance in colloid chemistry—a science whose applications in industry are difficult to enumerate. However, these measurements are not easy to make. The range of particle sizes of interest to colloid chemistry lies beyond the resolving power of the best microscopes. Therefore it may seem that only indirect methods can be available for this purpose. Recently, the ultramicroscope, the ultracentrifuge, and methods of measuring osmotic pressure have been used with considerable success to determine particle sizes. However, they all have one and the same drawback: with their aid it is impossible to distinguish a single particle from an aggregate of particles. If one leaves aside electron-diffraction methods, which have not yet received sufficient development, then the X-ray method will be the only method that makes it possible to determine particle sizes in the submicroscopic region independently of the aggregate state of the particles. Although this method has not yet been fully developed and there have been few successful applications of it, it nevertheless deserves attention because of the advantages indicated over other methods.
Even the earliest investigations by Debye and Scherrer showed that the diffraction lines of substances in the colloidal state are often broader than the diffraction lines of the same substances ground into powder. They quite correctly related this effect to the small sizes of the colloidal particles. In 1938, Scherrer1 published a formula relating the width of diffraction lines to particle sizes. This communication served as the basis for many works devoted to developing the theory of this effect, and a considerable number of determinations of particle sizes were made.
If the specimen is coarse-crystalline, the diffraction lines of the Debyegram will be discontinuous, since each crystallite will give its own spot. If the crystallite sizes are smaller than a certain definite value \((10^{-3}—10^{-4}\ \mathrm{cm})\), then their spots merge together and form continuous lines. Below this limit there lies a broad range of particle sizes for which the width of the diffraction lines is determined entirely by the geometry of the instrument (camera radius and specimen size, tube-focus size), by the degree of monochromaticity of the X-rays, and by absorption in the specimen. If the particle sizes become very small (less than \(1000\ \text{Å}\)), the lines begin to broaden. It is to such small particles that our further exposition applies.
Physical basis of the radiographic method for determining particle sizes. Before proceeding to consider methods of measuring particle sizes, it should be determined what
X-RAY DETERMINATION OF PARTICLE SIZES
we understand by a crystalline particle. The crystallites of which we shall speak consist at most of several million molecules, but may consist of only a few hundred. If desired, one may suppose that these particles are bounded by flat surfaces, similar to the faces of a single crystal. It is more probable, however, that in colloidal particles obtained in the laboratory, in a factory, or occurring in plants and animals, the corners and edges are rounded, and their shape may be regarded approximately as ellipsoidal or similar to it.
What diffraction pattern may be expected when X-rays are scattered by such small crystallites? A large crystal gives sharp diffraction lines at angles known to be connected with the principal vectors. The diffraction angles may be calculated by a method well known to all working in the field of X-ray analysis of crystals. At these angles the X-rays are scattered by the crystal in such a way that they reinforce one another. Rays scattered by individual groups of atoms or by individual cells, in all other directions, are not in phase and therefore cancel one another. In the case of small crystallites the scattered waves reinforce one another at the very same angles at which diffraction lines appear when large crystals scatter. However, in all other directions, owing to the small number of groups of molecules, few waves are scattered, and their cancellation is incomplete. This cancellation becomes the less complete the closer the angle under consideration is to the diffraction angle. As a result, in the diffraction of small particles, broad lines with maxima are obtained, located where sharp peaks should appear in the scattering by large crystals of the same substance. In this way one can explain the broadening of the lines observed by Debye and Scherrer.
It is possible to calculate exactly what the broadening of the maxima should be, and to relate this broadening to the sizes and shape of the particle. The interference function for particles of various shapes was calculated by Laue², Selyakov³, Murdock⁴, and Patterson⁵; they laid the foundation of the theory of particle-size determination.
Let us enumerate the various factors that may influence the broadening of the lines and that, consequently, must be taken into account when determining particle sizes.
First of all, let us note that in a pressed specimen, or in other preparations of a colloidal substance, many particles are present. These particles are usually oriented at random. However, if they have a special shape (graphite, bentonite), there may be some preferential orientation in their arrangement. Moreover, the sizes and shapes of the particles may in general be different in the specimen, and therefore X-ray analysis determines certain average sizes. It is natural to expect that the width of the lines should depend on the wavelength of the X-rays, on the type and parameters of the crystal lattice, and also on the angle at which the diffraction line appears. Unfortunately, we have no lens by means of which an X-ray beam can be made parallel; therefore the divergence of the beam must be taken into account. Insufficient spectral purity of the beam can distort our results. Absorption in the specimen may also influence the width of the line. In any case, the dimensions of the specimen cannot affect the width of the lines.
It may seem an almost hopeless attempt to determine a quantity depending on all these variables; however, by certain ingenious methods it has been possible to overcome all these difficulties.
The ideal experimental setup and its theory. All known up to now methods of determining particle sizes reduce to one of the four methods considered below, sometimes with minor modifications. In all these methods the following assumptions are made:
1) all particles have the same size and shape,
2) the particles are oriented at random,
3) the crystal structure is known,
4) the crystal lattice is not deformed,
5) the radiation is monochromatic.
Scherrer’s method. In his original 1918 paper on determining particle sizes, Scherrer^1,^6 gave the formula
\[ B=\frac{K\lambda}{\lambda \cos \Theta}+b, \]
where \(K\) is a constant equal to 0.94, and \(B\) is the width of the maximum, measured in the same units as \(\Theta\), at the point where the intensity is equal to one half of the maximum value. In deriving this formula, in addition to the above-listed assumptions Scherrer made the following assumptions: a) the rays are parallel, b) absorption may be neglected, c) the crystal lattice is cubic, d) the particles have the form of cubes with edge \(\Lambda\), parallel to the axes of the crystal, e) the correction for the dimensions of the specimen is adequately taken into account by an additive constant \(b\).
Selyakov^3 generalized Scherrer’s conclusions and showed that assumptions c) and d) are not necessary. He considered scattering by crystals of the triclinic system and, assuming that the external form of the crystal is similar to the form of the elementary cell, obtained a formula very similar to Scherrer’s formula. This result is interesting in itself, but its practical significance is small, since it is unlikely that the external form of a crystal would be similar to the form of the elementary cell. Cubic crystals are an exception, but this case had already been considered by Scherrer. Murdock^4 carefully recalculated Scherrer’s calculations for a cubic lattice. He first made the calculations for the cubic form of the particles, and then for the octahedral form. The results he obtained basically agree with Scherrer’s calculations. Murdock further showed that by this method one can determine the form of the particles as well as by Laue’s method (see below). In addition, Murdock took into account those errors that arose as a result of the approximations introduced into earlier theories in order to simplify the calculation of the integrals. Recently Patterson^5, with the aid of a new method, carried through to completion the calculations for a cubic lattice and a spherical form of the particles and obtained the required exact integrals. These calculations lead to the value \(K=1.11\), if \(\lambda\) is taken to be the diameter of the particle. However, if one takes into account the visible difference between the volume of a cubic and of a spherical particle, the value of \(K\), as Murdock showed, decreases to 0.89, which agrees well with the calculations of Scherrer and Murdock. Approximate calculations for particles of various shapes indicate that the width of the lines depends little on the form of the particle. Nevertheless, apparently, one can define a certain “equivalent ellipsoid,” which would be a rough approximation to the form of the particle. Then \(\Lambda\) will be the diameter of the ellipsoid in the direction of the normal to the reflecting plane, and \(K=1.1\) will be a suitable constant. It is useless to discuss the accuracy of this method, since the validity of assumption e) is highly doubtful. This question will be considered further in the section “Experimental verification of the theory and its practical applications.”
Laue’s method. In an extremely important work, which is the theoretical basis for all further approximations to the solution of the problem, Laue^2 investigated several types of theory. In addition to the assumptions listed above, he made the following assumptions: a) the bundle of X-rays is exclusively divergent, b) the actual source of radiation is a point diaphragm situated on the same cylindrical surface on which the film is placed, c) absorption may be neglected, d) the lattice may have a certain asymmetry, e) the form of the particle is ellipsoidal.
As a result of an exceedingly interesting analysis, Laue obtained an expression for the line width as a function of the radius of the camera and the radius of the specimen. For specimens whose radius may be neglected, Laue’s expression
takes the form of the Scherrer equation, although the value of the constant is somewhat different¹). This discrepancy in the constant is not surprising if one recalls that Laue defines the line width as the ratio of the total area under the peak to the maximum intensity. Of course, such a definition is equivalent to measuring the line width at half intensity only for peaks of triangular shape. Moreover, the results of Laue’s calculations clearly show that the correction for the dimensions of the specimen cannot be additive, except in very special cases. Laue also discussed the influence of the diaphragm dimensions on the line width and obtained an upper limit for the diaphragm dimensions at which this influence may be neglected. His investigations show that, with wide diaphragms, it is difficult to carry the analysis successfully through to the end. However, Cameron⁷ showed how the influence of the diaphragm dimensions on the line width can be taken into account experimentally.
Laue’s analysis of the problem of determining particle sizes is the most complete of all those available. Although the assumptions introduced by him may be criticized, at the present time they are the most acceptable. Laue’s analysis must undoubtedly serve as a model for all further investigations.
The method of Brill and Pelzer. Brill and Pelzer⁸˒⁹ devised an ingenious way to avoid the difficulties associated with obtaining cylindrical specimens whose absorption may be neglected. The substance under investigation was weighed into collodion, and a thin glass rod was coated with this suspension. The rod was then removed, and the hollow cylinder thus prepared was placed in a camera of the type used in Laue’s method for determining particle sizes. In general, such a specimen will give the double lines shown in Fig. 1. For very small particles the two maxima may merge. Using
Fig. 1. Method of the hollow specimen of Brill and Pelzer.
A — specimen camera. The dimensions of the latter are enlarged; B — formation of a double maximum; C — formation of a single maximum with very small particles.
Laue’s basic theory and making the same assumptions as Laue, Brill and Pelzer established a relation between the particle size and the distance between the two peaks or with the line width obtained when these peaks merge. The main advantage of this method is that it is much easier to measure the distance between two peaks than the line width.
The method of complete absorption of Brill and Pelzer. In this method Brill and Pelzer¹⁰ effected complete absorption in the specimen,
¹) Laue obtained the value \(K = 0.90\). However, in his calculations two errors were made. On p. 127 of his work² the value \(1/\omega^2\) should be equal to \(\sqrt[3]{\pi} = 1.46\) instead of \(\sqrt[3]{2\pi} = 1.85\), and on p. 41 it should be \(\eta = \dfrac{\lambda}{2\pi\Lambda}\) instead of \(\dfrac{\lambda}{4\pi\Lambda}\). If these corrections are made, we obtain \(K = 1.46\). The corresponding corrections should also be made in all subsequent works using Laue’s analysis. Note by A. L. Patterson.
coated the rod of lead glass under investigation with the substance. In this way it became possible to calculate the line broadening on the assumption that only a very thin layer of the specimen takes part in the formation of the line. For the rest they used Laue’s assumptions and his basic theory. Unfortunately, they were compelled to make additional assumptions.¹ Cameron’s experiments show that this may make their calculations unsuitable for a certain range of particle sizes. However, apparently this method represents one of the promising possibilities for the exact determination of particle sizes.
OTHER FACTORS AFFECTING THE DETERMINATION OF PARTICLE SIZES
Effect of the particle-size distribution. In all the methods considered above it is assumed that all particles are of the same size and shape. Patterson continued Laue’s calculations, generalizing them to the case of particles of different sizes but of the same geometrical shape. He found, as was to be expected, that, using Laue’s equation, we determine not the mean particle sizes, but the sizes of that particle which would give a line of the mean width. Of course, for a narrow range of particle distribution by size these sizes are close to one another, but for complex distributions it is impossible to establish a relation between them if the distribution function is unknown.
Effect of orientation. B. L. Brett showed that in the presence of partial ordering of the particles Scherrer’s formula remains unchanged. It must be borne in mind that any of the above methods may be used to determine the particle sizes of a substance with a preferred orientation of the particles.
Lattice deformations. The influence of lattice deformation on line broadening is well known. It should be noted, however, that line broadening caused by lattice deformation cannot be completely separated from line broadening caused by particle sizes. Therefore the possibility of errors caused by lattice distortions must always be taken into account when determining particle sizes.
EXPERIMENTAL VERIFICATION OF THE THEORY AND ITS PRACTICAL APPLICATIONS
The range of particle sizes that can be measured by the line-broadening method extends approximately from 10 to 1000 Å. Thus the field of applicability of this method begins where the lower limit of microscopic measurements lies, and ends where the particles are so small that they contain only a few crystallographic planes (Brill asserts that he measured diffraction lines obtained from scattering by 3–5 atomic layers). Many substances of great importance for industry have particle sizes lying in this region, for example: cellulose, rubber, soot, graphite, colloidal metals.
Determination of particle sizes in this range for substances can be carried out with the aid of the ultracentrifuge, the ultramicroscope, and measurements of osmotic pressure. Numerous authors¹, ⁶, ⁷, ¹¹, who set themselves the task of verifying the theories described, found satisfactory agreement between X-ray measurements and other measurements.
In order that Scherrer’s method could be applied, an ideally parallel beam and a nonabsorbing specimen in the form of a cylindrical rod are required. The first of these two conditions, i.e. parallelism of the beam, is by no means realized in the usual Debye–Scherrer apparatus. Genstenberg and Mark found that even if
¹ In choosing the limits of integration over the specimen they use the values derived for a parallel beam, whereas all the other calculations are made on the assumption that the beam is divergent.
X-RAY DETERMINATION OF PARTICLE SIZES
to place the camera at a distance of 60 cm from the X-ray tube, the effect of beam divergence remains noticeable. This deviation of the beam from parallelism is especially important to bear in mind when determining the sizes of large particles, when line broadening is small. It is probably for this reason that it was found that the Scherrer method is unsuitable for particles whose sizes exceed \(10^{-5}\) cm. In addition, in most investigations by this method it was impossible to neglect the absorption of X-rays in the specimen. To eliminate possible errors, it is desirable to make a control photograph of a specimen of the same composition, with particle sizes known to be too large (greater than 10 cm), in order to affect the line width. If, in this case, the width of the diffraction lines remains the same for all angles, then we have reliable grounds for applying the Scherrer equation.
Despite various objections to the Scherrer method, it has been shown quite convincingly that in many cases, in particular for substances with very small particle sizes and high atomic weight, this method is just as suitable as any other.
A great advantage of the Laue method and of the methods based on it is that by this method one can study not only the sizes of particles but also their shape. Hengstenberg and Mark applied the Laue method to measure the particle sizes of natural cellulose, artificial silk (viscose), and rubber. They found micelles in ramie fibers approximately 55 Å wide, of the same thickness, and at least 20 times as long. The fibers of viscose silk had a somewhat smaller transverse section and were almost half as short. The rubber particles appeared in the form of rectangular bars with the smallest dimensions 180 Å. Hofmann and Wilm investigated the sizes and shape of carbon particles prepared by several different methods—retort graphite, carbon obtained from carbon monoxide, benzene, and acetylene. In addition, they examined several activated wood charcoals. They found no evidence of the existence of amorphous carbon; on the contrary, all the substances examined had the structure of graphite, the particle sizes of which varied from 14 Å to a maximum of 196 Å. The smallest particles were found in activated wood charcoal; in one case the diffraction pattern was produced by two or three atomic planes.
In the case where the specimen can be made in the form of a hollow cylinder, the first Brill and Pelzer method is relatively simple and has all the advantages over the Laue method. The authors of this method reported the results of measurements of particles of magnesium oxide and lithium fluoride. Cameron investigated magnesium oxide, titanium dioxide, silver, and gold, but he was unable to obtain satisfactory roentgenograms for denser substances. He therefore concluded that the first Brill and Pelzer method is suitable only for substances of low atomic weight. The experimental technique in applying this method is considerably more difficult than when using other methods.
The complete-absorption method of Brill and Pelzer is probably the most acceptable experimentally, since it possesses all the advantages of the preceding methods and can be applied to any finely powdered substance, regardless of how it absorbs X-rays. Brill and Pelzer applied this method to the study of electrolytic deposited nickel. Cameron published the results of measurements of particles of colloidal platinum, which, however, in the case of small particles, agreed poorly with the theoretical curves.
In view of the indicated theoretical and experimental shortcomings of existing theories, and also because of insufficient photographic measurement of intensities, it must be thought that the actual particle sizes may differ from those found by \(1 \tfrac{1}{2} — 2\) times, i.e., the crystals may in reality turn out to be \(1 \tfrac{1}{2} — 2\) times larger or smaller. Relative measurements of the sizes of similar specimens can undoubtedly be
carried out with considerably greater accuracy; at best, probably, with an accuracy of up to 10%.
Which method is best to use depends on the nature of the substance being investigated and on the requirements imposed. In some cases it may be considered sufficient to arrange a series of samples of one and the same substance in a row according to increasing particle size. Then the peculiarities of the method employed and the character of the X-ray beam are not of great importance, provided only that the conditions remain constant for all the samples in the series.
Recently Warren[^16] reported measurements made by him at small scattering angles for substances with very small particle sizes (about 10 Å). This effect does not depend on the structure of the particles and offers the possibility of measuring particle sizes in a very interesting range.
LITERATURE
- P. Scherrer, Nachr. Ges. Wissen. Göttin., July 26, 1918.
- M. Laue, Z. Krist., 64, 115, 1926.
- N. Seliakow, Z. Physik, 31, 439, 1924; and corrections in Z. Physik, 33, 648, 1925.
- C. C. Murdock, Phys., 35, 8, 1930.
- A. L. Patterson, Phys. Rev., 49, 884, 1936.
- P. Scherrer, R. Zsigmondy, Kolloidchemie, 3. Aufl., Leipzig, 1920.
- G. H. Cameron, Physics, 3, 57, 1932.
- R. Brill u. H. Pelzer, Z. techn. Phys., 10, 663, 1929.
- R. Brill u. H. Pelzer, Z. Krist., 72, 398, 1929.
- R. Brill u. H. Pelzer, Z. Krist., 74, 147, 1930.
- U. Hoffmann u. D. Wilm, Z. physik. Chem., (B) 18, 401, 1932.
- R. Brill, Z. Krist., 68, 387, 1928.
- G. L. Clark, W. C. Astbury a. R. M. Wick, J. Am. Chem. Soc., 47, 2666, 1925.
- P. Scherrer u. H. Staub, Z. physik. Chem., (B) 154, 309, 1931.
- G. R. Levi et R. Haardt, Atti d. Reale Ac. Naz. Lincei, (6) 3, 91, 1926.
- B. E. Warren, Phys. Rev., 49, 885, 1936.