Diffraction of X-rays by Liquids

J. J. Trillat (Paris)

The structure of the liquid state is still very little known, and therefore attempts to bring clarity to this question by studying the diffraction of X-rays were natural. At the present time this problem is still far from being solved, but in any case the results obtained already make it possible to draw a whole series of very interesting conclusions, which will be discussed below.

Experimental investigations of the diffraction of X-rays by liquids

The first experiments on the diffraction of X-rays by liquids were carried out by Debye and Scherrer* in 1916, immediately after their work on the scattering of X-rays by substances with irregular orientation. These authors showed that if a narrow beam of monochromatic X-rays passes through a thin layer of liquid and then falls on a photographic plate, then on the latter there is obtained a more or less sharply outlined and narrow ring, separated from the central spot by a quite distinct zone (Fig. 1). It is interesting to note that in the case of benzene these authors used no vessel and made use directly of a thin jet of liquid, which is very important from the point of view that

* Chapter VI from the book by J. J. Trillat, Les Applications des Rayons X, translated by N. A. Shishakov. See also “Uspekhi fizich. nauk,” vol. XI, issue 3, p. 493; issue 4, p. 595; issue 6, p. 847; 1931.

tubes of glass, collodion, and other materials, even the thinnest ones, produce diffraction phenomena very similar to those observed in liquids.

In these investigations, the photographic method of Debye and Scherrer was used chiefly; the ionization method was used only by Hewlett and Stuart and by Morrow.

Fig. 1. 1. Spectra of liquids (Zogani): 1) benzene, 2) cyclohexane, 3) metaxylol, 4) hexane, 5) toluene, 6) acetic acid.

Debbern² studied other liquids, namely mercury, methyl iodide and methylene, benzene, and also some mixtures. The experimental results he obtained likewise show the presence of rings surrounding the central beam of rays.

Hewlett³, using the ionization method, studied benzene, octane, and mesitylene, placing them in small celluloid tubes and using the \(K\alpha\)-radiation of molybdenum \((0.707\ \text{Å})\). The curves he obtained for the dependence of intensity on the angle of diffraction show a distinct maximum, with the inner edge of the ring proving sharper than the outer one.

Wyckoff⁴ studied binary mixtures with molybdenum \(K\)-rays in order to determine whether the diffraction—

effect to vary proportionally to the composition of the mixture. In the absence of such proportionality, the cause of the diffraction would have to be sought in the atom, since the mutual influence of molecules of different kinds should lead to deviations; unfortunately, the results obtained proved not especially convincing, chiefly because most of the rings obtained in liquids turned out to be very diffuse and to overlap the positions of neighboring rings. In general, the observed effects may be regarded as a simple superposition of the effects produced by each individual constituent. Wyckoff, on the basis of his work, concludes that diffraction by liquids is of intermolecular origin. However, this hypothesis does not take into account the possibility of the association of several molecules.

Keesom and de Smedt, in their two papers5, 6, studied first liquid gases ($\mathrm{O_2}$, $\mathrm{N_2}$, $\mathrm{Ar}$, $\mathrm{H_2}$) and then a whole series of organic compounds, using the $K$ radiation of copper and molybdenum. The rings obtained at these different wavelengths proved to be exactly the same, if one takes into account the change in the diffraction angle accompanying the change in $\lambda$. In a subsequent paper de Smedt7 considered several organic compounds, for example benzaldehyde, benzoylbenzoic ester, etc.; in doing so he succeeded in observing also a second ring (oleic acid) and even a third (benzaldehyde).

Among the more recent works on this question we should note the investigations of Prins8, 49 on the diffraction of mercury; in this case the apparatus consisted of a dismountable metal tube which gave the $K$ radiation of copper, iron, or zinc; this tube could be rotated about a horizontal axis (Fig. 2). The X-rays, emerging through a thin window, thus fell at different angles onto the surface of mercury in a bath. It turned out that three blurred bands were thereby formed on the cylindrical film, the diffraction intensity at small angles being very weak.

Quite recently, Katz and co-workers9 carried out...

a large number of experiments on a whole series of organic substances, with the liquids being placed in small glass tubes, as in the works of Debye and Scherrer. In his first paper Katz found that the majority of the bodies he investigated give only one “amorphous” ring, corresponding, according to Ehrenfest’s formula,

\[ d=\frac{7.72\,\lambda}{4\pi \sin \frac{\theta}{2}}, \]

to an intermolecular distance of the order of \(5\)—\(2\ \text{Å}\), even for very complex substances. However, in his subsequent investigations this author provides evidence for the appearance of a second and even a third ring, whose diameter varies depending on the nature of the carbon chains, their branching, the state of polymerization, etc. Of course, this material, despite its complexity, already makes it possible to begin developing a theory that should explain the observed facts.

Fig. 2. Prince and Koster apparatus for the study of mercury.

Zogani \(^{10}\) carried out a large number of investigations of organic liquids and, as we shall see below, developed a fairly satisfactory theory. His work was then continued by Krishnamurti \(^{11}\). Further, Stuart and Morrow \(^{12}\) studied fatty acids and long-chain alcohols. They advanced a hypothesis of the spatial arrangement of molecules, which they call the state of “cybotaxis.”

Herzog and Jancke \(^{26}\) compared the diagrams of organic substances in the solid and molten states; it turns out that molten substances give broader bands than substances in the solid state, and that there exists some connection between the positions of the rings and the positions of the lines; the latter apparently proves the existence of a certain similarity between the liquid and solid states.

J. J. Trillat^40, using the wedge method and a tangential beam, as well as a transmitted beam, studied the surface and internal structure of liquid (and also solid) organic bodies, without using any vessel for the liquid. These investigations showed how important the question of the thickness of the irradiated layer of substance is; on the other hand, they made it possible to judge phenomena of surface orientation; in these experiments it was also possible to observe new rings that had previously been unknown.^1

Our knowledge concerning the liquid state was further enriched thanks to a large number of new works, of which the most important are the investigations of Prins and Koster^48,51, Debye, Bewilogua, and Ehrhardt^52, Krishnamurti^53,54,57, Mark^55, Shiba and Watanabe^56, Vaidyanathan^58, Stewart^12,59, and others.

Theories of the Diffraction of X-Rays by Liquids

The results of the works discussed above may be summarized as follows:

a) At small angles, the intensity of the scattered rays is very low.

b) The greater part of the scattered energy is distributed within a comparatively small range of angles, producing a ring quite distinctly separated from the direction of the primary beam.

c) In some cases, a considerable part of the energy also falls at somewhat larger angles, so that two or three rings are obtained.

In all these cases such rings are perfectly circular, and their diameter varies in proportion to the wavelength. This shows that the phenomenon must be ascribed to a certain diffraction, and that the diffraction centers possess a certain regularity in their arrangement.

If, from this point of view, one imagines a liquid on the basis of its constituent parts—electrons, atoms, and molecules—then we must successively consider their connect—

...namely: the atom, the molecule and, finally, an aggregate of molecules.

Therefore the study of the whole mass of liquids should be divided into three stages:

1) the arrangement of electrons in the atom,
2) the arrangement of atoms in the molecule,
3) the arrangement of molecules in the liquid.

We shall consider successively all these three cases in connection with questions of a theoretical order that can be applied to the phenomena of X-ray diffraction.

1. Arrangement of electrons in the atom. Until now this factor has not been considered at all in connection with diffraction in liquids. Nevertheless it is analogous to the structural factor that is considered in diffraction by crystals. Indeed, if the dimensions of the atom were small in comparison with the wavelengths of X-rays, then the oscillatory motions of all the electrons in the atom, excited by one and the same incident wave, would be almost exactly in one and the same phase, so that the amplitude of the field diffracted by the atom would be obtained simply by multiplying by \(N\) the amplitude of the field diffracted by one isolated electron. In reality, however, the atom has dimensions of the order of the wavelength of the X-rays employed, so that phase differences must appear between waves diffracted by different electrons, and these phase differences will be the cause of more or less considerable decreases in the intensities of waves diffracted in directions not too close to the initial direction.

If it were possible to consider simply the arrangement of electrons in each atom, it would not be difficult to determine the phase differences between the scattered waves and thus to compute the resulting amplitudes in a given direction. Unfortunately, the arrangement of electrons in the atom is not yet fully known; moreover, since the electrons are in motion, this problem becomes still more complex.

One way or another, but on the basis of certain hypotheses concerning the mean positions of the electrons in the orbits, Hartree^13 succeeded in calculating the magnitude of this factor \(F\) for various ions as a function of the diffraction angle. Thus for O and for \(\sin \beta = 0.15\) one obtains a value of \(F\) amounting to 64% of the value obtained when \(\sin \beta = 0\). This is precisely the quantity that interests us in the question of diffraction by liquids.

In the case of lighter ions the quantity \(F\) decreases with the angle \(\beta\) still more slowly. In this, apparently, lies the reason why in the case of organic liquids, where the constituent parts are light atoms such as C, N, O, this factor does not play an essential role, so that these atoms may be regarded as scattering uniformly in all directions of practical interest to us. Thus, since different atoms must be regarded as independent of one another, the diffraction effect may be attributed to the arrangement of electrons in the atom; moreover, this diffraction may be represented in the form of a curve sufficiently close to that which corresponds to the distribution of intensities in reflection from crystals^14 (Fig. 3).

Fig. 3. Diffraction by an atom.

1. The arrangement of atoms in the molecule. In this respect two attempts were made to construct a theory; one of them belongs to Debye and Scherrer^1, the other to Ehrenfest^15.

Debye and Scherrer examine in detail the case of radiant energy scattered by a group of atoms arranged irregularly, as in the case of amorphous bodies. It is assumed that the velocities of the electrons are sufficiently small, so that they may be considered immobile during one period of the incident wave; moreover, it is assumed that these corpuscles are arranged in the atom on a circle of radius \(a\). In this way it proves possible to show that the course of the phenomena will be different depending on the magnitude of \(a\).

If \(\lambda\) is large in comparison with \(a\), then the total intensity of radiation can be expressed as follows:

\[ I=\frac{1+\cos^2\delta}{2}\cdot \frac{\varepsilon^4}{\mu^2 c^4}\cdot \frac{N}{R^2}\,p^2 E^2, \]

where \(\varepsilon\) denotes the charge of the electron, \(c\) the velocity of light, \(p\) the number of electrons per unit volume, \(N\) the total number of atoms, \(\mu\) the mass of the electron, \(R\) the distance from the amorphous body, \(\delta\) the angle between the direction of the incident ray and the direction of observation, and \(E\) the amplitude of the electric force. It is obvious that the scattered energy is proportional to the square of the number of electrons and, moreover, is distributed symmetrically with respect to the direction of incidence (Fig. 4).

Fig. 4. Dependence of intensity on angle for \(\lambda \gg a\).

Conversely, if \(\frac{\lambda}{a}\) is small, one obtains Thomson’s formula, in which there is proportionality only to the number of electrons:

\[ I=\frac{1+\cos^2\delta}{2}\cdot \frac{\varepsilon^4}{\mu^2 c^4}\frac{N}{R^2}\,p\,E^2. \]

But in addition to this, in the intensity of the diffracted radiation one should expect the appearance of periodic changes (Fig. 4) depending on the angle between the direction of incidence and the direction of diffraction. As a result, light and dark rings are obtained, situated around the direction of incidence, which can be expressed by the complete formula:

\[ I=\frac{1+\cos^2\delta}{2}\cdot \frac{Np}{R}\cdot \frac{\varepsilon^4}{\mu^2 c^4} \sum_{0}^{p-1} \frac{ \sin\left(4\lambda a\sin\frac{\delta}{2}\sin\frac{n\pi}{p}\right) }{ 4\lambda a\sin\frac{\delta}{2}\sin\frac{n\pi}{p} }. \]

From the agreement between the appearance of the ring obtained in the case of benzene and one of the secondary maxima on the curve obtained, Debye derives the hypothesis that diffraction rings for liquids are caused by the atomic structure of the molecule. However, this hypothesis encounters one difficulty,

but it turns out precisely that monatomic liquids, such as argon and mercury, possess a ring analogous to the ring obtained in the case of more complex substances.

Ehrenfest’s theory considers the case of an ideal gas consisting of diatomic molecules, i.e., the case of a substance with two scattering points located at a constant distance from one another and oriented in all possible directions. Keesom then extended this theory to the case of liquids consisting of spherical molecules separated by an average distance \(A\). This element of regularity should be the cause of periodic changes in intensity (Fig. 5), which, however,

Fig. 5. Experimental intensity curves as a function of the angle of diffraction, in the case of liquids.

Fig. 6. Diffraction by a diatomic molecule.

is insufficient for a complete explanation of the observed phenomena (Fig. 6); in particular, the weak intensities diffracted at small angles (\(E\)) remain unexplained, as do the height and width of the maximum at \(A\) (Fig. 6). Both these facts indicate that for small angles of diffraction it is also necessary to take into account the relation between the positions of neighboring molecules. This is explained by considering the crystal. Indeed, in this case the positions of the various molecules are strictly connected with one another, and the first intense diffracted ray is clearly separated from the direct rays by a region with very weak darkening; conversely, for increasing diffraction angles the influence of such a correlation decreases, and the effect approaches what would be produced by molecules arranged at random, i.e., a gas*.

* Cf. Mark, Z. Physik, 54, issues 7 and 8, 1929; Prins \(^{48—51}\), Debye \(^{52}\).

The formula derived by Ehrenfest was used by many authors; moreover, on the basis of one argument or another, it was taken to be valid not only for the case of a diatomic molecule, in which the atoms were regarded as centers of diffraction, but also for the case of double groups of such molecules, the distances between the two molecules of each group being considered constant and equal to \(a\). Such a formula may be written in the following form:

\[ a=\frac{7.72\lambda}{4\pi\sin\frac{\varphi}{2}} =\text{the mean distance between the centers of the molecules} \]

where \(\lambda\) is the wavelength, and \(\varphi\) is the angle of diffraction.

The quantities calculated in this way turn out to be sufficiently close to those given by the Bragg equation:

\[ a=\frac{\lambda}{2\sin\frac{\varphi}{2}}; \]

the difference in the numbers obtained from these formulae proved to be of the order of \(23\%\).

Nevertheless, it appears very probable that the atomic structure also plays a role in the diffraction of X-rays by liquids. However, the mechanism of this action has still not been clarified; one could approach it only by studying monatomic substances, where the molecules are the atoms themselves. The study of gases, as Debye indicated,\(^1\) should also make it possible, if the necessary accuracy of investigation can be attained, to come closer to elucidating the mechanism of the occurrence of diffraction by individual molecules and atoms, since in this case there is no regularity whatever in the positions of the molecules, as occurs in the crystalline or liquid states.

It is precisely in this case that we are dealing with a truly amorphous body.

These various hypotheses now lead us to consider the arrangement of molecules in a liquid.

3. Arrangement of Molecules in a Liquid

The analogy between the rings obtained in the case of liquids and the rings corresponding to microcrystalline powders leads some authors to suppose that a liquid is formed of crystalline particles. Obviously, such a hypothesis is not exact, but nevertheless it must be examined carefully, as we shall do below in describing the results obtained by Stewart and Morrow, on the one hand, and by Cernicke and Prins on the other.

Debye1, who in 1921 obtained rings with various liquids (mercury, methyl iodide, methylene iodide, benzene, bromobenzene, etc.), was the first to propose a theory based on the existence of diffraction centers—atoms or molecules distributed at random, but satisfying the condition that neighboring centers are always at the same distance from one another, as, for example, the centers of identical spheres stacked one above another.

Let us therefore consider two adjacent centers, placed arbitrarily with respect to the direction of the incident ray.

In this case there will be interference of the waves from both centers, and the resultant amplitude in the plane of the incident ray will form with this ray an angle that is easy to calculate. It is easy to see, however, that this angle will pass through a minimum at a value \(\varphi\) determined by the equation

\[ \sin \frac{\varphi}{2}=\frac{\lambda}{2a}, \]

where \(\lambda\) denotes the wavelength, and \(a\) the distance between the centers.

It follows from this that, for each radiation, a considerable part of the diffracted energy is concentrated near the direction corresponding to the minimum.

Obviously, the formula thus obtained is analogous to Bragg’s formula.

As we have already indicated above, the theoretical calculations

Ehrenfest’s were also based on considering not a diatomic molecule, but the interaction of two adjacent molecules, regarded as centers of diffraction. Liken­ing the molecules to spheres tangent to one another, Keesom and Smedt5, 6 calculated a priori the distance \(b\) separating the centers of these molecules; namely, if \(d\) is the density of the liquid and \(M\) the molecular weight, then the following relation is easily obtained:

\[ b=1.33 \sqrt[3]{\frac{M}{d}} . \tag{1} \]

The distances calculated in this way, if one assumes that the molecules have a compact hexagonal packing, agree well with the experimental results obtained for certain liquids originally studied by these authors, and with those calculated by Ehrenfest’s formula; whence it follows that this formula indeed gives the mean distance between the centers of molecules.

However, it is rather difficult to prove the correctness of this hypothesis55. If we were to accept it, we would be obliged to admit that the diffraction rings approach the rings obtained in the case of crystalline powders, which are very sharp and well defined; in reality this is not so. Moreover, as the shape of the molecule departs more and more from spherical, the discrepancy between the results obtained from measurements of the rings and from formula (1) begins to become significant, as is also seen from the following table, borrowed from Katz17:

Alcohols	Ehrenfest	Keesom
Methyl	4.5	4.6
Ethyl	5.0	5.1
Propyl	5.3	5.6
Butyl	5.4	6.0
Hexyl	5.5	6.6
Octyl	5.5	7.2
Decyl	5.5	7.6

Finally, in a very large number of liquids, besides the principal ring, one or two other, sometimes rather intense, rings are also observed, whose origin cannot yet be regarded as clarified. Keesom attempted to give an explanation on the basis of the assumption of association of molecules or polymerization; then formula (1) becomes the following:

\[ b = 1.33\sqrt[3]{\frac{M}{nd}}. \]

Here \(n\) denotes the degree of association or polymerization. Thus it turns out that for paraldehyde and for benzyl benzoate \(n = 3\). This hypothesis, however, has to encounter a whole series of serious objections (see below—polymerization).

The theory of Raman and Ramanathan. The theory of Raman and Ramanathan is based on the principal idea that the molecules of a liquid are not scattered according to the law of chance, but that their arrangement possesses a certain degree of regularity, owing to which diffraction rings are obtained.*

Although the molecules of a liquid are not arranged with the same perfect regularity as in the case of a crystal, nevertheless it is highly probable that this arrangement is not as chaotic as in the case of a gas. In considering the specific volume of a substance in its three states, it is quite possible to say that, from the point of view of the arrangement of molecules in space, the liquid state approaches the solid state more than the gaseous. This regularity of arrangement, which can be derived from simple considerations, can also be justified thermodynamically, namely on the basis of considerations concerning the compressibility of liquids.

Proceeding from such ideas, Raman and Ramanathan extended to the region of X-rays the hypothesis of density fluctuations derived from the thermodynamic statistical theory of Einstein\(^{19}\) and Smoluchowski\(^{20}\)

* This theory was further developed by Zogani\(^{10}\), from whose work the following lines are also borrowed.

for the case of the scattering of light rays—a hypothesis which is justified in this latter case. Here we shall briefly consider how it proves possible to apply this hypothesis to the case of X-rays.

The only difference between the scattering of light and of X-rays consists in the wavelength, which in the latter case is considerably smaller than the mean intermolecular distances. What is essential in the ideas of Einstein and Smoluchowski is the consideration of a liquid as a continuous medium subject to local changes of density, which are determined on the basis of thermodynamic considerations. This hypothesis is justified in the case of visible radiation, for which the wavelength is considerably larger in comparison with molecular dimensions. The application of the theory depends above all on the possibility of dividing the medium into layers of thickness

\[ \frac{\lambda}{2\sin \frac{\theta}{2}}, \]

where \(\theta\) denotes the angle of diffraction, which is equal to twice the angle of scattering. These layers can again be divided into layers each containing many molecules, in such a way that the changes of density in each of them may be considered independent of the changes of density in neighboring layers, which is possible for a large wavelength (the optical case).

In the case of X-rays this could occur for very small values of \(\theta\), i.e. for small angles of diffraction. On the basis of data on the compressibility of benzene, Raman and Ramanathan conclude that, according to the Einstein–Smoluchowski theory, the energy scattered at \(2^\circ\) with Mo \(K\) rays, when the ring corresponds to \(8.5^\circ\), must be very small. For mean values of the diffraction angle, each layer of thickness

\[ \frac{\lambda}{2\sin \frac{\theta}{2}} \]

will contain in this thickness only a small number of molecules, so that it will no longer be possible to assume that, on the one hand, the wavelength is large in comparison with the distances between the centers of dif-

...ractions and that, on the other hand, when these layers are divided into thinner layers, the changes in density in the different layers will always be independent of one another.

Thus the theory of Einstein–Smoluchowski cannot be properly applied to the problem of the diffraction of X-rays, unless, of course, one excludes the case of small diffraction angles.

For large scattering angles the liquid may be regarded as having a granular structure. Applying the Fourier method to the analysis of the distribution of matter in a liquid, one can decompose it into a certain number of superposed periodic distributions of various fictitious wavelengths. In this way one obtains what Raman and Ramanathan call the “spectrum of structure,” or structural spectrum.

The law of this structural spectrum will evidently be determined by the form and structure of the molecules, by their mean distance, and by its variations from one side or the other at each instant. Each periodic distribution of matter with fictitious wavelength \(\lambda_1\) is, to a certain extent, comparable with a regular crystalline arrangement, so that the diffraction in a certain specific direction also depends on it, determined by the Bragg formula

\[ \lambda = 2 d_1 \sin \frac{\theta}{2}, \tag{1} \]

where \(\theta\) denotes the angle of diffraction.

Knowing the mean intermolecular distance and its fluctuations, determined by the compressibility of the liquid, one may derive an expression for the distribution of intensity:

\[ I = C_1 e^{-\frac{1}{16}\frac{N}{RT\beta}\lambda^3\left(1-\frac{\lambda_1^3}{\lambda_0^3}\right)^2}, \tag{2} \]

where \(N\) and \(R\) denote, respectively, Avogadro’s number and the gas constant referred to the gram-molecule, \(T\) is the absolute temperature, \(\beta\) the compressibility, \(\lambda_0\) the mean intermolecular distance, \(C_1\) a constant, and \(\lambda_1\) the mean intermolecular distance (\(C_1\) a constant and \(\lambda_1\) a quanti-

...a quantity determined by equation (1), where \(\theta\) denotes the wavelength of the incident rays).

From equation (2) it is easy to see that the intensity maximum will occur at \(\lambda_1=\lambda_0\), and its direction will be determined from the equation

\[ \lambda = 2\lambda_0 \sin \frac{\theta_0}{2}, \]

where \(\lambda_0\) is equal to the mean intermolecular distance.

The curve in Fig. 7 shows the course of this phenomenon (upper part).

Equation (2) was applied by the authors to the case of benzene; the agreement with Hewlett’s ionization curve (Fig. 7, lower curve) is very remarkable, if one excludes the region of small angles, where this curve does not agree with Uyikov’s photometric curves. The quantities given by this theory are lower than those derived from Hewlett’s curves; they agree better with Uyikov’s measurements.^4

Fig. 7. Diffraction by benzene. Upper drawing—theory (Raman and Ramanathan), lower—Hewlett’s experiment.

These results show, in the case of benzene, that the arrangement of molecules in a liquid is probably the most important factor or, in any case, a more important factor than the internal structure of the molecule.

Collins^21 indicated some applications of the Raman and Ramanathan theory, in which, in accordance with the theory, he finds the diffraction ring for ether (whose coefficient of compression is very high) to be strongly blurred. However, he notes that the agreement here is only qualitative; this is not surprising, bearing in mind that the theory has an approximate character.

It would be entirely possible to apply this theory to determining the effect of high temperatures on the fineness of the ring; experiments of this kind, which are of very great interest, have recently been carried out by Vaidyanathan ^58.

Prins and Cernike ^22, on the basis of the same ideas, developed a theory of the diffraction of X-rays by liquids, in which they take into account the intermolecular arrangement. According to this theory, the molecule is regarded as having a definite shape, and the law of the mean free path from the kinetic theory of gases is applied to the free space between molecules. If one imagines a mesh plane whose elements are parallel lines moving in a certain space (Spielraum), illuminated by a given wavelength, it is possible to determine the intensity of the diffracted rays if the probability distribution of these elements is known. From this one can pass to the case of three-dimensional space. The expression obtained gives, for the diffraction rings of liquids, a certain number of maxima, whose distances from the center are proportional to the square roots of successive integers; moreover, the maxima tend to disappear as the compressibility increases.*

This theory explains the dispersion at small angles fairly well, but it is not justified in the case of larger angles, since even in the case of very weak compressibility the secondary maxima, which alone are observed, do not have intensities comparable with the principal maximum.

EXPERIMENTAL RESULTS

A considerable number of results confirm the theory of Raman and Ramanathan. The most important of them are the results of the work of Zogani ^20 and Krishnamurti ^25, ^53, ^54, ^57, concerning various organic liquids.

Arrangement of molecules in a liquid. Zogani established, first of all, that the principal ring can be attributed—

* The reader will find a survey of these theories in the papers of Prins ^59 and Stewart ^59.

describe the arrangement of molecules in a liquid, and that this arrangement, with the exception of a few unexpected cases, corresponds to the regular hexagonal grouping whose existence was assumed by Keesom and Smedt; the intermolecular distances deviate from the value calculated by the formula

\[ a = 1.33 \sqrt[3]{\frac{m}{d}}, \]

the more, the more the shape of the molecules deviates from the shape of a sphere. In any case, it is quite possible to determine the intermolecular arrangement in the following way:

For a liquid or gas, the distance between neighboring molecules is of the order \(n^{\frac{1}{3}}\), where \(n\) is the number of molecules per unit volume.

Consequently, one may write:

\[ a' = k \sqrt[3]{n} = k \sqrt[3]{\frac{m}{d}}, \]

where \(m\) is the mass of the molecule, \(d\) is the density, and \(k\) is a constant.

The value of \(k\) in some cases can be calculated theoretically (Raman \(^{23}\)), and it varies from 1.123 (regular cubic grouping, as compact as possible) to 0.554 (chaotic arrangement of a perfect gas); for a liquid the value of \(k\) must lie between 0.8 and 1, depending on the nature of the liquid and on the conditions of temperature and pressure.

The table placed on p. 233 gives the values of \(k\), calculated from the ratio (\(a\)—Bragg formula; \(a'\)—Raman formula).

From this table it is seen that \(k\) changes in passing from one liquid to another, i.e., that the distribution of molecules changes depending on their shape. Between aromatic and aliphatic molecules there is a very clear difference. It is very interesting that hexane and cyclohexane (Fig. 1), which have the same number of carbon atoms, but in one of which the chain is linear and in the other cyclic, have differences in the value of \(k\) of 15%, and this apparently means that the molecules of cyclohexane are grouped

much more compactly than hexane molecules. The elongated form of hexane molecules does not allow them to group as closely as occurs in the case of cyclohexane molecules, so that in the former case a certain part of free space remains between the molecules; the difficulties experienced by these molecules in tending toward close packing may be compared with the case of a heap of long nails arranged at random.

	$\theta$	$a=\dfrac{\lambda}{2\sin \dfrac{\theta}{2}}$	$\alpha'=\sqrt[3]{\dfrac{m}{d}}$	$k$
Pentane	18°1	4,90 Å	5,27 Å	0,847
Hexane	18°3	4,85	6,01	0,808
Heptane	17°8	4,97	6,24	0,797
Octane	18°1	4,90	6,47	0,758
Oleic acid	18°3	4,85	8,05	0,603
Ether	19°2	4,63	5,56	0,834
Glycerin	19°8	4,49	4,95	0,906
Benzene	18°1	4,90	5,29	0,926
Cyclohexane	17°1	5,20	5,64	0,924
Phenol	18°1	4,90	5,23	0,927
Toluene	16°5	5,37	5,61	0,957
Aniline	17°6	5,03	5,34	0,942
Water	27°3	3,27	3,11	1,05
Butyric acid	19°2	4,63	5,35	0,865

Thus the shape of the molecules exerts a profound influence on their grouping (Fig. 1).

It should be noted that the gradual elongation of the chain is accompanied by an equally gradual decrease in $k$. For oleic acid (a chain with 18 carbon atoms) $k=0.607$, a value close to that which corresponds to the absolute chaos of the molecules of a perfect gas ($k=0.554$).

Appearance of the ring as a function of chain length. Examination of Fig. 1 shows that, apart from the thinness or width of the ring, almost no changes occur in it with change in chain length. They are all such that the effective distances between the centers of diffraction remain, as it were, one and the same, whatever the length of the molecule. This is undoubtedly due to the circumstance that

that for long molecules the probability of end-to-end connection is very small in comparison with the probability of side-to-side connection; the cross-section of the molecules remains almost constant, as was shown in other works, whence the constant appearance of the ring is obtained.

Relation between the blurring of the rings and compressibility. When a group of molecules is subjected to compression, the distance between each of them decreases, as does also the distance between the layers which play the role of reflecting planes.

And indeed, the Raman and Ramanathan equation may be written in the following form

\[ I=C_1 e^{-\frac{1}{16\,kT\beta}\,\frac{N}{\nu}\lambda^3 \left(1-\frac{\sin^3 \frac{\theta_0}{2}}{\sin^3 \frac{\theta_1}{2}}\right)^2}, \]

where \(\theta_0\) denotes the angle corresponding to the principal ring, and \(\theta\) the angle of some diffraction. If one calculates \(I\) as a function of \(\vartheta\), or better of \(\frac{\theta_1}{\theta_0}\), then, evidently, a curve should be obtained whose form will depend only on the magnitude

\[ \frac{N\lambda_0^3}{16kT\beta}; \]

the maximum will be the sharper, the larger this quantity is, i.e. the weaker the compressibility. Observations of the diffraction rings of simple fatty compounds fully confirm this hypothesis, since the substances studied have rings that are the more blurred the greater the compressibility of these substances. In the case of cyclic compounds this law is less applicable, since here, apparently, the shape of the molecules and their branching must also play a role. Apparently, the best means for studying these two phenomena (compressibility, structural factor) will be the study of liquids at various temperatures and the observation of whether the ring changes with increasing temperature, and consequently also with increasing value of \(\beta\) (Vaidyanathan \(^{58}\)).

Influence of the shape and molecular structure. The diffraction ring of a liquid changes very distinctly as the complexity of the molecule increases.

1) Orthoaminobutyric acid (drop)

2) Lead oleate on heating (Trillat)

3) Lead oleate in the cold (Trillat)

4) Interaction of mercury and hippuric acid (Trillat)

5) Palmitic acid (interior of the drop)

6) Palmitic acid (surface of the drop)

Fig. 8. Study of surfaces by the drop method (Trillat):

The slightest differences in the spatial arrangement of atoms, or groups of atoms, leading to the formation of different isomers, are reflected in changes in the external appearance of the ring.

The number of experiments devoted to this question is very large. Katz, in particular, dealt with it extensively, investigating, chiefly, the most varied organic compounds.

In liquids with only slightly differing compressibility, Zogand found that the symmetry of the benzene molecule (a sharp ring) could change upon the introduction of light groups, for example OH (phenol), NH$_2$ (aniline), CH$_3$ (toluene), etc. (Fig. 1). Molecular asymmetry increases with an increase in the number of diffracting electrons introduced together with the group. Thus, for example, ethylbenzene and nitrobenzene have very broadened rings in comparison with the molecules indicated above. If one considers the case of simultaneous introduction of several groups, then their mutual arrangement proves also to play an important role. In this case a second ring often appears, and the X-ray diffraction patterns no longer have the same appearance, for example, for the ortho-, meta-, and para-positions (Krishnamurti $^{53,54,57}$, Stuart and Morrow $^{12}$).

Katz $^9$, who studied the influence of the shape of molecules on the diffraction rings, showed that when two rings are obtained, the inner ring, generally speaking, satisfies Keesom’s equation, whereas the outer one gives a value of $a$ the larger, the longer the side chains are (Fig. 8); and indeed this phenomenon is obtained especially in the case of organic molecules containing several identical groups. This value of $a$ agrees fairly well with the value given by the attached radical. For example, tripropylamine gives two rings corresponding (according to Ehrenfest’s formula) to the following values:

\[ a_1 = 8.1\ \text{\AA}, \quad a_2 = 5.3\ \text{\AA}, \]

whereas propyl alcohol gives a ring corresponding to

\[ a' = 5.3\ \text{\AA}. \]

Thus there exists a period of identity, determined by the average distances of the side chains. Similarly, two rings are observed in the case of paraldehyde, where there are several similar groups, but joined in the form of a ring. However, the appearance of the second ring may depend on the combination of several molecules forming larger units, or even on complex chemical compounds, more or less stable or short-lived. In this connection, Trillat and Tibaud have recently shown that the second ring, which is very intense, may often be attributed to the continuous spectrum emitted by the tube, if the thickness of the liquid being penetrated is sufficient.*

Finally, there are cases (fatty acids, normal alcohols) where a single unbranched molecule can give two rings (Katz, Stewart, and Morrow; see above); thus the problem of the influence of molecular form is still far from being solved.

Fig. 9. Diffraction by normal alcohols with a long carbon chain (Stewart and Morrow).

Study of liquid compounds with a long chain and statistical orientation. In their recent works Stewart, Skinner, and Morrow \(^{12,59}\) studied normal alcohols and fatty acids with a long carbon chain, using an ionization chamber; the X-rays were obtained in a tube with a molybdenum anticathode.

The ionization curves (Fig. 9) show two maxima.

* J. Tibaut et J. J. Trillat, C. R., 4 et 75 nes 1929.

One of them is practically independent of the length of the chain; the other varies linearly with the number of carbon atoms. These maxima correspond to values analogous to those that were obtained by Katz by the photographic method; moreover, the “main ring” has an order of \(4.6\ \text{Å}\), while the “variable ring” ranges from \(9.6\ \text{Å}\) (normal propyl alcohol) to \(15.8\ \text{Å}\) (heptyl alcohol). These changes are linear in character, with each new carbon atom corresponding to \(1.3\ \text{Å}\), which is in good agreement with the values obtained by Adam in the case of monomolecular layers on water, and by Müller, Shearer, and Trillat (loc. cit.) for alcohols and acids in the solid state.*

Stuart and Morrow come to the conclusion that, as in the solid state, the molecules of alcohols and normal fatty acids are linked by their two active groups. The distance \(4.6\ \text{Å}\) agrees very well with the transverse dimension of the molecules, which was found in previous work, so that it may be asserted that it represents the mean distance between molecules of the liquid. Consideration of the densities also gives very good agreement with the theory. However, the structure of the solid state is not preserved intact in the liquid state; thus, for example, the distances \(35.6\ \text{Å}\) and \(4.2\ \text{Å}\) in solid lauryl alcohol become, respectively, \(24.7\ \text{Å}\) and \(4.6\ \text{Å}\) in the liquid alcohol. The structures of the two states may have some analogy, but they obviously cannot be similar. Thus, the groups of molecules of a liquid are not simply pieces of crystals of the solid body, which is also proved by the absence of higher-order reflections.

Such results can readily be explained if it is assumed that the molecules are in a state of mobile orientation. The fixed arrangement of molecules in the solid state becomes a mobile arrange—

* See the bibliography to the article “Organic Compounds,” U. F. N., issue 3, 1931.

by the positions in the case of the liquid state. Throughout the entire mass of the liquid at a given moment there is a considerable number of groups possessing the spatial arrangement that has been found; the probability of the existence of such groups will be maximal, but this does not exclude the possibility of the existence of somewhat different arrangements, which also explains the breadth of the maxima or rings.

Thus, at every moment in the liquid there will exist short-lived arrangements of molecules placed parallel to one another; the active groups are arranged in parallel planes and, apparently, give rise to the reflection of X-rays, these planes possibly having some inclination with respect to the direction of the chains. This state, for which Stewart and Morrow devised the name “cybotactic,” presupposes the existence of mobility, but it will not be random motion. It is precisely this that will characterize the liquid state. Thus the molecules may be regarded as possessing the same orientation, while the groups formed by them are too small to give rise to optical anisotropy.

Stewart^24 showed that saturated normal hydrocarbons (paraffins) give only one maximum, namely the one that does not change with the number of carbon atoms; undoubtedly, it should be attributed to the absence of active groups.

Katz (loc. cit.), in his work on the study of fatty acids and alcohols, arrives at an analogous conclusion; according to his views, the molecules of these liquids in most cases combine into very small groups, in which a predominant orientation exists, while these groups themselves are arranged according to the law of chance. Such orientation must arise because, under the influence of high internal pressures, the molecules are arranged in narrow spaces. It should be noted here that Trillat also arrived at the same conception in studying the question of the influence of pressure on the orientation of hydrocarbon molecules forming part of oils in contact with metallic-

with walls; the molecules are oriented and arranged in parallel, so as to offer resistance to external pressure (see Trillat’s article on organic compounds, Uspekhi fizicheskikh nauk, issue 3, 1931).

Very recently, Prins\(^ {50}\) has also succeeded in confirming these results; however, the increase in chain length per carbon atom, especially in the case of fatty acids, proves to be weaker than was found by Stewart.

Be that as it may, we can state that all modern theories of the structure of liquids assume a fine structure in the arrangement of molecules—a statistical structure, changing from one moment to another, but accessible to interpretation on the basis of X-ray diagrams. From the molecular point of view, this mobile structure may be regarded as anisotropic, but in general a statistically isotropic medium is obtained; in this is reflected the profound difference between a molecular phenomenon considered separately and a complex of molecular phenomena.

Thus, the main ring in the case of liquids apparently arises from the proper distribution of intermolecular distances. How one could relate a molecule to some center of diffraction and how one could explain the appearance of other rings—this constitutes a problem not yet solved; moreover, only qualitative considerations are possible here, while exact numerical data are as yet unavailable. Indeed, for the present one has to rely now on one, now on another understanding of this problem. It is clear that the difficulties of this question consist in the fact that the case of a liquid is not as simple as the case of a crystal with its perfect regularity, or even the case of a gas with its absolute chaos. Our knowledge in the field of the liquid state is still far from complete, so that for a more perfect understanding of the phenomena, deeper experimental investigations are naturally required. All that can be said at the present time is that the liquid state is an intermediate state between the solid ...

...crystalline state (completely oriented molecules) and the gaseous state (a distribution according to the law of chance); moreover, strictly speaking, the expression “amorphous state” should be reserved only for the latter case.

The relation between the liquid and crystalline states. On the basis of the conclusions presented above, one may now ask oneself: will not the liquid state in some sense “resemble” the crystalline state, or, in other words, is there not some connection between the position of the amorphous rings of a liquid and the Debye–Scherrer rings that are obtained in the case of microcrystalline bodies?

Herzog and Jancke^26 studied for this purpose a large number of solid and molten organic compounds, and in order to obtain the diagrams the same distance was maintained in all cases. Comparison of the photographs led them to the following empirical rule: an intense amorphous ring is located either near a particularly intense crystalline ring, or near a group of many crystalline rings. When amorphous rings are located near the central spot, they are displaced outward relative to the crystalline interference; and if they occupy the periphery, then, on the contrary, they are displaced inward relative to the crystalline ring.

Thus, near the most intense crystalline ring of a given substance there is always an amorphous ring corresponding to the molten state, but the converse is not always the case. This circumstance apparently confirms that in the liquid state there is something characteristic of the crystalline state.

Intermediate states of matter. The transition from the solid state to the liquid state is not always abrupt. There may exist a series of successive stages giving rise to more or less definite structures. At present it is believed that these stages are more numerous and more particular than had previously been supposed. Examples of this are found in the stretching of colloidal gels, surface structures, etc.

It is known that there is a series of organic bodies, studied mainly by Lehmann, for which there exists a definite range of temperatures between the melting point and the point at which the liquid becomes clear—namely, the range corresponding to the turbid state. Observations with a polarizing microscope show that such a turbid liquid possesses special optical properties, namely molecular anisotropy.

Friedel, in his remarkable article[^27], established by optical means the existence of three mesomorphic anisotropic phases, whose character depends on the temperature of the principal structures of liquid crystals. Mauguin also studied the phenomena of orientation observed in numerous bodies belonging to this class; this orientation is obtained under the influence of a magnetic field or owing to the formation of deposits on the surface of glass or crystals[^28].

These intermediate states are defined by Friedel as follows: 1) the nematic state, where the molecules are distributed according to the law of chance, but where they all have one common direction; 2) the smectic state, where the molecules, having one common direction, are, in addition, distributed over equidistant parallel surfaces; 3) cholesteric bodies, which form a third group, are nematic bodies endowed with twist.

If these ideas are correct, then neither nematic nor cholesteric bodies should give pure diffraction of X-rays. This was indeed shown by Göckel[^29] and van der Lingen[^30]. The Debye–Scherrer diagrams of the nematic state completely resemble the diagrams of the liquid amorphous state, although the one and the other are formed in somewhat different ways. Conversely, smectic bodies, possessing a periodic distribution of molecules, should act on X-rays as a system with parallel lattice planes of a crystal. In particular, a known fraction of a smectic substance, which is characterized by all possible orientations of these paral-

liquid surfaces, should diffract monochromatic rays and yield a spectrum formed of concentric rings.

M. de Broglie and Friedel[^31] and subsequently Piper[^32] attempted to confirm this on certain soaps, for example the oleates of sodium, potassium, and ammonium. However, the character of the X-ray photographs does not make it possible to assert the presence of an amorphous ring, unless one counts the presence of successive orders of reflection on planes separated by large distances. On the other hand, these results were disputed by MacBain,[^32] who admits the existence in soaps and their solutions of five different forms. Be that as it may, soaps apparently must exist either in a microcrystalline form or in an anisotropic, of course smectic, form (sodium oleate in the experiments of Perrin and Wells[^34]), depending on their concentration.

Fig. 10. Kast apparatus for the X-ray study of paraoxy-anisole. Collimator; \(P—C\)—specimen heated electrically; electromagnet.

Further experiments by Friedel[^35] with typical smectic bodies (ethyl azoxybenzoate and ethyl azoxycinnamate), carried out by the rotating-crystal method, showed that below a certain temperature \(T_1\) X-ray photographs of crystalline bodies are obtained; between \(T_1\) and \(T_2\) (anisotropic phase), an X-ray photograph of a completely special type is obtained, revealing a single direction of equidistant planes characteristic of the smectic state; and below the temperature \(T_2\) (isotropic melting) no diagrams are obtained. Consequently, these substances crystallize in one direction, and are amorphous in the perpendicular direction, so that the production of an amorphous ring must correspond to the mean distance of the molecules in the direction perpendicular to the direction of the periodic distribution.

Soon thereafter Kast,[^36] working with paraoxy-anisole in the solid and anisotropic states (Fig. 10), showed that (Fig. 11) nematic melting gives

...the diagram of an ordinary liquid; whereas, if an anisotropic liquid is placed in a magnetic field perpendicular to the X-rays, so that the long molecules are oriented in this direction, the amorphous ring splits into two rings giving two sectors. In general, here one obtains an imperfect fiber diagram.

Fig. 11. X-ray photographs of para-azoxyanisole (Kast). I. A rod cooled in a brass tube of diameter 1 mm and solidified in a magnetic field parallel to the axis of the tube. II. Tube of diameter 10 mm. Molecules oriented parallel to the axis of the tube. III. A rod cooled in a brass tube of diameter 1 mm and solidified (effect of the walls). IV. Rotation of a single crystal about an axis.

The ratio of the intensities of the rings in the direction of the magnetic field and perpendicular to it is, moreover, a measure of the degree of perfection of the magnetic orientation (Ornstein ^60). In one of the figures, for example, it can be seen that the intensity of the rings in the direction of the field is equal to zero (\(H = 3000\) gauss), whence it follows that for this value of the field the orientation will already be perfect, which is in agreement with electrical measurements.

The effect of the walls on the orientation was also studied by Kast. For this purpose he cooled para-oxyanisole in the nematic phase, contained in a small brass tube, in the presence or absence of a magnetic field. In the first case he observed that the orienting effect extends rather far from the walls, and that the \(C\) axis of the crystals tends to arrange itself along the axis of the tube. In the second case, if a magnetic field is taken parallel to the axis of a brass tube of radius \(10\ \mathrm{mm}\), and the X-rays are directed perpendicular to the tube, then in the central part of the substance a very strong orientation of the crystallites is observed, with the Debye–Scherrer rings breaking up into spots (Fig. 11). Here the molecules are oriented parallel to the axis. Thus it turns out that, starting from a nematic state of orientation, one can also obtain a polycrystalline oriented state. Consequently, the anisotropic phase is characterized by destruction of the lattice, but not of the intermolecular cohesion, which disappears only at higher temperatures.

Are there, between the liquid and solid states, besides those considered above, also other intermediate states? At present this question may be answered in the affirmative. Clark\(^{37}\) introduced for this purpose two new terms, namely: the “paracrystalline” state and the “metacrystalline” state. The first corresponds to a transitional state adjacent to the crystalline one; the second corresponds to a random symmetrical arrangement of diffraction centers connected with certain external conditions, which does not constitute a stage of transition to the crystal; this includes, for example, rubber and stretched gelatin, which were discussed in another article.

In studying strongly stretched gels of nitro- and acetocellulose, the author\(^{47}\) was also able to show that the structure of these gels, initially amorphous (in the sense of a liquid), changes upon stretching: the circular rings become elliptical, the major axis of the ellipse coinciding with the direction of stretching, and more or less distinct intensifications appear, which exactly coin-

give, according to their position, spots from the same bodies in the crystalline state. Thus stretching can give rise to molecular orientation, then to a grouping of molecules close to that which corresponds to the crystalline state. The crystalline phase is not perfect, and the molecules may have some motion about their mean position. Moreover, for each value of the stretching this crystalline phase is in equilibrium with the amorphous phase, both structures being, in a certain sense, complementary to one another.*

STUDY OF INTERFACES

The drop method. A very large number of studies on the orientation of organic long-chain bodies, carried out in recent years, has shown that the possibility of studying the interfaces between these bodies and the surfaces orienting them (glass, metal, water, etc.) is of enormous interest; in most cases the rotating-crystal method makes it possible to investigate these bodies and to detect orientation phenomena.

It is obvious that it would be still more interesting to study liquid–air, solid–air, and liquid–liquid interfaces, since these surfaces are of particular interest from the point of view of adsorption phenomena, surface tension, and perhaps catalysis. It was precisely for this purpose that the author used a method based on the following principle \(^{40}\).

It is known that the rotating-crystal method requires this rotation in order to obtain successively different angles of incidence of the beam of X-rays on the surface. If, instead of this flat surface, a curved crystal is taken, then without any rotation one automatically obtains a series of diffraction spots corresponding to reflections from a family of lattice planes.

* We also point to Katz’s work on this question, “Similarity in the X-ray spectra of liquid-crystalline and liquid phases of a substance” \(^{61}\). The reader is also referred to the fundamental works of Vorländer \(^{63}\).

It was precisely this method that M. de Broglie used in studying a flake of curved mica. If now, instead of a curved crystalline surface, one considers the surface obtained when a drop of liquid is placed on a plane, and if the slit of the collimator is made horizontal, then the shape of the drop, for one and the same curvature, should make it possible to obtain angles (between the surface and the ray) varying from \(0^\circ\) to a value determined by the contact angle.

Fig. 12. Method of the lying drop. \(A\)—collimator with a horizontal slit, \(C\)—cup, \(F\)—electric furnace, \(E\)—drop (adjustable in height by means of a screw under cup \(C\)), \(G—G\)—lead shields, \(H\)—photographic plate, \(B\)—optical bench.

If, therefore, the surface obtained in this way represents the phenomenon of molecular orientation at a certain depth, it is easy to foresee that diffraction spectra should be obtained without displacement or rotation of either the X-ray tube or the drop. Obviously, such a method can also be applied to the case of a solidified drop obtained by cooling, or to the case of a solid or liquid drop situated on a layer of liquid capable of enveloping it (Fig. 12).

All this was established by the author as a result of a very large number of experiments concerning the cases considered above. The adjustment in these experiments was obtained

very simply by raising the droplet and the vessel supporting it, so that it was twice traversed by the beam of X-rays. Registration was carried out on films, which were placed at various distances from the droplet. Experiment showed that the center of the droplet may be taken, with sufficient approximation, as the origin.

Application to the study of the orientation of fatty acids by mercury. The earlier investigations were continued by the author with the aid of this method for the case of the orientation of fatty acids by mercury. For this purpose a small quantity of fatty acid was placed on a drop of mercury, after which slight heating was carried out so that the acid would melt if it was solid. In this case the mercury became covered with a shell of small thickness (Fig. 8, photograph 4).

The spectrograms obtained in short intervals of time (less than one hour) reveal intense spectra. The results may be briefly summarized as follows:

1. Despite the liquid state of mercury, a perfectly oriented layer is obtained, as in the case of both solid and liquid fatty acids. Thus it becomes entirely possible to study liquid—liquid and liquid—solid interfaces.

2. The spectra obtained show a regular decrease in the distance between the lines, which corresponds to a linear increase in the distance of the lattice planes with the number of carbon atoms, as was also found in the case of solid metallic surfaces.

3. An excess of acid gives either an amorphous ring (if the acid is liquid), or a pure spectrum (if the acid is a solid body).

4. The influence of temperature was traced approximately up to 100°. Even at this temperature there is perfect orientation, despite the oscillations of the molecules of mercury or of the molten acid.

Application to the study of lead oleate. Lead oleate is a body which, upon melting

passes into a turbid state and decomposes at 150°. Study of its surface as a function of temperature shows that, when cold, the substance is finely crystalline and oriented, like a fatty acid, the molecules becoming oblique to the surface. In the hot state, beginning at about 60°, a new orientation diagram appears, the more intense the higher the temperature (Fig. 8, photographs 2 and 3). This diagram contains one amorphous ring instead of Debye–Scherrer rings, as occurs in the first case. Apparently, when cold this substance is finely crystalline, whereas on heating it is amorphous in the sense that it gives only one ring. These results show that the surface layer of viscous oleate, i.e., the surface between the liquid and the air, consists of molecules oriented normally to the interface, and that, moreover, the molecules are distributed between planes parallel to this surface. Thus we have a whole series of curved leaflets superposed one upon another, which constitutes an analogy with the smectic state. This structure is gradually destroyed as one moves inward into the drop. Such a phenomenon is reversible: after repeated heating and cooling the first diagram is again obtained.

In addition to the conclusions concerning the form of the molecule that may be drawn from this,^40 it will be interesting to point out that the use of polarized light gives nothing, perhaps because the birefringence is too weak. Further, liquid–air interfaces may be associated with an orientation extending to a considerable depth. This was found by the same method for liquid fatty acids, triglycerides, and alcohols, and also for slowly solidified fatty acids. In this last case the acid–air surface is formed of oriented molecules, as occurs in the case of supporting surfaces (see the article on organic compounds), but here the CH₃ groups are situated on the outside. Consequently, the solidified drop must repel water, which

does not wet it, which is also well confirmed experimentally (Fig. 8, photographs 5 and 6).

General conclusions on surface orientation. These phenomena of surface orientation are of great importance not only from a purely physical point of view, but also from chemical and biological points of view. There is no doubt that when the position of molecules changes, and hence also that of the various groups of which they are composed, the surface chemical properties also change. Thus, for example, artificial silks made of cellulose acetate, when subjected to strong stretching, fix dyes much better than silk in the ordinary state. Furthermore, stretched rubber does not possess the same chemical properties as rubber at rest; fatty acids melted on water have one side of their molecules that is held by the water, and another neutral side that is repelled by the water (Dzuo); the same is observed in the case of fats, whence the interest that such phenomena present for the process of soap manufacture is understandable. Adsorption is usually accompanied by orientation; the latter will be the more intense, the more the adsorbed body encounters active groups of the adsorbent capable of holding the molecules of that body. Consequently, vibrations or rotation of the molecules of the adsorbent must reduce the probability of fixation of the molecules of the adsorbed substance, as happens, for example, in the case of stretched silk. Frey's work has shown that the dyes impregnating plant fibers and leading to the appearance of double refraction and pleochroism are not in the state of small crystals, but that their molecules undergo orientation under the action of adsorption forces. Absolute measurements of adsorption, carried out by Max Boehm[^42] on substances such as paratoluidine on the surface of their aqueous solutions, show that molecularly oriented chains of the dissolved substance sink into the solution until they are destroyed by thermal vibrations. The surface of a pure liquid is represented as formed from

of a large number of unstable chains directed into the interior of the liquid, which is in complete agreement with the author’s experiments mentioned above. Sols of Fe(OH)₃, V₂O₅, benzopurpurine 4B, etc., also undergo spontaneous regular arrangement, as was shown optically by Zocher ⁶².

It appears, finally, probable that many phenomena of adsorption on catalysts are also accompanied by an orientation of molecules; surface tension is evidently likewise connected with this surface structure, as was shown by Langmuir and also by Lecomte du Noüy ⁴³.

These briefly described examples show that there is a new, highly fruitful path of research, following which it will be possible in the future to achieve theoretical and practical results of great importance.

Study of the polymerization of solid and amorphous organic substances. To conclude this article, let us add a few words about the study, by means of X-rays, of amorphous polymerized organic substances, such as gelatin, rubber, plastics, etc. In the article devoted to these substances, we have already pointed to the results obtained, which are very close to what liquids give. No complete explanation of these results has yet been given, so that one may consider such gels to be identical with liquids that have suddenly become very viscous. In such a case there should exist a state of statistical isotropy, so that the phenomena of X-ray diffraction may lead here to what is observed in the case of a liquid. Clark ⁴⁴ studied rubber and nitrocellulose films as a function of their purity, their variability upon aging, the nature of the solvent, etc.; moreover, he succeeded in following changes in intermolecular distances, calculated by Bragg’s formula. Similar results were obtained with solid films of polymerized China wood oil, with resins, polymerized varnishes, etc.

In connection with this an important question arises: will it not be

can a more or less strong state of polymerization be reflected in any way in the character of the X-ray photographs?

According to Katz[^9], amorphous polymerized organic products possess a molecule formed from several simple molecules, which corresponds to a certain regular repetition of elementary groups. The X-ray photographs obtained often (but not always) consist of two broad rings, which apparently are “amorphous” rings; in some cases the inner ring is very intense, in other cases it is weak or barely perceptible. The outer ring, generally speaking, coincides with the ring of the unpolymerized substance. When the inner ring is absent and the outer ring coincides for the polymerized substance and the liquid substance, the paradoxical conclusion is obtained that these two substances give identical X-ray photographs, despite considerable differences in density, solubility, volatility, etc. Such a phenomenon is found, for example, in isoprene and isoprene rubber, which both have one intrinsic period of 6 Å (Ehrenfest’s formula), and also in bakelites in states $A$, $B$, $C$, as was observed by the author himself.

In the case of solid polymerized substances there exists, evidently, a certain uncertainty, which cannot be said of liquids; namely, it is not known whether there will exist a more or less well-formed crystalline phase capable of giving rise to the inner ring, which Katz calls the “polymerization ring.” At present it is still difficult to judge this possibility, since there can be no certainty in studying a solid polymerized substance by means of X-ray spectra. However, in many cases it could be asserted that in solid colloids a crystalline phase exists in equilibrium with the amorphous one: this was found, for example, in the case of cellulose acetate in powder and various nitrocelluloses, which give crystalline rings[^39], or, to a somewhat lesser degree, in gels, but precisely of these substances.

The possibility of assuming a crystalline structure upon stretching, characteristic of these substances (aceto-

and nitrocellulose), confirms this hypothesis; the same should also be said of the recent discovery of purely crystalline forms of these substances.

However, some polymerized substances sometimes show significant differences from the corresponding unpolymerized bodies. For example, Katz gives the following results, calculated with the aid of Ehrenfest’s formula:

Substance	Value
Liquid styrol	\(a = 5.9\,\text{\AA}\)
polymerized (metastyrol)	\(a_1 = 12.5;\quad a_2 = 5.8\)
Liquid indole	\(a = 6.5\)
polymerized (metaindole)	\(a_1 = 12.45;\quad a_2 = 6.3\)
Liquid coumarone	\(a = 6.4\)
Coumarone resin	\(a_1 = 12.7;\quad a_2 = 6.15\)
Acetaldehyde	\(a = 4.8\)
Paraldehyde	\(a_1 = 7.5;\quad a_2 = 4.6\)
Propionaldehyde	\(a = 5.4\)
Parapropionaldehyde	\(a_1 = 9.1;\quad a_2 = 5.1\)

Consequently, the outer ring approximately coincides with the outer ring of the unpolymerized substance, whereas the inner ring agrees with the mean distance of the polymerized molecules[^45]. Thus, in the case where the polymerized substance is represented by two rings, one of them may be ascribed to those groups which constitute the polymer; as for the other ring, its origin still remains unknown. In future investigations of polymerization it will be necessary to determine whether the “polymerization ring” depends on the nature of the polymerization, and whether this nature of the polymerization coincides with a lowering of the freezing point.

The problem of determining the degree of polymerization in amorphous organic bodies has not yet led to interesting results. Here one may simply point to Smedt’s experiment[^46], based on an arbitrary theory concerning spherical molecules, according to which the position of the rings depends on the degree of polymerization. But we have seen that, except for certain special cases, it would be impossible to ascribe this spherical form to molecules that prove to be elongated. Let us note only that for the present-

of a symmetric molecule, such as the paraldehyde molecule, to which one may assign the formula

\[ \begin{array}{c} \mathrm{H}\backslash\ \ \ \ \ \mathrm{O}\ \ \ \ /\mathrm{CH_3}\\ \ \ \mathrm{C}\ \ \ \ \ \ \ \ \mathrm{C}\\ \mathrm{CH_3}/\ \ \ \ \ \ \ \ \backslash\mathrm{H}\\ \ \ \ \ \mathrm{O}\ \ \ \ \ \ \mathrm{O}\\ \ \ \ \ \ \ \ \ \mathrm{C}\\ \ \ \ \ \ \ /\ \ \backslash\\ \ \ \ \mathrm{H}\ \ \ \mathrm{CH_3} \end{array} \]

in the expression

\[ a = 1.33 \sqrt[3]{\frac{M}{nd}} \]

(where \(a\) denotes the distance calculated by Ehrenfest’s formula, \(M\) the mass of a simple molecule, \(d\) the density, and \(n\) the degree of polymerization), the value \(n = 3\) fits very well, which is in agreement with what is known about the polymerization of acetaldehyde.

In the article on colloidal substances* we saw that attempts to determine the degree of polymerization were made in the region of crystalline organic substances; from this point of view, the works of Ott, Meyer, Staudinger, and Hengstenberg (see that article) apparently give more interesting results than we have in the case of amorphous substances or colloidal gels, where this problem is still far from being solved.

In conclusion, let us note that a whole series of recent works—for example, by Prins \(^{50}\), Krishnamurti \(^{53}\), and Shiba and Watanabe \(^{56}\)—has been devoted to questions of the degree of association in liquids and colloidal substances, based on X-ray analysis. Since they do not remove the uncertainties in this field, we confine ourselves here to merely mentioning them.

* Cf. “Advances in the Physical Sciences,” 11, 595, 1931. Ed.

LITERATURE

1. P. Debye and P. Sherrer, Nachrichten, Göttingen, 16, 16, 1916.
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2. Mac Bain, Nature, p. 554 and 49, 1124.

3. Perrin et Wells, Ann. de Phys., 69, 1921.

4. E. Friedel, C. R. Acad. Sci., 180, 269, 1925.

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6. Clark, Applied X-Rays, p. 176.

7. J.-J. Trillat, Ann. de Phys., mars-avril-septembre 1928; 7, juil.-août 1926.

8. J.-J. Trillat, Revue des colloïdes, mars, avril et sept. 1928; C. R. Acad. Sci., p. 859, 1928.

9. J.-J. Trillat, J. de Phys., p. 32—43, 1929; C. R. Acad. Sci., p. 168, 1928; Congrès de Chimie physique, Paris, octobre 1928.

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11. Mac Bain, Nature, 10, sept. 1927.

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14. Katz, ZS. f. phys. Chem., 125, p. 321.

15. De Smedt, Bull. Chem. Soc. Belg., 10. 366, 1924.

16. J.-J. Trillat, C. R. Acad. Sci., 188, 1246, 1929.

17. Prins, Nature, 15 jun. 1929. X-ray data on the question of intermolecular forces in liquids.

18. Coster et Prins, J. de Phys., mai 1928, p. 153, Study of liquids by the rotating-crystal method.

19. Prins, ZS. f. Phys., 56, H. 9/10, 1929. Diffraction of X-rays in liquids and solutions.

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22. Krishnamurti, The Indian J. of Phys., 2, 492, 1928. X-ray diffraction and its relation to the complexity of molecules in the liquid state.

23. Krishnamurti, The Indian J. of Phys., 2, 355, 1928. The relation between chemical composition and X-ray diffraction in liquids.

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25. Shiba and Watanabé, Inst. of Phys. and Chem. Research, Tokio, 20 march 1929. Diffraction rings in aqueous solutions of electrolytes.

26. Krishnamurti, The Indian J. of Phys. 15 oct., 1928. X-ray diffraction by aqueous solutions of cane sugar, levulose, and glucose.

Ibid., 15 dec. 1928. Diffraction of X-rays in liquid mixtures.

Ibid., 15 Feb. 1929. Diffraction of X-rays in liquids and solutions and the factor of molecular structure.

Ibid., 30 Nov. 1928. On the nature of dextrin, gelatin, and sodium oleate solutions on the basis of X-ray diffraction.

1. Vaidyanathan, Indian J. of Phys., 15 Feb. 1929. The influence of temperature on the X-ray rings of liquids.

2. Stewart, Phys. Rev., 32, 558, 1928. Theory of the diffraction of X-rays in liquids.

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4. Katz, Naturwiss., H. 40, 738, 1928.

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6. Vorlander, Chemische Kristallographie der Flüssigkeiten, Akad. Verlagsgesell., Leipzig, 1924.

Vorlander und Selkё, ZS. f. Phys. Chem., 129, 435, 1927. Orientation of molten crystalline masses and liquid crystals.

1. Debye. ↩