Advances in Understanding the Structure of Silicate Glasses

N. A. Shishakov, Moscow

The chief signs of the truly solid state of a substance are invariability of form and crystalline structure. The most diverse glasses occupy, in this respect, a not entirely definite position. In their external features they very strongly resemble true solids, but at the same time they possess the characteristic peculiarity that in them neither by means of heating curves, nor by means of the microscope, nor even by means of X-ray investigations has it been possible to ascertain crystalline structure with sufficient certainty. This fundamental contradiction was the cause of the difficulties associated with attempts to construct theories of the vitreous state, and also the reason for the existence at the present time of several such theories. The purpose of the present review is, first, to compare these theories with one another here; second, to reveal their positive features, i.e., those features that are not in contradiction with any facts; and, finally, as far as possible to give a simple answer to the question: what, then, are silicate glasses?

Fig. 1

Tammann’s theory regards glass as a supercooled liquid. An essential property of glass, according to Tammann, is the presence in it of greater internal energy than in a crystalline substance of the same composition. This is vividly illustrated by the halt, occurring at the melting point \(T_s\), in the course of the temperature during the heating of a crystalline body or during its cooling (curve \(ABCD\) in Fig. 1). An analogous halt also occurs in the case of substances capable of giving supercooled liquids (curve \(A'B'C'D'\) in Fig. 1). The only difference is that the halt in the temperature course is observed here at a lower tempera-

ture than the melting temperature \(T_s\), in accordance with which the very jump in the energy released upon solidification becomes smaller than in the case of readily crystallizing substances.

In the case of glasses this phenomenon is of still less noticeable character. However, since glasses must be regarded as supercooled liquids, it follows from this that they possess somewhat greater internal energy than the corresponding crystalline substances, i.e., they are in a less stable state. Confirmation of this is seen, first, in the phenomena of devitrification and, second, in the results of experiments comparing the heats of dissolution of quartz glass and crystalline quartz in hydrofluoric acid, which lead to the conclusion that the internal energy in the former case is considerably greater than in the latter. However, as will be shown below, such a conclusion is not confirmed by other methods.

Another consequence of Tammann’s theory is that the internal structure, and together with it the external physical properties, must change with temperature in a continuous manner, without any jumps. Thus, for example, the change in viscosity \(\eta\) upon cooling of a liquid, even taking into account the phenomena of molecular association, must proceed according to the following law:

\[ \eta = A_0 e^{\frac{B'}{T-T_0}}, \tag{1} \]

i.e., proceed continuously down to the temperature \(T_0\), at which the viscosity becomes equal to infinity. However, these assertions too are in contradiction with the observed facts.

When temperature changes, by no means all the physical properties of glasses undergo a smooth change. In particular, it may be considered that various glasses, at least quartz glass and complex silicate glasses, are characterized by a certain comparatively narrow temperature region which separates the plastic and brittle states of the glass. True, plasticity and brittleness here can be spoken of only in a relative sense, for a glass that exhibits brittleness at a given temperature under rapid external actions becomes plastic at the same temperature under sufficiently slow actions. But at those rates of cooling and of external action with which one has to deal in practice, this transition displays a clearly expressed jump-like character. This transition takes place at the temperature \(T_g\), lying between the temperature \(T_0\) mentioned above and the melting temperature \(T_s\). Above the temperature \(T_g\), the change in viscosity follows Tammann’s law, applicable to any liquid, including a supercooled one; below it, it follows another law (Fig. 2).

There are other facts confirming the existence of such a transition point in glasses. Tammann himself found that near the temperature \(T_g\), at which brittleness appears in a solidifying glass, there also occurs a sharp jump in the temperature coefficient of specific volume, the coefficient of thermal conductivity, specific heat, and other temperature coefficients. Analogous

jumps were also observed by other authors. To explain this abrupt change of properties, Tammann put forward the hypothesis that the transformation of a viscous liquid into glass is caused by the cessation of the rotational motion of molecules as a result of their close approach and an increase in the forces of interaction between them.

In the case of quartz glass the transformation point lies near 900°, as is indicated by the presence at this temperature of a maximum on the curve for the torsional modulus and by a sharp change in the logarithmic decrement of damping. The empirical formula of Tammann and Hesse

$$ T_g = T_s \left(1 - \frac{C}{\sqrt[3]{M}}\right), \tag{2} $$

where \(M\) is the molecular weight and \(C\) is a constant, gives a temperature \(T_g\) from 640 to 900°. However, as will be seen below, at present there are grounds for doubting the correctness of the molecular weight of quartz glass, \(M = 60.3\), adopted by these authors, which also indicates the need to revise Tammann’s views.

Fig. 2. Double logarithmic dependence of the viscosity of three different technical glasses on temperature (Le Chatelier)

Together with Tammann, other investigators also refused to recognize glass as a true solid substance. It was observed, however, that since the course of the energy with temperature below the point \(T_g\) is different from that above it, glass cannot be assigned to the liquid state of matter either. In accordance with this, a number of investigators (Tool, Turner, MacBain, Parks, and others) proposed that glass be regarded as a colloidal state of matter, in which the role of micelles is played by molecular aggregates (or, instead of them, there is a rigid spongy skeleton of silica), while the role of the intermicellar liquid is played by a substance with non-coarsened particles. The similarity of glasses to gels is confirmed, for example, by the fact that when the former are treated with acids they lose their bases and often leave a skeleton of silicic acid of porcelain-like appearance.

In developing these views, Berger suggested that the main cause of the appearance of the jump at the temperature \(T_g\) is a rapidly occurring process of molecular aggregation. This is indicated by the similarity in the dependence of viscosity on temperature both in the case of supercooled liquids and in the case of associated liquids. In both cases the viscosity is expressed by one and the same Tammann formula. Therefore it may be considered that in both cases, upon cooling, the same aggregation of molecules takes place. This is also indicated by the dependence of the properties of glass on its thermal history. Strongly

quenched glass possesses properties indicative of its instability, which may be regarded as the result of an incomplete formation of molecular complexes owing to rapid cooling.

If such glass is subjected to a certain annealing, then, depending on the degree of the latter, a gradual formation of molecular complexes should take place, and along with this a definite change in the properties of the glass, which is in fact observed. By means of suitable annealing it would even be possible to attain a constant equilibrium in the glass. However, the time required for this proves, generally speaking, to be too great, and for this reason such equilibrium is not observed in practice. Thus, for example, if according to Berger 16 hours are required to reach an equilibrium state at the temperature \(T_g\), then already at a temperature \(100^\circ\) below \(T_g\) about 40 days would be required, and at a temperature \(300^\circ\) below \(T_g\)—about 200 years. In the case of silicate glasses the temperature \(T_g\) is approximately \(500^\circ\), so that observing changes in the properties of the glass at room temperature is, of course, practically impossible.

In opposition to Tammann’s views, Berger, on the basis of calculations of the differences in glass–crystal energies from the latent heats of melting and the temperatures \(T_s\) and \(T_g\), comes to the assertion that at temperatures below \(T_g\) the glassy state possesses no greater reserve of internal energy than the corresponding crystalline state. This means that the glassy state is just as stable as the crystalline one. This conclusion is confirmed, according to Berger, by the fact that the process of devitrification at temperatures below \(T_g\) has never been observed, except perhaps only in cases that were caused by chemical influences and that have no bearing on the present considerations.

The question of the nature of molecular aggregates in glasses remains open, but Berger nevertheless considers it possible that these aggregates are precisely those crystallites whose existence is hinted at by X-ray analysis.

At present, perhaps the greatest success has fallen to the theory of Zachariasen and Warren, which regards glass as a continuous atomic network devoid of symmetry and periodicity. These views are based on the similarity of the mechanical properties of glasses and the corresponding crystalline forms. The differences between glass and crystal, which X-rays make it possible to establish, consist in the fact that, with the atomic bonds and coordination being the same, the polyhedra identical in both cases are arranged in the crystal in a definite order, and in the glass—in a disordered manner.

In the case of silicate glasses such a three-dimensional network is built of \(\mathrm{SiO}_4\) tetrahedra; in the case of borosilicate glasses, \(\mathrm{BO}_3\) triangles must also take part in the structure of the network. Not all bonds in such a network are saturated; therefore some of them are accounted for by cations, which remain in the free interstices between the basic polyhedra and are distributed in the glass in a statistical manner.

X-ray structural analysis, based on the application of the Cernike and Prins method and making it possible to determine the mean radial distribution of atoms around a given atom, fully confirms Zachariasen’s hypothesis1. However, the application of this method is not connected

Fig. 3

a — SiO₂ glass, b — cristobalite, c — silica gel

with the requirement that the object under study necessarily be amorphous; the method is equally applicable also to a polycrystalline substance. But precisely for this reason it, as Warren himself admits, “does not give an answer to the specific question—whether the given substance can be considered crystalline or not.”

Warren approaches the solution of this question by analyzing microphotograms from glassy silica, cristobalite, and dry silica gel (Fig. 3). If one assumes, as some authors have done (see below), that SiO₂ glass consists of crystals of cri-

if they are cristobalite crystallites, then their average dimensions must be

\[ L=\frac{0.89\lambda}{B\cos\vartheta} =\frac{0.89\cdot 1.54}{0.181\cdot 0.98}=7.7\text{ Å}. \tag{3} \]

(here \(B\) is the width of the maximum, expressed in radians, at half the maximum intensity).

Since the edge length of the elementary crystal cell of cristobalite is also \(7\) Å, in the present case the term “crystals” is not appropriate, because there is no regular repetition of the cell in space. However, apart from these formal considerations, the very character of the microphotometric curve for \(\mathrm{SiO_2}\) glass also contradicts the crystallite theory; it differs sharply in its initial part from the curve for silica gel. The principal maximum in the latter is the same as in the glass, but in addition to this there is strong scattering at small angles, which is not observed in the glass. According to Warren, this strong scattering is a consequence of the presence of inhomogeneities, so that silica gel may be imagined as consisting of discrete particles \((10\text{–}100\text{ Å})\), with ruptures and voids between them. Conversely, the absence of scattering at small angles in \(\mathrm{SiO_2}\) glass shows that it is a continuous medium in which there are no discrete particles or breaks in the bonds between them.

To explain the high viscosity of glass melts, Zachariasen and Warren assume that even in the melt there is a strong tendency for each silicon atom to retain four oxygen atoms around itself, and hence also a tendency for each oxygen atom to serve as a linking element between two silicon atoms. The easy bendability of these bonds is, according to Warren, the cause of the formation of an aperiodic asymmetric lattice, which, however, is just as stable as the lattice of a crystal.

Substantial additions to the Zachariasen–Warren theory were made by Huggins\(^2\). In considering the ability of substances to form glasses, he dwells in great detail on the characteristics of the bonds between ions and radicals—for example, \(\mathrm{K^+}\) and \(\mathrm{NO_3^-}\)—at the melting temperature, and between the atoms within the radicals themselves. The latter bonds are stronger than ionic bonds, as a result of which these radicals continue to exist also in the molten substance. The salt \(\mathrm{KNO_3}\) does not form a glass because of the small size and great mobility of the ions.

The situation is different for compounds of boric acid and for silicates. Boron has a strong tendency to coordinate three oxygen atoms, and this tendency can be realized only in the case when the oxygen triangles have common corners. In this way, straight inorganic chains are formed from oxygen triangles with boron atoms at their centers.

These chains, having the composition \(n\mathrm{BO_2}\), are arranged parallel to one another, and they, of course, may be regarded as infinite radicals \(n(\mathrm{BO_2})^{-2}\), in which the atoms are held together by comparatively strong bonds. The crystalline lattice

of calcium metaborate as a whole and is constructed from such large radicals and \(Ca^{+2}\) ions. Upon melting calcium metaborate, the bonds between the \(Ca^{+2}\) ions and the chain radicals are destroyed, but the chains still continue to exist in the liquid owing to the presence in them of stronger bonds. It is considered probable that the former length of the chains is not preserved—they break up into smaller parts, and the latter may even prove to be deformed to one degree or another. It is precisely the existence of such fragments of chains, whose length, of course, depends on the temperature, that accounts for the viscosity of the given liquid and, together with this, its capacity for glass formation, since on solidification the correct orientation of the chains is greatly hindered. The oxides of certain other metalloids, including silica, should behave in an analogous manner.

The nature of the radicals in the case of silica is determined, according to Bragg, by the ratio of the numbers of atoms \(Si:O\) in the substance. At \(Si:O = 0.25\) we have isolated \(SiO_4\) tetrahedra, characteristic of orthosilicates. At a lower relative oxygen content, tetrahedral coordination is preserved, but already at the expense of the formation of complex radicals with free valences. Thus, for example, groups \((Si_2O_7)^{-6}\) made up of two tetrahedra, or rings \((SiO_3)_n^{-2n}\), preserve this coordination. If we have the ratio \(Si:O = 1:3\), then infinite chains of composition \(n(SiO_3)^{-2}\), characteristic of the pyroxene group, may be formed. Similar chains, having only a more complex structure and the composition \(n(Si_4O_{11})^{-6}\), characterize the structure of amphiboles. At a still higher ratio \(Si:O\), namely 0.4, the number of oxygen atoms capable of entering into tetrahedral coordination is so small that radicals with unlimited extension in two dimensions are formed (see the scheme, Fig. 4). These two-dimensional radicals, having the composition \(n(Si_2O_5)^{-2}\), occur in micas, in clay minerals, in talcs, and similar layered lattices.

Depending on the starting substance, either one or another whole radical, or their fragments, must exist in the melt. The character of the connection of the tetrahedra in these radicals probably remains the same as in the original radicals, but, owing to the disruption of ionic bonds that have a directional character, the radicals in the melt become more or less deformed.

Hägg does not exclude the possibility of the existence in the melt of three-dimensional, of course distorted, radicals, which would not contradict Zachariasen’s theory; but he also notes the possibility that these radicals may be close to layers \(n(Si_2O_5)^{-2}\). This is indicated by the fact that in technical glasses free of boron the ratio \(Si:O\), as was noted by Zachariasen himself, is 0.40, which also corresponds to radicals of the two-dimensional type. Zachariasen (1935) agreed with these additions by Hägg.

If silicate glasses are characterized by the presence in them of two-dimensional radicals of composition \(Si_2O_5\), then this could be directly demonstrated by means of X-ray and electron-diffraction investigations. Indeed, according to Laue’s theory\(^3\), an object consisting of randomly arranged two-dimensional crystal-

of crystals, should give a diffraction pattern differing very little from the diffraction pattern given by three-dimensional crystals properly composed of such two-dimensional formations. In the first case only \((hk0)\)-reflections will arise, very sharp on the side of the central beam and, depending on the thickness of the crystal sheets, more or less blurred toward larger angles.

Since glass in essence represents precisely the case envisaged by Laue’s theory, and since the structure of two-dimensional silicate layers (Fig. 4) and the corresponding interatomic distances are known, one can predict in advance what kind of diffraction effect should be obtained in the case most closely approximating the ideal—for example, in the case of pure quartz glass taken in the form of an extremely fine powder, so as to reduce as much as possible the deformation of the crystal sheets. The structure of the two-dimensional crystals \(\mathrm{Si_2O_5}\) is hexagonal.

Fig. 4

The lattice constant \(a\) may be determined in advance from the mean value of the same constant \(a\) for micas and clays, which varies within the limits from 5.12 to 5.26 Å. Let us take it equal to \(a = 5.16\) Å. Since, according to Laue’s theory, all possible \((hk0)\)-reflections^4 must arise, then from the quadratic form of Bragg’s equation for a hexagonal lattice one can compile their table in advance (Table 1).

Table 1

Roman numerals I denote the strongest reflections, and numerals II the next in intensity. Reflections not marked by numerals are comparatively still weaker.

However, a considerable part of the two-dimensional crystals must be in union with one another, so that embryos of three-dimensional crystals, probably of the cristobalite or tridymite type, must exist in the glass. That such modifications of silica indeed consist of stacks composed of two-dimensional layers was shown quite clearly by X-ray diffraction by Nieuwenkamp,^5 which once again confirms the ideas about the high

strength of the bonds within a given two-dimensional layer and the lesser strength of the bonds between individual layers.

In one way or another, this ability of two-dimensional crystals of silica to merge, probably realized to a certain degree at least in SiO₂ glass, may well prove to be the cause of the appearance, in diffraction patterns, also of \((hkl)\)-reflections from three-dimensional formations. Therefore, to the list of \((hk0)\)-reflections given above one should first of all add the most intense in all cases \((111)\)-reflection of cristobalite, corresponding to the periodicity \(d = 4.1\) Å. Secondary in comparison with it, any reflections of cristobalite and of all other modifications of silica cannot play an essential role here.

Fig. 5

All these considerations should facilitate the examination of the results of X-ray and electron-diffraction analyses listed below. True, these results are still very scanty; the reflections are for the most part strongly blurred and sometimes overlap one another, but their systematic consideration nevertheless indicates that there is no contradiction between these facts and what theory predicts.

Fig. 6

Curve 1—quartz, 2—cristobalite, 3—fused quartz

The first indications of the crystalline structure of glasses are found in Lebedev⁶, who drew attention to abrupt changes in certain physical constants of glasses at definite temperatures and connected them with structural transformations. An example of such abrupt changes is provided by Fig. 5. The presence of a discontinuity at a temperature of \(500—600^\circ\) led one to think that the glass contains crystalline quartz, for which the transformation point of the \(\alpha\)- into the \(\beta\)-modification, associated with a sharp change in many properties, is precisely \(565^\circ\) (Fig. 6).

Apparently, the first to undertake X-ray investigations was Kiropulos (1917). X-ray photography of powder and of a rod of fused quartz, carried out with platinum rays, made it possible

to obtain only one blurred reflection. In a similar way, no crystals were found in glasses either by Hall and Meeter (1922) at the Bragg laboratory, or by Selyakov, Strutinskii, and Krasnikov (1925), who used the characteristic radiation of a copper anticathode for their investigations, or by Lazarev, Trapeznikov, and Simanov (1927), who sought a parallelism between birefringence and crystallinity and for this purpose used quenched borax glasses.

More systematic X-ray investigations of the structure of glasses were carried out by Wyckoff and Morey[^7]. For both quartz and complex silicate glasses they for the most part obtained diffuse reflections, similar to those obtained in the case of liquids. However, in addition to diffuse bands, a whole series of glasses also showed sharp reflections, which should have been characteristic only of crystals. They were unable to establish any regularities, but the considerable sharpness of the lines shows that glasses which are completely pure and, moreover, have been checked under the microscope for the absence of weathering—that is, glasses completely homogeneous in appearance—must contain considerable quantities of crystals. In the SiO₂ corner of the ternary diagram, both broad bands and lines are usually observed, which must now be explained by the comparatively more favorable conditions for studying glass close in composition to pure silica. Unfortunately, the authors give no numerical characteristics of these reflections, indicating only that different glasses often give the same reflections.

Apparently, we owe the first quantitative data to Clark, Parmalee, and Badger[^8]. They succeeded in obtaining from quartz glass two quite distinct reflections with identity periods \(d\) of about 7.15 and 2.5 Å. At first they regarded this fact as confirmation of Sosman’s hypothesis, according to which quartz glass consists of threadlike molecular aggregates of silica, the first periodicity corresponding to the length of an individual chain molecule and the second to its thickness. Later Clark[^8] abandoned this explanation.

Further very careful X-ray investigations of various glasses were carried out by Randall, Rooksby, and Cooper[^9]. From quartz glass they obtained several other, though likewise blurred, reflections, namely with periodicities of 4.3 and 1.5 Å. They explained these reflections by the presence in the glass of true crystals, so small, however (\(10^{-7}\) cm), that sharp interference maxima could not be obtained. Their supposition found confirmation in the fact that theoretically calculated X-ray diagrams of deliberately finely divided crystallites of cristobalite proved to be more or less similar to X-ray diagrams from quartz glass. An analogous similarity was obtained for borax and for certain complex silicate glasses.

According to this work, there is no sharp boundary between the glassy and crystalline states. Weathering represents a process of gradual growth of crystallites. Complex glasses consist of crystallites of varied composition.

However, quite apart from the fact that Randall’s theory generally does not provide an explanation of many of the most important properties of glasses, its very weak point is the discrepancy between the calculated and the observed density of quartz glass. If the latter consisted of cristobalite crystals with normal lattice parameters, then the periodicity corresponding to the strongest observed reflection would be approximately \(4.1\ \text{Å}\) (cf. p. 414). Randall, however, obtained the value \(4.33\ \text{Å}\). From this followed the conclusion that the cristobalite crystals in the glass, as compared with normal crystals, have a 17% larger specific volume, i.e. that their density is \(1.92\) instead of the normal density of cristobalite, \(2.35\). However, quartz glass itself has a density of \(2.20\). To explain this excessively large difference Randall had to make the additional assumption that, in addition to cristobalite crystals, quartz crystals are also present in the glass, whose density, as is known, is \(2.65\). Subsequently Randall \({}^{9}\) abandoned his theory and accepted the Zachariasen–Warren theory.

The next attempt to investigate the structure of quartz glass by the X-ray method was undertaken by Levin and Ott \({}^{10}\). In addition to one distinct reflection, \(4.27\ \text{Å}\), they were also able to observe another, very diffuse maximum, consisting of three narrower maxima, to which the periodicities \(1.61\), \(1.24\), and \(1.12\ \text{Å}\) correspond. These authors drew no conclusions about the nature of quartz glass.

One of the most recent X-ray studies is the work carried out at the State Optical Institute by Valenkov and Porai-Koshits \({}^{11}\).

The results of their investigations, performed with very small X-ray cameras, may be summarized as follows. Both quartz glass prepared from cristobalite or from quartz, and silica gel, give four or five reflections located approximately at those positions where the groups of strong reflections of a mixture of crystals of low-temperature cristobalite and tridymite are found. Of these, four maxima agree very well with the maxima obtained for silica gel by Levin and Ott \({}^{10}\). The sharpness of the pattern increases in passing from glass obtained from quartz to glass obtained from cristobalite, and then to silica gel. The structure of the latter is therefore ascribed the most ordered character. These experiments show that the structure of the glass depends on the method of preparation, which contradicts the Zachariasen theory of a noncrystalline network. In Table 2 are compared the results obtained by these and by other authors.

The authors draw the following conclusions: fused quartz consists chiefly of crystals of low-temperature cristobalite, perhaps with an admixture of high-temperature cristobalite and tridymite; silica gel, however, consists of crystals of low-temperature cristobalite. In an analogous manner, the authors detected the presence of crystals of sodium metasilicate and cristobalite in binary glasses.

In connection with the Zachariasen—Warren theory it is of interest to consider also the work of Hartleif1, who carried out his

Table 2

X-ray data for quartz glass and silicalite

X-ray measurements by the direct ionization method. He, like Warren, used monochromatic copper \(K\alpha\)-radiation. Table 3 presents the results of his observations on the diffraction of X-rays by quartz glass.

Table 3

Hartleif’s X-ray data1

According to Hartleif, the structure of the glasses investigated cannot be explained by means of the Zachariasen theory for the following reasons: 1) the existence of a new maximum \(d = 8,33\) indicates a higher degree of ordering than the theory predicts; 2) in potassium glasses there is no uniform distribution of potassium ions, but there exist at least two structurally different parts, one of which corresponds to quartz glass, while the other may be attributed to the compound \(\mathrm{SiO_2}\) with \(\mathrm{K_2O}\), which, however, cannot be regarded as “crystallites,” as Lebedev and others consider it. Therefore Hartleif regards both the crystallite theory and the theory of the noncrystalline lattice as insufficiently substantiated and too narrow.

The chief shortcoming of the X-ray method is the excessively high penetrability of the rays through so light a substance as glass. Therefore, from any powder, and still more from a solid piece of glass, there must arise diffraction of X-rays both from surface undeformed, two-dimensional or lamellar crystals, and from those crystals which are located in the very volume of the glass and which, probably, are sufficiently compressed and therefore deformed. As a result, diffraction patterns from both the surface and the volume must be superposed on one another, the former, of course, being relatively very weak. This, apparently, is precisely what may explain why, as a rule, very blurred reflections, often merging with one another, are obtained on radiographs, and why analysis is difficult. From this point of view one might hope that the less penetrating electron rays would give sharper patterns.

The first electronographic work apparently belongs to Dixit[^13], who found that polished glass gives, on reflection of electrons, a diffuse ring corresponding to the periodicity \(d = 1.5 \ \text{Å}\), and that inside this ring a spot is observed corresponding to the periodicity \(1.55 \ \text{Å}\). He explains this meager pattern by the fact that polishing destroys the structure and that therefore the only periodicity that can be observed remains the constant distance between silicon and oxygen atoms, \(\mathrm{Si}—\mathrm{O} = 1.55 \ \text{Å}\).

More complete results were obtained by Maxwell and Mosley[^14], who transmitted fast electrons through thin films of quartz glass, obtained by rapidly blowing a small globule of the molten mass and subsequently etching the resulting film with hydrofluoric acid. They consider the essential feature in the results of their experiments to be the presence, on the electronograms, of the following reflections:

\[ \begin{aligned} \text{Distinct ring} \ . . . . . . . \quad & d = 4.17 \ \text{Å} \\ \text{Diffuse ring} \ . . . . . . . \quad & d = 2.5 \ \text{»} \\ \text{Strongly diffuse ring} \ . . . \quad & d = 1.25 \ \text{»} \end{aligned} \]

In their opinion, the presence of these rings testifies to the amorphous nature of the state of quartz glass in the sense in which Warren understands it. However, among these diffuse rings Maxwell and Mosley also observed indications of thin rings and arcs, of which they mention only the most distinct:

\[ \begin{aligned} \text{Strong arc} \ . . . . . . . \quad & d = 4.2 \ \text{Å} \\ \text{» \quad »} \ . . . . . . . \quad & d = 3.7 \ \text{»} \\ \text{Weak arc} \ . . . . . . . \quad & d = 2.52 \ \text{»} \end{aligned} \]

Unfortunately, no numerical characteristics of the other reflections are given in the communication of Maxwell and Mosley. The authors ascribe the presence of these thin arcs on the electronograms, along with the diffuse rings, to the presence of some impurities, perhaps fluorosilicates, which could have formed under the action of hydrofluoric acid.

However, one can hardly agree with such a reference to accidental impurities, since approximately the same patterns were observed by Hiroshi Kamogawa^15 also in electron diffraction of oxidized thin films of silicon, obtained by distilling it onto the polished surface of a sodium chloride crystal and isolated by dissolving the latter in water. Oxidized in air at room temperature, these films gave a pattern consisting of three diffuse rings, corresponding to the periodicities:

Kamogawa attributes these diffuse rings to the amorphous character of such a silica film.

When such films are heated to various temperatures from 410 to 800°, the diffuse rings gradually become sharper and sharper. At a temperature of 450° the pattern becomes such that it can no longer be explained on the basis of Warren’s ideas, and therefore Kamogawa is compelled to admit that it comes from true crystals^1).

Of the other reflections observed by Kamogawa, only the characteristic ones may be mentioned here: 2.50; 1.73; 1.47 and 1.24.

A careful consideration of the entire body of the data listed, taking into account, of course, the low accuracy of the X-ray determinations, leads to the conclusion that the strongest reflections from glasses, 4.4; 2.5; 1.65; 1.47, etc., coincide strikingly with the strongest $(hk0)$ reflections from the layered lattices of micas and clay minerals, and consequently also from ideal two-dimensional crystals of silica.

As was indicated above, these two-dimensional crystals may be quite free in the melt, adhere to one another during cooling of the glass, and sometimes give rise to three-dimensional lattices; hence there also arises the appearance of additional $(hkl)$ reflections, 4.1; 3.6; 2.0, etc. The 7–8 Å reflections observed by Clark^8 and Hartleif^12 may quite well be assigned to the planes $(001)$, especially since approximately such values are invariably observed also in the layered lattices of many clay minerals, where the indices $(001)$ are ascribed to them. The remaining doubtful reflections are difficult to analyze because of their slight intensities and their accidental occurrence with different investigators. Nevertheless, it may still be insisted that, despite the sometimes observed

^1) Almost simultaneously with Maxwell and Mozzi, and several years before Kamogawa, the author of this review succeeded in obtaining very good electron diffraction patterns from quartz glass^16. Among these three groups of data there proves to be agreement within the limits of experimental error of 10%, which was also noted by Kamogawa. The exception, the 3.85 Å reflection observed by Kamogawa, is explained, without any exaggeration of the circumstances, by the fact that, unlike the others, it was diffuse, so that it may be regarded as a doublet of the 4.1 and 3.6 Å reflections.

significant differences in the numerical characteristics of the reflections, the contradictions can in no way be attributed to shortcomings of the theoretical premises; instead, they are simply explained by the extremely low accuracy, in this case, of the X-ray measurements (cf., for example, the reflections 2.49 and 2.70 in Table 2; cf. also the intensities of the doubtful reflections, etc.).

Thus, the basis of the structure of silicate glasses must be considered to be two-dimensional crystals of composition $\mathrm{Si_2O_5}$. The reason for the very limited successes of X-ray analysis is the incorrect piling-up of these two-dimensional crystals, which is especially pronounced in the case of complex glasses, where metallic ions also begin to play a role in the formation of the structure; the arrangement of these ions, of course, cannot be made correct under the conditions in which the glass is produced. However, in both cases one may expect that the three-dimensional aggregates growing in this way still retain a plate-like form. Confirmation of this point of view can be found in the literature on optical analysis.

In Matossi’s work[^17] on the reflection of infrared spectra from glasses, arguments are given in favor of the existence, in all silicate glasses, of $\mathrm{SiO_4}$ groups. It is indicated, however, that these groups cannot form an aperiodic lattice, but are part of crystals. In particular, according to Matossi, certain crystals are also found in quartz glass. These considerations are also confirmed by the results of investigations of Raman spectra obtained by Koymcelis[^18] and Landenberg[^19]. The latter authors even go so far as to assert the planar structure of these crystals, which is also in agreement with the conclusions drawn above.

Thus, the question of crystals of silicate glasses and of the paths of their aggregation may be considered, in the main, satisfactorily resolved. However, all this constitutes only part of the theoretical information that must be available in order to proceed to the practical use of the results. Practice deals with real bodies, among which glass must also be counted. Up to this point the structural features of glass have been considered as those of an ideal body. In accordance with this, attention must be paid to one most important feature of silicate glasses, which compels us not to regard them as ideal bodies and requires careful consideration. This is the sorption capacity of glasses.

It is well known that all silicate glasses are capable of absorbing, releasing, and even under very severe treatment continuing to retain within themselves enormous quantities of vapors and gases. When referring to the fate of these occluded gases, one usually confines oneself to the use of the term “solubility of gases in glass”; no explanation of the mechanism of this solubility is usually given anywhere, although without it much remains unclear.

At present the assertion seems quite appropriate that there is no true solubility of gases in glasses connected with the penetration of gas molecules through the interatomic spaces in glasses. On the contrary, there are grounds for thinking that this

the penetration of molecules proceeds through very spacious channels, whose existence therefore must in one way or another be explained.

For this one may start from a consideration of the origin of glass. It is believed that the various vapors and gases contained in the batch and arising from it during glass melting do not have time to escape from the melt because of its high viscosity and remain walled up in the glass even after its solidification. A further conception of the fate of these occluded gases can be obtained if one admits the existence of two-dimensional silicate radicals in the melt. Upon solidification of the glass these radicals should rather coalesce directly with one another than through the mediation of gaseous inclusions that happen to lie between them. As in cases of true crystallization, upon solidification of glass we should expect that foreign inclusions will be squeezed out by the three-dimensional formations that arise. However, since the number of gaseous inclusions in glass is enormous and since their distribution is apparently uniform, one may expect that the final stage in the formation of the solid vitreous state will be the appearance of three-dimensional (not isodimensional) aggregates, approximately equal in size and free of gaseous inclusions, surrounded in one way or another by the expelled gas molecules. This also means that glass is a real solid body, devoid of homogeneity, having a granular structure and containing gaseous inclusions along the grain boundaries. When glasses are heated, the gas content in them decreases, but healing of the gaps between the grains apparently nevertheless does not occur, owing to the enormous viscosity of the solidified glass; so that after degassing these gaps are preserved, although they become more or less emptied, i.e. the glass continues to retain its granular structure. Numerous facts point to the existence in glasses of cracks, and moreover cracks having a more or less regular distribution.

The experimentally observed mechanical strength of glasses is many times lower than the theoretical strength that can be calculated on the basis of various physical constants. If, for this calculation, one uses, for example, the total heat of vaporization, then the theoretical value of the internal cohesion found from it shows that the ratio of the experimental strength to the theoretical is only about 1:20. That the excessively small value of the technical strength is determined by defects in the structure of glass was shown by Griffith’s experiments (1920), in which, by rapid drawing at high temperatures, he succeeded in obtaining thin quartz filaments exhibiting a strength very close to its theoretical value. Incidentally, quartz filaments of high strength had been obtained earlier by Boys (1889).

Griffith explains the comparatively small actual strength of glass in tensile tests by the fact that fine cracks are always present in glass, distributed throughout the whole volume of the glass.

In contrast to Griffith, Joffe (1929) showed that the anomalous mechanical strength of both amorphous and crystalline solid bodies could be explained by assuming the existence of surface cracks. This is confirmed by a whole series of experiments carried out at the Leningrad Physico-Technical Institute on rock salt, bismuth single crystals, and other substances. It is considered that the small strength, as compared with the theoretical strength, is due to the combined weakening action of surface cracks, which are always present on specimens, and secondary defects arising on the surface in the course of plastic deformation. The elimination of both kinds of defects by dissolving the surface of rock-salt specimens in water during the deformation itself causes a considerable increase in strength.

In an analogous manner, Zhurkov ²⁰ showed at the same institute that the removal of surface defects in glass threads—for example, by etching them with hydrofluoric acid—also leads to a considerable increase in strength. However, Zhurkov’s experiments with deep etching forced him to recognize that the inhomogeneities leading to premature fracture of glasses are distributed throughout the entire volume of the glass, although the surface inhomogeneities nevertheless remain the most dangerous.

To explain the superstrength of thin quartz threads, Griffith assumed that in glass there exist molecular complexes having the form of chain-like or layered formations, which are oriented along the thread when it is drawn, and this entails an increase in strength. As can be seen, this conception of the layered structure of the complexes is in agreement with the conclusions from ideas about crystalline structure.

In addition to the phenomenon of increasing strength in thin threads, confirmation of the idea of the orientation of complexes along the direction of drawing may be provided by the occurrence of double refraction in strongly drawn threads, observed by Rayleigh ²¹, and also by the breaking of glass tubes under pressure into long pieces whose surface is densely covered with deep longitudinal cracks (Aleksandrov, Zhurkov, Yanushevsky ²²).

The further development of the view of glasses as inhomogeneous bodies was undertaken in the works of Smekal ²³, who, together with his collaborators (Möller, Apelt, Tirbach, Mangler, Wirz, Mengelkoh, Eichler), carried out for this purpose, at the Institute of Theoretical Physics at the University of Halle over a number of years, detailed investigations of the phenomena of glass fracture. Smekal proceeds from a comparison of the phenomenon of brittle fracture with the phenomena of fracture of plastic material occurring as a result of the development of fatigue in certain machine parts subjected to stresses alternating in direction.

Among the many works devoted to the study of plastic fracture, one should note here the review paper by Thum and Oschatz ²⁴. It has been established that fracture from fatigue begins in one of those defects, invisible to the eye, which in large numbers are found

are seen on the surface of the metal. The fracture then propagates inward into the metal and leads to the formation of a flat surface, situated, generally speaking, perpendicular to the direction of greatest stress. The dullness of the fracture surface that is formed depends on the grain size of the material, but the surface itself, irrespective of its size, is almost always bounded by a certain arc (Fig. 7, a). Beyond this surface there is a fracture giving an uneven surface with grooves arranged radially with respect to the point F.

If glass had a homogeneous structure, the character of the fracture would be quite different. It turns out, however, that in this case there is the same kind of fracture as

Fig. 7. Schematic representations of the fracture surface

a — on the right, a mirror, which begins to form at the flaw F; on the left, a grooved surface; b — torn pieces of the rod: 1 and 2 are joined along the mirror; at the end of the mirror the fracture surface bifurcates and a wedge-shaped piece 3 springs away.

and the fracture due to fatigue of a metal. Here too the fracture begins from some flaw F on the surface of the rod, continues further into the glass, and in the end gives the same surface bounded by an arc, and a field with radial grooves. The surface corresponding to the first stage of the fracture, in the case of glass, is optically smooth, i.e. it constitutes a mirror, and always lies perpendicular to the direction of tension. The only difference from the fatigue fracture of a polycrystalline material consists merely in the fact that beyond the boundary of the mirror, in brittle fracture, two grooved surfaces arise, as a result of which a wedge-shaped piece 3 springs away from the glass upon fracture (Fig. 7, b). This similarity between fatigue-fracture surfaces and brittle-fracture surfaces led to the supposition that the mechanism of fracture is the same in both cases.

The theory of brittle fracture was developed by Griffith (1920). It is based on an energy conception of an elliptical crack expanding during fracture in an ideal homogeneous body, the origin of which is some macroscopic flaw. According to Griffith, the formation of the fracture surface proceeds at the speed of sound; the fracture surface itself must entirely take on the appearance of a mirror.

Numerous experiments by Smekal and his co-workers showed that the mirror in reality has limited dimensions, and that the surface with a mirror indicated in Figs. 7, 8a, and 8b is obtained in glass specimens also without macroscopic flaws. It was further shown that the fracture proceeds by no means at the speed of sound, since both the size of the mirror and the fracture strength itself, as it turns out, depend on the rate of increase of the load even in slowly conducted experiments. And indeed, recently Shardin and Struth²⁵, who studied the cinematography of the destruction of glass under punc-

of his bullets found that the speed of propagation of a crack in glass is approximately \(\frac{1}{3}\) of the speed of sound in glass \((5000\ \text{m/sec})\). In quartz glass this speed of crack formation is equal to \(2200\ \text{m/sec}\). Smekal explains this fact by saying that the speed of propagation of a crack is not the speed of propagation of macroscopic elastic disturbances in a quasi-homogeneous material, but is the speed of propagation of submicroscopic processes through a truly inhomogeneous medium.

Since, in contrast to the action of a macroscopic flaw considered by Griffith, the fracture of glass, according to Smekal, begins

Fig. 8a

Fig. 8b

at a point containing merely a flaw smaller in magnitude, namely a submicroscopic flaw, it follows from this that invisible flaws of this kind exist in every glass and that the presence in it of such inhomogeneities determines the character of fracture considered above. Such a view does not contradict the views of Griffith and Ioffe, for in the presence of a macroscopic flaw, for example on the surface of a body, it will simply exert an additional strong action, with respect to which the body will behave as a quasi-homogeneous body, i.e., in the way predicted by Griffith’s theory.

A mirror-like fracture surface in glass, when flaws are present in it, could arise only in the event that these flaws were sufficiently small and, moreover, were distributed there in such enormous quantity that a sufficient number of them would occur in any direction. According to Smekal, these flaws have the form of very thin lenses, and they are oriented in space in a completely random manner.

The course of fracture of glass may be imagined in the following way. If the glass is taken in the form of a round rod and if the direction of the macroscopic stress in it goes along the axis, then the contour of such

the lens will be under a somewhat higher stress. According to Griffith’s theory, the greatest stress must occur at those cracks which are situated perpendicular to the axis of the rod or, generally speaking, to the direction of the macroscopic stress.

When the rod as a whole is under an axial elastic stress, then on any of the flaws—mostly on those located on the surface of the rod—there falls, as indicated above, an increased stress. If such flaws have approximately identical dimensions, then the greatest stress must fall on that one of them which is situated perpendicular to the direction of tension, i.e. to the axis of the rod. At some definite load \(L\), when the tensile force uniformly distributed over the whole section \(q\),

\[ Z=\frac{L}{q}, \]

is obtained, expansion of the crack begins, i.e. a diminution of the cross section of the rod, and at the same time a shift of the maximal elastic stresses to neighboring cracks, which in turn are also carriers of excess stresses.

The further course of the rupture and the character of the surface formed in this rupture will depend on to which of the possible internal cracks the rupture from the initial crack will propagate. According to Smekal, only two cases can exist.

In the first case the rupture proceeds by means of those cracks which lie in the plane of the initial crack, i.e. perpendicular to the direction of the macroscopic stresses. Since the rupture surface obtained in this way remains macroscopically even and mirror-like, it follows from this that the irregularities arising on it are, in magnitude, less than the wavelength of light. In Smekal’s opinion, the length of the cracks forming in glasses an entire network system must be of the order of \(0.0001\) mm or less.

In the second case the rupture proceeds by means of those cracks which lie outside the plane of the initial crack and which, in the case of a particularly favorable orientation, have on their contours somewhat greater stresses than the stress at the edge of the mirror that has reached them. It is precisely in this way that the switching of the rupture into the second stage takes place; this stage proceeds practically instantaneously and ends in the branching of the initially single surface into two grooved surfaces and in the formation of a wedge. The unevenness of the two surfaces formed is explained by the simultaneous independent action of very many cracks located in different places.

According to Smekal’s theory, the actual tensile strength \(Z_0\) is determined by the ratio of the load \(L\) not to the whole area of the cross section \(q\), but to the same area after subtracting the area of the mirror \(s\),

\[ Z_0=\frac{L}{q-s}=\frac{\frac{L}{q}}{1-\frac{s}{q}}=\frac{Z}{1-\frac{s}{q}}, \]

which is also confirmed by numerous experiments for

\[ 0.10 \leq \frac{s}{q} \leq 0.40. \]

Smekal’s theory finds further confirmation in the following facts. In the case of ground, polished rods having rectangular ribs, the initial point of the mirror always lies on the surface of the rod (Fig. 9). Conversely, in the case of round, fire-polished rods, internal mirrors are very often encountered (Fig. 10). The probability of formation of an internal mirror increases by approximately 20% when the surface of the glass is etched. These facts apparently mean, first, that under the action of etching or fire-polishing the surface inhomogeneities are reduced and, second, that in the case of ground rods these surface inhomogeneities are considerably more active than the internal ones.

The coarser the flaw present on the surface of the glass, the more strongly its action outweighs the action of the internal cracks inherent in the glass; consequently, the fracture of glass becomes more similar to the fracture of a homogeneous solid body. Griffith was able to confirm his theory experimentally precisely because he used artificial scratches ranging in length from 3.8 to 22.6 mm. Such an experimental situation is too complex, and therefore, on the basis of the results obtained by Griffith, one cannot draw conclusions about the nature of the fracture of undamaged glass.

Fig. 9

Fig. 10

The next fact confirming Smekal’s theory is that the size of the mirror decreases with increasing rate of loading. This shows that the formation of the mirror is a time-dependent phase of fracture, i.e., that glass does not obey Griffith’s theory, and this in turn means that it is not a homogeneous substance.

With regard to the nature of flaws in glasses, Smekal draws the following conclusions. On the basis of statistical regularities in the sizes of mirrors, the comparatively low rate of their formation, and the dependence of the fracture phenomenon on temperature, one may consider that the inhomogeneities present in glass form an irregular branching lattice, but not cracks isolated from one another. In Smekal’s opinion, further light can be shed on the nature of cracks in glass—in particular, the conditions of their occurrence can be clarified—by

would be by studying the influence of the chemical composition of glasses on these cracks.

The abrupt change in the physical properties of glass at the transformation point \(T_g\) is connected, according to Smeĸal, not with molecular changes, but with the appearance of cracks as a result of the emergence of stresses within the glass during cooling. On reheating, these cracks fuse again, and the glass acquires the properties characteristic of the viscous state.

The average length of a crack in the network structure of pores, according to Griffith’s calculations, is \(0.0015\)—\(0.01\) mm. As was indicated, the value \(0.0001\) mm found by Smeĸal is more probable. The latter also finds confirmation in the results of investigations of the Raman spectra of optical glasses. From such investigations Krishnan \(^{26}\) concludes that glasses consist of molecular complexes which are not small in comparison with the wavelength of light, i.e., consequently, they are larger than \(0.0001\) mm \(^{1}\).

That the entire volume of glass is indeed penetrated by cracks, and that they do indeed pass into one another, forming a whole network system, is shown by experiments on the through permeability of glasses to helium, argon, nitrogen, and other gases even at room temperature. From the sizes of the molecules of these gases one can also obtain an initial idea of the thickness of the cracks. A somewhat better idea of this can be obtained from the fact that, on thorough heating, glasses release such a quantity of gases, i.e., such a total volume of molecules, as corresponds to approximately \(1\%\) of the total volume of the glass. However, such considerations allow one to take into account only part of the total volume of voids and, consequently, to obtain from this too small a crack thickness.

The best idea of the thickness of the crack is provided by the results of observations on the permeability of glasses to liquids at high pressures \(^{28}\). It turns out that molecules of alcohol (size \(5\) Å) and ether (\(7\) Å) easily penetrate into glass, while oil and glycerin do not penetrate. Since the length of the normal chain of the very lowest of the hydrocarbons of oil, \(\mathrm{C}_{12}\mathrm{H}_{26}\), is \(17\) Å, apparently the thickness of the cracks must lie between \(7\) and \(17\) Å \(^{2}\).

\(^{1}\) The author of this review \(^{27}\) has had occasion to make observations of the peptization of various glasses by water. Determination of the size of the smallest particles arising in this process from the rates of settling of suspensions shows that their dimensions range from \(700\) to \(3000\) Å, which is apparently in good agreement with the conclusions mentioned, obtained by other methods. The very fact of peptization of glasses indicates that the particles in question have a discrete existence also in the glass itself.

\(^{2}\) According to the calculations of the author of this review \(^{27}\), it is \(12\) Å for quartz glass. For the calculation, use was made of the difference between the observed density of quartz glass, \(2.20\), and the density characteristic of crystals of the cristobalite type, \(2.30\)—\(2.35\) \(^{16}\). Since \(5\%\) of the volume of the glass is thus accounted for by the voids, then, taking the form of the mosaic blocks of the glass to be cubic and assuming their linear size to be \(1000\) Å, for the thickness of the crack we find \(12\) Å.

Thus, numerous data show that quartz glass and complex silicate glasses consist of two-dimensional crystals which, upon cooling, interlock with one another and with metallic ions. The large aggregates obtained in this way have approximately the same size, perhaps owing to identical conditions of their origin. Proceeding from these apparently sufficiently modern data, one should now approach the explanation of the known properties of glasses and the prediction of new facts.

Lack of space does not permit us to dwell here on this practical side of the matter. One may, however, insist on the assertion that the simple views set forth here make it possible freely to explain all known properties of glasses, including mechanical, chemical, sorption, and electrical properties, or at least do not lead to contradictions with any of the known facts. The former incomplete theories of the glassy state had incomparably less success in this respect. It must therefore be assumed that the further development of views on glass as an inhomogeneous solid body will not only benefit practice, but will also play an essential role in the further development of theoretical views on the nature of solids in general.

LITERATURE

1. Some details concerning Fourier analysis are contained in the review by A. I. Kitaigorodsky, Uspekhi fizich. nauk, 19, 201, 1938. See also, for example, B. E. Warren, J. Appl. Physics, 8, 645, 1937.
2. G. Hägg, J. Chem. Phys., 3, 42, 1935.
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4. S. B. Hendriks, Z. Krist., 95, 247, 1936.
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6. A. A. Lebedev, Trudy Gos. optich. in-ta, 2, No. 10, 1922; 3, 24, 1924.
7. R. W. G. Wyckoff and G. W. Morey, J. Soc. Glass Techn., 9, 265, 1925.
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9. J. T. Randall, H. P. Rooksby and B. S. Cooper, Z. Krist., 75, 196, 1930; J. T. Randall, Diffraction of X-Rays and Electrons by Amorphous Solids, Liquids and Gases, London, 1934.
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11. N. N. Valenkov and E. A. Porai-Koshits, Nature, 137, 273, 1936; Z. Krist., 95, 195, 1936; see also A. A. Lebedev, Izv. Akad. nauk SSSR, Ser. fiz., No. 3, 381, 1937.
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1. Lord Rayleigh, Proc. Opt. Convention, 1, 41, 1926.

2. A. P. Aleksandrov, S. N. Zhurkov, and I. P. Yanushevskii, Journal of Technical Physics, 7, 1111, 1937.

3. A. Smekal, Glastechn. Ber., 13, 141, 222, 1935; 15, 259, 1937; Z. Physik, 103, 495, 1937.

4. A. Thum and H. Oschatz, Metallwirtsch., 13, 1, 1934.

5. H. Schardin and W. Struth, Glastech. Ber., 16, 219, 1938.

6. R. S. Krishnan, Proc. Ind. Ac. Sci., A 3, 211, 1936.

7. N. A. Shishakov, Journal of Technical Physics, 8, 347, 1938.

8. T. C. Poulter and R. Wilson, Phys. Rev., 40, 877, 1932.

1. With additions of potassium oxide to silica, a gradual change in this diffraction pattern occurs. ↩↩↩