DIFFUSION IN SOLID METALS¹

R. F. Mehl, Pittsburgh, USA

Among the various important problems of physical metallurgy, few questions are of greater significance than that of diffusion in solid metals. The phenomena of diffusion are closely connected with many fundamental problems of the metallic state; in addition, the process of diffusion is also of primary practical importance. The formation of alloys by sintering metallic powders, first applied by Faraday and Stodart² ⁸¹ in 1820 and in later years investigated by Mazing¹⁸⁷, ¹⁸⁹ and others, gave rise to the development of special alloys such as carboloy, elkon⁶⁸, etc., and has recently been applied to the production of alloys of noble metals for electrical contacts¹⁸².

Earlier work on cementation⁸,⁹⁷ likewise gave rise to similar work on improving metallic surfaces by means of diffusion into them of various metals and metalloids, and to the development of such processes as sherardizing, chromizing, nitriding, and processes for the formation of bimetals²¹⁸.

However, the process of diffusion has an even broader significance: the homogenization of segregated alloys, the rates of transformation and precipitation and, in connection with this, for example, the rate of hardening of steel or the aging of alloys of the duralumin type depend to a greater or lesser extent on the rate of diffusion. Since diffusion is, evidently, connected with atomic motions, the rate of diffusion is determined by the possible energy states of the atom and is closely related to questions of solid-state physics. Judging from the low rate of diffusion in inorganic salts and from the absence of diffusion in minerals and rocks over geological periods, it was believed that diffusion in the solid state must proceed infinitely slowly in comparison with diffusion in the liquid state. However, in 1884 Spring²⁵⁵–²⁵⁷ pointed out that metals can penetrate into one another by diffusion; and indeed, although

¹ Trans. A. I. M. E., 122, Inst. Metals Division, 1936, p. 11, lecture at the meeting of the Institute of Metals Division, 1936, translated under the editorship of S. T. Konobeevsky.
² The bibliography is placed at the end of the article.

it had been shown still earlier that C penetrates rapidly into γ-Fe; however, the high coefficient of diffusion of Au in Pb, measured by Roberts-Austen in 1896, was unexpected. In the following years the study of diffusion was rather sporadic, but the growing interest in the electrical conductivity of crystalline salts prompted investigations that soon yielded a number of significant works on the diffusion of metals. This, together with the rapid success of physical metallurgy in all its branches and the recognition of the practical importance of the question of diffusion, has provided a large amount of information which, although not sufficiently complete to be regarded as exhaustive, is nevertheless sufficiently interesting and valuable to be subjected to critical review.

Methods

The modern quantitative treatment of diffusion in solid metals is based on Fick’s law $^{82}$, derived by analogy with Fourier’s law for heat flow. In it the spreading of a dissolved substance is regarded as analogous to the motion of heat. Stefan, in order to solve Fick’s equation, applied it to measurements of diffusion in liquid solutions carried out by Graham $^{99}$, proving that the course of diffusion obeys Fick’s law. Stefan compiled tables facilitating the calculation of the diffusion coefficient under those experimental conditions in which Graham’s data had been obtained. Later these tables were supplemented by Kawalki $^{167}$ and Jost $^{154}$. The mathematical treatment of diffusion was also somewhat supplemented by the authors indicated below. Fick’s law states

\[ dm = DA \frac{dc}{dx}\,dt, \tag{1} \]

where $dm$ is the amount of dissolved substance, measured in grams, diffusing during the time $dt$ in the presence (and in the direction) of the concentration gradient $\frac{dc}{dx}$ through the area $A$, expressed in square centimeters. Thus $D$ is a proportionality factor, denoted the diffusion coefficient. It expresses, in grams, the quantity of substance diffusing in one second through an area of $1\ \mathrm{cm}^2$ at a concentration gradient of unity. It has the dimension of the square of length divided by time, as is evident from equation (1), but may also be expressed in $\mathrm{cm}^2/\text{day}$ or $\mathrm{cm}^2/\text{sec}$.

From equation (1) we derive the differential equation

\[ \frac{dc}{dt} = D \frac{d^2c}{dx^2}, \tag{2} \]

in which $D$ is assumed to be independent of concentration—an assumption justified neither for liquid nor for solid solutions. Equation (2) can be solved for various

boundary conditions imposed by experiment. As an example, we may take Stefan’s original method (using Cavalcanti’s tables), in which diffusion takes place from a layer of solution of width \(2x\) into a pure solvent of width \(6x\) (the total width of the layer is \(8x\)). Under these conditions, after a certain period of time the dissolved substance spreads through the separating surface, and the concentration in the initial \((2x)\) layer decreases. Cavalcanti’s tables give the values

\[ \frac{x}{2\sqrt{Dt}} \]

(the notation is as before) for the concentration of each of the layers \(x\), as determined by analysis1. If the corresponding value of

\[ \frac{x}{2\sqrt{Dt}}, \]

has been found, then \(D\) can be computed, since \(x\) and \(t\) are known.

The divergence in the values of \(D\) for the different layers indicates that \(D\) changes with concentration. Van Ostrand and Dewey \(^{210}\) solved equation (2) for diffusion from a saturated solution containing undissolved substance (the concentration remains unchanged throughout the entire experiment) into a pure solvent, and obtained the equation

\[ c = c_0 \left[ 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{\frac{x}{2\sqrt{Dt}}} e^{-\omega^2}\, d\omega \right], \tag{3} \]

where \(c\) is the concentration of the dissolved substance at any point in the direction of diffusion, at a distance \(x\) from the separating surface; \(c_0\) is the saturation concentration; and \(\omega\) is the variable of integration. The quantity to the right of the minus sign is Gauss’s error function \(\Phi\), denoted in mathematical tables as the probability integral; its values may be found in tables. We may rewrite equation (3) as

\[ c = c_0 \left[ 1 - \Phi\!\left(\frac{x}{2\sqrt{Dt}}\right) \right]. \tag{4} \]

Thus, chemical analysis of the composition at a series of points should give values of \(c\), which, together with \(c_0\), may be used for the numerical determination of the error function \(\Phi\), and consequently, when \(x\) and \(t\) are known, \(D\) can be calculated. Fig. 1 shows a concentration–depth diagram of the layer, that is, the distribution curve for a specified value of \(D\) and \(t\); deviations from this type of curve must represent the dependence of \(D\) on \(c\) (often designated in the literature as deviations from Fick’s law) or indicate phase transformations accompanying diffusion (Fig. 7).

Diffusion through the separating surface from a column of solution of infinite length, i.e. actually in a system,

in which the concentration at the far end of the column of solution remains unchanged, while at the opposite end of the column (the solvent) it remains equal to zero, can be determined by means of the equation

\[ c=\frac{c_0}{2}\left[1-\Phi\left(\frac{x}{2\sqrt{Dt}}\right)\right]. \tag{5} \]

None of these equations provides for variation of \(D\) with concentration, although in reality the systematic change in the value of \(D\) with increasing \(x\) indicates this. However, it is scarcely necessary to specify the value of \(D\) for the concentrations occurring at one or another point \(x\): rather, one may approach the true value of \(D\) for individual concentrations by taking sufficiently small differences of concentrations \(c-c_0\). The variation of \(D\) with concentration was usually not taken into account in works on diffusion in solids, and the published data in most cases should be regarded as averages for concentrations between the observed values \(c\) and \(c_0\) ^1).

Fig. 1. Behavior of the function in equations (3) and (4), representing the concentration-distribution curve

Matano \(^{195}\) gives equation (2) the form

\[ \frac{dc}{dt}=\frac{d}{dx}\left(D\frac{dc}{dx}\right), \tag{6} \]

which directly takes into account the dependence of \(D\) on \(c\). He gives a solution of this equation for the boundary conditions of equation (5). For this it is necessary to make an assumption about the dependence on the variables either \(c\) or \(D\). Matano assumes that \(c\) is a function of \(\dfrac{x}{\sqrt{t}}\), and obtains a solution that enables him to calculate

^1) Nernst recently found a change of \(D\) with concentration in liquid systems \(^{207}\).

a single value of \(D\) for diffusion of both the solute and the solvent at any concentration.

From the data of Grube and Jedele \(^{107}\) (Fig. 2) he derived the curve of \(D\) as a function of concentration, shown in Fig. 3. In Fig. 2 Matano indicates the position of the boundary surface in order to show that the number of Cu and Ni atoms that have diffused from each side is the same. This is apparently necessary for substitutional solid solutions, since unequal transfer leads to absurd consequences. The selection of solutions of equation (2) on the basis of the accuracy of agreement with experimental data, without taking account of boundary conditions, is an entirely unjustified method, although it is used. Of primary importance is exact observance of the boundary conditions. Many concentration curves cannot be used for calculating the diffusion coefficient because of ambiguity in the nature of the boundary conditions, especially if the latter relate to concentration, and also because of phase transformations, which cannot be properly treated mathematically \(^{105,108,109}\).

Fig. 2. Distribution curves of copper and nickel, showing compositions at different distances from the dividing surface \(^{107}\)

Fig. 3. Diagram of the diffusion coefficient as a function of concentration for the copper—nickel system \(^{195}\)

Certain special technical details deserve particular attention.

Dann \(^{70,71}\) determined \(D\) for diffusion of Zn in \(\alpha\)-brass by evaporating Zn in vacuum and measuring the loss in weight \(^{1}\). Dann assumed that the concentration of Zn at the surface is zero and that only the rate

\(^{1}\) An analogous [[unclear: word obscured]] was carried out in our country by V. S. Bugakov and V. D. Neskuchaev \(^{41}\). —Ed.

diffusion of Zn to the surface determines the rate of loss of Zn (but not the evaporation rate itself). This method is not entirely reliable owing to shrinkage of the specimen and, consequently, displacement of the section surface. Cracking of the surface during dezincification, in all probability, also affects the calculated diffusion coefficient ^79,274,284, although Dunn’s data agree well with measurements made by other methods. The study of the diffusion rate of C or N in Fe by reaction with carburizing or nitriding gases, or of the rate of removal of these elements by reaction with decarburizing or denitriding gases, involves a similar assumption concerning the rate of the surface reaction (carburization or decarburization), which is taken to be almost instantaneous in comparison with the rate of diffusion in the solid metal; in some cases confirmation of this assumption would have been desirable. The work of Brouwer, Larsen, and Chen is an interesting example of determining the diffusion rate (of O in γ-Fe) from the rate of removal of the diffusing element by means of reaction with H at the surface. It is quite obvious that these methods are of little use for an exact analysis of \(D\) as a function of \(c\), for which concentration-distribution curves with depth are more suitable. The radioactive-isotope method (see the numerous works of Hevesy and Seith) is likewise a special technique ^1), as is the technique of thermionic emission, which will be discussed later. These methods are usually used for small concentrations of the dissolved substance.

Methods connected with averaging—for example, those in which the changes accompanying diffusion in electrical conductivity ^7,38,39,269 or in the lattice parameter ^192,193 are measured in specimens prepared by alternate electrodeposition of successive layers of two diffusing metals—prove insufficiently reliable, since the results do not coincide with those obtained by more orthodox methods ^2).

For constructing curves of the concentration distribution with depth and, thus, of the diffusion coefficient, any method of concentration determination may be used. Usually simple chemical (or spectroscopic) analysis is applied, and it deserves preference; but determination of the lattice parameter with the aid of X-rays is also quite satisfactory, especially in those systems where the parameter changes sharply with concentration. Special methods, in which only a single

^1) The development of artificial radioactive elements greatly expands the possibilities of this method; up to the present time it has been limited to Pb, Tl, Bi, and their alloys.

^2) The discrepancy here often depends not on the unreliability of the method, but on the special conditions under which diffusion proceeds in thin layers. It has been shown that in thin-layer specimens the diffusion coefficient increases greatly owing to plastic deformation arising during heating. —Ed.

concentration at a definite point (and the corresponding value \(x\)), for example, on the outer surface of the diffusing pair, by means of X-rays or by means of a preassigned limit of corrosion resistance, are accurate, but do not give the complete distribution curve and therefore cannot provide information about the variation of \(D\) with \(c\).

Comparatively few diffusion coefficients among those that have been published deserve confidence. Those of them which apparently are not in doubt are collected in Table 1.

TABLE 1

Deshman–Langmuir equation as applied to experimental diffusion data

Solvent	Dissolved element	\(Q\) From diffusion–temperature curves	\(Q\) From the Deshman–Langmuir equation
Cu	Zn \((9.580\%)^{70}\)	41,700	41,000ᵃ
Cu	Zn \((29.080\%)^{7}\)	41,700	38,000ᵃ
Pb	Sn \((10\%)^{196}\)	40,200	40,000
Pb	Pb (self-diffusion)\(^{239}\)	27,900	24,400
Pb	Sn\(^{244}\)	24,000	23,200
Pb	Tl\(^{244}\)	21,000	22,380
Pb	Bi\(^{244}\)	18,600	21,900
Pb	Cd\(^{238}\)	18,000	19,970
Pb	Ag\(^{244}\)	15,200	15,700
Pb	Au\(^{220–222, 210, 237}\)	13,000	13,300
Ag	Au\(^{156}\)	57,600	57,000
W	Th (W grains small)\(^{85}\)	94,000	96,700
W	Th (W grains large)\(^{85}\)	94,450	118,200
W	U\(^{77}\)	100,000	100,500
W	Ce\(^{75}\)	83,000	82,700
W	Zr\(^{75}\)	78,000	77,400
W	Yt\(^{75}\)	68,000	70,100
γ-Fe	N\(^{24}\)	34,600	38,100
γ-Fe	C\(^{213}\)	36,000	36,700

Calculated by Dunn, \(Q\) is incorrect owing to an error in the use of units. The correct values are given in the table.

It is quite clear to the metallurgist that close contact between two diffusing bodies at the boundary surface is of great importance in determining the degree of penetration, especially when penetration is achieved by annealing a metal placed in a powder, as in chromium plating, or in the manufacture of bimetal. Any obstacles at the boundary surface, even thin oxide films, if they do not dissolve immediately during diffusion upon annealing, or slag particles, reduce

the separating surface of contact [the factor \(A\) in equation (1)], giving incorrectly low values for the degree of penetration and for the calculated diffusion coefficients (Fig. 4). The recently published extensive work of Bardengeier and Müller\({}^{14}\) on the diffusion of sprayed metallic coatings illustrates this fact. The presence of a reducer or a flux is expressed in counteracting the retarding influence of an oxide or slag, although, of course, a single flux can only partially oppose this influence. Usually, actual atomic contact is necessary for the transfer of atoms, but this is not always easy to achieve. Electrodeposition is the most expedient method, if precautions are taken to prevent the formation of oxide films, but this method is not always applicable, especially when diffusion from one alloy into another is required. When the diffusing metal has an appreciable vapor pressure, the transfer of atoms is facilitated by passage through the gas phase, and this mode of transfer undoubtedly occurs in many cases in which it is not even suspected, although it can hardly be thought, as has sometimes been asserted, that transfer through the vapor phase is essential for all diffusion processes. Of course, diffusion in alloys during homogenization or during decomposition takes place without participation of the gas phase. Mechanical methods, in which two surfaces are brought into close contact by rolling or drawing, have been little used, although they have special advantages by destroying oxide barriers. The diffusion coefficient varies considerably with temperature. This variation may be represented by the following illustrative equation:

Fig. 4. Diagram showing the factors influencing diffusion from a metallic powder into a massive metal.

\(a\)—diffusing material in powder form, \(b\)—secondary diffusion, \(c\)—oxide or nonmetallic contamination, \(d\)—primary diffusion, \(e\)—poor contact, \(f\)—good contact.

\[ D = Ae^{-\frac{b}{T}}, \tag{7} \]

where \(T\) is the absolute temperature, and \(A\) and \(b\) are constants. In logarithmic form one obtains

\[ \ln D = \ln A - \frac{b}{T}. \tag{8} \]

This equation gives a straight line if \(\ln A\) is plotted against \(1/T\), with \(b\) determining its slope. This is clearly shown in Fig. 5, where the data of measurements of the diffusion of Pb in Au are plotted. All results,

obtained so far for diffusion in metallic systems obey this law¹), although these data may still be considered insufficiently complete and limited to too narrow a temperature interval to be entirely reliable²).

PHYSICS

Since diffusion consists in an exchange of atoms between two crystal lattices, the mechanics of the process must take into account the crystal lattices and their energy. In order to represent the picture of the necessary transfer of atoms, many theories have been proposed. Rosenhain²²⁴ assumed a coordinated slip occurring in such a way that neighboring crystal blocks are displaced, and atoms that had previously been neighbors in one atomic plane move far apart from one another. However, this picture apparently does not lead to the directed displacement required in the present case, and the experimental data do not confirm this theory. Frenkel⁸⁸, seeking to avoid the difficulties that may arise on the basis of a simple rearrangement of lattice atoms, supposed that atoms must be torn (“dissociated”) from their positions in the lattice and become mobile in the interstices of the lattice, eventually becoming established again (“associated”) in new positions. Smekal²⁴⁸ assumed that atoms can migrate only in the interstices, and especially within distorted and mosaic regions in the lattice, and he indicated a number of separate processes which, as a whole, are based on transport processes within the lattice. However, as we shall see later, the role of mosaic structure in diffusion is still very uncertain. Langmuir¹⁴⁸, ²²⁴, on the other hand, proposed a simple cyclic displacement of atoms, including also atomic rearrangement at individual nodes of the lattice; and Braunbek³³, in order to explain the electrical conductivity of salts, developed a theory based on a simple rearrangement of atoms at the nodes of the lattice. At the present time it is hardly necessary to go beyond this conception in order to explain the still very limited data available.

The vibrations of atoms about their mean positions in the lattice, the distribution of frequencies and the limiting maximum frequency, as well as the increase of their amplitudes and energies with temperature, constitute the necessary dynamical basis for the mobility of atoms. Deshman and Langmuir⁷⁶ applied, by analogy with the treatment of the kinetics of chemical reactions, the idea that an atom must possess a minimum energy in order to jump over the potential barrier surrounding it, and that the probability of a change of position for some

¹) Owen and Pickup derived similar equations for the mutual diffusion of mixed metallic powders²¹¹, ²¹².

²) Data on the electrical conductivity of solid salts indicate a dependence of \(A\) on \(T\), but for diffusion in metallic systems there are no indications of such a dependence²⁴⁸.

atom (in the analogous case of chemical reactions, the probability of a reactive collision) is proportional to the expression

\[ e^{-\frac{E}{kT}}, \tag{9} \]

where \(k\) is the Boltzmann constant. For one gram-atom this gives

\[ e^{-\frac{Q}{RT}}, \tag{10} \]

where \(R\) is the gas constant, and \(Q\) is the transfer energy of 1 g-atom. Proceeding from this, Deshman and Langmuir proposed the following general diffusion equation:

\[ D=\frac{Q}{Nh}\delta^{2}e^{-\frac{Q}{RT}}. \tag{11} \]

Here \(Q\) denotes the heat of diffusion, analogous to the heat of activation in chemical reactions, \(N\) is Avogadro’s number, \(h\) is Planck’s constant, \(\delta\) is the interatomic distance; the remaining notation is as before1. Equation (11) is often written in the simplified form:

\[ D=Ae^{-\frac{Q}{RT}}, \tag{12} \]

which is another form of expression (7).

Verification of formula (11) is difficult because of the insufficiency of reliable diffusion coefficients. It should be noted, however, that the data deserving the greatest confidence are always in agreement with formula (11), with the exception of systems of noncubic metals11, to which equation (11) is not applicable. Table 1 contains values of \(Q\) derived graphically from the dependence of the logarithm of the diffusion coefficient on the reciprocal absolute temperature. The agreement between the calculated values and those obtained from measurement confirms formula (11). Deviations are large only in those cases where the experimental data are doubtful. The application of the equation is not limited to solid substitutional solutions, where diffusion can occur only by rearrangement of atoms, but also extends well to solid interstitial solutions (C in \(\gamma\)-Fe), where diffusion occurs by penetration into the gaps between atoms. The latter result seems somewhat surprising in view of the substantial difference in the manner of atomic transport in these two cases. Although it cannot be asserted definitively that the Deshman—

Langmuir is justified in all cases in view of the insufficiency of experimental data; however, one cannot but feel astonishment at its scientific and practical significance1. Looking at the formula, we see that knowledge of only a single diffusion coefficient at some one temperature is sufficient to calculate \(Q\), since the remaining quantities are known constants (\(\delta\) can easily be found from data for pure components). Conversely, if \(Q\) is determined, then \(D\) can be calculated for any temperature. Thus, for example, the process of cementation or homogenization can be calculated for any temperature on the basis of studying the process at some one temperature.

Relating the rate of atomic rearrangement to the activation energy by the same method as Deshman and Langmuir, Braune[^34] brought this energy into agreement with Lindemann’s theory of melting and derived for the diffusion coefficient the expression

\[ D = Ae^{-\frac{3b^2T_s}{T}}, \tag{13} \]

where \(b\) is a constant, whose value ranges from 1 to 2 for different metals, \(T_s\) is the absolute melting temperature, and \(A\) is also a constant.

F. Limit[^183,^184] developed this theory and derived the “jump time,” the fraction of atoms that exchange places in one second, and also the number of vibrations performed by an atom before leaving its place. He derived the equation

\[ D = \frac{8}{3\pi}\nu\delta^2 e^{-\frac{3b^2T_s}{T}}, \tag{14} \]

in which \(\nu\) is the characteristic (monochromatic) frequency. Thus the value of \(\nu\) can be calculated from \(D\); it is found to agree, in order of magnitude, with the value of \(\nu\) derived from Lindemann’s equation2. The equation derived by Polanyi and Wigner[^216] is in complete agreement with the Deshman–Langmuir formula. Further development of these formulas encounters difficulties because of the incompleteness of the physical theory of the metallic lattice and the lack of good experimental data on diffusion3.

Self-diffusion

The simplest type of diffusion is self-diffusion, i.e., the diffusion of atoms in their own lattice. The technique for determining the coefficient of self-diffusion was developed by Hevesy \(^{102,103,131,136,239}\) and consisted in contaminating the surface of Pb with the radioactive isotope of lead ThB and following the diffusion of the isotope into the lead by means of a simple measurement of the radioactivity of the specimen surface with an electroscope. The radioactivity decreases when ThB diffuses from the surface; the diffusion coefficient is determined by the method developed by Fours \(^{901}\). The data obtained are given in Tables 1 and 2.

Rate of diffusion

Many of the ideas underlying modern conceptions of diffusion in solid metals originate in the work of Tubandt, Joffe, Jost, Hevesy, and others on the electrical conductivity of solid inorganic compounds. It was shown that the electrical conductivity of quartz \(^{150,494}\), and especially of such salts as \(\mathrm{PbJ}_2\), is wholly or partly electrolytic in character and is accompanied by transport of ions, so that the electrical conductivity proves to be proportional to the rate of diffusion of an ion or ions. In this way we can represent the change of electrical conductivity with temperature by means of an equation similar to equation (12), where the electrical conductivity \(\lambda\) replaces \(D\), and the constant \(A\) becomes another constant. Such an equation may be applied where only one ion is mobile, as in \(\alpha\mathrm{AgJ}\), \(\alpha\mathrm{CuJ}\), etc. When both ions are mobile, two exponential terms are required, each with its own characteristic value. For example, for \(\mathrm{PbJ}_2\) at a temperature not far from the melting point we have \(^{132,133,134}\)

\[ \lambda = 9.70 \cdot 10^{-4} e^{-\frac{8760}{RT}} + 1.5 \cdot 10^{-5} e^{-\frac{30000}{RT}} . \tag{15} \]

The quantity \(Q\) in the equation of electrical conductivity or diffusion represents the energy required in order to tear a gram-atom from the lattice and make it mobile (the energy of “loosening”). This “loosening,” of course, is chiefly caused directly by thermal vibrations, but it was pointed out by Fajans \(^{80}\) and Hevesy \(^{129,130}\) that the normal polarity of such ionic lattices partly—

—to eliminate this influence very high pressures are required. A combination of the technique of high pressures by Bridgman and the radioactive method for measuring diffusion would give positive results here.

\(^{1}\) This method was also applied to determine the rate of self-diffusion of Pb ions in solid salts \(^{131,132,133}\). When radioactive radiations are used, values of \(D\) down to \(10^{-12}\ \mathrm{cm}^2\) per day can be measured.

neutralized, and the electrostatic forces of the lattice prove to be weakened as a result of the increase in the number of collisions (approaches) of the vibrating ions and the resulting mutual penetration of the electron shells. This gives a weakening of the bonds in the lattice, depending on the greater or lesser polarizability and deformability inherent in the ions. The degree of “loosening” is high when the deforming force of the cation and the capacity for deformation of the anion are large, i.e., when the electron affinity of the cations is large and that of the anions small, as in AgJ. The mobility of Ag in its salts increases when we pass from the nitrate through a series of anions decreasing in strength to the telluride. Since in all molten salts the bonds are completely freed, we may expect only a slight difference in their electrical conductivity, which is what is observed. But the degree of loosening (liberation) in the solid state varies within wide limits. As a qualitative indicator of the degree of liberation in a given solid substance one may take the ratio of the electrical conductivity above and below the melting point. When this ratio is large, the solid substance is little liberated; when it is small, the liberation is considerable. In the case of AgJ the bonds in the solid substance are so strongly weakened that its electrical conductivity is even 10% greater than that of the melt; but in obviously polar salts this ratio is very large, as in NaCl, where it is \(10^5\).

What is observed in alloys apparently agrees with this point of view. Although the influence of the formation of an alloy, and especially of a solid solution, on the distribution of the valence electrons, as well as on the state of ionization and deformation, is extremely difficult to study, it is nevertheless well known that such an effect exists. The classic work of Kremann\(^{173}\) on the electrolysis of liquid alloys and the recently published work of Seith on the electrolysis of solid alloys are direct experimental proofs of partial ionization in alloys\(^{1}\). Experimental data on diffusion in Pb are very instructive. The rate of self-diffusion in Pb above the melting point is much greater than the rate of self-diffusion below the melting point (coefficient 40,000); thus the lattice is very little weakened. Since it is known that the electronic structure of ThB is identical with that of Pb, diffusion of ThB is not accompanied by any chemical effect. However, when Sn diffuses in Pb, its atoms move in an atomic lattice somewhat different in its chemical nature, and the effect caused by chemical affinity—polarization and deformation of the atom—is expressed in a loosening of the lattice and, correspondingly, in large values of \(D\) and smaller values of \(Q\). This effect becomes still more noticeable when the chemical difference increases, as is clearly shown by the data—

\(^{1}\) Seith and Kubaschewski showed that solid solutions of C in \(\gamma\)Fe, Au in Pb, and Au in Pd can be subjected to electrolysis, with C moving toward the cathode and Au in both cases toward the anode\(^{162}\).

… of Geveshi and Seith4 are presented in Table 2 and in Fig. 8.

TABLE 2

Diffusion of metals in lead

Metal	Au	Ag	Cd	Bi	β-Tl	β-Sn	Self-diffusion
\(D\), cm\(^2\)/sec at 285°C	\(4.6 \cdot 10^{-6}\)	\(9.1 \cdot 10^{-8}\)	\(2 \cdot 10^{-9}\)	\(4.4 \cdot 10^{-10}\)	\(3.6 \cdot 10^{-10}\)	\(1.6 \cdot 10^{-10}\)	\(7 \cdot 10^{-11}\)
\(Q\)	13,000	15,200	18,000	18,600	21,000	24,000	28,000
Crystalline system of the dissolved substance	f.c.c.	f.c.c.	hex. close-packed	rhomb.	f.c.c.	tetr.	f.c.c.
Maximum solid solution, atomic %	0.05	0.12	1.7	35	79	29	100
Atomic radii, Å	1.44	1.44	1.52	1.82	1.71	1.58	1.74
Melting points, °C	1063	960	321	271	303	232	327

We may note that the diffusion rate increases and the heat of diffusion decreases when the various diffusing elements become more and more chemically different from Pb, and1 when, at the same time, the solid solubility decreases (although it does not vanish), the difference in melting points increases, and the difference in atomic radii decreases. F. N. Rhines6 has recently shown that the rate of diffusion in solid \(\alpha\)-solutions in the systems Cu—Sn, Cu—Si, Cu—Al, and Cu—Zn decreases in the indicated order, i.e., decreases as the chemical relationship of the two diffusing elements increases. This general approach to solving the problem, encompassing diffusion in systems structurally analogous to Cu, Ag, and Au with elements of subgroup \(B\), seems exceptionally attractive, especially if one takes into account the correlations recently discovered by Hume-Rothery[^145] between the concentration of valence electrons and the liquidus and solidus temperatures, as well as in connection with the latest classification of metals[^146] and the thermodynamic treatment of solid solutions[^246].

When the Pb lattice is disturbed by foreign atoms in a solid solution, the rate of self-diffusion increases. Thus, the Au atom is small, and its valence electron is firmly bound; when alloyed with Pb,

the atoms of which are large and exhibit a smaller electron affinity, the Pb lattice is weakened, and the rate of self-diffusion increases[^134]. In a similar manner the mobility of Ag increases in an alloy with Sn[^130]. The relation between the melting point and \(D\) and \(Q\) is of special interest. Equation (13) was derived by Braune for a quantitative determination of this relation. In Braune’s opinion, melting points may be taken as the corresponding

Fig. 5. Diffusion of gold in lead. 1 — Roberts-Austen; 2 — Van Orstrand and Dewey; 3 — Seith and Ewald

temperatures for comparing diffusion data, provided only that the melting points are not too close to one another. F. G. Hevesy approximately determined the coefficients of self-diffusion in the elements Ag, Au, and W[^135]1, which, together with the measured diffusion coefficient for Pb, show that the diffusion coefficient at constant temperature is inversely related to the absolute melting temperatures.

With respect to binary systems, the observation was made that the diffusion coefficient in both terminal solid solutions is greatest in the solid solution with the lower melting point; thus the diffusion coefficient of Au in Pb is 26,000 times greater than that of Pb in Au[^134]2. Metallurgists know that the solid solubility in a binary system is usually greater in the metal with the higher melting point, and thus it may be expected that the rate of diffusion in these systems will increase as

as the melting points diverge and the solid solubility decreases. The relation between the melting point and the diffusion coefficient is confirmed by Jost’s investigations^160 on the systems Ag, Au with Pt and Pd, by Matano’s investigation^194 (the systems PdCu, PdAg, PdAu), and especially by the investigations of Iedele^147,194 on the systems AuPt, AuPd, and AuNi, according to which the greatest change in \(D\) with change in concentration occurs in the AuNi system, where the melting points of the alloy components show the greatest divergence. Sen’s recent work^247 may be interpreted in the same way. Fig. 3 illustrates this relation for the Cu—Ni system.

The influence of the melting point has also been demonstrated in the recent work of Seith, in which it is shown that Bi, which lowers the melting point of Pb, increases the rate of self-diffusion, whereas Tl, which raises the melting point, does not increase this rate. These investigators showed that Danna’s data^70 on the diffusion rate of Zn in \(\alpha\)-brass of different composition (and with different melting point) can be reduced to the ordinary diffusion rate if these data are referred to a correspondingly lowered temperature. We may conclude that: 1) the rate of self-diffusion at a given temperature is the smaller, the higher the melting point of the metal; 2) in a whole series of alloys with a common base metal, such, for example, as the lead alloys described above, the diffusion rates are the greater, the higher the melting point of the diffusing metal; and 3) in binary systems the diffusion rate is greater in the metal with the lower melting point. However, when the melting points in a binary system do not differ too greatly from one another, as in the Cd—Pb system, atomic polarization is apparently the decisive factor, and large differences in the diffusion coefficients may occur here. Diffusion in metals of low symmetry, with low coordination numbers, apparently proceeds more slowly than in cubic metals^55, and in this case the symmetry relations themselves apparently play a role^129,130,134,135,148.

It is quite likely that high diffusion coefficients are associated with a large difference in melting points, a large difference in atomic sizes, and a small degree of solid solubility, reflecting the substantial influence of the polarizability of the atoms and of the resulting disorder and asymmetry; but here, as in similar attempts to establish the factors affecting solid solubility, the true form of this relation eludes precise definition.

Determination of the influence of a third element on the rate of diffusion opens interesting possibilities. It has repeatedly been established that Cr does not diffuse into Fe from powder at temperatures below \(1100^\circ\)C if Si is not present in the alloy^14. It is not clear, however, whether a genuine effect of the third element is expressed here, since it is possible that Si acts here as a deoxidizer. Grube^107 noted that 0.5% Mn in Ni decreases the rate of diffusion of Cu in Ni

to \(1/4\) of the value of the diffusion rate in pure Ni. And although the grain size increases in Mn-containing Ni, which may explain the decrease in diffusion rate, the influence of the third element is nevertheless present here; irrespective of the cause, this influence has been noted and must be taken into account in the homogenization of cupronickel castings. Since the third element, if it is present in sufficient quantity, considerably changes the characteristics of the lattice—for example, the cell dimensions—the effect of this influence is readily understood. The influence of a small percentage of a third element is, however, unexpected. Grube\({}^{109}\) found that the diffusion rate of W in vacuum-melted Fe is four times greater than in ordinary electrolytic Fe. Yamaguchi\({}^{299}\) found that Pb, Sn, Zn, and Fe increase the diffusion rate of Zn in \(\alpha\)-brass, whereas Mn decreases it. Tammann and Schönert found that the diffusion rate of C in \(\lambda\)-Fe changes with the purity of Fe; the same was found by Bramley\({}^{23-31}\). He and his co-workers discovered many unexpected effects, among which we may mention the effect produced by O, which reduces the diffusion rate of C in \(\gamma\)-Fe; S produces the same effect. Further, C reduces the diffusion rate of P, and, remarkably, O increases the diffusion rate of N. Fry\({}^{89}\) states that P and S diffuse more rapidly in \(\gamma\)-Fe when they are present simultaneously than when they are present separately. Fritsch found that simultaneous diffusion of Mg and Si in Al in the ratio corresponding to the compound \(\mathrm{Mg}_2\mathrm{Si}\) proceeds faster than would correspond to the normal diffusion of both components separately. It is possible that some of these cases are caused by the influence of the third element on grain size, for this possibility is usually not taken into account. This question should be studied more thoroughly, since it may yield important results for the technique of surface improvement, especially carburizing and nitriding, as well as for homogenization and, indirectly, as we shall see, for controlling the rate of reaction in the solid state.

Metallography

A condition for diffusion to be able to take place is the formation of a solid solution, as was proposed by Guinier\({}^{116}\) in 1914. This leads to a number of substantial metallographic consequences\({}^{1)}{}^{18}\). Two metals diffuse into one another, forming layers equal in number to the stable phases in the binary system at the given temperature. The rate at which these layers form varies from layer to layer and from system to system according to the corresponding diffusion coefficients\({}^{2)}\). Thus copper, in contact—

\({}^{1)}\) Numerous interesting individual observations on diffusion are very difficult to bring together. Desch collected many of the earlier observations in an interesting review published in 1912.

\({}^{2)}\) There are very few data on the diffusion rate in intermetallic phases. Keller\({}^{171}\) reports that Zn diffuses in \(\beta\)-brass faster than

concerning pairs of Zn at 400°, forms layers of α-, β-, γ-, ε- and η-brass, as was indicated by Ilam5. The ε phases, as well as γ, are brittle and break out during polishing; this property of theirs, in combination with the difficulty of etching these phases, creates difficulty in recognizing these phases in the specimen, so that only the α, β and γ phases are visible in Fig. 6. Such alloy layers were found in zinc chill castings coated with copper; these layers were detected even after “annealing” at room temperature[^47][^217]; Ring[^219] showed that four layers formed when Fe—Zn pairs were heated to a temperature 100° below the melting point of Zn. Applying this technique, Goodson[^143] showed that the β phase in the Cu—Zn system forms during diffusion below 470°C and is thus stable below this temperature; in a similar way, some uncertainty regarding the continuity of the phase field1 in the Al—Zn system[^230] can easily be clarified. It is clear that diffusion cannot occur in phases that have no region of solid solubility, and this circumstance can serve as a check on solid solubility, as Hume-Rothery[^144] made use of. The composition of diffusion layers is not easy to determine, although even in qualitative determination this method is more useful in studying the composition of metallic—

Fig. 6. Penetration of zinc into copper[^205]. Cu, cemented in Zn powder; held 150 hours at 400°; quenched. Not etched. ×50. In the upper part there is an α phase; in the narrow band near the central part, a β phase; white γ phase at the base.

Fig. 7. Curve of the distribution of chromium concentration in iron, 96 hours at 1200°[^137].

chemical systems than is usually assumed. Within each diffusion layer there is the entire set of concentrations of the solid solution permitted by the diagram; at each boundary between layers there occurs a discontinuous change in composition corresponding to the size of the intermediate heterogeneous region. The diffusion curve of Cr in γ-Fe (Fig. 7), obtained by Hicks^137 by determining the lattice parameter, illustrates this fact. The jump on the curve corresponds to the α—γ interface at the diffusion temperature. Of interest is the fact that, as Ilam showed, the relative orientations of the crystallites obtained on these interfaces are similar to those in analogous cases during phase reconstruction of crystals, or in oxide films of metals^200, and are also very similar to those observed in Widmanstätten figures. These facts provide a simple method for determining the limits of solid solubility, since, obviously, during diffusion the pure metal will absorb only that amount of dissolved substance which can enter the solid solution. This simple method of establishing solid solubility may be recommended; using it, Seith and Eschold^237 determined the very slight solubility of gold in lead, and Ziegler^301 studied the solubility of oxygen in solid iron. When the diffusion of one metal into another is accompanied by the formation of a new phase, it leads to the appearance of columnar grains. The first explanation of this phenomenon belongs to Green^100, who proposed it to explain the phenomenon occurring during the decarburization of austenite between the temperatures \(A_1\) and \(A_3\). It was later accepted by Jeffries, Hattren, and Benedict in a discussion of the work of Kelly^170. The first center of the new phase, formed on the surface, grows inward, gradually absorbing the underlying crystals of the original phase, in the same way as columnar crystals grow from the liquid state during solidification^2, ^5, ^13.

Anisotropy

The success of the generalized diffusion equations [equations (11) and (14)], which include the factor of distance between lattice points and assume that this diffusion occurs by discrete jumps from one point to another, compels one to suppose that in many of its features diffusion in the solid state depends on the geometry of the crystal lattice, and this supposition has been confirmed. Having at first excluded all extraneous influences, we may consider diffusion in a single crystal or in one grain. Further, the study can be extended also to a polycrystalline aggregate by including here the investigation of the role of grain boundaries, as well as surface effects and the effects of distortion.

It was found that the magnitude of the electrical conductivity of PbJ\(_2\) and PbCl\(_2\), and therefore also the diffusion coefficient, depends on the crystallographic direction^234 (several years ago the observation was made that the electrical conductivity of quartz exhibits a similar

same dependence $^{150,294}$). At $360^\circ\mathrm{C}$ the electrical conductivity of $\mathrm{PbJ}_2$ parallel to the $C$ axis is $1/30$ of the electrical conductivity perpendicular to this axis; the lattice is a complex non-cubic layered lattice$^{1)\,234}$. A similar anisotropy was observed in self-diffusion in rhombohedral Bi$^{234}$. Although it may be asserted that the structurally sensitive nature of Bi and its mechanical brittleness may deprive the experiment of part of the reality that it would have in application to an actual single crystal, anisotropy in diffusion may nevertheless be expected here, in view of the low symmetry of the lattice. Fig. 9 shows the result of Seitz’s measurements$^{234}$. The rate of diffusion at $269^\circ\mathrm{C}$ perpendicular to the rhombohedral axis $B$ is greater than the rate of diffusion parallel to the same axis $A$ by approximately a factor of a million. This is the only example of anisotropy now available in metallic crystals, although it may be expected in all non-cubic crystals. Although the mechanism of diffusion apparently requires the exchange of atoms at lattice sites, i.e. the transfer of motion along

Fig. 8. Diffusion of various metals in lead$^{242}$

Fig. 9. Self-diffusion in bismuth

$^{1)}$ The self-diffusion coefficients of Pb in $\mathrm{PbJ}_2$ and $\mathrm{PbCl}_2$ were measured by Geveshi and Seitz$^{132,133}$, who used $\mathrm{TlB}$; they found that an ion having the smaller value of $Q$ (J and Cl) shows a smaller dependence on direction$^{214}$; diffusion of the Pb ion does not depend on orientation.

directions of the crystal; however, in cubic metals there is no resulting macroscopic anisotropy of diffusion.

Ilam5 found that Zn diffuses into a single crystal of Cu uniformly in all directions, and this result was recently confirmed.6

Diffusion of C in $\gamma$-Fe in a coarse-grained Fe specimen, as found by C. Wells,6 reveals a uniform penetration of C into each crystalline grain, as may be judged from the depth at which the last traces of the precipitated carbide can be found. The penetration does not depend on the orientation of the grain. The same results were obtained for the diffusion of N in $\alpha$-Fe. The remarkable photographs of Cu—Si alloys whose surfaces had been oxidized, presented several years ago by Dr. Smith,7 show that, as far as can be judged from the appearance of the $SiO_2$ particles, oxygen penetrated to an equal depth in each grain, irrespective of the orientation of the individual grains. Although a certain inhomogeneity of the microstructure along the diffusion zone in the specimens has been interpreted as evidence of anisotropy,8,9,10 this evidence is nevertheless not sufficiently weighty. From the data at our disposal we may conclude that the rate of diffusion in cubic crystals does not depend on direction. This seems somewhat to contradict the stated assumption that atoms diffuse by jumps along preferentially selected crystallographic directions, but calculation shows that joint (collective) motions along several equivalent directions must result in macroscopic isotropy, and not anisotropy.6

Distortions

Many authors, especially Smekal,11–13 have repeatedly pointed to the existence of a mosaic in crystals and to its role in diffusion. The increase of the rate of diffusion with temperature [represented by the dependence of $Q$ on $T$ in equation (11)] is reduced to a loosening of the lattice, as has already been said above.

This conception of thermal liberation (“loosening”) of the lattice was further developed by various authors, especially by Geveshi, and included mechanical liberation of the lattice as a result of cold working. Less justified is Smekal’s idea of an inherent weakening of the lattice connected with lattice defects and mosaic structure. It is possible that imperfections of the lattice in many cases accelerate diffusion, creating, as it were, additional grain boundaries where diffusion proceeds faster than within grains. But up to the present there is no experimental basis for the assumption of a special participation of periodic mosaic

surfaces during the process of diffusion. Ioffe^161 doubts the importance of Smekal’s assertion, while Jost^19,129 showed that the electrical conductivity and the rate of diffusion of salt crystals do not depend on the thickness down to thicknesses smaller than that usually assumed for the mosaic period, and also^156 showed that the rate of diffusion in thin \((10^{-6}\ \text{cm})\) layers of metal is just as great as in large specimens; in layers of such thickness periodic planes are evidently absent, and the phenomena must be typical for crystals without a mosaic. Moreover, the creation of homogeneous solid solutions by diffusion will sooner or later require atoms to penetrate into all parts of the lattice, and not only into mosaic or imperfect surfaces. Thus, apparently, the various functions \(e\) proposed by Smekal in order to transfer diffusion from lattice sites to places of disorder along distorted surfaces are unnecessarily complicated. The problem of the role of lattice distortions in connection with diffusion is in the same uncertain state as the problem of the mechanical properties of crystals. At present it seems more expedient to obtain good experimental data than to develop hypotheses too far.^1)

It was observed that the electrical conductivity of salt crystals increases noticeably upon cold working. Pressed powders and rapidly solidified (and consequently stressed) crystals of inorganic salts possess an electrical conductivity 5 times greater than carefully prepared single crystals. Undoubtedly, cold working can often increase the rate of diffusion, but this fact, together with the supposed necessity of lattice disturbances, has led many authors to believe that diffusion cannot take place in a real crystal free from stresses (deformations).^65–67,79 The latter point of view is definitely incorrect, since diffusion has been observed in single crystals of metals freed, as far as possible, from stresses.

A metallurgist will immediately recognize the necessity of diffusion in a single-crystal grain, since the formation of precipitates inside the grain is often observed. Seith and Keil presented indisputable evidence that self-diffusion occurs in single crystals of lead; other authors showed that Mo^184, C^6 and Th^181,85 diffuse into single crystals of W, and that H diffuses through single crystals of Fe. Several years ago it was shown that N diffuses rapidly in very large crystals of \(\alpha\)-Fe. The observation made by Ilam^79 that Zn does not diffuse from \(\alpha\)-brass into a single crystal of Cu until the specimen has been mechanically deformed is undoubtedly,

^1) Rosenhain’s suggestion^223 should be used, namely that diffusion should be studied in single crystals obtained by different methods, i.e. by cooling the melt, on the one hand, and by annealing—recrystallization—on the other. Dehlinger^68 observed that such crystals exhibit different plastic properties.

should have been interpreted simply as proof of the presence of an oxide film formed during casting; her own observation that Zn diffuses from the gas phase into a Cu single crystal is sufficient proof of this.

These results were confirmed in my laboratory. True, diffusion in single crystals is often extremely slow and may escape observation or measurement in comparison with the more rapid diffusion along grain boundaries and, consequently, in a polycrystal. Nevertheless, there is no doubt that diffusion is a structure-sensitive property. Fonda, Jucker, and Young[^85] showed that the rate of diffusion of Th into W single crystals (obtained by Pinch’s method), measured by the electron-emission method used by Langmuir,[^179],[^181] increases upon twisting of the crystal even when no visible grain refinement occurs. During annealing the distortion was removed, and the diffusion coefficient again returned close to its original value. Earlier observations by Zwikker[^302] showed that W single crystals (Pinch), which, as is known, are less dense and thus less perfect than crystals grown from the gas phase, exhibit a diffusion coefficient for C 10–30 times greater than that for more perfect crystals. For Pb, Seith and Keil[^239] could not establish an effect of cold working, evidently owing to the low temperature of recrystallization and recovery of lead; however, it turned out that 0.3% Au raises the temperature of recrystallization and recovery to such an extent that the effect of deformation becomes apparent and is reflected in an increased diffusion rate.

Of great interest from this point of view is the recent work of Finch, Quarrell, and Roebuck[^83],[^273] on electron diffraction from metallic layers deposited on a polished and etched metal surface. Although, according to Germer,[^96] the nature of the polished layer is not entirely clear, contrary to the opinion of some English authors who consider the existence of the amorphous Beilby layer to have been proved, we shall not enter into this question here. The observation was made that the electron-diffraction spectrum of layers deposited by condensation of a metallic vapor on a polished surface disappears with exceptional rapidity; whereas when deposition takes place on an etched surface (not deformed) of the same metal, the spectrum is preserved for as long as desired. For example, a very thin layer of Zn deposited on a polished Cu disk began to fade after 1 sec., and after 10 sec. was no longer visible. Successively deposited layers disappeared more and more slowly; finally, the twelfth layer was visible after 4 hours. In the end the Cu disk acquired the color of brass, which indicated the formation of a solid solution of Cu and Zn. A comparative experiment, performed with a single layer on an etched disk, showed that the Zn spectrum remained visible even after 1.5 hours. Similar experiments on the deposition of Zn, Sn, Pb, and Ag on polished and etched surfaces of mild steel, Pb, Au, Cu, and Zn

gave the same results. The time between deposition and disappearance of the layer varied for different combinations of these elements. This time, apparently, corresponded to the order of the expected diffusion coefficient, although no complete comparison was made. The special significance of this work lies in the fact that it shows that the rate of diffusion in deformed metals and metals subjected to cold working1 is considerably greater than in undeformed metals.

Since the experiments took place at room temperature, the effect of deformation can be studied without simultaneous recovery and recrystallization. This method is one of the most promising in the study of diffusion phenomena and should be fully exploited. Valuable qualitative data can be obtained on such questions as the anisotropy of diffusion of metals in noncubic metals, the influence of grain on the rate of diffusion (without the danger of a simultaneous change in grain size), and the influence of a third metal on the rate of diffusion; it is possible that this method can be made at least semi-quantitative.

It should be expected that diffusion proceeds more rapidly in deformed metals than in undeformed ones, since deformation causes the formation of a less dense structure, in which the interatomic forces are not so large (compressibility, for example, increases), and also causes the formation of a less symmetrical force field (or charge-density distribution); both these phenomena produce a higher rate of diffusion. Usually, however, it is difficult to demonstrate this effect, since, as is evident from the recent work of Muradyan and Norton[^206], the rate of recovery (relaxation) is much greater than the rate of the measured diffusion. This effect can be demonstrated only by applying techniques of special sensitivity, such as thermionic emission, radioactivity, or electron diffraction, or when the metal shows an unusual reluctance to recrystallize or recover. The formation of a solid solution, which always accompanies diffusion (with the exception of self-diffusion), creates two types of distortion: a microscopic one, characteristic of the solid-solution state and in itself detectable in the broadening of X-ray diffraction lines2, and a macroscopic one, produced as a result of the change in volume accompanying the penetration of diffusing atoms into regions of low concentration, as a result of which stresses arise. This type of deformation can even lead to recrystallization and grain growth[^118,^119,^188].

When Zn vapors diffuse into a Cu bicrystal, the initially sharply delineated and almost straight grain boundary may be replaced by an irregular boundary and by new crystals, and in some cases twins may form, as shown in Fig. 10^204. Absorption of Zn causes an increase in volume, and the internal compression creates stresses that lead to recrystallization and the formation of twins. It has been established that twins appear more often in annealed cast brass and bronze than in unannealed castings; in this case the local increase in volume is accompanied by displacement of atoms just as in a large bicrystal. Usually, however, homogenization of dendrites in segregated castings is not accompanied by large changes in grain size, judging from the work of Mathewson^197, Adcock^224, and Mazing^188; diffusion at high temperature favors some grain growth, while diffusion at low temperature favors grain refinement^5,141,196. This question still requires more detailed investigation.

Fig. 10. Penetration of zinc into copper bicrystals.^209 The Cu bicrystal is placed in shavings of α-brass; annealed for one week at 775°. Etchant: ammonia + hydrogen peroxide. ×100. α-brass at the base (etched), Cu above; the grain boundary runs from the apex to the base; there is no preferential orientation along the grain boundaries.

Diffusion along grain boundaries

Deshman and Kohler^181 showed that the rate of diffusion of Th in W depends on grain size, and Clausing pointed out that diffusion in fine-grained W filaments at ordinary incandescent-filament temperatures occurs almost exclusively along grain boundaries. Fan-Limpt^184 observed that the rate of diffusion of Mo in W at 1600° is ten times greater in polycrystalline W with an average grain diameter of 20 μ than in a single crystal. Langmuir^181 and Fonda, Uoker, and Young^85 showed that for polycrystals a constant rate of diffusion of Th in W can be obtained if all data are referred to a single grain, in order to eliminate the effect of diffusion along grain boundaries. These investigators calculated the rate of diffusion of Th in W in a single crystal and along grain boundaries^181:

for volume diffusion

\[ D = 1.00 \cdot e^{-\frac{120000}{RT}}, \tag{16} \]

for diffusion along grain boundaries

\[ D = 0.74 \cdot e^{-\frac{90000}{RT}}. \tag{17} \]

DIFFUSION IN SOLID METALS

Thus, at \(2400^\circ\) K the diffusion rate is approximately 100 times greater along grain boundaries than within the grain\({}^{1}\). The heat of diffusion \(Q\) is considerably higher in the grain, since here the atoms are more compressed and a greater activation energy is required to liberate them\({}^{2}\). In many respects, as Langmuir\({}^{181}\) and Fan Limpt\({}^{180}\) have indicated, and as metallurgists have long supposed, grain boundaries are very similar to regions of distortion inside the crystal, and in view of this one should expect higher diffusion coefficients\({}^{3}\). Zwickker found that the diffusion of C in W proceeds four times faster in fine-grained W than in a single crystal. Edmunds\({}^{77}\) reports that diffusion of Cu into Zn single crystals is not observed, although the process proceeds rapidly in polycrystalline Zn; the later experiments of Rayns showed that diffusion of Cu into Zn single crystals can be effected, but diffusion of Cu into polycrystalline Zn proceeds approximately six times faster. When Zn is extracted from a bicrystal of \(\alpha\)-brass, it disappears from the boundary much faster than from within the grains\({}^{4}\), as is shown in Fig. 11. Dix\({}^{69}\) establishes that diffusion in Al alloys is often observed along grain boundaries, which, for example, occurs in the annealing of alclad (diffusion from duralumin into pure Al). Preferential penetration here and in similar cases is accelerated by the effect of stresses. Similar results were reported by Pilling\({}^{215}\) for the diffusion of Cr from a Cr—Ni alloy into Ni. Recently it was reported that copper penetrates along the grain boundaries of \(\gamma\)-Fe in bimetallic

\({}^{1}\) The opinion has been expressed\({}^{184}\) that boundary diffusion, rather than intragranular diffusion, obeys exact regularities and depends on the mutual orientation of neighboring grains. However, experimental data on this interesting question are lacking. The discovery of such a phenomenon would mean the existence of anisotropy for boundary diffusion, which for cubic crystals seems unlikely.

\({}^{2}\) The conclusion that the work of Fond and co-workers establishes an effect of grain size only on \(A\), and not on \(Q\), is, of course, incorrect.

\({}^{3}\) The work of Deshman and Langmuir was subjected to serious criticism by Geiss and F. Limpt\({}^{92,94}\), who present data on the temperature coefficient of W containing Th and conclude that Th does not enter into a solid solution in W, but is located along grain boundaries, and also by Hertz\({}^{91}\), who attempted to explain the thermionic properties of thoriated filaments on the basis solely of boundary diffusion. Deshman and Fond\({}^{74}\), on the contrary, report that wires prepared from a mixture of W and Th, when heated in a nonoxidizing atmosphere, show values of electrical conductivity and of the temperature coefficient considerably higher than pure W; similar results were obtained for thoriated “pinch” wires annealed under reducing conditions with partial reduction to metallic Th and dissolution of Th in W. In view of this, the analytical methods used by Langmuir and Deshman for finding the diffusion coefficients seem indisputable. Recent observations by Becker\({}^{17}\) with the electron microscope indicate that Th atoms reach the surface of W rather as if by jumps than by slow and gradual movement, as follows from the diffusion equations; no satisfactory explanation of this effect has yet been found.

\({}^{4}\) Desh\({}^{66}\) establishes that Zn, during evaporation from brass, disappears faster from twin planes than from within the grain.

in places during annealing^229. In all these examples it is established that diffusion along grain boundaries proceeds faster than within them^1).

On the other hand, Seit and Keil^239 were unable to detect any effect of grain size on the rate of self-diffusion in lead. Ilim^79 reported that Zn shows no preferential penetration along the grain boundaries of copper, and this was confirmed by Rhines^204 ^2). The same absence of preferential penetration of N along the grain boundaries of α-Fe is clearly confirmed by Wells’s^204 recent observations, under the influence of the effect of grain size on the depth of nitrogen penetration^3), as may be judged

Fig. 11. Loss of zinc from a bicrystal of α-brass.

Left: A bicrystal of α-brass (70 : 30) was held for one hour in vacuum at 790°. Etchant the same as in Fig. 11; ×100; reduced by 1/3 in reproduction. Cracks along the grain boundary.

Right: the same treatment as on the left, but stained during etching. Loss of Zn around the crack (the copper-saturated portion is lighter)

from the appearance of nitride “needles.” A specimen of Armco iron purified in hydrogen, in the form of a rod 31.2 mm in diameter, with about 200 grains per 1 mm², and another of the same dimensions, with 1 grain per 1 mm², were nitrided in ammonia at 525° C for 24 hours; in both, nitride “needles” were

^1) It has been observed that electrical conductivity and self-diffusion in CaCO₃ and NaNO₃ increase as grain size decreases; thus the rate of diffusion along grain boundaries is greater than within the grain^134.

^2) Yamaguchi^299, however, observed the opposite; the reasons for this are unclear.

^3) Rowland and Appelgraw^226 recently showed that commercial steels with different sizes of austenitic “inherited” grain are decarburized in the temperature interval between A₁ and A₃ in moist hydrogen at different rates, and found, surprisingly, that coarse-grained steels decarburize faster. This, however, is not a true indication of the influence of grain size, since, as the authors point out, the fine-grained steels contain larger amounts of SiO₂ and Al₂O₃, which should impede diffusion.

visible at a depth of \(1\ \mathrm{mm} \pm 0.05\ \mathrm{mm}\), from which it may be concluded that the grain-size effect should not be more than \(5\%\). Thus, in these cases there is no influence of grain size on the rate of diffusion and acceleration of diffusion along boundaries.

If diffusion along boundaries proceeds considerably faster than within the grain, the diffusion coefficient measured in a polycrystal by the ordinary method represents something intermediate between intra- and intergranular diffusion, and not a single constant. If we consider the majority of the measurements of diffusion coefficients that have been made, then—apart from a few exceptions only just noted—no attention is paid to the grain-size effect; therefore one must reflect on the true meaning of these coefficients. It may be said that ideal diffusion is that which takes place between two single crystals, for only in this case is there no grain-size effect making the measurements unreliable; van Liempt\(^{184}\) was the only one to carry out such measurements, although, incidentally, work on the diffusion of gaseous metals into single crystals also seems quite satisfactory.

Surface Diffusion

Folmer\(^{286–289}\) and his collaborators showed that H atoms, condensing on Hg crystals, do not remain at the point of attachment, but retain a mobility that permits them to move along the surface of the crystal; that is, something analogous to surface self-diffusion occurs. This leads one to suppose that similar movement may also occur for foreign atoms, and such movements have indeed been observed in a number of cases.

Fig. 12. Types of diffusion in crystalline aggregates.

hours. The three types of diffusion are presented in the form of diagrams in Fig. 12, which is also intended to illustrate anisotropy1.

The study of activated thermionic filaments has yielded a great deal of information about true surface diffusion. Surface films of various elements on W obey the laws of diffusion and have temperature coefficients of the diffusion rate quite analogous to those for volume diffusion; this process may be understood as two-dimensional diffusion. Taylor and Langmuir \(^{270,271}\) showed that Cs is mobile on W, and found for a monatomic layer the value \(Q = 14\,000\) cal; when a second layer is added, the diffusion rate at \(500^\circ\) K increases ten thousandfold, and the value of \(Q\) is only 2,500 calories, which is in full agreement with the fact that Cs atoms, moving over a Cs film, move in a much less intense force field than when moving over the surface of W.

It was established that the diffusion coefficient changes with concentration, as in solid alloys. Becker \(^{16}\) also studied films of Cs and Ba on W. In an astonishing experiment \(^{186}\) it was shown that Th, adsorbed on one side of a filament, moves around it to the other side. Bozorth \(^{21}\) studied Na on W and found evidence of grain fragmentation accompanying the diffusion of Na, which is a very remarkable fact. The work of Langmuir and his collaborators, especially on Th on W \(^{32,181,271}\), is of particular interest because of the completeness of the data on volume and boundary diffusion in this system. Th diffuses from one W grain and reaches the grain boundary, where its motion is accelerated one hundredfold, and finally reaches the surface. At this point it either evaporates or, if the temperature is sufficiently low, diffuses along the surface at a rate increased by another factor of 10, forming a monatomic layer with one Th atom for every two underlying W atoms. This high rate is again characterized by a small value of \(Q\), since here again the atoms move while being less compressed by the force field. For Th on W \(^{180,181}\), for surface diffusion we have

\[ D = 0.47 \cdot e^{-\frac{66400}{RT}} . \tag{18} \]

The relation among the three types of diffusion of Th in W is shown graphically in Fig. 13; it has been compiled from the three equations given above and shows an increase in the diffusion rate in the sequence: volume—grain boundary—surface, as well as a decrease in the slope, or a decrease in the value of \(Q\). This is the most complete picture currently available for diffusion in any system, and may be regarded as a classic example of work in this field. Schwarz \(^{231}\) asserts that Po is mobile on the surface of Ag, but does not noticeably penetrate inward \(^{305}\); it is quite possible that the difference may consist only in the rate, since, as we have seen, in the case of Th on W surface diffusion occurs much more rapidly than diffusion along boundaries or volume diffusion. It may be supposed that mobility on the surface can occur when volume diffusion is entirely absent, but no information on this question, apart from that cited above, is available \(^{1}\). It would be very interesting to have data on surface diffusion for better-known metal systems, but methods of measurement are generally lacking here, since information has been obtained almost exclusively from the field of thermionic phenomena.

Fig. 13. Diffusion of thorium in tungsten, 1 — surface, 2 — grain boundary, 3 — volume.

Metallurgical Processes

From the foregoing it becomes quite clear that diffusion plays a very important role in many metallurgical processes. Most metallurgical processes are too complex to permit an analysis of special problems from the standpoint of diffusion,

\(^{1}\) Some relation can be found between the tendency to form oriented growth figures (orientation during crystallization on a crystalline substrate. Ed.) and surface diffusion. The deposition of isomorphous atomic networks during oriented growth is, as it were, a type of two-dimensional solid solubility; it requires surface diffusion, just as the formation of a solid solution requires volume diffusion.

but cases in which diffusion plays a role are so frequent and promise so much that some of these problems must, if not be solved, at least be posed. The carburizing process, which requires diffusion in order to be able to proceed, is especially complex. Bramley’s works \(^{23\text{--}31}\), although not entirely satisfactory, give a large amount of information about this process. Bramley derives the diffusion coefficient from experiments both on carburizing and on decarburizing, with identical results if carburizing is carried out by means of CO and a hydrocarbon, or decarburizing in “moist” H; if, however, mixtures of CO and \(\mathrm{CO_2}\), on the one hand, and “moist” H, on the other, are used, different diffusion curves are obtained. Apparently, in some cases the rate of the process is determined rather by the rate of reaction at the surface than by the rate of diffusion itself, as is usually assumed. It has been established in practice that not only Fe and C take part in carburizing and decarburizing, but also many other elements, especially O and N (and sometimes H), present in carburizing gases, and O, N, P, and alloying elements in solid steel. Moreover, the chemical identity of compounds arising in the gas phase at the surface separating the gas and the metal with those which form on the very interface in reaction with the metal has not yet been established. A physicochemical analysis of the carburizing process (and of nitriding) is very necessary, since many peculiarities have been noted here \(^{115,139}\). From the point of view of physical chemistry this field has scarcely been investigated. Such an analysis must take into account the nature and rate of surface reactions and the influence of the additional elements present initially in the steel and added during the process.

The uncertainty of the existing theories of decarburizing is evident from the discussion of the recently published paper by Rowland and Lantergo \(^{226}\). Gudremon and Schroeder \(^{140}\) have recently carried out an investigation of the rate of carburizing of a series of alloy steels and concluded that Cr, Mo, and W reduce the rate of diffusion; C, Al, and Si have an uncertain effect, while Ni, Co, and Cu increase it; these conclusions were drawn on the basis of concentration curves and chemical analysis of the surface—these experiments were not intended to determine true diffusion coefficients, and the conclusions seem not entirely well founded. The work of Baukloh and Gutmann \(^{15}\) on the decarburizing of alloy steels is likewise of little use for clarifying the role of diffusion. Tammann’s work \(^{262}\) on the carburizing of Fe alloys seems too qualitative to be genuinely useful, although his investigation should be continued with a more careful analysis of the carburizing conditions and of the rate of diffusion; properly conducted experiments would add much valuable information about carburizing, as well as about reactions in steel, as is evident from the discussion presented. Homogenization of alloys obviously requires diffusion,

and the rate of homogenization, at least in pure binary systems, can be readily explained on the basis of the diffusion rates of the metals entering into them. Weiss[^295], following Tamman’s proposal[^261], found that the rate of homogenization in segregated castings of AgSb alloys with 14% Sb, containing primary precipitates of Ag$_3$Sb, changes exponentially with temperature, and therefore—

Fig. 14. Change in electrical resistance during aging of an aluminum–silver alloy with 38% silver

Fig. 15. Initial rates of aging at different temperatures of an aluminum–silver alloy with 38% silver

therefore proposed an exponential law for the diffusion rate. Thus the time of “equivalent homogenization” at different temperatures can be predicted in advance on the basis of a single experiment, using equation (11), instead of determining it by numerous experiments.

In ternary alloys we encounter a more complicated case, owing to the fact that the diffusion rates of the constituent metals vary within wide limits; but it is quite evident that the more slowly diffusing element determines the controlling rate, as, for example, in segregated FeMnC alloys, in which the rate of dissolution of Fe$_3$C containing Mn is much lower than that for Fe$_3$C in simple carbon steels, apparently owing to the lower diffusion rate of Mn. These cases, however, are complex and have not been studied in detail.

Deschmann[^73] in 1929 pointed out that many metallurgical processes, such as work hardening during cold working, and the general ability of alloys to react, must depend on the state of energy

*

of the atom, and therefore must also obey, apparently, exponential laws similar to those given for diffusion in equation (11). From the aging theory of Merica, Waltenberg, and Scott\(^{208}\) it follows directly that diffusion takes place in the aging process\(^{148,202}\). But there has as yet been no quantitative analysis of its role. Seig and Goss\(^{138}\) in 1933 showed that the initial rate of precipitation of the \(\gamma\)-phase from the terminal solid solution of Ag in Al (measured by means of electrical resistance) is an exponential function of temperature and can be expressed by an equation of type (11). The aging curves of the alloy with 38% Ag are presented in Fig. 14, and the logarithms of the initial rate plotted against the reciprocal of the absolute temperature (Fig. 5) are shown in Fig. 15. The value \(Q\) derived from this diagram is of the order of 22,000 cal; it is quite acceptable for the given system.

It is very likely that the rate of diffusion is the determining factor for the rate of aging, but this argument would be more convincing if it could be shown that the value of \(Q\) found is identical with that derived from direct measurements of the rate of diffusion; work in this direction is now being carried out. In the past year Jenkins and Bucknall\(^{149}\) have shown that the time required to bring an alloy being treated by aging to its maximum hardness or strength, when plotted in logarithmic form against the reciprocal of the aging temperature, gives a straight line, as shown in Fig. 16, thus linking the rate of aging with the rate of diffusion\(^{1}\). Here again it is desirable to compare the derived values of \(Q\) with those determined from measurements of the rate of diffusion. The values of \(Q\) derived from these graphs seem too low. Bucknall extrapolated the curve for duralumin to room temperature and calculated that at this temperature duralumin should show no “overaging” (decrease in strength) for a million years.

Fig. 16. Aging rates at different temperatures for Lautal alloy (Lautal), measured by the time required to attain maximum tensile strength\(^{149}\)

\(^{1}\) The rates of precipitation during aging are connected with questions of the stability of supersaturated solid solutions and therefore are not determined solely by diffusion coefficients. The latter, however, is true for dissolution, at elevated temperatures, of the phase precipitated during aging. See Izv. Akad. Nauk, Chemical Series, No. 5, p. 1209, 1937. —Ed.

It is very important to achieve an exact correlation between the rate of aging and the rate of diffusion, since information established in one field supplements information from the other. At the same time, a systematic program of investigation of aging systems with predetermined behavior could be developed. It can be shown qualitatively that this path is very promising, since, for example, systems consisting of refractory metals, such as FeMo, FeW \(^{259}\), and CuBe, in which the diffusion coefficient could be high only at high temperatures, require high temperatures for an appreciable aging rate, whereas systems consisting of low-melting metals, such as, for example, duralumin, age appreciably already at room temperature. We may also note that the rate of aging of \(\alpha\)-Fe alloys with C and N, as was already indicated by Bain and Davenport \(^{12}\), has an order of magnitude of their diffusion coefficients in \(\alpha\)-Fe, and consequently, as may be supposed, also in \(\gamma\)-Fe. We may also suppose that “overaging” in \(\alpha\)-Fe, which, as we now know, is caused by O, is also a precipitation process requiring deformation to accelerate precipitation, since the rate of diffusion of O in \(\alpha\)-Fe is small \(^{1}\).

However, we must proceed with great caution. If we treat Koster’s data \(^{172}\) on the rate of aging of \(\alpha\)-Fe alloys with C by the method presented in Fig. 16, we obtain a value of \(Q\) of 5,000 cal. At the same time, Bailey and Roberts \(^{10}\) plotted on a diagram the time of complete spheroidization of pearlite and concluded that the linearity of the curves obtained proves that spheroidization is a simple process of agglomeration controlled by the rate of diffusion \(^{297,298}\); however, the value of \(Q\) derived from Bailey’s data is 28,000 cal, i.e., a value five times greater than that derived from Koster’s data on aging. This discrepancy is too serious to be disregarded.

Can we compare such diverse data; can we directly compare hardness data and electrical-conductivity data; are we justified in assuming that equal degrees of change in these properties mean equal degrees of precipitation? Of course, a discussion of the details of the aging process \(^{254}\) is hardly necessary here; however, it may be assumed that preferential precipitation of the aging phase at grain boundaries may be associated with a high rate of diffusion along grain boundaries, and that this effect should be absent in systems in which diffusion along grain boundaries proceeds no faster than diffusion within the grains. On superficial consideration this idea seems attractive; however, we must admit one uncertainty—the grain boundary, precisely because of its characteristic disorder, itself

\(^{1}\) Here again it may be assumed that the rate of diffusion of O in \(\alpha\)-Fe is less than the diffusion of C or N, since it is less in \(\gamma\)-Fe.

contributes to the formation of grains, and although grain formation requires diffusion, we cannot conclude that the number of nuclei is proportional only to the rate of diffusion.

The aging of ternary alloys is accompanied by complex diffusion effects. If a compound is to precipitate from a solution—for example, such as Mg$_2$Si from Al—then it is clear that the two atoms, in order to make up the compound, must diffuse with the appropriate relative rates, whatever their normal rates may be. Fricke $^{87}$ assumes that in this case the diffusion rate of the more slowly diffusing metal controls the overall rate of precipitation. It is also possible in such cases that, at the beginning of precipitation, a phase appears different from what might have been expected, as was shown by Mehl, Barrett, and Rhines $^{199}$ for the case Al—Mg—Si. It should be noted that such experiments make it possible to test the presence or absence of molecules in a solid solution. For example, diffusion into pure copper of a solution of the “compound” Ni$_2$Sn in copper may furnish such evidence, for if Ni$_2$Sn molecules really exist in the solution as such, they must diffuse as a whole, and the distribution curves should show this.

We may note here that an increase in the rate of aging as a result of cold working can be only one of the causes, since we must also not forget the acceleration of the nucleation of grains.

In transformations of the austenite–pearlite type we have less that is reliable, although an understanding of the mechanism of these reactions is of fundamental importance for the heat treatment of steel $^{1}$). These processes differ from the aging process by an allotropic transformation of the phase that is the solvent. This first causes an increase in the reaction rate at low temperature and then a decrease, as shown in Bain’s work $^{11}$. This is undoubtedly the result of the combined action of two processes, as Austin $^{9}$ suggested: one proceeding at a rate that decreases with decreasing temperature—perhaps the diffusion of C in austenite during the formation of the carbide Fe$_3$C, as I indicated in the discussion of Austin’s paper—and another proceeding at a rate that increases with decreasing temperature, perhaps at a rate determined simply by the change in free energy, although an orthodox thermodynamicist may insist that there is no formal correlation between the change in free energy and the reaction rate. It is quite clear that the formation of pearlite requires diffusion $^{2}$), forma-

$^{1}$) The capacity for reaction in solid inorganic systems has attracted much attention, and it has been shown that it is precisely the rate of diffusion that determines the effect $^{120–123,280,285}$. The rate of formation of superstructures is likewise governed by the rate of diffusion.

$^{2}$) The influence of a decrease in grain size on the increase in the rate of decomposition of austenite can hardly be attributed to an increase in the rate of diffusion at grain boundaries, for, as indicated above in this article, the rate of diffusion of C along the boundaries of $\gamma$-Fe grains does not exceed that within the grains.

the formation of martensite does not require diffusion, since the distribution of C in the initial “white” stage of martensite is exactly the same as in austenite, i.e., statistically homogeneous.

The influence of alloying elements that slow the rate of decomposition of austenite is undoubtedly associated with a decreasing rate of diffusion. Recent experiments carried out in my laboratory on the rate of decarburization of Hatfield Mn-steel and of ordinary C-tool steel with the same C content revealed the same rate in both steels, so that we can hardly attribute the retarding influence of Mn on the rate of decomposition of austenite to an influence that slows the rate of diffusion of C in austenite; we are more inclined to explain the retardation by the necessity of simultaneous diffusion of Mn, required for the formation of carbide containing Mn^296, and by the slow diffusion of Mn in austenite as compared with C^213,296. However, Ni likewise delays the decomposition of austenite, but it apparently does not form carbides in steels, so that our first explanation is scarcely applicable here. In Mn-steels the carbide contains Mn in excess as compared with austenite, and thus Mn must diffuse, for example, from the comparatively thin carbide layer in pearlite, whereas in Ni-steels the carbide contains little nickel, or none at all, and it must diffuse from the relatively small regions occupied by the carbide into relatively thick layers of ferrite. These phenomena are very complex from the standpoint of the rate of diffusion, but properly performed determinations of diffusion coefficients in alloy steels should nevertheless lay the foundation for a scientific substantiation of the factors governing the rate of decomposition of austenite.

Conclusion

From all that has been presented above it is clear that, in the question of diffusion in solid metals, much experimental work still remains to be done, both to determine the diffusion coefficients themselves, which is a relatively simple matter, and to elucidate the behavior of alloys, which in many cases may be extremely complex. The study of so important a question should also be recommended to metallurgists, since the development of physical metallurgy is of enormous importance for them. Metallurgy is in the fortunate position that everything still lies ahead of it; it still has much to discover in the way of simple facts and theories, and the development of physical science in its field must lead to practical applications of broad significance.

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1. Surface mobility occurs in many systems, and in a number of cases the phenomenon should rather be described not as diffusion but as spreading. For example, it is known \(^{248}\) that ThB is mobile on quartz and that a layer of benzophenone is mobile on the surface of glass, mica, and diamond, the rate of spreading in the latter case being characterized by a high temperature coefficient. The spreading of a drop of mercury over a metallic surface is the same kind of phenomenon \(^{4,95,204}\). In this case the texture produced by cold rolling of the foil is the cause of the formation of ellipses by the spreading mercury. In all these and similar cases, of course, what is involved is not diffusion at all, but surface propagation (spreading) in a liquid or liquid–solid system. ↩↩↩↩↩↩↩↩

2. This phenomenon could be observed only at very large distortions. In ordinary cases what is observed is not broadening, but a weakening of line intensity. — Ed. ↩↩↩

3. Some investigators have tried to prove the influence of pressure on the rate of diffusion[^54], but without success[^210,^237]; it is possible that for demon- ↩

4. Superscript references visible on the page: 134, 135, 238, 242, 244. ↩

5. Ilam. ↩↩

6. Reference 204. ↩↩↩↩

7. Smith. ↩

8. Reference 105. ↩

9. Reference 108. ↩

10. Reference 109. ↩

11. Smekal. ↩↩

12. Reference 249. ↩

13. Reference 250. ↩