ON THE SOLUBILITY OF ELEMENTS IN DISPERSION-HARDENING ALLOYS

A. D. KLEMENT'EV

(Presented by the scientific seminar of the Department of Metal Technology, TPI)

The question of the solubility of elements in dispersion-hardening alloys arose in connection with the study of alloying of iron–cobalt–tungsten alloys [1]. The introduction of alloying elements into dispersion-hardening alloys, in addition to influencing the concentration of the solid solution, leads to a change in the mutual solubility of the elements with temperature. The solubility conditions of the elements in these alloys, naturally, do not fully determine the process of dispersion hardening, but they are a most important condition for the latter (to the extent that the amount of the precipitating phase is connected with the strengthening of the alloy).

The question of the temperature dependence of solubility, which is of primary importance in alloying dispersion-hardening alloys, is one of the unresolved questions in the problem of estimating the solubility of elements. Here it is possible for the present to clarify only qualitative regularities, but these too are important milestones in the selection of alloying elements. For an approximate, qualitative estimate of the solubility of elements in iron–cobalt–tungsten alloys, the hypothesis of I. I. Kornilov [2], which applies to solid solutions of iron and its analogues, may be used.

In the present article, the temperature dependence of the solubility of elements is considered chiefly. As for estimating the degree of solubility of elements, it may be carried out with the aid of a small number of simple, practically convenient criteria, such as: the volume factor, the type of crystal lattice, compressibility, etc. It is also possible to use other solubility criteria (heat capacity, electron density, ionization potential, etc.).

Since the mutual solubility of elements is connected with the difference in their atomic diameters, which characterizes the degree of distortion of the crystal lattice of the solid solution, the influence of temperature on solubility must, according to I. I. Kornilov, be due to the corresponding change in the difference in atomic diameters (provided there are no transformations in the alloy that exert a sharp influence on solubility). As an indicator of the change in atomic diameters with temperature, in a first approximation, the coefficient of linear expansion [2] may serve; it reflects the thermal state of atoms at different temperatures.

Thus, the temperature dependence of the mutual solubility of elements is connected with the influence of temperature on the volume factor. For dispersion-hardening alloys, a favorable change in the volume factor consists in a decrease in the difference between the atomic diameters of the elements of the solid solution with increasing temperature (an increase in solubility). This regularity also expresses the basic requirement imposed on alloying elements from the standpoint of their solubility in dispersion-hardening alloys.

Fig. 1 presents the graphical interpretation proposed by us of the temperature dependence of the mutual solubility of elements. Points $A$, $B$, and $C$ represent the values of the atomic diameters, respectively,

Fig. 1.

of the solvent and the additions. The curves represent the change in atomic diameters with temperature, as determined by the coefficients of linear expansion of the elements. The convergence or divergence of the curves characterizes an increase or decrease in the difference between the atomic diameters of the elements and the corresponding change in the conditions of solubility. Hence follows a rule determining the conditions for favorable solubility of elements in dispersion-hardening alloys: with increasing temperature, the curves of change in atomic diameters of the corresponding elements must converge. In practice, the assessment of the solubility of elements reduces to a comparison of the values of the atomic diameters and the coefficients of linear expansion of the solvent and additions at ordinary temperatures. If other elements are present in the solid solution that substantially affect solubility, it is advisable to make an analogous comparison with the coefficients of linear expansion of these elements as well.

The simplicity of estimating the temperature conditions of solubility of elements follows from an analysis of the regularities of the change in the coefficient of linear expansion with temperature [2, 3]. The coefficient of linear expansion of various elements increases with increasing temperature, and to a greater degree the larger its value at ordinary temperature. Further, if the coefficient of linear expansion of one element is greater than that of another at ordinary temperature ($t_0$), then this fundamental difference in the values of the coefficients of linear expansion is preserved also at higher temperatures ($t_1$). Consequently, for a qualitative comparison of the change in solubility with temperature,

the values of the coefficients of linear expansion at ordinary temperature may be used.

The course of the curves (Fig. 1) qualitatively reflects the actual change in the atomic diameters of the elements with temperature, whereas the inclination of the curves to the temperature axis (at \(t_0\)) corresponds to the values of the coefficients of linear expansion at ordinary temperature (the true coefficient of linear expansion is the tangent of the angle of inclination of the tangent drawn to the curve of change in the linear dimensions of a specimen at the temperature for which the value of the coefficient of linear expansion is determined). To simplify the graph, the curves may be replaced by straight lines (and then the angle of inclination of the straight lines to the temperature axis will correspond to the mean values of the coefficients of linear expansion.

The graphical representation of the conditions for the solubility of elements makes the evaluation and selection of alloying elements clear and simple, which is especially important in cases where the solubility of many elements in a complex solid solution is being compared, and also makes it possible to analyze the solubility of elements in complex alloy compositions (as many modern engineering alloys are).

Let us turn to an evaluation of the solubility of alloying elements in iron–cobalt–tungsten alloys.

Table 1

According to Fig. 1 and the data of [3, 4] in Table 1, which gives the values of the atomic diameters (for coordination number \(K=12\)), the differences in atomic diameters from iron, and the coefficients of linear expansion for a number of elements, the condition outlined above for favorable solubility in a complex solid solution is fully satisfied (that is, with respect to all components of the system: iron, cobalt, and tungsten)

for such elements as molybdenum, chromium, vanadium, and beryllium. For titanium, tantalum, niobium, and silicon this condition is satisfied only with respect to iron and cobalt, and for manganese—only with respect to tungsten.

However, I. I. Kornilov’s hypothesis can be valid only under the condition of equivalence of the elastic interaction of atoms in the solid solution, which, as is known [5], does not occur in reality. This explains the “anomalies” that arise in connection with the use of this hypothesis in explaining certain solubility phenomena, in particular the solubility of elements in systems of dispersion-hardening alloys. Consequently, I. I. Kornilov’s hypothesis must be supplemented by taking into account the essential features of the interaction of atoms in the solid solution that are associated (which is especially important for dispersion-hardening alloys) with the temperature dependence of solubility.

Analysis of the conditions of mutual solubility of elements indicates the presence of solubility “anomalies” for elements of relatively low elasticity. At the same time, it is known [5] that the solubility of elements depends on their compressibility, which reflects the elastic energy of distortion of the crystal lattice of the solid solution. If the additive has a greater compressibility than the solvent, then in the force field of the solvent atoms its atomic diameter will decrease, and to a degree that is greater the greater the compressibility. In a solid solution this will affect the change in the volume factor and, consequently, the conditions of mutual solubility of the elements.

With increasing temperature, compressibility increases. Besides direct experimental data [3, 6], this follows from the relation of compressibility to other elastic constants [7].

From the relation between solubility and compressibility and the dependence of the latter on temperature follows the influence of compressibility on the temperature dependence of solubility. A significant influence of compressibility on solubility, not only at ordinary temperature [6] but especially at high temperatures, follows from analysis of experimental data [3, 6]. Thus, for aluminum, which is characterized by low elasticity, upon heating over an interval of 400° the linear dimensions increase (in accordance with the coefficient of linear expansion) by 1.37%, while the compressibility increases by approximately 15% [6].

Fig. 2 illustrates the influence of compressibility on the volume factor and its change with temperature. The curves \(Aa\) and \(Dd\) represent the change in atomic diameters, determined by the coefficients of linear expansion, of the solvent and of the additive, respectively, with increasing temperature. If the additive has a greater compressibility than the solvent, then its atomic diameter will decrease, and to a degree that is greater the greater the compressibility and the higher the temperature; graphically this is expressed in the displacement of the curve \(Dd\) to the position \(D_1d_1\), and, with still greater compressibility of the additive, to the position \(D_2d_2\). The resulting convergence of the curves of the solvent and the additive at ordinary \((t_0)\) and especially at high \((t_1)\) temperatures indicates a corresponding change in the volume factor and, consequently, in the solubility conditions. Improvement of solubility in connection with increased compressibility will be observed for all additives with an atomic diameter larger than that of the solvent. Similarly, it can be shown that for additives of increased compressibility with an atomic diameter smaller than that of the solvent there will occur an unfavorable, from the standpoint of solubility, change in the volume factor, especially at high temperatures, as follows from the positions of the curves \(Ee\) and \(E_1e_1\) (for an additive with increased compressibility) relative to the curve \(Aa\).

Owing to the high compressibility of manganese, chromium, and aluminum (according to the data of Table 1), their solubility conditions in iron–cobalt–tungsten alloys must differ from those outlined above (without taking the compressibility factor into account).

Thus, the elements listed in Table 1 may be classified as follows according to their solubility conditions in iron–cobalt–tungsten alloys. Manganese, nickel, and vanadium are elements of favorable solubility. The solubility of manganese in iron–cobalt–tungsten alloys is a characteristic example of the influence of the compressibility factor on the conditions of mutual solubility of elements. According to Fig. 2, if curve $Aa$ corresponds to iron and cobalt, then the curves corresponding to manganese are: $Dd$—without taking compressibility into account (unfavorable solubility), and $D_2d_2$—with compressibility taken into account (favorable solubility). The situation is different with the solubility of manganese in a complex solid solution relative to tungsten. If, according to Fig. 2, curve $Aa$ corresponds to tungsten, then manganese corresponds to—

Fig. 2.

$Ee$—without taking compressibility into account, and $E_1e_1$—with compressibility taken into account (deterioration of solubility). Consequently, the advisability of alloying with manganese increases as the tungsten content in the alloys is reduced. Titanium, niobium, and tantalum are elements with a profound and complex influence on solubility. These elements satisfy the requirements of the temperature dependence of solubility with respect to iron and cobalt, but do not meet these requirements with respect to tungsten. The unfavorable nature of the volume factor is further aggravated by the presence in the alloys of tungsten, with its large atomic diameter. Consequently, titanium, niobium, and tantalum must be distinguished by very limited solubility in iron–cobalt–tungsten alloys, which can be increased by lowering the tungsten content in them (which is of interest from the standpoint of replacing tungsten). The presence of tungsten in the alloys complicates and lowers the solubility of silicon (unfavorable temperature dependence) and beryllium (unfavorable volume factor), bringing their solubility conditions closer to the preceding group of elements. The high compressibility of chromium indicates a worsening of the temperature conditions of its solubility relative to tungsten (according to Fig. 2, tungsten corresponds to curve $Aa$, and chromium to $Ee$ when compressibility is not taken into account and to $E_1e_1$ when compressibility is taken into account). However, a very

the favorable volume factor relative to iron and cobalt makes chromium a promising alloying element, provided that the tungsten content in the alloy is reduced. The unfavorability of the volume factor and of its temperature variation for aluminum is compensated by its very high compressibility, which permits one to count on the use of aluminum as an alloying element in iron–cobalt–tungsten alloys. Molybdenum is analogous to tungsten, which forms the basis of the alloys. The unfavorable solubility conditions for boron and zirconium indicate the possibility of using these elements mainly as modifiers.

Let us turn to an analysis of some systems of dispersion-hardening alloys from the standpoint of the correspondence of their solubility to the regularities that have been outlined.

Let us first consider the solubility conditions of the components of the iron–cobalt–tungsten system. As is known, binary iron–tungsten alloys are distinguished by a high degree of dispersion strengthening; the addition of cobalt promotes their further strengthening. According to Fig. 3 and the data of Table 1, the curves of variation of the atomic diameters

Fig. 3.

of iron and tungsten approach each other with increasing temperature, which indicates a favorable change in the volume factor and, consequently, in solubility. With an analogous change in temperature, the cobalt curve approaches the curves for iron and tungsten. The closeness of the atomic diameters of iron and cobalt ensures their high mutual solubility. Thus, from the standpoint of solubility, the introduction of cobalt into the base of iron–tungsten alloys receives a clear justification.

Let us turn to iron–titanium alloys. Both components of the iron–titanium system satisfy the condition of a change in the volume factor with temperature determined by the course of the curves \(Aa\) and \(Cb\) (Fig. 1, Table 1). According to the study [8], silicon and nickel increase the dispersion strengthening of iron–titanium alloys, whereas manganese, chromium, and aluminum have no effect on it. The positive influence of silicon and nickel on the hardening effect follows from the conditions of their solubility in the solid solution at low and high temperatures. If, according to Fig. 1 and the data of Table 1, the change in the atomic diameter of iron with temperature corresponds to the curve \(Aa\), then the analogous curves for silicon and nickel will be, respectively, \(Cb\) and \(Bc\). Consequently, with increasing temperature the curves converge (solubility improves). It is natural

is also the fact that in the study [8] no significant influence of chromium, manganese, and aluminum on the process of dispersion hardening was found. With a titanium content in the alloy of not more than 3% [8, p. 51], the introduction of the indicated elements of high solubility cannot ensure a noticeable increase in the supersaturation of the solid solution. The above considerations are confirmed by data from the study [9] of iron—titanium—silicon alloys, in which the solubility of titanium in the solid solution is reduced by the addition of silicon. The introduction of chromium into these alloys leads to a substantial improvement in dispersion strengthening.

In iron—titanium—silicon—aluminum alloys, dispersion strengthening is increased by chromium, tungsten, molybdenum, and vanadium [9]. The alloying elements, as well as the elements of the alloy base, have a favorable change of the volume factor with temperature (according to Fig. 1 and the data of Table 1, the change in the atomic diameters of these elements follows curve $C_{\delta}$).

The positive influence of chromium and silicon on the conditions of dispersion hardening of iron–beryllium alloys [10, 11], as well as the components of this system themselves, is in complete agreement with the requirement for the volume factor and its temperature dependence for dispersion-hardening alloys. The elements nickel and beryllium follow curve $B_{\delta}$, which converges with curve $C_{\delta}$, followed by chromium and silicon.

The examples given are a good illustration of the condition of mutual solubility of elements in dispersion-hardening alloys.

In studying the alloying of iron–cobalt–tungsten alloys, the regularities obtained above make it possible to create a definite direction in the study of the influence of alloying elements on alloys, to classify elements by solubility, and also to reduce significantly the scope of the experimental part of the investigation.

The estimate outlined in the present article of the solubility of elements in iron–cobalt–tungsten alloys is in good agreement with experimental data [1]. Thus, for example, when alloys with 20% cobalt and 20% tungsten are alloyed, the optimum content of additions of titanium, niobium, and silicon, owing to the high tungsten content in the alloys, is indeed small (up to 1%), while causing a substantial increase in dispersion strengthening. The role of the compressibility factor is very clearly manifested in the assessment of manganese as an alloying element. As was to be expected, the introduction of manganese into the alloy (its content up to 4% was studied) is highly favorable to dispersion hardening. Without taking compressibility into account, the estimate of the influence of manganese would have been directly opposite, being in contradiction with experiment.

Despite the good agreement of the qualitative regularities of solubility established in this work with the experimental material considered, it seems to us that, because of the complexity of the real conditions of interaction of elements, the character of the temperature dependence of the solubility of individual elements will differ from that assumed. This should especially be expected in those cases when the change in the thermal state of atoms, determined by the coefficient of linear expansion, or their elastic interaction, characterized by compressibility, is overlapped by the factor of chemical interaction of the elements or by their influence on the gamma–alpha transformation in the alloy.

Conclusions

1. A qualitative estimate of the solubility of elements in dispersion-hardening iron-base alloys can be made on

on the basis of I. I. Kornilov’s hypothesis (which relates the volume factor to the coefficient of linear expansion), supplemented by the compressibility factor.

1. The use of the compressibility factor and its temperature dependence makes it possible to obtain an estimate, consistent with experiment, of the solubility of elements with reduced elasticity (in comparison with the solvent—iron), which is not the case when the elastic properties of the elements are not taken into account.

2. The paper gives a graphical interpretation of the conditions for the temperature dependence of solubility and formulates the resulting requirements for the solubility of elements in dispersion-hardening alloys.

3. The qualitative estimate of the solubility of alloying elements in dispersion-hardening alloys outlined in the paper is well confirmed in various systems of iron-based dispersion-hardening alloys and in an experimental study of the alloying of iron–cobalt–tungsten alloys.

References

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2. I. I. Kornilov, Iron Alloys, vol. 2, Moscow–Leningrad, Publishing House of the Academy of Sciences of the USSR, 1951.

3. M. P. Savinskii, Physicochemical Properties of the Elements, Moscow, Metallurgizdat, 1952.

4. Metal Science and Heat Treatment. Handbook, Moscow, Metallurgizdat, 1956.

5. K. K. Umanskii and Ya. S. Umanskii, Physical Metallurgy, Moscow, Metallurgizdat, 1955.

6. F. Birch et al., Handbook for Geologists on Physical Constants, Moscow, IL, 1949.

7. Ya. I. Frenkel, Introduction to the Theory of Metals, Moscow, Publishing House of Physico-Mathematical Literature, 1958.

8. R. Wasmuhl, Archiv für das Eisenhüttenwesen, No. 1, 45–56, 1931.

9. I. E. Kontorovich, “Vestnik metallopromyshlennosti,” No. 8, 43–50, 1935.

10. I. S. Gaev and R. S. Sokolov, “Metallurg,” No. 4, 42–48, 1937.

11. H. Wentrup and P. Schafmeister, “Archiv für das Eisenhüttenwesen,” No. 12, 613–620, 1932.