Interaction of chromium with solute carbon and solute nitrogen in iron
鉄中の固溶炭素・固溶窒素とクロムの相互作用
NUMAKURA Hiroshi
Department of Materials Science, Graduate School of Engineering,Osaka Prefecture University
1. Introduction
Quantitative knowledge on interaction between alloying elements and C and N in iron is importantin steelmaking and in controlling the properties of engineering steels. Effects of tertiary components
on the solubility of carbon and that of nitrogen in molten iron were extensively studied in the 1950s,
and the accumulated data of ‘interaction coefficient’ [1] have been utilized in producing, for example,
high purity steels. For solid phases, effects on the solubilities of C and N in γ iron were also studied
for some important alloying elements such as Mn, Cr, and V, together with solubility products [1]. For
α iron, on the other hand, much fewer data are available, and not many of them are reliable, eventhough the interaction between the solute species plays critical roles in properties of ferritic steels. The
experimental difficulties for α iron are (1) very limited solubilities of C and N, and (2) slow kinetics,
which requires a long time to attain true equilibrium.
We have been studying the interaction between the interstitial (i) solutes (C and N) and alloying
elements, most of which are substitutional (s) solutes, in α iron through the influence of the latter on
the Snoek relaxation of the former [2, 3, 4]. The Snoek relaxation is a well-known anelastic relaxation
effect due to stress-induced diffusional motion of the interstitial solute atoms [5]. If the substitutional
solute atom affects the potential energies of the interstitial sites around it, the rates of diffusional
jumps of the interstitial solute atom become different from those in unalloyed iron. This can be usedconversely to evaluate of the interaction between the i and s solute atoms, or ‘i–s’ interaction [6]. Thismethod does not suffer much from the two difficulties mentioned above: the mechanical relaxation cansensitively detect C and N of concentrations as low as 10 at. ppm, and the method is to observe single
atomic jumps of C and N in the paraequilibrium state distributed in the host alloy crystal.
In a previous work we studied the interaction between N and Cr in α iron studied by measuring the
Snoek relaxation in a series of dilute Fe–Cr–N alloys [3, 4]. We continue to work on the importantalloying element Cr; in the present paper we report the interaction between C and Cr studied by thesame method. In this study, in addition, we have evaluated the N–Cr and C–Cr interactions by thethermochemical method, which was employed in many of the past studies on liquid iron and γ ironmentioned above: measuring the dependence of the equilibrium solubility of C or N on theconcentration of Cr. We discuss how these two experiments compare with each other in theory and inpractice.

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2. Interaction coefficient and interaction energy
Traditional work on evaluating the interaction between different components is based on Wagner’s
theory of the thermodynamics of multicomponent solutions [7]. Here we show the relation between
the thermodynamic interaction coefficient and the atomistic interaction energy [8, 9, 10, 11], which are
the central parameters characterizing the influence of the tertiary element on the solubility and atomic
jumps, respectively, of the interstitial solute.
In multicomponent alloys, the activity of component i in the solution, which can be written asai = γi xi(1)

for dilute solutions, where γi is the activity coefficient and xi is the concentration (mole fraction) of i.
The interaction coefficient, which describes the influence of component j on the activity of i, isdefiend as⎛ ∂ ln γ i ⎞,⎟⎝ ∂x j ⎠ xi ,x j →0ε i( j ) ≡ ⎜(2)

where xj is the concentration of j. This can be evaluated by measuring the equilibrium solubility (notthe solubility limit) of i under an identical activity of i in a series of dilute alloys differing in theconcentration of j. For dilute solutions where the effect of j can be approximated as linear on
concentration, the activity may be expressed asln γi = ln γi° + εi(j) xj ,(3)

where γi° is the activity coefficient in the absence of j. In this case, the effect of j on the equilibrium
solubility is written as()

xi = xi ° exp − ε i( j ) x j .

(4)

The procedure of evaluation described above is evident from this expression. When εi(j) is negative, forexample, the solubility of i is increased by the addition of j, indicating attractive interaction.
To connect the phenomenological thermodynamics to an atomistic theory, we consider variations of
the potential energies for the interstitial solute atom, ‘i’, around a substitutional solute atom, ‘s’, asshown in Fig. 1. For simplicity we assume that only the energy of the interstitial site closest to the satom is changed by ΔE. Generally the energies at farther sites may be influenced as well as thesaddle-point energies, but we consider here a model with only two distinct sites, referred to as a
two-level system. The i atoms distribute over the two groups of sites, one with a lower energy (trapsites) and the other, normal sites, with fractions depending on the energy and population of the trapsites. By applying Fermi–Dirac statistics to the distribution, a relation between the chemical potential

μi (= kT ln ai) and the interaction energy ΔE can be derived. Then the expression for the interaction
coefficient is obtained from Eq. (2). The result is

ε i( j ) =z ⎡⎛ ΔE ⎞ ⎤.
1− exp ⎜ −⎢⎝ kT ⎟⎠ ⎥⎦m⎣(5)－ 92 －

Fig. 1

Bottom: An interstitial solute atom ‘i’ at the nearest
neighbour octahedral interstitial site of a substitutional
solute atom ‘s’ in a body-centred cubic crystal. Cross
symbols indicate farther interstitial sites. Top: Profile ofthe potential energy for the ‘i’ atom. Numbers indicate theclassification of the sites according to the distance fromthe ‘s’ atom.

Here, z is the number of trap sites per ‘s’ atom, m is the number of interstitial sites per host atom site (3for octahedral interstitial sites in a bcc crystal), k is Boltzmann constant, and T is temperature.
3. Experimental procedure
3.1. Materials
Pure iron and the five dilute Fe-Cr alloys were used in this work. The alloys prepared from highpurity iron and chromium by argon arc melting. Their nominal compositions are 0.3, 0.6, 0.9, 1.2, and
1.8 at. % Cr, and the Cr concentrations analysed by the inductively-coupled plasma analysis are 0.24,
0.60, 0.85, 1.20, and 1.77, with error margins of ± 0.01, all in at. %. Wires of 1 mm in diameter and
50 mm in length were prepared by cold-rolling and drawing from button-shaped ingots. To eliminate
deformation textures possibly formed by the cold-work, the wires were annealed at 1223 K (above theα↔γ transformation temperature) for 1 h in a vacuum. The wires were then subjected to annealing in
an atmosphere of wet hydrogen (hydrogen gas of 1 atm fed through water of 310 K) at 973 K for 24 h.
The DC electrical resistivity and the Snoek relaxation proved that the contents of C and N werereduced to below the detectable limit, a few at. ppm, by this treatment.
3.2. Nitriding and carburizing
Either nitrogen or carbon was introduced to the hydrogen-treated wires by placing them in a flowof a mixture of high purity hydrogen and ammonia or methane [12, 13, 14]. The total pressure was
1 atm and the flow rate was about 200 ml min–1. The activity of nitrogen or carbon was controlled by
the proportions of the gas components and temperature. A typical setting was 98% hydrogen and 2%
ammonia, 96% hydrogen and 4% methane. The absolute flow rate was electronically regulated within±1 ml min–1 but was different each time by 10 or 20 ml min–1, for the sake of conveniently specifying
the desired proportions of the gas components.
3.3. Mechanical relaxation measurements
Mechanical loss at small strains was measured using a low-frequency inverted torsion pendulum
instrument by the free-decay method. The resonance frequency and the logarithmic decrement were
measured with an initial strain amplitude of 10–5, while temperature was changed continuously with－ 93 －

time. The frequency of vibration was about 1 Hz. To suppress magnetomechanical damping, a static
magnetic field of 1.6 × 104 A m–1 was applied in the direction parallel to the wire axis.
4. Results and discussion
4.1. Effect of Cr on solubility of N
Wire samples of 0Cr, 0.3Cr, 0.6Cr, and 0.9Cr, were nitrided at 836 K for 11.5 h under threedifferent N activities, (1) 2.69 × 10–4, (2) 3.96 × 10–4, and (3) 5.13 × 10–4. The hydrogen : ammonia
proportion was 990 : 10, 985 : 15, and 980 : 20, respectively. The nitriding time was chosen to realize
almost uniform distribution of N in the wire sample: it was taken as twice the time required for the
diffusion of N atoms to raise the concentration at the centre of the wire to 95% of the concentration at
the surface.
The N contents of the nitrided samples were analysed by combustion method. The results areshown in Fig. 2. This diagram indicates the enhancement of the solubility by Cr that conform to
Eq. (4). The N concentrations attained are all well below the solubility limit of N against the
precipitation of Fe4N in unalloyed iron at 836 K, which is over 3,000 at. ppm [15]. In the presence ofCr, however, the solubility limit is reduced: it is about 800 at. ppm for 0.9 at. % Cr at 836 K according
to the solubility product of Cr and N reported by Imai et al. [16]. The uppermost point in Fig. 2 isclose to but still lower than this limit. We thus assume that in none of the samples occurred
precipitation of nitrides; they are of single-phase solid solution. The smooth variations of the solubilityon the Cr concentration seem to support the assumption.
The same data are plotted against the activity aN in Fig. 3 (a). A straight line can be drawn from theorigin for each of the data points from the same Cr concentration. In reference to Eq. (1), the slope ofthe line is the reciprocal of the activity coefficient γN. Its values are then plotted agains Cr
concentration xCr in Fig. 3 (b). The four points from the four Cr concentrations form a straight line, as
expected from Eq. (2). By linear regression, the slope, which is the interaction coefficient εN(Cr), is
determined to be –58.5 ± 3.2. This is similar in magnitude to those reported previously by the same
method by Taneike and Okamura but the error range is much smaller.

Fig. 2－ 94 －

Concentrations of N in pure iron and dilute
Fe–Cr alloys nitrided at 836 K. Curves 1, 2,
and 3 are those from the three sets nitrided
under the N activities described in the text.

(a)Fig. 3(b)(a) The same data as Fig. 2 plotted against N activity, and (b) the activity coefficient of N as
a function of Cr concentration.

The interaction energy is given from the interaction coefficient by inverting Eq. (5):m ⎞⎛ΔE = −kT ln ⎜ 1 − ε ⎟ .
⎝
z ⎠

(6)

If the first nearest neighbour octahedral sites to the ‘s’ atom are the low-energy positions (i.e., trapsites), z is 6. It is 12 for the second neighbour sites, and 18 if both the first and the second neighbour
sites are similar trap sites. The values of ΔE derived from the interaction coefficient are listed inTable 1, together with those reported by Guan et al. [4] by analysing the influence of Cr on the Snoek
relaxation of N for Cr concentrations up to 0.2 at. %. The two sets of evaluations agree only if both thefirst and second nearest neighbour sites are assumed to be trap sites of similar importance. However,since the Snoek relaxation is measured at temperatures around 300 K, attention should be paid topossible effects of temperature on the interaction energy; it is not guaranteed if ΔE could be regarded
constant for the range of temperatures.
Table 1 Interaction energy ΔE between N and Cr in α iron evaluated from the solubility[this work] and from the Snoek relaxation [4].
ΔE / eV

Trap site*

From solubility at 836 K [This work]

From Snoek relaxation [4]1–0.246 ± 0.004–0.1802–0.198 ± 0.004–0.165

1 and 2–0.171 ± 0.004

1: –0.17
2: –0.14* 1: first nearest neighbour octahedral sites, 2: second nearest neighbour octahedral sites.

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4.2. Effect of Cr on solubility of C
Samples of 0Cr, 0.6Cr, 1.2Cr, and 1.8Cr were carburized at 998 K for 4.5 h, aiming at C contents ofabout 100 at. ppm and 200 at. ppm, in a flow of hydrogen and methane composed of 197 : 3.1 and
173 : 9.1, respectively. The concentrations of Cr of the alloys used are twice as high as those fornitriding, since previous studies [17, 18] suggested weaker influence of Cr on C than on N.
The C contents analysed by combustion method are shown in Fig. 4. In contrast to the case of N
(Fig. 2), the amounts of C absorbed in Fe–Cr alloys are virtually equal to that in unalloyed iron. As the
C concentration is independent of interaction within the experimental errors, we conclude that theinteraction coefficient εC(Cr) is essentially zero, and so is the interaction energy ΔE.

Fig. 4

Concentrations of C in pure iron and dilute
Fe–Cr alloys nitrided at 998 K. Curves 1
and 2 are those from the two sets carburizedunder the two conditions described in thetext.

Wada [19] made a similar experiment on a series of Fe–Cr alloys up to 8.7 at. % Cr at 985 K andobtained –72 ± 2 as the value the interaction coefficient. It yields –0.31 to –0.22 eV (for z = 6 to 18)
for the the interaction energy, which is larger in magnitude than the present result on N–Cr. Nishizawa
et al. [20] attempted to evaluate the interaction coefficient using a ‘diffusion multiple’, which is a setof diffusion couples connected in series, of Fe–Cr–C alloys differing in Cr concentration. During
annealing carbon atoms redistribute over neighbouring sections by diffusion, while Cr atoms are muchless mobile, so that paraequilibrium is realized. By analysing the C concentration in an annealed
multiple, which is not uniform but is a function of the local concentration of Cr, they obtained theeffect of Cr solubility of C, indicating attractive interaction with ΔE of about –0.2 eV. In view of thepresent result, these data in the literature appear too large in magnitude. A possible origin of thisdiscrepancy is clustering of Cr atoms driven by the presence of C atoms, i.e., to form clusters
consisting of quite a few atoms of Cr and C.
Recent experiments on the Snoek relaxation of N atoms in Fe–Cr alloys containing larger amountsof N strongly suggest that such clustering does occur [21]. It is a gradual, or stepwise transition from
ideal paraequilibrium (in which the distribution of Cr solutes is frozen) to orthoequilibrium. The
transition is to proceed significantly if the annealing time is inappropriately long. This could be thecase for the experiments of Wada and Nishizawa et al., who both held the samples at the equilibration－ 96 －

temperature for a much longer time than in the present experiment, more than 100 hours. A variety ofmetastable solute-atom clusters may appear during the transition, and the effective interaction energy
would vary depending on the nature of the clusters. To determine the interaction energy between asingle ‘i’ and ‘s’ solute atoms, or the formation energy of an isolated i–s atom pair, experimental
conditions should be carefully set up to avoid formation of those higher order solute-atom clusters.
4.3. Effect of Cr on Snoek relaxation due to C
The wire samples carburized at 998 K were solution-treated at the same temperature for 45 minusing a vertical vacuum furnace and were quenched into oil of 273 K without breaking the vacuum.
Each of the samples, of supersaturated solid-solution, was mounted to the torsion pendulum
instrument, cooled down to 150 K and then the mechanical loss and resonant frequency were
measured as a function of temperature by heating or cooling at a rate of 1 K min–1.
Figure 5 shows the results of the measurements for samples containing about 110 at. ppm C. Unlike
the case of Fe–Cr-N alloys [3, 4], effects of Cr on the Snoek relaxation is hardly discernible. The
results for the other set of samples, with 200 at. ppm C, were essentially the same.

Fig. 5The mechanical loss Q–1 (lower) and the square of the resonance frequency f of carburized
samples containing 110 to 120 at. ppm C. The frequency is normalized to the value at 200 K.

These results are in agreement with the previous study [18] carried out for lower Cr concentrations－ 97 －

employing the same method of carburizing. This indicates that the interaction between C and Cr in αiron is very weak, and is consistent with the little influence on the C solubility. It was reported, on theother hand, that addition of Cr did not give rise to a new component in the relaxation profile but
systematically reduced the relaxation strength [22]. We suspect that solute atom clusters weredeveloped, as the samples were prepared by decarburization from materials containing larger amounts
of C through extended heat treatments. Similar trend (lower relaxation strength in alloys of higher Cr
concentrations) was in fact observed in the present experiment for the second set of samples
containing about 200 at. ppm C. It is possible that the higher concentration of C promoted formationof Cr–C atom clusters.
5. Conclusions
The effects of Cr on the equilibrium solid solubility on N and C in α iron have been studied by thestandard thermochemical equilibration method, by subjecting a set of dilute Fe–Cr alloys in a flow ofgas mixture (hydrogen and ammonia, and hydrogen and methane, respectively). The results are
summarized as follows:(1) The addition of Cr up to 0.9 at. % appreciably increases the solubility of N at 836 K, and the effectcan be described by Wagner’s theory with the interaction coefficient εN(Cr) = –58.5 ± 3.2. Theinteraction energy ΔE is estimated to be –0.17 to –0.25 eV, depending on the trap sites assumed. This
result agrees fairly well with the evaluation from the effect on the Snoek relaxation in dilute Fe–Cr-N
alloys [4].
(2) No significant effect of Cr was observed on the solid solubility of C at 998 K for Cr concentrations
up to 1.8 at. %, and the same is true for the Snoek relaxation for solution-treated Fe–Cr–C alloy
samples.
Since Cr is one of the carbide forming elements in iron, such a weak interaction of Cr and C was
unexpected and is against the common knowledge established for liquid iron and γ iron. Moreexperiments are desired to ensure this finding, with different carburizing conditions, pre- and post heat
treatments, deliberate long-time ageing for cluster evolution, as well as direct observation of
nano-scale microstructures by TEM or atom-probe.
Acknowledgements
This work was carried out in collaboration with TANAKA Yuki (formerly graduate student of
Osaka Prefecture University, now at Kobe Steel Co. Ltd.) and CHEN Ying (Department of
Nanomechanics, School of Engineering, Tohoku University). The financial support by the JFE 21stCentury Foundation is gratefully acknowledged.
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