Diffusion of Carbon in Niobium and Molybdenum

Jun-ichi Imai1, Osamu Taguchi2,+, Gyanendra Prasad Tiwari3 and Yoshiaki Iijima1

1Department of Materials Science, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan2Department of Materials Science and Engineering, Miyagi National College of Technology, Natori 981-1239, Japan3Department of Information Technology, Ramrao Adik Institute of Technology,Vidya Nagri, Nerul, Navi Mumbai 400709, India

Diffusion coefficients of carbon in niobium and molybdenum have been determined by the residual activity method with radioactive tracer14C in the temperature ranges between 1168 and 1567K for niobium and between 1271 and 1669K for molybdenum. The temperaturedependences of the diffusion coefficient of carbon in niobium and molybdenum are expressed by D/m2 s¹1 = 2.2 © 10¹6 exp(¹152 kJmol¹1/RT) and D/m2 s¹1 = 5.2 © 10¹6 exp(¹163 kJmol¹1/RT), respectively. Since the solubility of carbon in molybdenum is very small, the diffusionof carbon in molybdenum is strongly influenced by carbide precipitation at lower temperatures. [doi:10.2320/matertrans.M2014277]

(Received July 31, 2014; Accepted September 30, 2014; Published November 8, 2014)

Keywords: carbon diffusion, niobium, molybdenum, carbon solubility, precipitate effect

1. Introduction

Iron, nickel and cobalt based superalloys appear to haveachieved full potential in relation to their use as structuralmaterials for corrosive environments as well as hightemperatures.1) The strength of these alloys comes partlyfrom solid solution strengthening and partly from precip-itation hardening. For applications at temperatures higherthan 1200K and under intense radiations encountered in fastand fusion reactors, precipitation hardening loses its sheen asthe source of strengthening for the matrix. Titanium andzirconium are ruled out because of phase transformation andhigh diffusion rates. Hence, niobium, vanadium, tantalumand molybdenum are the available choices as suitablematerials for structural components of fast and fusionreactors. The combination of high melting point and highstrength possessed by these metals enhances their usefulnessin severe environments. However, these metals show highpropensity to absorb interstitials like carbon, nitrogen andoxygen. Because of its tendency to form carbides, carbonhas a profound influence on the mechanical properties oftransition elements. In view of this, knowledge of thediffusion properties of carbon assumes great significance. Inthe present paper, we present results of the radioactive tracerdiffusion in niobium and molybdenum. It may be mentionedhere that the previous studies of diffusion of carbon inniobium were carried out during the years 1950­1972.2­7)

Similarly, the diffusion of carbon in molybdenum wasstudied in the years 1964­1978.4,8­14) This is first reportedinvestigation of carbon diffusion in these metals after morethan three decades. It is also pertinent to recall here manyof the earlier investigations were performed via indirecttechnique without the use of 14C radioactive tracer. Theobject of the present article is to present new data on thediffusion of carbon in pure niobium and molybdenummatrices and make a comparative study of present resultswith the earlier ones.

2. Experimental Procedure

2.1 MaterialNiobium metal rod arc-melted and machined to 12.5mm in

diameter was supplied by Materials Research Corporation,USA. Main impurities, in mass ppm, shown by chemicalanalysis are C-40, O-20, N < 5, Ti < 30, Fe-20, Si-25, Cr-30and Ta-250. To induce grain growth, as received rod wasannealed by an electron beam heating at 2173K for twohours under a vacuum of 1 © 10¹5 Pa. The resultant grainsize was about 4mm. The rod was cut to make 7mm thickdisc specimens. One of the flat faces of the discs wasmetallographically polished followed by electropolishing in aH2SO4 solution containing 10% HF.

Six mm diameter “low carbon molybdenum rod” wasobtained from Climax Molybdenum Company, USA. Ac-cording to the supplier, maximum nominal impurities, inmass ppm, are as follows: C-50, N-20, O-15, Si-80, Fe-80and Ni-20. Three such rods were melted together by anelectron beam heating to produce a rod of ten mm diameter.The grain size was about 3­4mm. These rods were sliced tosix mm thick disc specimens. One of the flat faces of the discwas metallographically polished and finished with electro-polishing in a 10% H2SO4 solution.

2.2 Radioactive tracerRadioisotope 14C (¢-ray 156MeV, half-life 5730 years)

was supplied in the form of fine carbon particles of less thanone µm diameter by The Radiochemical Centre, Amersham,UK having a relative activity of 3.3 TBq/kg. A few drops ofthe suspension of the particle of 14C in CCl4 were put on theflat and polished surface of the specimen with the help of amicropipette and dried in air.

2.3 Diffusion annealingThe sealed specimens in evacuated quartz tubes were

diffusion annealed at temperatures between 1168 and 1567Kfor varying periods from 8.40 © 103 to 3.46 © 105 secondsfor niobium. The annealing temperatures for molybdenumranged from 1271 to 1669K and the corresponding annealing

+Present address: Professor Emeritus, Miyagi National College ofTechnology. Corresponding author, E-mail: mniotaosm@yahoo.co.jp

Materials Transactions, Vol. 55, No. 12 (2014) pp. 1786 to 1791©2014 The Japan Institute of Metals and Materials

periods varied from 9.00 © 102 to 8.73 © 105 s. All thetemperatures were controlled within «2K. After the diffusionannealing, the cylindrical surface of the specimen wasmachined in a precision lathe to reduce the diameter byabout 1.8mm in order to eliminate the possible contributionof surface diffusion to diffusion inside specimen matrix.

2.4 Concentration profilingThe flat surface of the specimen was removed successively

through grinding. The thickness of each layer after grindingranged from 5­50 µm which was estimated from weight lossmeasured in a precision balance, surface area of the specimenand density of niobium or molybdenum as the case may be.Residual activity of the specimen after each grinding wascounted in a windowless Q-gas flow counter having 2³geometry. The background of the counter was 20­30 cpm. Toreduce statistical uncertainty of the counting, Q-gas flowedfor 2min before the beginning of 5min counting.

2.5 Analysis of the dataFor one dimensional diffusion of a tracer from a thin film

into a sufficiently long rod analyzed by the residual activitymethod,15) the solution of Fick’s second law is given by

®In �dIndXn

¼ const:CðXnÞ ð1Þ

Here ® is the absorption coefficient (in m¹1) of the matrixfor ¢ radiation from radioisotope 14C, In is the surface activity(in counts per set time) after a thickness Xn is removed fromthe original surface. C(Xn) is the concentration of radio-active tracer in the matrix at a distance Xn from the originalsurface. The values of ® for 14C ¢-ray are 140000m¹1 and170000m¹1 for niobium and molybdenum, respectively. As aresult, the value of the parameter ®In/(¹dIn/dXn) º 100 forall the cases, so the term dIn/dXn in eq. (1) can be neglectedwithout introducing any significant error. Thus, C(Xn) isproportional to In. The solution of Fick’s law for aninstantaneous source of thin film geometry diffusingunidirectionally through the lattice is given by

CðX; tÞ ¼ M=2ffiffiffiffiffiffiffiffiffi

³Dtp

expð�X2=4DtÞ: ð2ÞHere M denotes the total mass of the diffusing substance atX = 0 and at time t = 0. When the surface concentration ismaintained constant through the period of diffusion, eq. (2) istransformed as below:

CðX; tÞ ¼ C0 erfcðX=2ffiffiffiffiffiffi

Dtp

Þ; ð3Þwhere C0 is the constant concentration at the surface definedby X = 0. If ¯(X) represents the probability function fornormal Gaussian distribution, eq. (3) is transformed to

�ðX=ffiffiffiffiffiffiffiffi

2Dtp

Þ ¼ 1� CðXÞ=2Co ð4Þ

3. Results

3.1 Diffusion of carbon in niobiumFigures 1 and 2 show the concentration profiles of 14C

in niobium. The plot of {1 ¹ C(X)/2C0} versus X showsa linear relationship at all temperatures. Here, influence ofgrain boundary diffusion on the diffusion profile is negligible

(a)

(b)

(c)

Fig. 1 Examples of penetration profiles for diffusion of carbon in niobiumat 1168, 1269 and 1369K.

1 -

C/2

Co

0.50

(a)

(b)

Fig. 2 Examples of penetration profiles for diffusion of carbon in niobiumat 1478 and 1567K.

Diffusion of Carbon in Niobium and Molybdenum 1787

because the grain size is much larger than the length ofconcentration profiles. The diffusion coefficients calculatedfrom the slopes in Figs. 1 and 2 are listed in Table 1. TheArrhenius plots of diffusion coefficient of carbon in niobiumare shown in Fig. 3. The figure also shows experimentalresults obtained by earlier workers.4­7) As seen in Fig. 3, thediffusion coefficients obtained by us are somewhat higherthan those of earlier workers but fairly consistent withinthemselves. The temperature dependence of diffusioncoefficients determined in course of the present investigationcan be expressed by the following equation

D=m2 s�1 ¼ 2:2� 10�5 expð�176 kJmol�1=RT Þ: ð5ÞIt is important to note here more than one technique havebeen employed by different workers as shown in Table 2. D0

and Q are the preexponential factor and the activation energyin the Arrhenius relation. Their results are plotted along withour own data in the Fig. 3. Smallest diffusion coefficientare those reported by Hörz and Lindenmaier7) obtained by

studying the decarburization kinetics. The results of Schmidtand Carlson6) by diffusion couple method and those ofNakonechnikov et al.4) and Son et al.5) by 14C tracer methodappear consistent. Our own results match satisfactorily withthose of Nakonechnikov et al. at lower temperatures. Thedifferences go up marginally with the increase in temper-ature. The overall differences do not amount to more than10%.

3.2 Diffusion of carbon in molybdenumFigures 4, 5 and 6 show the concentration profiles of 14C

in molybdenum. The plot of {1 ¹ C(X)/2C0} versus X showsa linear relationship at all temperatures. Then, influence ofgrain boundary diffusion on the measured diffusion coef-ficient is negligible because of the same reason as describedabove. The diffusion coefficients calculated from the slopesin Figs. 4, 5 and 6 are listed in Table 3. The Arrhenius plots

Table 1 Diffusion coefficient of carbon in niobium.

Temperature/K Diffusion time/s Diffusion coefficient/m2 s¹1

1567 8.40 © 103 3.55 © 10¹11

1567 8.40 © 103 2.59 © 10¹11

1478 2.47 © 104 1.23 © 10¹11

1478 2.47 © 104 1.32 © 10¹11

1369 8.64 © 104 4.12 © 10¹12

1369 8.64 © 104 4.60 © 10¹12

1269 1.07 © 105 1.16 © 10¹12

1269 1.07 © 105 9.83 © 10¹13

1168 3.46 © 105 2.58 © 10¹13

1168 3.46 © 105 3.52 © 10¹13

Nakonechnikov et al.

Diff

usio

n co

effic

ient

, D/ m

2s-1

Fig. 3 Arrhenius plot of diffusion coefficients of carbon in niobium.

Table 2 Previous data on diffusion of carbon in niobium.

AuthorsTemperaturerange/K

D0/m2 s¹1 Q/kJmol¹1 Method

Wert (1950) 323­413 1.5 © 10¹6 113 internal friction

Powers and Doyle(1959)

403­503 4 © 10¹7 138 internal friction

Nakonechnikovet al. (1966)

1373­1673 9.3 © 10¹7 146 14C tracer

Son et al. (1967) 1203­2073 3.3 © 10¹6 159 14C tracer

Schmidt and Carlson(1972)

2173­2573 2.6 © 10¹6 158 diffusion couple

Hoerz andLindenmaier (1972)

1873­2393 1.8 © 10¹6 159 decarburization

1376 K

(a)

(b)

Fig. 4 Examples of penetration profiles for diffusion of carbon inmolybdenum at 1271 and 1376K.

J. Imai, O. Taguchi, G. P. Tiwari and Y. Iijima1788

of diffusion coefficient of carbon in molybdenum are shownin Fig. 7. This figure also includes the experimental results ofprevious workers.4,9,11­14) The temperature dependence of thediffusion coefficients determined in the course of presentinvestigation below 1515K is expressed by the followingequation

D=m2 s�1 ¼ 2:2� 10�5 expð�367 kJmol�1=RT Þ: ð6ÞThe large distribution in the magnitudes of preexponential

factors and the activation energies reported by differentauthors listed in Table 4 is truly surprising. The preexponen-tial factors range from 7.3 © 10¹11 to 4.0 © 10¹6m¹2 s¹1.The activation energies range from 73 to 382 kJmol¹1. Thisvariation could be attributed to the following factors:(1) The differences in the techniques used for measure-

ments by different authors.(2) Variations in the impurity level and contents of the

specimens used by different authors.(3) Because of the high reactivity of molybdenum, the

results could also be influenced by the presence of

hydrocarbons, oxygen and nitrogen in the annealingatmospheres.

(4) In most of the cases, there is a paucity of data points inthe diffusivity plots. In view of this, determination of atruly representative of preexponential factor andactivation energy is difficult.

Above 1540K, the temperature dependence of diffusioncoefficients obtained by Schmidt and Carlson,12) Kunze andReichelt11) and Lorang and Langeron14) are consistent withthe present work. On the other hand the Arrhenius plotobtained by Rudman9) and Nakonechnikov et al.4) are littlelower than above results. However, the results of Lesageand Huntz13) are significantly lower than all other authorsyielding very high activation energy. Present results below

0 1 2 3 4 5 6 7 8 9

0.60

0.70

0.80

0.90

0.95

X / 10-4 m

0.50

1 -

C/2

Co

1515 K

0 1 2 3 4 5 6 7 8 9

0.60

0.70

0.80

0.90

0.95

X / 10-4 m

0.50

1 -

C/2

Co

1469 K

0 1 2 3 4 5 6 7 8 9

0.60

0.70

0.80

0.90

0.95

0.50

1 -

C/2

Co

X / 10-4 m

1419 K

(a)

(c)

(b)

Fig. 5 Examples of penetration profiles for diffusion of carbon inmolybdenum at 1419, 1469 and 1515K.

Table 3 Diffusion coefficient of carbon in molybdenum.

Temperature/K Diffusion time/s Diffusion coefficient/m2 s¹1

1669 9.00 © 102 4.90 © 10¹11

1669 9.00 © 102 4.71 © 10¹11

1569 7.20 © 103 2.60 © 10¹11

1569 7.20 © 103 3.08 © 10¹11

1515 2.05 © 104 1.18 © 10¹11

1515 2.05 © 104 1.24 © 10¹11

1469 5.76 © 104 3.57 © 10¹12

1469 5.76 © 104 6.09 © 10¹12

1419 1.48 © 105 3.10 © 10¹12

1419 1.48 © 105 2.76 © 10¹12

1376 1.73 © 105 7.04 © 10¹13

1376 1.73 © 105 3.20 © 10¹13

1271 8.73 © 105 2.84 © 10¹14

1271 8.73 © 105 6.12 © 10¹14

1669 K

(a)

(b)

Fig. 6 Examples of penetration profiles for diffusion of carbon inmolybdenum at 1569 and 1669K.

Diffusion of Carbon in Niobium and Molybdenum 1789

1515K are similar to that of Lesage and Huntz13) in onerespect. There is break in the diffusivity plots suggestive ofa bimodal diffusion behavior indicating that the diffusionparameters may be different in different temperature regions.

4. Discussion

In a situation when we are faced with more one expressionfor diffusion coefficient, it is imperative to find an expressionwhich represents, as closely as possible, the parametersdefining the process of diffusion under consideration. Thechoice becomes easier if the diffusion coefficients coveringfive to six decades of diffusion coefficients over extendedtemperature ranges are available.16) As seen in the Tables 2and 4, the temperature range of most measurements does notexceed 500K. In case of niobium, Powers and Doyle3) havepointed out that the internal friction peak observed by Wert2)

is due to oxygen atoms. Schnizel8) has obtained very smallactivation energy for carbon diffusion in molybdenum byinternal friction measurements. This is probably caused bythe fact that broad internal friction peaks observed sometimesarise from the interactions between oxygen, nitrogen andcarbon atoms. It is not easy to identify the intrinsic peakcaused by carbon diffusion. Hence, the diffusion dataobtained by internal friction measurements are excluded.

In case of niobium, if disregard the results of Hörzand Lindenmaier,7) the present results along with those ofNakonechnikov et al.4) Son et al.5) and Schmidt andCarlson6) constitute a consistent set of data. There is a gooddegree of consistency at low and high temperature region andvariations in the middle range are restricted to a few percentpoints only. A least mean square fit giving equal weight to alldata points may yield a truly representative expression tocharacterize the diffusion of carbon in niobium as follows:

D=m2 s�1 ¼ 2:2� 10�6 expð�152 kJmol�1=RT Þ: ð7ÞIn case of molybdenum, there is break in diffusivity plot.

Hence, we must have two different expressions for low andhigh temperature regions. Prior to this, we would like todiscuss the results of Lesage and Huntz.13) These dataobtained through decarburization technique are smaller thanour results in the overlapping temperature region by morethan an order of magnitude. The specimen employed byLesage and Huntz13) contained 300mass ppm of carbon andtherefore contained a fine dispersion of carbides. Some ofthese precipitates may not contain carbon in full stoichio-metric ratio. This kind of precipitates may interfere withflux of diffusing carbon atoms by trapping them. Such aphenomenon can lead to low apparent diffusivity. Thus whendiffusing species is chemically reactive in nature, a strictcontrol on the composition of the matrix is essential. Thesolubility of carbon in molybdenum has been measured byGebhardt et al.17) and by Rudman.9) The temperaturedependence of the solubility Cs (mass ppm) is expressed byGebhardt et al.17) as

lnCs ¼ 15:64� 2:075� 104=T ð8Þand by Rudman9) as

lnCs ¼ 16:78� 2:29� 104=T : ð9ÞThus, the carbide precipitates in the specimen containing300mass ppm of carbon dissolve above 2088K (eq. (8)) or2067K (eq. (9)). Therefore, if Lesage and Hunts13) extendedthe temperature range of measurements above the precip-itation temperature, the observable diffusion coefficients areconsistent with the diffusivity data at higher temperatures. Asseen in Fig. 7, if the Arrhenius line obtained by Lesage andHunts13) is extended to higher temperature, it crosses theArrhenius lines obtained by Scmidt and Carlson,12) Lorangand Langeron14) and Kunze and Reichelt11) at about 2000K,which is near to the carbide dissolving temperature estimatedas above. On the other hand, Kunze and Reichelt11) used themolybdenum specimen containing 130 at ppm of carbon.Hence, the precipitation temperature of carbide in thespecimen is estimated to be 1613K (eq. (8)) or 1635K

Fig. 7 Arrhenius plot of diffusion coefficients of carbon in molybdenum.

Table 4 Previous data on diffusion of carbon in molybdenum.

AuthorsTemperature

/KD0

/m2 s¹1Q

/kJmol¹1 Method

Schnitzel (1964) 300­673 7.3 © 10¹11 73 internal friction

Nakonechnikovet al. (1966)

1473­1873 2.0 © 10¹6 172 14C tracer

Rudman (1967) 1783­2243 3.4 © 10¹6 172 carburization

Shchelkonogovet al. (1968)

293­673 2.8 © 10¹8 167 internal friction

Kunze and Reichelt(1970)

1508­2033 4.0 © 10¹6 164 decarburization

Schmidt and Carlson(1976)

2163­2593 3.3 © 10¹6 153 diffusion couple

Lesage and Huntz(1976)

1079­1513 4.16 © 10¹8 382 14C tracer

Lorang andLangeron (1978)

1533­2283 1.04 © 10¹6 139 decarburization

J. Imai, O. Taguchi, G. P. Tiwari and Y. Iijima1790

(eq. (9)). These temperatures are close to 1618K, from wheredownward deviation from Arrhenius line begins in exper-imental results reported by Kunze and Reichelt.11) Further-more, the activation energy observed below 1515K by thepresent experiment is 367 kJmol¹1 which is close value to382 kJmol¹1 observed by Lesage and Hunts.13) This suggeststhat the same diffusion mechanism is operative on the bothcases. The bending temperature of the Arrhenius line of thepresent results is 1563K. Therefore, the solubility of carbonin the present specimen can be estimated to be 10.6mass ppm(eq. (8)) or 8.4mass ppm (eq. (9)). This is consistent with thevalue of 10mass ppm observed by Suezawa and Kimura18)

in “low carbon molybdenum rod” supplied by ClimaxMolybdenum Company. As mentioned earlier, the temper-ature dependence of the diffusion coefficients of carbonin molybdenum can be represented by combining theexperimental results obtained above 1540K by Schmidtand Carlson,12) Kunze and Reichelt11) and Lorang andLangeron14) including the present results as follows:

D=m2 s�1 ¼ 5:2� 10�6 expð�163 kJmol�1=RT Þ ð10ÞIt is important to emphasize here that there exists a paucity

of data of sufficient precision on the diffusion of metalloids inmetals. On the other hand, for the diffusion of substitutionalsolutes in metallic systems good quality data on a wideranging system are available.19) The situation is complicatedby two main reasons. As evident from the Table 2 and 4, awide variety of techniques have been employed and theresults from different techniques are inconsistent with eachother. We need radioactive tracer diffusion coefficientmeasurements over extended temperature ranges so that thecurvature in the diffusivity plots can be delineated andidentified. Because of the reactive nature of metalloids, it isalso essential to use only high purity materials in suchstudies.

5. Conclusions

The diffusion coefficient of carbon in niobium observed bythe present experiments is consistent with those of previousauthors. The experimental data on diffusion of carbon inmolybdenum by previous authors and the present work areanalyzed in view of carbon content in the specimens. Thesignificant influence of carbide precipitates on diffusion ofcarbon in molybdenum is emphasized. Tracer diffusion studiesin high purity materials are needed to establish true coefficientsof carbon, nitrogen and other metalloids in metallic systems.

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Diffusion of Carbon in Niobium and Molybdenum 1791