ANALYTICAL MODELING OF CARBON TRANSPORT
ROCESSES IN HEAT TREATMENT TECHNOLOGY OF STEELS
Jürgen Gegner
SKF GmbH, Material Physics, Ernst-Sachs-Str. 5, D-97424 Schweinfurt, Germany
ABSTRACT
Thermochemical edge zone processes of steels induced by solid-state diffusion of carbon represent one of the economically and scientifically closest points of contact of
materials science to industrial application: carburization is the main step of case
hardening, whereas out-diffusion of carbon, though sporadically used (e.g. decarburization of cast iron), is primarily relevant to heat treatment and hot-working
processes, like austenitizing or forging, as undesirable side effect that impairs the
mechanical properties of the rim region. In the present paper, the technological background of both thermodynamically inverse processes is explained. Realistic modeling
of the carbon transport, which can be controlled by diffusion and/or surface reactions,
for profile prediction is important to reliable carburization control and failure analysis
of decarburized parts or defective plants. Suitable analytical solutions of Fick's law
are summarized and a realistic computer model for the consideration of simultaneous
carbide dissolution during decarburization is developed. The effect of influencing
parameters (e.g. carbon potential, mass transfer coefficient, carbon solubility in
austenite) on the depth profiles is discussed in detail. Selected concrete examples of
multi-step carburization and decarburization processes are simulated and compared
with experimental data.
INTRODUCTION
As the first operation step of case hardening, carburization of low-alloyed steels, i.e.
carbon enrichment in the edge zone to typically 0.6 (maximum martensite hardness)
to 0.85 m.% C, is one of the oldest industrially utilized surface refinement processes,
which is based on controlled inward mass transport by solid state diffusion. This
widely used technique permits the production of mechanically and tribologically highly loadable components with hard rim (58 to 67 HRC) and tough core (30 to 50 HRC).
The costs are considerably lower, if compared, for instance, with carbonitriding, nitrocarburizing or nitriding. Core hardenability (thick components) can be improved by
alloying elements (e.g. Cr, Mo). With a market share of today more than 30 percent,
case hardening of low-carbon grades (0.07 to 0.3 m.% C) is the most important heat
treatment procedure of steels since almost 60 years. It is preferentially applied to motor, gear, machine or jet engine parts that require high fatigue, wear and shock resistance, like cogwheels, journals, bolts, shafts (e.g. camshafts), and bearings (e.g. wheel
bearings).
In certain applications like forgeability improvement of tempered casting, the inverse
thermochemical process of case decarburization is also used in engineering technology. However, this out-diffusion of carbon caused by surface oxidation reactions,
possibly accompanied by scaling and/or internal oxidation, is more relevant as undesirable side effect to heat treatment and hot-working processes particularly of through
hardenable steels, e.g. austenitizing, soft annealing, forging or upsetting. Since the
martensite start temperature increases locally with the decreasing carbon concentra1 - 95
tion, transformation-induced tensile residual stresses are formed in the affected edge
zone. Apart from the resulting higher crack sensitivity and lower fatigue strength, also
the hardness near the surface and thus the wear resistance is reduced.
Although the essential neglect of the composition dependence of the carbon diffusivity in austenite leads to characteristic deviations from real concentration profile
shapes, mathematical modeling with analytical process simulation permits prediction
of the main operation quantities with the same accuracy as numerical methods (see
e.g. Fig. 4). These parameters are the surface carbon content and the carburization
(generally ranging from 0.05 to more than 10 mm, i.e. few hours to over one week) or
decarburization depth that are defined as surface distance at 0.35 m.% C and 0.92c0,
respectively. Here, c0 denotes the initial carbon concentration. Analytical methods,
which are most suitable for the evaluation of the fundamental effect of (changing)
control parameters on process flow, have often proved superior to numerical techniques for the purpose of solving diffusion problems in metallurgy (1). In the present
paper, appropriate mean values of the carbon diffusivity in austenite between limiting
concentrations c1 and c2 are calculated as follows (2):
Dγ 
1 c2
D γ (c C ) d c C
c 2  c1 c1
(1)
Data derived from several experimental investigations into the binary Fe–C system is
inserted (3). cC denotes the carbon concentration. For the sake of simplicity, D is used
instead of Dγ in the following.
GAS CARBURIZATION OF LOW-ALLOYED STEELS
Today's state of the art is the well-established two-step gas carburization process at
customary temperatures between 1120 and 1250 K with continuous control by the C
potential (or C level) cp. It is defined as the carbon content of a pure iron sample (e.g.
thin foil) in equilibrium with the furnace atmosphere and usually expressed in m.%.
Analytical process simulation
The main carburizing reaction in industrially applied, so-called fast mixtures of carrier
and enrichment gas containing CO and H2 is the heterogeneous water gas equilibrium,
COH2CH2O. C stands for carbon dissolved in steel. The C level can be derived
from thermodynamic data by applying the activity-concentration relation (4, 5):
log
cp
m.%
 0.15
cp
m.%
 log
pCO  pH 2
p H 2O

4800
 5.286
T /K
(2)
T and p denote the temperature and the partial pressure of the gas components, respectively. The influence of the steel composition on the carbon activity-concentration
relation is taken into account by the dimensionless alloy factor ka that typically ranges
from 0.9 to 1.1:
cp(corr)  cp  k a
(3)
1 - 96
The corrected C level cp(corr) corresponds to the desired surface content and is usually
also called cp. The mass flux density jC, i.e. the carbon amount transferred from gas to
steel per unit time and area, can be derived as reaction rate from the kinetics law (6):
jC  k
pCO
pH O
1 KO 2
pH2
 as 
p  pH2
1    k  CO
 a p  a s   β c p  c s 
 a 
p H 2O
p 

(4)
Here, KO denotes the adsorption constant of oxygen at the surface. Due to the low
carbon content of steels, activities (ap, as) are replaced with corresponding concentrations. The rate constant and its modified value are referred to as k and k′, respectively.
The surface carbon concentration cs varies with time t. For constant temperature, the
mass transfer coefficient  is thus a function of partial pressures and, as the small H2O
content does not change significantly, approximately proportional to the CO-H2 product. Based on the predominating heterogeneous water gas equilibrium, effective control parameters cp and  can be derived that additionally consider the slower carbon
releasing surface processes of the Boudouard reaction and the methane decomposition. The solution of Fick's law for constant initial carbon concentration c0 under the
third-kind boundary condition of Newton's reaction law in Eq. (4) describes the boost
period of gas carburization (2):

 x
 βx  β 2 t 
x
t 





cC  c0  (c p  c0 ) erfc
 exp
erfc
β
 D



D

2 Dγ t
2
D
t
γ
γ 
γ





(5)
Here, x denotes the distance from surface and erfc is referred to as the error-function
complement. The one-dimensional approach holds in almost all cases of practical interest. Figure 1 shows resulting carbon profiles for cp1.2 m.%, c00.2 m.%, t12 h
and several  values, which
typically range from 1105 to
4105 cm/s in industrially applied atmospheres. The concentration line cC0.35 m.%
corresponding to the carburization depth dc is drawn in.
The inset reveals the development of the surface carbon
content cs with time and thus
illustrates the accepted expressions fast or slow gas
mixture. Figure 2 demonstrates the influence of the
steel composition on the
Figure 1. Carburization profiles according to Eq. (5). profile and the carburization
depth. This diagram clarifies
that the C level must be chosen appropriately with respect to the desired cs value.
According to Eq. (3), the alloying elements affect the mass flux and thus the carburization profile without impact on the diffusivity.
1 - 97
In order to smooth the sharp nearsurface concentration gradient (cf.
Figs. 1, 2), subsequent to the boost
period of duration tb a shorter diffusion anneal (time td: 10 to 25 % of
tb) with reduced carbon level from
c pb (0.8 to 1.2 m.% C) to c pd (0.6 to
0.85 m.% C) is performed (two-step
procedure) by changing the gas
composition in the furnace. Usually,
the mass transfer coefficient  remains almost constant and an isothermal process is applied. Solving
Fick's law for c0const. and ttb then
yields (7):
Figure 2. Effect of the alloy factor ka on the
carburization profile (ka1: Fe–C).



 βx  β 2 t 
x
 erfc  x  β t 
cC  c0  (c pb  c0 ) erfc
 exp 
 D

2 D t
Dγ 

2 Dγ t
γ
γ






 βx  β 2 (t  t b ) 
x
b
d


 (c p  c p ) erfc
 exp 


D
2 Dγ (t  t b )

γ



(t  t b ) 
x

 erfc 
β
 2 D (t  t )

D
γ
γ
b


(6)
Application of protective atmosphere or vacuum (uncommon) instead of C potential
control during the diffuse stage in
order to avoid re-decarburization
(boundary condition of the second
kind: jC0 at x0, employed e.g.
for doping of semiconductors)
would result in significantly longer
equalization times. Figure 3 reveals the carbon profile of a typical two-step gas carburization process ( c pb 1.2 m.%, c pd 0.7 m.%,
tb10 h, td2 h; c00.15 m.%) calculated in accordance with Eq. (6),
where ttbtd12 h is inserted. The
cs-t curve is shown in the inset. As
can be seen, proper process control
Figure 3. Analysis of a two-step boost-diffuse leads to almost uniform concentration (cs0.05 m.%) in the edge
gas carburization process.
zone after the diffusion period up
to a surface distance of around one third of the carburization depth as precondition of
optimal mechanical properties. For comparison, both corresponding one-step boost
1 - 98
operations are also computed according to Eq. (5) with tb10 h and tb12 h, respectively. Whereas the profile shape in the near-surface region is significantly changed by
the diffusion anneal that is obviously associated with some carbon loss (re-decarburization), no noticeable effect on the carburization depth dc occurs, which is thus,
under these customary conditions, controlled by the total process time t (here 12 h)
alone. This result also explains why a parabolic kinetics law, dct, holds for common two-step boost-diffuse gas carburization treatments.
Experimental verification
Since industry demands for high
quality procedures, target quantities must be observed accurately.
Consequently, up-to-date process
simulation should be able to particularly predict the carburization
depth as main desired parameter
within narrow scatter bands of
about 0.1 mm even for large dc
values. For a high temperature gas
carburization (T 1233 K) experiment performed at 1243 K in a
modern bulk production plant,
Fig. 4 shows the carbon profile Figure 4. Comparison between analytical and
numerical solution.
calculated numerically (FDM:
finite difference method) by the
furnace control software. The mass transfer coefficient and the C level, measured in
both process steps boost (tb56.6 h) and diffuse period (td13.2 h) by means of the foil
method, reached the following values: c pb 1.20 m.%, c pd 0.80 m.%, 3.010–5 cm/s
(bd). Figure 4 also presents the analytical solution according to Eq. (6). The
transition from the boost to the diffuse period occurred within less than 5 min by controlled air admission to the atmosphere and can thus be neglected.
Note that the diffusivity expression
used by the FDM program is not
known. The calculated carburization depth, as well as the surface content, agrees well with the
prediction of the furnace control
unit. The deviations from the typical S profile shape (see also Fig. 6)
stem from the neglect of the
composition dependence of the
carbon diffusivity in the austenite
phase, which increases with concentration cC.
Figure 5. SIMS measurement of the
carburization profile.
1 - 99
Figure 5 shows the result of the
microchemical cross-section analy-
sis by a novel high-accuracy SIMS technique (secondary ion mass spectrometry). The
measured carbon profile deviates significantly from the prediction of the furnace control software.
The actual carburization depth of
5.48 mm is around 10 % higher
than the desired value, d cdes 5.02
mm, among other things resulting
in application and quality problems
(overstepping of specification). For
economical aspects, the corresponding marked exceedance of the required process time of about 20 %
is also unacceptable. One obvious
cause of this huge discrepancy
could be found in the used diffusion coefficient D. For instance,
the influence of alloying elements
Figure 6. Analytical fits with respect to
on the carbon diffusivity in the
carburization depth.
austenite phase of the applied steel
is not considered. Figure 6 presents analytical fits according to Eq. (6): the D values
yielding d cdes and d cSIMS deviate by around 20 %.
DECARBURIZATION OF LOW-ALLOYED STEELS
In reactive atmospheres, e.g. contaminated protective gas or (wet) air, the following
decarburizing processes may occur: 2CO22CO, CO2CO2, CCO22CO
(Boudouard reaction), and CH2OCOH2 (heterogeneous water gas equilibrium).
The resulting carbon removal from the surface generates the concentration (activity)
gradient as driving force of (outward) diffusion. Mathematical examination and prediction of decarburization profiles is of special importance to quality assurance (e.g.
grinding allowance) of heat treatment processes, like austenitizing, and failure analyses of components and industrial plants.
Decarburization of through
hardenable bearing steel
In order to estimate potential edge
zone damage during austenitizing
(typical temperatures between
1100 and 1170 K for around 30
min, high-speed steels up to 1500
K for only few minutes) of standard bearing steel 100Cr6 (SAE
52100, measured initial carbon
content: c01.02 m.% C), two isothermal experiments at 1163 K
are performed. The course of decarburization in the phase dia- Figure 7. Phase diagram of bearing steel 100Cr6.
1 - 100
gram (intersection through the ternary Fe–Cr–C system at 1.5 m.% Cr) is shown in
Fig. 7 (8).
The first experiment describes the
worst case, i.e. annealing in air for
30 min. Figure 8 represents the
result of the SIMS analysis that
yields a decarburization depth,
ddx(cC0.94 m.% C), of 350 µm.
The inset reveals the decarburized
microstructure with scaling. As
the atmosphere offers oxygen in
excess, it can be assumed that the
surface carbon concentration cs remains constant at 0 m.% during
the whole anneal (9). Under the
simple initial and boundary conditions c0const. and csconst.,
Figure 8. Decarburization in ambient air.
the depth profile is given by the
Van Ostrand-Dewey equation.
Figure 8 confirms that the decarburization induced austenite-ferrite () phase transformation in the edge zone (cf. Fig. 7), which leads to the formation of a penetrating
 interlayer and thus to the development of a near-surface inflexion point in the
measured concentration-distance curve as for the diffusivities D100D(cC0) is
valid, can be considered by shifting the origin in order to model the inward moving
outset of the  region (10):
cC  cs  (c0  cs )  erf
xξ
(7)
2 Dγ t
No blocking effect of the porous, poorly adhering scale (see Fig. 8, inset) on carbon
out-diffusion is observed.
In the second experiment, after
20 min annealing in protective
gas, dosed increasing admission
of air occurs in order to simulate
furnace leakage. Figure 9 demonstrates that the highly accurate
SIMS technique reveals the
preserved carbide segregation,
which stem from steel production, in a depth of around 60 µm
(see microstructure, arrow). The
decarburization depth reaches
100 µm. No appreciable scaling
is observed. As shown in the
inset of Fig. 9, the controlled air
contamination of the neutral gas
Figure 9. Controlled admission of air for 10 min.
1 - 101
atmosphere for 10 min is realistically modeled by linearly decreasing surface concentration:
cs  c0  mt
(8)
The positive slope parameter m reaches 6. 6 10–4 m.%/s. The solution of Fick's law
under the first-kind boundary condition of Eq. (8) for c0const. can be expressed as
follows (2):
 x 2
 x 2 
1 
x
x


cC  c0  4mt 
 erfc

exp  




8
D
t
4
4
D
t
2
D
t
4
π
D
t
 γ
γ 
γ
γ



(9)
Figure 9 supports that the measured decarburization profile quantitatively agrees with
the result of this analytical simulation.
Computer model for decarburization with simultaneous carbide dissolution
If the initial content c0 exceeds the solubility of carbon in austenite, c γM3C , the characteristic phase diagram in Fig. 7 shows that decarburization, i.e. cs c γM3C , is accompanied by the dissolution of M3C carbide particles (metal fraction M: Fe, Cr; general
stoichiometry MC, e.g. 3, 7/3). To consider this background process, which is
neglected in the computations presented in the previous section, an existing Fortran
code is upgraded that can also be used to calculate carburization or decarburization
profiles of complex multi-step processes and solves the plane-sheet diffusion problem
(here: thickness l2√Dt, i.e. no center effects) for an arbitrary, discretely defined
(evaluation points xi) initial concentration distribution fC(x) under the boundary condition csconst. (2):
Dγ t 

(2n  1) π x
exp  (2n  1) 2 π 2 2 
l
l 
n 0

Dγ t  l

2 
n πx
n π x
  sin
exp  n 2 π 2 2    f C ( x) sin
d x
l n 1
l
l  0
l

cC  cs 
4 cs
π

1
 2n  1 sin
(10)
In Eq. (10), n is a non-negative integer. All the following exemplarily simulations
refer to rolling bearing steel 100Cr6 at 1123 K with cs0.1 m.% C (no  transformation, cf. Fig. 7). c Cl and cCcarb denote the carbon content in the austenite lattice,
where (volume) diffusion only occurs, and the carbides, respectively: cC(tot)  c Cl  cCcarb .
Firstly, a homogeneous distribution of the M3C particles is assumed. The initial values
are taken from Fig. 7: c 0l  c γM3C 0.70 m.% C, c0carb 0.32 m.% C ( c 0l  c0carb c01.02
m.% C). Carbide dissolution is described by a first-order kinetics law with the positive
rate constant k (11):
d cCcarb
 k  cCcarb
dt
(11)
1 - 102
In the proposed extended computer-aided iterative process simulation, Eq. (11) is
computed after any time step t at each evaluation point (e.g., xi1xi1 µm). The
calculated amount,  c Ccarb (xi)k cCcarb (xi)t, is then totally (if possible: cCcarb 0,
c Cl  c 0l ) or partly (maximum release from MC carbon store) added to c Cl (xi) and
correspondingly (mass conservation) subtracted from cCcarb (xi). Note that therefore in
this material model, between two diffusion steps of same duration t in the  lattice
that are respectively evaluated according to Eq. (10) with c Cl fC, the described local
carbon redistribution from
the carbides (source) to the
austenite matrix phase (sink)
occurs up to their complete
dissolution, i.e. cCcarb (xi)0,
changing the concentration
Figure 10. Carbide dissolution during decarburization.
profile c Cl (x) continuously.
Figure 10 represents the microstructure of decarburized through
hardened steel 90MnCrV8 after
faulty austenitizing. The sharp
boundary of carbide dissolution
that has been formed, points to a
large k value in Eq. (11).
Figure 11. Decarburization with fast carbide
dissolution.
Figure 12. Decarburization with stable carbides.
1 - 103
This metallographic finding is
taken into account in the first
simulation shown in Fig. 11: it
illustrates the effect of including
carbide dissolution in the decarburization model even if diffusion
remains the rate-controlling reaction step. For comparison, the onephase calculation for pure austenitic material () according to
the Van Ostrand-Dewey equation,
cCcs(c0cs)erfx/(2Dt),
is
drawn in. Applying the same process time of t1 h, Fig. 12 demonstrates the other limiting case, k0.
This supposition corresponds to
the unrealistic assumption of
stable carbides during decarburization of the  matrix. An intermediate value of the control parameter k is considered in the simulation of Fig. 13. The gradual carbide dissolution leads to a broad
inward moving transition region.
In Fig. 13, the resulting total
carbon profile, cC(tot)  c Cl  cCcarb , is
exemplarily drawn in. From these
cC(tot) –x curves, the decarburization
depth dd can be determined.
Figure 13. Decarburization with medium
carbide dissolution rate.
Figure 14 reveals the time development of dd as the result of this
evaluation. For comparison, the
one-phase computation is also included. Validity of the parabolic
kinetics law for the decarburization depth, ddt, points to a
diffusion controlled process.
According to Fig. 14, consideration of simultaneous carbide dissolution yields lower dd values that
decrease further with increasing
rate constant k. Note that the effect
on the mechanical properties, e.g.
hardness loss in the edge zone,
mainly depends on the decarburization level of the austenite matrix that can be characterized, for
instance, by the surface distance
x( c Cl 0.92 c 0l ).
Figure 14. Influence of carbide dissolution rate
on decarburization depth.
In the above computations (cf. Figs. 11 to 14), a uniform initial cCcarb profile is assumed, i.e. cCcarb (x0,t0) c0carb const. However, with the presented simulation tool,
also real heterogeneous carbide distributions (e.g. segregations) within the austenitic
matrix can be included in the extended decarburization model. An example is given in
Fig. 15. The calculation involves fast carbide dissolution according to Eq. (11), i.e.
k. In order to emphasize the effect on the decarburization depth-time curve, dd(t),
the longer process duration of t10 h is considered.
The original one-dimensional distance distribution of the carbides, taken from a lightoptical micrograph of rolling bearing steel 100Cr6 by digital image analysis, covers a
range of 0.530 mm. As usual for engineering materials, the detected arrangement
reveals pronounced accumulation and depletion regions around the average value. It is
recurrently repeated (period p) to provide the complete initial carbide distribution. At
1123 K, c 0carb amounts to 0.32 m.% C.
1 - 104
Figure 16 finally presents the
resultant development of the
decarburization depth with time.
The drawn comparison with the
curve describing the corresponding uniform particle arrangement,
cCcarb (t0)0.32 m.% C, evidently
illustrates the influence of the considered heterogeneous carbide distribution on the program output.
Figure 15. Decarburization with real initial
carbide distribution.
SUMMARY AND
CONCLUSIONS
Analytical simulations of industrially highly relevant carburization and decarburization processes of low-alloyed steels are
presented and discussed in detail.
Comparisons with experimental
data confirm their applicability.
Consideration of steel dependent
carbon diffusivities is particularly
recommended to optimize the
process control of state-of-the-art
Figure 16. Effect of heterogeneous carbide distwo-step boost-diffuse gas carbutribution on decarburization depth.
rization heat treatments. The new
SIMS technique (secondary ion
mass spectrometry) for measuring carbon profiles with high accuracy should be used.
The presented computer model of decarburization with simultaneous carbide dissolution illustrates the potential of analytical simulation methods, as realization of this
complex process scheme with less flexible standard numerical software systems is difficult.
ACKNOWLEDGEMENT
The SIMS measurements were performed in cooperation with the Institute of Material
Physics, University of Göttingen (Germany). The author is grateful to Dr. Peter J.
Wilbrandt and Prof. Dr. Reiner Kirchheim for support and helpful discussions.
1 - 105
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