UDK 621.793:669.14	ISSN 1580-2949
Original scientific article/Izvirni znanstveni članek	MTAEC9, 48(2)237(2014)
ESTIMATION OF THE BORON DIFFUSION COEFFICIENTS IN FeB AND Fe2B LAYERS DURING THE PACK-BORIDING OF A HIGH-ALLOY STEEL
DOLOČANJE KOEFICIENTA DIFUZIJE BORA V PLASTEH FeB IN Fe2B MED BORIRANJEM VISOKO LEGIRANEGA JEKLA V
SKRINJI
Zahra Nait Abdellah1'2, Mourad Keddam1
1Laboratoire de Technologie des Materiaux, Departement de Sciences des Materiaux, Faculte de Genie Mecanique et Genie des Procedes,
USTHB, B.P N°32, 16111 El-Alia, Bab-Ezzouar, Alger, Algerie 2Departement de Chimie, Faculte des sciences, Universite Mouloud Mammeri, 15000 Tizi-Ouzou, Algerie
keddam@yahoo.fr
Prejem rokopisa - received: 2013-03-31; sprejem za objavo - accepted for publication: 2013-06-07
In this work we propose a diffusion model to estimate the boron diffusion coefficients in FeB and Fe2B layers during the pack-boriding of AISI M2 steel in the temperature range 1173-1323 K for a treatment time of 4-8 h.
The proposed model is based on the mass-balance equations at the two growth fronts - FeB/Fe2B and Fe2B/substrate - under certain assumptions. The estimated values of the boron activation energies in the FeB and Fe2B layers were compared with the literature data. The present model was extended to predict the thickness of each boride layer for the borided samples at different temperatures for 10 h. Iso-thickness diagrams were established to be used as a tool for predicting the thickness of each boride layer as a function of the two parameters: temperature and time. Finally, a simple equation was proposed to estimate the required time to obtain a single Fe2B layer by diffusion annealing.
Keywords: boriding, incubation times, Fick's laws, simulation, growth kinetics, annealing
Predstavljeno delo predlaga model difuzije za določanje koeficienta difuzije bora v plasteh FeB in Fe2B med boriranjem v skrinji jekla AISI M2 v temperaturnem območju 1173-1323 K pri spreminjanju trajanja postopka od 4 h do 8 h. Predlagani model temelji na enačbi masne balance na dveh rastočih mejnih ploskvah (FeB/Fe2B) in (Fe2B/osnova) pri določenih predpostavkah. Določena vrednost aktivacijske energije bora v FeB- in Fe2B-plasti je bila primerjana s podatki iz literature. Predstavljeni model je bil razširjen, da bi lahko napovedal debelino vsake od obeh boridnih plasti za borirane vzorce pri različnih temperaturah in trajanju do 10 h. Postavljeni so bili diagrami enake debeline, ki so uporabni kot orodje za napovedovanje debeline vsakega od boriranih slojev v odvisnosti od dveh parametrov (temperature in časa). Predlagana je preprosta enačba za določanje potrebnega časa za nastanek plasti Fe2B z difuzijskim žarjenjem.
Ključne besede: boriranje, inkubacijski čas, Fickovi zakoni, simulacija, kinetika rasti, žarjenje
___ _	gained much attention to simulate the boriding kinetics
1 INTRODUCTION	during r^^^nt decades.2-26
One of the surface-modification methods for improv- In the present work an original diffusion model is
ing the surface properties of ferrous and non-ferrous proposed to estimate the boron diffusion coefficients in
alloys is boriding. According to the Fe-B binary system,	the FeB and Fe2B layers grown on AISI M2 steel by
two kinds of iron borides, i.e., FeB and Fe2B, with a	considering the boride incubation times. A non-linear
narrow range of composition can be identified.^ The	boron-concentration profile is assumed through the
boriding process applies in the temperature range	boride layers. The mass-balance equations were applied
1073-1323 K between 1 h to 10 h and it can be carried
to the two diffusion fronts: the FeB/Fe2B and Fe2B/sub-
out in solid, liquid or gaseous media. The possible for-	strate interfaces in the temperature range 1173-1323 K.
mation of the FeB and Fe2B iron borides depends upon	In addition, a simple equation was proposed to estimate
various factors, such as the boron activity of the boriding	the required time to obtain a single Fe2B layer by medium, the chemical composition of the substrate, the	diffusion annealing. process temperature and the treatment time. The morphology of the boride layers is influenced by the pre-	2 THE DIFFUSION MODEL sence of alloying elements in the matrix. Saw-tooth-
shaped layers are obtained in low-aHoy steels, whereas in	The model takes into account the FeB/Fe2B bilayer
high-alloy steels, the interfaces tend to be flat. The	growth on fhe saturated substrate with boron atoms, as
modelling of the boriding kinetics is considered as a	showiiBin Figui-^B1.
suitable tool to match the case depth with the intended	C up' and C Jw (= 16.23 % B) are the upper andlower
industrial applications for this borided steel. So, the	boron masSBconcentrations in the FeB, while C(= 9 %
modelling of the growth kinetics for boride layers has	B) and C l„ew (= 8 83 % B) are, resPectively, the uPPer
Figure 1: Boron concentration profile through the FeB/FeiB bilayer Slika 1: Profil koncentracije bora skozi plasti (FeB/FeiB)
and lower boron concentrations in the FeiB. Cads denotes the adsorbed concentration of boron,15 while u is the position of the FeB/FeiB interface, and v is the position of the FeiB/substrate interface. Co is the boron solubility in the matrix and is equal to 35 ■ 10-4 % B.2 The upper boron content in the FeB phase (C,FeB), imposed by the boriding medium, gives rise to the two iron borides: FeB and FeiB. From a thermodynamic point of view, the FeB phase exhibits a narrow composition range (of about the mole fraction x = 1 % B or the mass fraction w = 0.2 % B), as identified by Massalski.27 The upper boron content in the FeB phase was taken in the composition range of mass fractions 16.25-16.43 % B to obtain a bilayer configuration consisting of the two iron borides, FeB and Fe2B.
The following assumptions are considered during the formulation of the diffusion model:
•	The kinetics is dominated by the diffusion-controlled mechanism
•	The growth of the boride layers is a consequence of the boron diffusion perpendicular to the sample surface
•	The range of homogeneity of the iron borides is about x = 1 % B
•	The iron borides nucleate after a certain incubation time
•	The boride layer is thin in comparison to the sample thickness
•	A local equilibrium occurs at the phase interfaces
•	A planar morphology is assumed for the phase interfaces
•	The volume change during the phase transformation is ignored
•	The diffusion coefficient of boron in each iron boride does not vary with the boron concentration and follows an Arrhenius relationship
•	A uniform temperature is assumed throughout the sample
•	The alloying elements have no effect on the boron diffusion
• The presence of porosity is neglected during the boron diffusion.
The initial conditions of the diffusion problem are set up as follows:
C FeB {X(t > 0) = 0}= 0 C pe2B {X(t ^ 0) = 0}= 0	(1)
C Fe {X(t ^ 0) = 0}= 0
The boundary conditions are given by the following equations:
C FeB {x[t = tFeB(r)] = 0} = C:peB for C ads ^ 16.23wt.%B (2)
C FeB {x[t = tFeB(r)] = 0} = C lFf for C ads ^ 16.23wt.%B and with the FeB phase:	(3)
CFe2B{x[t = t':'"(T)] = 0} = C:pe2B for 8.83wt.%B ^ Cads ^ 16.23wt.%B and without the FeB phase:	(4)
CFe2B{x[t = tFe2B(T)] = 0} = Ci;;:2b for Cads ^ 8.83wt.%B and without the FeB phase:	(5)
C FeB(x(t = t) = U) = C C Fe2B(x(t = t) = u) = C
^ u(x(t = t) = v) = C
FeB low . Fe2B up
.FejB low
(6) (7)
(8)
C Pe(x(t = t) = v) = C 0	(9)
The mass-balance equations28 are given by the equa-
tions (10) and (11):
(du dt
dv
w
'jFeB - ./!e2B
w
dt
+w'
'du v dt,
j
(10) (11)
with
W FeB =1
w
Fe B =
[05x(C :;B - C-)+(C lFwB -C :;2B)_ x(C :;2B - C F^)+(C r -C 0^ w=05x (C :;2B -C r)
The boron flux through a given boride layer is obtained from Fick's first law as follows: , dC (x, t)
JB =-DB —d-with i = (FeB or Fe2B)	(12)
D
FeB and D!e2B are, respectively, the diffusion coeffi-
^B and db
cients of boron in the FeB and Fe2B phases. The boron concentration profile in the FeB layer is given by:
FeB
C FeB( x, t) = C r +
(C — C P®B)
(C low C up '
erf
u
2. /DB^
• erf
2. Dr
For0 < x < u	(13)
In the same way, the boron concentration profile in the Fe2B layer can be obtained as follows:
C t)=C +
V'-' low	up
r
I erf

- erf

r
erf
/ X		/ X
u	-erf	x
1 t J		[J
For u < x < v
(14)
u = k FeB [t-t FeB( T )]1
The FeB layer thickness u grows parabolically according to equation (15), where kFeB represents the parabolic growth constant at the FeB/FeaB interface:
(15)
The distance v is the location of the FeaB/substrate interface and k its parabolic growth constant (equation (16)) and the difference (l = v - u) denotes the layer thickness of the FeaB (equation 17):
v = k[t -1 „( T)]1/2	(16)
l = v - u = k[t -1 „( T)]1/2 - k FeB [t -1 FeB( T)]1/2 (17) with tFeB (T) ^ 10 (T) and k >■ kFeB
where to(T) is the boride incubation time of the total boride layer and t FeB (T) is the boride incubation time of the FeB layer. To take into account the effect of the boride incubation times when solving the mass-balance equations, it is necessary to define the two parameters /3FeB(T) and ß(T):
and
ß FeB (T) =
ß( T) =
r
t F
1-
T) 10
1-
t o( T)
(18)
(19)
The layer thickness of the FeB (u) is related to the ßFeB(^ parameter by equation (20):
u = k FeB ß FeB( T)^t	(20)
In the same way, the layer thickness of the Fe2B (l) is expressed using equation (21):
l = [kß(T) - k FeB ß FeB( T)]^t	(21)
3 ESTIMATION OF THE BORON DIFFUSION COEFFICIENTS IN THE FeB AND FeiB LAYERS
To estimate the boron diffusion coefficients in the FeB and Fe2B layers, the experimental results published by Campos-Silva et al.29 on borided AISI M2 steel were used. In this reference work, the powder-pack boriding was carried out at four temperatures, (1173, 1223, 1273 and 1323) K, for three exposure times, (4, 6 and 8) h, using the B4C Durborid as a boriding medium. Eighty measurements were performed on different cross-sections of the borided samples from the AISI M2 steel to determine the thickness of each boride layer.
Tables 1 and 2 list the experimental parabolic growth constants for each phase interface with the corresponding incubation times. The experimental values of the parabolic growth constants at each phase interface were obtained from the slopes of the curves relating the squared boride layer thickness to the boriding time. The boride incubation times were deduced for a null boride layer thickness.
Table 1: Experimental values of the parabolic growth constants at the FeB/Fe2B interface in the temperature range 1173-1323 K with the corresponding boride incubation times
Tabela 1: Eksperimentalne vrednosti konstant parabolicne rasti na stiku (FeB/Fe2B) v temperaturnem območju 1173-1323 K, z ustreznim inkubacijskim časom borida
T/K	Experimental growth Constants: kFeB/(^m s-1/2)	tFeB( T )/s	
1173	0.065		10131
1223	0.121		6085.7
1273	0.179		4347.8
1323	0.238		3815.5
Table 2: Experimental values of the parabolic growth constants at the Fe2B/substrate interface in the temperature range 1173-1323 K with the corresponding boride incubation times
Tabela 2: Eksperimentalne vrednosti konstant parabolicne rasti na stiku (Fe2B/podlaga) v temperaturnem območju 1173-1323 K, z ustreznim inkubacijskim časom borida
T/K	Experimental growth Constants: k/(^m s-1/2)	t0(T)/s
1173	0.168	8806.2
1223	0.305	4729
1273	0.448	4323
1323	0.589	3742.7
It was demonstrated that the higher boriding temperatures involve the shorter incubation times,24 as shown in Tables 1 and 2. The two respective parameters ßFeB(r) and ß(T) are linearly dependent on the boriding temperature and can be approximated by equations (22) and (23) from a linear fitting of the experimental data displayed in Figure 2:
ß FeB( T) = (1-39 x10 -3 T - 0.8579)	(22)
and	ß(T) = (1-40x10-3 T - 0.8470)	(23)
For this purpose, a computer code written in Matlab (version 6.5) was used to estimate the boron diffusivity in each boride layer. This program requires the following input data: the time, the temperature, the lower and upper boron concentrations at each phase interface as well as the two parameters ßFeB(r) and ß(T). By solving the mass-balance equations (equations (10) and (11)) via the Newton-Raphson method,30 it is possible to determine the boron diffusion coefficients in the FeB and Fe2B layers. Table 3 summarizes the estimated values of the boron diffusion coefficients in the FeB and Fe2B layers for an upper boron content equal to w = 16.40 % in the FeB phase.
Figure 3 depicts the temperature dependence of the boron diffusion coefficients in the FeB and Fe2B layers according to the Arrhenius equation. The value of the
1
v
u
1
Figure 2: Evolution of the two parameters as a function of the boriding temperature: a) ;SFeB(I) and b) ß(T)
Slika 2: Razvoj dveh parametrov v odvisnosti od temperature bori-ranja: a) ßFeB(^ in b) ß(T)
Figure 3: An Arrhenius relationship between the boron diffusion coefficient and the temperature: a) FeB layer, b) FeaB layer Slika 3: Arrheniusova odvisnost med koeficientom difuzije bora in temperaturo: a) FeB-plast, b) FeaB-plast
Table 3: Determination of the boron diffusion coefficient in each boride layer for an upper boron mass fraction content of w =16.40 % in the FeB layer
Tabela 3: Določanje koeficienta difuzije bora v vsaki boridni plasti za zgornji masni delež vsebnosti bora 16,40 % v plasti FeB
T/K	D!eB(m2s-1) x10	
1173	0.376	B 0.462
1223	1.282	1.502
1273	2.797	3.227
1323	4.915	5.539
-220. 2 kJ/mol
= 2.8x10-3 exp
(m2 s -1)
(24)
d!'2" = 1.6 x10-
B
exp
(m2 s -')
(25)
boron activation energy in each boride layer can be easily obtained from the slopes of the corresponding curves. So, the boron diffusion coefficients in the FeB and Fe2B layers are, respectively, given by equations (24) and (25):
where R is the universal gas constant (= 8.314 J/(mol K)), and T represents the absolute temperature in Kelvin.
The reported values of the activation energies24,29,31-33 of the borided steels are listed in Table 4 together with the values from this work. The obtained values of the activation energies are found to be dependent on the boriding method and on the chemical composition of the substrates.
In Table 5, a comparison was achieved between the experimental boride layer thicknesses and the simulated ones at different temperatures for 10 h of treatment. The simulated results were obtained from equations (20) and (21). The present model was able to predict the boride layer thickness (FeB or Fe2B) for the given boriding conditions.
Table 4: Values of the boron activation energies obtained for different borided steels Tabela 4: Vrednosti aktivacijske energije bora, dobljene iz različnih boriranih jekel
Material	Boriding method	Activation energy of FeB £/(kJ mol-1)	Activation energy of Fe2B £/(kJ mol-1)	Reference
AISI M2	Paste	283	239.4	32
AISI4140	Paste	-	168.5	33
AISI H13	Powder-pack	-	186.2	31
AISI 316L	Powder-pack	204	198	24
AISI M2	Powder-pack	223	207	29
AISI M2	Powder-pack	220.2	213	Present study
-213 kJ/mol
Table 5: Experimental (exp.) and simulated (sim.) values of the boride layer thickness in the temperature range 1173-1323 K for 10 h of treatment, with an upper boron mass fraction of content of w = 16.40 % in the FeB phase
Tabela 5: Eksperimentalne (exp.) in simulirane (sim.) vrednosti za debelino plasti borida v temperaturnem območju 1173-1323 K, za 10 h obdelave pri gornjem masnem deležu vsebnosti bora 16,40 %, v FeB plasti
T/K	FeB (pm)	FeB (pm)	Fe2B (pm)	Fe2B (pm)
	exp.	sim.	exp.	sim.
1173	10.17	10.30	19.66	16.70
1223	20.98	17.89	32.81	28.23
1273	28.30	29.75	51.83	45.81
1323	40.24	47.60	72.28	71.67
T/K	Time	FeB (pm)	FeB (pm)	Fe2B (pm)	Fe2B (pm)
	(h)	exp.	sim.	exp.	sim.
	4	4.24	6.47	8.31	10.53
1173	6	6.96	7.92	12.04	12.89
	8	8.88	9.15	14.87	14.89
	4	11.03	11.32	18.96	17.85
1223	6	15.07	13.86	24.54	21.86
	8	18.23	16.00	29.08	25.24
	4	17.94	18.81	27.02	28.97
1273	6	23.51	23.04	35.37	35.48
	8	28.00	26.60	42.09	40.97
	4	24.48	24.89	36.31	35.90
1323	6	31.73	32.27	46.97	46.43
	8	37.61	38.25	55.61	54.98
concentration gradient of boron in the FeB is null (i.e., C uF;;B = C FwB = 16.23 %), the FeB layer will be converted into an Fe2B layer. The time required to eliminate the FeB layer during the diffusion annealing can be obtained from equation (26):
1 FeB /-^FcjBn
t
uxlx(Cr -Cup

UFeB = 0
ö!e2B(c
Fe2B
up
C low /
(26)
In Table 6, the predicted values of the boride layer thicknesses are compared with the experimentally determined values in the temperature range 1173-1323 K for a treatment time varying from 4 h to 8 h. Good agreement was observed between the experimental data and the simulation results for an upper boron content equal to w = 16.40 % in the FeB phase.
Table 6: Experimental (exp.) and simulated values (sim.) of the boride layer thickness in the temperature range 1173-1323 K for different treatment times with an upper boron content w = 16.40 % in the FeB phase
Tabela 6: Eksperimentalne (exp.) in simulirane (sim.) vrednosti debeline plasti borida pri temperaturah 1173-1323 K za različne čase obdelave in zgornjo vsebnostjo bora w = 16,40 % v FeB-fazi
where u is the FeB layer thickness (pm), l the Fe2B layer thickness (pm) and represents the boron diffusion coefficient in Fe2B. It is clear that the annealing time depends on the boron diffusion coefficient in Fe2B, and also on the thickness of each boride layer. During the diffusion annealing, an infinitesimal reduction of the FeB layer is related to the infinitesimal growth of the Fe2B layer by equation (27):
Au = -
w
Fe2B
Al =-05493Al (27)
The value of the Fe2B layer thickness	l' (pm) after diffusion annealing becomes:
Au
l'=l+05U-3	(28)
Table 7 presents the simulation results obtained from equations (26) and (28) to estimate the Fe2B layer thickness after diffusion annealing and the time required
Figure 4 displays the iso-thickness diagrams describing the evolution of the boride layer thickness as a function of the time and the boriding temperature. The results derived from Figure 4 can be used as a tool to predict the boride layer thickness in relation with its practical use in an industrial area.
4 OBTAINING OF A SINGLE LAYER OF FeiB BY DIFFUSION ANNEALING
In industrial practice it is possible to reduce the brittleness of boride layers by controlling their micro-structure. It is known that a single Fe2B boride layer is more desirable than a dual FeB-Fe2B layer.34 This makes it possible to reduce the FeB layer thickness by applying a diffusion annealing in a hydrogen atmosphere. During this stage, the supply of boron is stopped since the
Figure 4: Iso-thickness diagrams describing the evolution of the boride layers: a) FeB, b) Fe2B
Slika 4: Diagram enakih debelin opisuje razvoj boridnih plasti: a)
FeB, b) Fe2B
to eliminate the FeB layer in the case of the borided samples treated at different temperatures for 10 h. The obtained annealing times are increased with an increase of the boriding temperature since the boride layer becomes thicker. In this context, Kulka et al.26 have experimentally determined the annealing time using a hydrogen atmosphere to obtain a single FeaB layer on gas borided Armco Fe at 1173 K for 2 h in a gas mixture (Ha-BCls). They found that the total elimination of the FeB layer took about 1 h.
Furthermore, it was shown by Dybkov et al.35 that annealing of a borided Fe-Cr sample for 6 h resulted in the disappearance of the FeB layer.
Table 7: Estimation of the Fe2B layer thickness and the time required to eliminate the FeB layer for the borided samples at different temperatures for 10 h
Tabela 7: Določanje debeline plasti Fe2B in cas, potreben za odpravo FeB plasti, za vzorce, borirane 10 h pri različnih temperaturah
T/K	FeB sim.	Fe2B sim.	Fe2B (^m) After diffusion annealing Equation (28)	Annealing time Equation (26)
1173	10.30	16.70	35.45	4.15
1223	17.89	28.23	60.79	4.99
1273	29.75	45.81	99.96	5.92
1323	47.60	71.67	158.32	6.93
5 CONCLUSIONS
In this work an original diffusion model was proposed to estimate the boron diffusion coefficients in the FeB and Fe2B layers grown on AISI M2 steel. To determine the boron activation energy in each boride layer, the mass-balance equations were formulated, including the effect of the boride incubation times. The estimated boron activation energies were compared with the literature data. The present model was extended to predict the thickness of each boride layer for the borided samples at different temperatures for 10 h. Iso-thickness diagrams were established to be used as a tool to predict the thickness of each boride layer as a function of the two parameters (temperature and time). The required time to obtain a single Fe2B layer by diffusion annealing was estimated on the basis of a simple equation. The formation of a single Fe2B layer on AISI M2 steel depended on the boriding parameters.
Acknowledgements
This work was carried out in the framework of the CNEPRU project under code number J0300220100093 of the Algerian Ministry of Higher Education and Scientific Research.
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