- Email: [email protected]
Growth kinetics of nitride layers during microwave post-discharge nitriding
Surface & Coatings Technology 200 (2005) 1458 – 1463 www.elsevier.com/locate/surfcoat
Growth kinetics of nitride layers during microwave post-discharge nitriding J.L. Bernal a, A. Fraguela b, J. Oseguera c,*, F. Castillo c a
ITESM-TOL, E. Monroy C 2000, Toluca Edo de Mex. 50110, Me´xico BUAP, CU, Rı´o Verde y Av. San Claudio, San Manuel Puebla, Mexico ITESM-CEM, Carretera al Lago de Guadalupe, km. 3.5 Edo de Mex. 52926, Me´xico b
c
Available online 16 September 2005
Abstract An algorithm is presented to estimate the nitrogen diffusion coefficient during the growth of nitrided concomitant layers produced by microwave post-discharge nitriding. Diffusion coefficients in each phase are estimated by setting the inverse problem associated with growth of a compact nitrided layer gV-Fe4N 1x and the formation of an austenite layer. Both layers grow over a nitrogen diffusion zone in ferrite. The associated direct problem is a moving boundary one with conditions of Stefan type, where the diffusion coefficient in ferrite is assumed to be known. In this frame, the evolution of the layer width is studied from the initial states of the process. From the very beginning of the diffusion process, a nitrogen profile in ‘‘supersatured’’ ferrite is considered. The evolution of nitrogen profile concentration from supersatured ferrite to the formation of compact nitride layers is described. Nitrogen concentrations in each phase and diffusion zone are not considered to be bounded by their solubility limits. Evolution for large periods (quasi-steady periods), coincides with layer growth evolution considered in mass balance models. D 2005 Elsevier B.V. All rights reserved. Keywords: Post-discharge nitriding; Diffusion; Inverse problem; Diffusion coefficients
1. Introduction Microwave post-discharge nitride flow is mainly composed of neutral and excited species [1 – 4]. Several materials have been treated using these active molecular and atomic species in the post-discharge, improving their mechanical, tribological and corrosion properties [5,6]. Mathematical simulation of the growth of concomitant nitride layers in post-discharge nitriding has been performed by models considering mass balance in each interface and nitrogen concentration on surface [7,8]. The rapid evolution of surface nitrogen concentration in this process, in contrast to the ammonia gas mixture process [9 –12], develops a compact nitride layer and a parabolic growth from the very beginning [13]. Nevertheless, it has been pointed out [14] that in some models of the analogous type, the interfaces may move as t n/2 for small time, where n 2. * Corresponding author. E-mail address: [email protected] (J. Oseguera). 0257-8972/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2005.08.058
The diffusion coefficients in each phase have been calculated assuming parabolic growth of the interfaces [7,8]. The mathematical simulation from mass balance assumes the growth of layers in a moving flat front parallel to the surface, with equal specific volumes. Thermodynamic equilibrium is assumed between the growing phases. The nitrogen concentrations in the interfaces are represented from phase diagram nitrogen solubility at each temperature. Mass transport in the solid has been described by Fick’s equations, where intrinsic diffusion coefficients are independent of concentration and follow an Arrhenius relationship [7,8]. Based on experimental results obtained from iron nitriding in an atmosphere produced by post-discharge microwaves, we compare results from mass balance mathematical simulation and a moving boundary model with Stefan type conditions which considers a quasistationary state for large nitriding periods [15]. To develop the numerical algorithm, a functional to be minimized is built. This functional measures the deviation between the data theoretically obtained from the approximate solution of
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
where [15]. The general sequence of the nitriding experiments started with heating of the sample to 893 K in a tubular resistance furnace in a non-oxidized and nonnitriding atmosphere composed of 26Ar – 80H2 at a total pressure of 900 Pa. The applied and reflected power were 200 and 65 W and the distance from the discharge point was 7 cm. Upon reaching the prescribed temperature, the atmosphere was switched to a mixture of 300 N2 – 26 Ar – 80 H2 at 1200 Pa and recording of the nitriding time started. After the nitriding time was completed, the atmosphere was switched back to the initial non-oxidizing, non-nitriding atmosphere. Fig. 1 presents a cross-sectional view of a sample nitrided for 120 min under the conditions described previously. The nitrides below the compact nitride layers precipitated during cooling of the sample, due to the desaturation of the ferrite. The width of this zone is between 40 and 50 Am, and the diffusion zone of nitrogen in the ferrite is near 3 mm.
γ’- Fe4N1-x Transformed austenite
Diffusion Zone
1459
5 µm
Fig. 1. Cross-sectional view of a nitrided iron sample in microwave post-discharge.
the model and the experimental data. Experimental results and some experiments are shown.
2. Experimental details 3. Mathematical model Samples were obtained from an ARMCO iron bar (25.4 mm diameter and 7 mm thick. Mn 880 ppm; C and P, 200 ppm; and S 150 ppm). Nitriding was carried out in a postdischarge microwave-generated plasma described else-
(a)
Fig. 2a –d shows a representation of nitrogen concentration profile during the growth process. Fig. 2d corresponds to the growth associated to the mass balance models.
(c) Cs
t*
1
C
t2
1
C
0
min
min
C2max
C2max C2min
C2min
C3max
C
3
C
3
C
max
3
min
0
X
1
0
X
min
X1 X2
Xα
2
(b)
Xα
(d) Cs
Cs
t1
C1min C2max
C2max
C2min
C2min
C3max
C3max
C3min
C3min X1 X02
t(st)
C1min
Xα
Xγ’ Xγ
Xα
Fig. 2. Schematic representation of nitrogen profiles. Mass balance model. (a) The formation of compact layers begins to take place, (b) nitrogen concentration in gV layer has attained a value that corresponds to the equilibrium with austenite, (c) the growth of nitride layers has attained the nitrogen concentration according to the Fe – N phase diagram, (d) shows that the growth of the nitride layers follows from the mass balance model assumptions.
1460
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
The interface position as a function of time has a parabolic growth. The layer growth is described by displacement of discontinuity in each interface. This growth has been determined by balance between inward and outward flux through the interface. The mass balance equation is written as [7]:
i iþ1 Cmin Cmax
dni BCi ¼ Di ð x; t Þjx¼ni0 dt Bx BCiþ1 ð x; t Þjx¼niþ0 ; þ Diþ1 Bx
A certain time t 0 should pass for the concentration to reach the threshold value C S, which follows from:
where C i (x,t), i = 1, 2 ,3, represents the nitrogen concentration at each layer and diffusion zone, C imin, i = 1, 2 is the minimum nitrogen concentration at each layer, C i+1 max is the maximum value of the nitrogen concentration at the austenitic layer and the diffusion zone respectively. Finally, D i , i = 1, 2, 3 stand for the diffusion coefficients at each layer and diffusion zone. Considering one flow balance equation for each interface and the classical mathematical solutions for Fick’s equations [16], a system of coupled conditions for each interface is obtained [7,8]. Table 1 summarizes values of diffusion coefficient obtained from experimental data in post-discharge nitriding by the mass balance approach. The diffusion is considered to evolve as a flat front parallel to the surface, with equal specific volumes. The temperature in the specimen is identical during the whole process. The evolution of nitrogen concentration at the beginning is described by a normal diffusion process
C ð x; 0Þ ¼ C0 ;
0 < x < þ V;
t>0
ð1Þ
0
lim C ð x; t Þ ¼ C0 ;
ð2Þ
t > 0:
xYþV
The evolution of the nitrogen concentration on the surface may be described by [17]: BC k ð0; t Þ ¼ C Ceq x¼0 ; Bx D
t>0
! C Ceq x ¼ erf C0 Ceq 2ð DtÞ1=2 ! k k2 x k 1=2 x þ t erfc þ t þ exp : D D D1=2 2ð Dt0 Þ1=2 ð4Þ
i ¼ 1; 2
BC B2 C ¼D 2 ; Bt Bx
concentration. The analytical solution of (1) –(3) is given by [18]:
ð3Þ
where k is the kinetic reaction coefficient, C eq is the equilibrium concentration and C(x,t) represents the nitrogen Table 1 Values of diffusion coefficient obtained from experimental in postdischarge nitriding by the mass balance approach Obtained coefficients diffusion from reference at 843 K [m2/s]
Calculated diffusion coefficient from mass balance at 843 K [m2/s]
D 3 = D a = 1.83122 10 11 [19] D 2 = D g = 1.27401 10 14 [20] D 1 = D gV= 3.02204 10 13 [21]
D 3 = D a = 1.83122 10 11 [19] D 2 = D g = 2.186 10 13 D 1 = D gV= 1.0259 10 13
2 C Ceq k k 1=2 t0 erfc ¼ exp t : C0 Ceq D D1=2 0
ð5Þ
Let us put
f ð xÞuC ð x; t0 Þ ¼ Ceq þ C0 Ceq
( erf
x
!
2ð Dt0 Þ1=2
!) k k2 x k 1=2 x þ t0 erfc þ 1=2 ðt0 Þ þ exp D D D 2ð Dt0 Þ1=2
where f(x) denotes the initial profile of concentration when the layer formation begins. The first step corresponds to a normal diffusion process which takes place before the formation of the layers gVand the austenite. For this period, labeled as time t 0 in Fig. 2a, a concentration profile is shown corresponding to the solution of nitrogen in ferrite, corresponding to f(x) and marked with a dashed line in Fig. 2a to d. In the model, this is equivalent to a ‘‘supersatured’’ ferrite. During this period, the formation of compact layers begins to take place, as is shown in the same Fig. 2a, the points x 1 and x 2 x 1 represent the width of the layer. Note that in this case, the position of the discontinuity does not correspond to the thermodynamic phase equilibrium associated with the limits of solubility of each phase in the Fe –N phase diagram. Nitrogen concentration of phase gVin equilibrium with austenite has a value determined by the intersection with f(x) ‘‘supersatured’’ ferrite. However, the nitrogen concentration in the ferrite coincides with the initial nitrogen concentration profile. The beginning of the layer and interface formation are modeled by Fick’s law at each phase. The nitrogen diffusivity in gVFe4N 1x , g-austenite and a-ferrite are denoted by D 1, D 2 and D 3, respectively. BCi B2 C ¼ Di 2 ; t > t0 ; ni1 ðt Þ < x < ni ðt Þ; i ¼ 1; 2; Bx Bt t > t0 ; n2 ðt Þ < x; i ¼ 3
ð6Þ
Ci ð x; t0 Þ ¼ f ð xÞ; x0i1 < x < x0i ; i ¼ 1; 2; x02 < x; i ¼ 3
ð7Þ
i Ci ðni0 ðt Þ; t Þ ¼ Cmin ; t > t0 ; i ¼ 1; 2
ð8Þ
C1 ð0; t Þ ¼ CS ; t > t0;
ð9Þ
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
iþ1 Ciþ1 ðniþ0 ðt Þ; t Þ ¼ Cmax ; t > ti ; i ¼ 1; 2
i Cmin Fi ðt Þ
ð10Þ
dni BCi ¼ Di ð x; t Þjx¼ni0 dt Bx BCiþ1 ð x; t Þjx¼niþ0 ; þ Diþ1 Bx
lim C3 ð x; t Þ ¼ C0 ; t > t0
ni ðt0 Þ ¼ x0i ; i ¼ 1; 2:
decreasing and reaches its maximum and minimum values at the beginning and at the end of the phase, respectively. 3.1. Simulation of the concentration profile in the quasisteady stage We propose
i ¼ 1; 2 xYþV
1461
Ci ð xÞ ¼ ai ðx˜ i xÞmi þ bi ðx˜ xÞ þ ci ; i ¼ 1; 2; 3
ð12Þ ð13Þ
Note that this model describes the three different stages of the process, characterized by the following facts: When t Z [t 0,t 1 we have F i (t) = f(n i (t)), i = 1, 2. For t Z [t 1,t 2) we have F 1(t) = C 2max, F 2(t) = f(n 2(t)) and finally, for t t 2 we have F i (t) = C i+1 max, i = 1, 2. To be consistent with the common notation, we define n 3 (t), t t 0 as the depth at which 3 : C3 ðn3 ðt Þ; t Þ ¼ 0:1Cmax
The growth of each layer is shown in Fig. 2b, for a time labeled t 1. In the schematic representation, the nitrogen concentration in gV layer has attained a value that corresponds to the equilibrium with austenite. The width of gVis x 1. The growth of this layer follows from the mass balance model; in contrast, the austenite width is x 2 x 1, and the maximum nitrogen concentration in austenite has not been attained. In the same schematic representation, it is shown that the nitrogen concentration in austenite has not yet attained equilibrium with ferrite, which is determined from the Fe – N phase diagram. Fig. 2c, labeled as time t 2, shows that the growth of nitride layers has attained the nitrogen concentration according to the Fe – N phase diagram. This is the equilibrium which corresponds to the nitrogen concentration considered in mass balance models and is taken from the Fe –N phase diagram [7,8]. Fig. 2d, labeled as t st, shows how the growth of the nitride layers follows from the mass balance model assumptions. The last step corresponds to a period previous to the layer growth ‘‘stabilization’’, where the layers and interfaces present the behavior of a moving boundary problem as in the standard mass balance model. This last step is finished after a large time t 4. The layer flow balance becomes negligible, which is justified from experimental essays and also analytically: pffi interfaces move with velocities proportional to 1= t , which becomes smaller for large values of time. Thus, the process reaches a quasi-stationary stage, where variation (flC/flt) is very small. Moreover, from the fact in every stage diffusion equation fulfills (flC i /flt ) = D i (fl2C i /flx 2) and the rate of change of the concentration is positive, then (fl2C i /flx 2) 0 and C i for every i is a convex function in x, which is also
ð14Þ
where x˜ i is the depth at which each layer reached the quasisteady state (experimental observation). m i (integers greater or equal than 2), a i , b i , c i , i = 1, 2, 3 are constants to be determined. Following the usual notation for layers, we define: x˜ i ¼ xcV ; x˜ 2 ¼ xc ; x˜ 3 ¼ xa : The form of (14) guarantees that the last two terms also satisfy the quasi-steady state equation (fl2C i /flx 2) = 0, while the first term should be such that m i (m i 1)a i (x˜ i x)m i 2 be very small for x in the i-th phase. Form (14) is inspired by Goodman’s method [16]. The problem is then to determine the relationships between m i , a i , b i , c i , i = 1, 2, 3 such that conditions for decreasing concentration and convexity at each phase, and a concentration jump and null net flow at the interfaces are fulfilled. More exactly, we write down: m 1 C1 ð xÞ ¼ a1 xcV x þ b1 xcV x þ Cmin n 2 C2 ð xÞ ¼ a2 xc x þ b2 xc x þ Cmin C3 ð xÞ ¼ a3 ðxa xÞk þ b3 ðxa xÞ
C min3
because guarantees
ð15Þ
could be considered null. Form (15)
1 2 C1 xcV ¼ Cmin ; C2 xc ¼ Cmin ; C3 ðxa Þ ¼ 0:
From the null net flow in the third interface, it follows that b 3 = 0, so (15) takes the form m 1 C1 ð xÞ ¼ a1 xcV x þ b1 xcV x þ Cmin n 2 C2 ð xÞ ¼ a2 xc x þ b2 xc x þ Cmin C3 ð xÞ ¼ a3 ðxa xÞk :
ð16Þ
In a straightforward fashion, we obtain a3 ¼ ð 1Þk
3 Cmax
xa xcV
k
b2 ¼ k
3 D3 Cmax D2 ðxa Þ xc
ð17Þ
xc xcV D3 3 1 2 2 n Cmax Cmin k C a2 ¼ xa xc D2 max xcV xc
ð18Þ
" # D2 ðn 1Þk D3 3 n 2 2 C þ C Cmin b1 ¼ D1 xa xc D2 max xcV xc max
ð19Þ
1462
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
(
"
ðn 1Þk D3 3 C CS ðxa xc Þ D2 max #) 2 n 2 Cmax : þ Cmin xcV xc
1 a1 ¼ m xcV
1 Cmin
D2 þ xcV D1
To reach the quasi-steady state, we should have B Ci ð xÞ 0 Bx2 2
for all x, i = 1, 2, 3 and large values of time. If we assume that the concentrations decrease following a convex profile as functions of depth, we immediately have that BCi ð xÞ B2 Ci ð xÞ V0; 0 Bx Bx2
for all x and i = 1, 2, 3. These two conditions mean that m1 BC1 ð xÞ ¼ ma1 xcV x b1 V0 Bð xÞ
BC3 ð xÞ ¼ ka3 ðxa xÞk1 V0 Bx m2 B2 C1 ð xÞ ¼ mðm 1Þa1 xcV x 0 Bx2 n2 B2 C2 ð xÞ ¼ nðn 1Þa2 xc x 0 2 Bx B2 C3 ð xÞ ¼ k ðk 1Þa3 ðxa xÞk2 0: Bx2
D3 V D2
2 Cmin
3 kCmax
D2 D3 ; D1 D2
2 2 ¼ a1 C1 ð2Þ Cˆ 1 ð2Þ þ a2 C1 ð4Þ Cˆ 1 ð4Þ þ a3 C1 ð6Þ Cˆ 1 ð6Þ 2
xa xc xc xcV
1 3 CS Cmin xc xcV D1 kCmax þ 2 2 2 2 D2 xcV ðn 1Þ Cmax Cmin Cmax Cmin D3 n : D2 n1
xcV ¼ 7 lm; xc ¼ 10 lm; xa ¼ 50 lm 1 2 2 CS ¼ 19:2; Cmin ¼ 19; Cmax ¼ 9:625; Cmin ¼ 6:5 3 3 Cmax ¼ 0:365; Cmin ¼ 0 D1 ¼ 1:0259 1013 m2 =s; D2 ¼ 2:186 1013 m2 =s D3 ¼ 1:987 1011 m2 =s
J
Then m, n, k 2, a i 0, i = 1, 2, 3. After some manipulations, it follows 2 Cmax
To find diffusion coefficients D i , i = 1, 2, 3 we use the approximate expression (15) of the quasi-steady problem and a nonlinear optimization algorithm to solve the leastsquare problem. Experimental measures of the quasi-steady depth of the layers and diffusion zone are taken as well as the experimental values of C S, C imax, C imin. All these values were obtained in the laboratory by the procedure described in the second section above. The values of the diffusion coefficients D i , i = 1, 2 were obtained from experimental measures in post-discharge nitriding by the mass balance approach, while D 3 was chosen from references.
From this data, theoretical expressions for concentration profiles at each layer and diffusion zone are computed, using expressions (16) and relations (17) – (20). Then a simulation process is performed, generating artificial data from the theoretical expressions of the profile (adding random errors) in order to test the feasibility and sensitivity of the solution of the inverse problems. Due to technical difficulties in obtaining real measurements, the artificial data Cˆ i ( y j ) in the first two layers are generated only at depths y j = 2, 4, 6, 8 Am, corresponding to those layers and at depths y j Am, j = 5, . . . N corresponding to the zone of diffusion. Afterwards, we use the generated data to solve the leastsquare problem of minimizing the function
n1 BC2 ð xÞ b2 V0 ¼ na2 xc x Bx
4. Numerical experiments
2 N 2 þ a4 C2 ð8Þ Cˆ 2 ð8Þ þ ~ aj C3 yj Cˆ 3 yj : j¼5
xc xcV xa xc
These relationships establish bounds for the diffusion coefficients D i , i = 1, 2, 3 under the assumption of boundedness in the second derivatives (fl2C i /flx 2), convexity and decreasing profile of C i (x). These bounds may be used to check whether the diffusion coefficients initially given are reliable. If so, we may use them to obtain corrections of the coefficients through an algorithm based on an inverse problem. Numerical experiments have been conducted to choose suitable values for m, n, k in (15).
Computing was performed with a PC at 900 MHz, using MATLAB regression program version 6.5. The results of the numerical experiments yield: – Estimates are not sensitive to the increase of the number of measurements in the diffusion zone in the case when the artificial data does not have random errors. – Estimates are very sensitive to the initial vector in the optimization process. It seems that an initial value of (D 3/D 2) near to the order of the actual values is the best option, while for (D 2/D 1) the initial value does not matter when random errors are not considered. – Estimates are also sensitive to measurements errors. A regularization process is needed to improve results. – More numerical experiments are needed to confirm these preliminary conclusions.
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
5 . Conclusions & Based on the proposed moving boundary model with Stefan conditions, an algorithm which allows the estimation of approximate values for the diffusion coefficients in austenite and phase gV has been built. This algorithm assumes the diffusion coefficient of the ferrite to be known and also considers a gradual process of layer formation until solubility at each phase is reached, after the layer quasi-stabilization. & The optimization of the functional allows the analysis of whether numerical and experimental results are congruent. & The approach presented for obtaining diffusion coefficients through an inverse moving boundary problem with Stefan conditions may be generalized to the study of phenomena associated with mass transport and compact layer formation, e.g. cementation, boriding and oxidation processes. Acknowledgement The authors wish to acknowledge Dr. Juan Alfredo Gomez (UFRO, Chile) for his inspiring and fruitful collaboration on various parts of the work presented in this article.
References [1] A. Ricard, J.E. Oseguera, L. Falk, H. Michel, M. Gantois, IEEE Trans. Plasma Sci. 18 (6) (1990) 940. [2] Falk, J. Oseguera, A. Ricard, H. Michel, M. Gantois, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 139 (1991) 132.
1463
[3] T. Belmonte, C. Jaoul, J.N. Borges, Surf. Coat. Technol. 188 – 189 (2004) 201. [4] A. Ricard, C. Joaul, F. Gaboriau, N. Gherardi, S. Villeger, Surf. Coat. Technol. 188 – 189 (2004) 287. [5] A. Medina-Flores, J. Oseguera, P. Santiago, J.A. Ascencio, Surf. Coat. Technol. 188 – 189 (2004) 7. [6] A. Medina-Flores, C. Arganis, P. Santiago, J. Oseguera, Surf. Coat. Technol. 188 – 189 (2004) 140. [7] U. Figueroa, J. Oseguera, P.S. Shabes-Retchkiman, Surf. Coat. Technol. 86 – 87 (1996) 728. [8] Ivan Campos, J. Oseguera, U. Figueroa, E. Mele´ndez, Surf. Coat. Technol. 102 (1998) 127. [9] M.A.J. Somers, Mittemeijer, Metall. Mater. Trans. 26A (1995) 57. [10] H. Du, J. Agren, Metall. Mater. Trans., A Phys. Metall. Mater. Sci. 27A (1996) 1073. [11] L. Torchane, P. Bilinger, J. Dulcy, M. Gantois, Metall. Mater. Trans. 27A (1996) 1823. [12] Lifang Xia, Mufu Yan, Acta Metall. Sin. (English Edition), Ser. B 2 (1) (1989) 18. [13] Mufu Yan, Jihon Yan, T. Bell, Model. Simul. Mater. Sci. Eng. 8 (2000) 491. [14] L.N. Tao, Acta Mech. 45, 49 – 64. [15] J.L. Bernal, O. Salas, U. Figueroa, J. Oseguera, Surf. Coat. Technol. 177 – 178 (2004) 665. [16] T.R. Goodman, Adv. Heat Transf. 1 (1964) 51. [17] J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1956. [18] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd edR, Clarendon Press, Oxford, 1965. [19] J.D. Fast, M.B. Verrijp, J. Iron Steel Inst. 176 (1954) 24. [20] P. Grievson, E.T. Turkdogan, Trans. Met. Soc. AIME 230 (1964) 1604. [21] A. Marciniak, Surf. Eng. 1,4 (1985) 283.
Growth kinetics of nitride layers during microwave post-discharge nitriding J.L. Bernal a, A. Fraguela b, J. Oseguera c,*, F. Castillo c a
ITESM-TOL, E. Monroy C 2000, Toluca Edo de Mex. 50110, Me´xico BUAP, CU, Rı´o Verde y Av. San Claudio, San Manuel Puebla, Mexico ITESM-CEM, Carretera al Lago de Guadalupe, km. 3.5 Edo de Mex. 52926, Me´xico b
c
Available online 16 September 2005
Abstract An algorithm is presented to estimate the nitrogen diffusion coefficient during the growth of nitrided concomitant layers produced by microwave post-discharge nitriding. Diffusion coefficients in each phase are estimated by setting the inverse problem associated with growth of a compact nitrided layer gV-Fe4N 1x and the formation of an austenite layer. Both layers grow over a nitrogen diffusion zone in ferrite. The associated direct problem is a moving boundary one with conditions of Stefan type, where the diffusion coefficient in ferrite is assumed to be known. In this frame, the evolution of the layer width is studied from the initial states of the process. From the very beginning of the diffusion process, a nitrogen profile in ‘‘supersatured’’ ferrite is considered. The evolution of nitrogen profile concentration from supersatured ferrite to the formation of compact nitride layers is described. Nitrogen concentrations in each phase and diffusion zone are not considered to be bounded by their solubility limits. Evolution for large periods (quasi-steady periods), coincides with layer growth evolution considered in mass balance models. D 2005 Elsevier B.V. All rights reserved. Keywords: Post-discharge nitriding; Diffusion; Inverse problem; Diffusion coefficients
1. Introduction Microwave post-discharge nitride flow is mainly composed of neutral and excited species [1 – 4]. Several materials have been treated using these active molecular and atomic species in the post-discharge, improving their mechanical, tribological and corrosion properties [5,6]. Mathematical simulation of the growth of concomitant nitride layers in post-discharge nitriding has been performed by models considering mass balance in each interface and nitrogen concentration on surface [7,8]. The rapid evolution of surface nitrogen concentration in this process, in contrast to the ammonia gas mixture process [9 –12], develops a compact nitride layer and a parabolic growth from the very beginning [13]. Nevertheless, it has been pointed out [14] that in some models of the analogous type, the interfaces may move as t n/2 for small time, where n 2. * Corresponding author. E-mail address: [email protected] (J. Oseguera). 0257-8972/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2005.08.058
The diffusion coefficients in each phase have been calculated assuming parabolic growth of the interfaces [7,8]. The mathematical simulation from mass balance assumes the growth of layers in a moving flat front parallel to the surface, with equal specific volumes. Thermodynamic equilibrium is assumed between the growing phases. The nitrogen concentrations in the interfaces are represented from phase diagram nitrogen solubility at each temperature. Mass transport in the solid has been described by Fick’s equations, where intrinsic diffusion coefficients are independent of concentration and follow an Arrhenius relationship [7,8]. Based on experimental results obtained from iron nitriding in an atmosphere produced by post-discharge microwaves, we compare results from mass balance mathematical simulation and a moving boundary model with Stefan type conditions which considers a quasistationary state for large nitriding periods [15]. To develop the numerical algorithm, a functional to be minimized is built. This functional measures the deviation between the data theoretically obtained from the approximate solution of
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
where [15]. The general sequence of the nitriding experiments started with heating of the sample to 893 K in a tubular resistance furnace in a non-oxidized and nonnitriding atmosphere composed of 26Ar – 80H2 at a total pressure of 900 Pa. The applied and reflected power were 200 and 65 W and the distance from the discharge point was 7 cm. Upon reaching the prescribed temperature, the atmosphere was switched to a mixture of 300 N2 – 26 Ar – 80 H2 at 1200 Pa and recording of the nitriding time started. After the nitriding time was completed, the atmosphere was switched back to the initial non-oxidizing, non-nitriding atmosphere. Fig. 1 presents a cross-sectional view of a sample nitrided for 120 min under the conditions described previously. The nitrides below the compact nitride layers precipitated during cooling of the sample, due to the desaturation of the ferrite. The width of this zone is between 40 and 50 Am, and the diffusion zone of nitrogen in the ferrite is near 3 mm.
γ’- Fe4N1-x Transformed austenite
Diffusion Zone
1459
5 µm
Fig. 1. Cross-sectional view of a nitrided iron sample in microwave post-discharge.
the model and the experimental data. Experimental results and some experiments are shown.
2. Experimental details 3. Mathematical model Samples were obtained from an ARMCO iron bar (25.4 mm diameter and 7 mm thick. Mn 880 ppm; C and P, 200 ppm; and S 150 ppm). Nitriding was carried out in a postdischarge microwave-generated plasma described else-
(a)
Fig. 2a –d shows a representation of nitrogen concentration profile during the growth process. Fig. 2d corresponds to the growth associated to the mass balance models.
(c) Cs
t*
1
C
t2
1
C
0
min
min
C2max
C2max C2min
C2min
C3max
C
3
C
3
C
max
3
min
0
X
1
0
X
min
X1 X2
Xα
2
(b)
Xα
(d) Cs
Cs
t1
C1min C2max
C2max
C2min
C2min
C3max
C3max
C3min
C3min X1 X02
t(st)
C1min
Xα
Xγ’ Xγ
Xα
Fig. 2. Schematic representation of nitrogen profiles. Mass balance model. (a) The formation of compact layers begins to take place, (b) nitrogen concentration in gV layer has attained a value that corresponds to the equilibrium with austenite, (c) the growth of nitride layers has attained the nitrogen concentration according to the Fe – N phase diagram, (d) shows that the growth of the nitride layers follows from the mass balance model assumptions.
1460
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
The interface position as a function of time has a parabolic growth. The layer growth is described by displacement of discontinuity in each interface. This growth has been determined by balance between inward and outward flux through the interface. The mass balance equation is written as [7]:
i iþ1 Cmin Cmax
dni BCi ¼ Di ð x; t Þjx¼ni0 dt Bx BCiþ1 ð x; t Þjx¼niþ0 ; þ Diþ1 Bx
A certain time t 0 should pass for the concentration to reach the threshold value C S, which follows from:
where C i (x,t), i = 1, 2 ,3, represents the nitrogen concentration at each layer and diffusion zone, C imin, i = 1, 2 is the minimum nitrogen concentration at each layer, C i+1 max is the maximum value of the nitrogen concentration at the austenitic layer and the diffusion zone respectively. Finally, D i , i = 1, 2, 3 stand for the diffusion coefficients at each layer and diffusion zone. Considering one flow balance equation for each interface and the classical mathematical solutions for Fick’s equations [16], a system of coupled conditions for each interface is obtained [7,8]. Table 1 summarizes values of diffusion coefficient obtained from experimental data in post-discharge nitriding by the mass balance approach. The diffusion is considered to evolve as a flat front parallel to the surface, with equal specific volumes. The temperature in the specimen is identical during the whole process. The evolution of nitrogen concentration at the beginning is described by a normal diffusion process
C ð x; 0Þ ¼ C0 ;
0 < x < þ V;
t>0
ð1Þ
0
lim C ð x; t Þ ¼ C0 ;
ð2Þ
t > 0:
xYþV
The evolution of the nitrogen concentration on the surface may be described by [17]: BC k ð0; t Þ ¼ C Ceq x¼0 ; Bx D
t>0
! C Ceq x ¼ erf C0 Ceq 2ð DtÞ1=2 ! k k2 x k 1=2 x þ t erfc þ t þ exp : D D D1=2 2ð Dt0 Þ1=2 ð4Þ
i ¼ 1; 2
BC B2 C ¼D 2 ; Bt Bx
concentration. The analytical solution of (1) –(3) is given by [18]:
ð3Þ
where k is the kinetic reaction coefficient, C eq is the equilibrium concentration and C(x,t) represents the nitrogen Table 1 Values of diffusion coefficient obtained from experimental in postdischarge nitriding by the mass balance approach Obtained coefficients diffusion from reference at 843 K [m2/s]
Calculated diffusion coefficient from mass balance at 843 K [m2/s]
D 3 = D a = 1.83122 10 11 [19] D 2 = D g = 1.27401 10 14 [20] D 1 = D gV= 3.02204 10 13 [21]
D 3 = D a = 1.83122 10 11 [19] D 2 = D g = 2.186 10 13 D 1 = D gV= 1.0259 10 13
2 C Ceq k k 1=2 t0 erfc ¼ exp t : C0 Ceq D D1=2 0
ð5Þ
Let us put
f ð xÞuC ð x; t0 Þ ¼ Ceq þ C0 Ceq
( erf
x
!
2ð Dt0 Þ1=2
!) k k2 x k 1=2 x þ t0 erfc þ 1=2 ðt0 Þ þ exp D D D 2ð Dt0 Þ1=2
where f(x) denotes the initial profile of concentration when the layer formation begins. The first step corresponds to a normal diffusion process which takes place before the formation of the layers gVand the austenite. For this period, labeled as time t 0 in Fig. 2a, a concentration profile is shown corresponding to the solution of nitrogen in ferrite, corresponding to f(x) and marked with a dashed line in Fig. 2a to d. In the model, this is equivalent to a ‘‘supersatured’’ ferrite. During this period, the formation of compact layers begins to take place, as is shown in the same Fig. 2a, the points x 1 and x 2 x 1 represent the width of the layer. Note that in this case, the position of the discontinuity does not correspond to the thermodynamic phase equilibrium associated with the limits of solubility of each phase in the Fe –N phase diagram. Nitrogen concentration of phase gVin equilibrium with austenite has a value determined by the intersection with f(x) ‘‘supersatured’’ ferrite. However, the nitrogen concentration in the ferrite coincides with the initial nitrogen concentration profile. The beginning of the layer and interface formation are modeled by Fick’s law at each phase. The nitrogen diffusivity in gVFe4N 1x , g-austenite and a-ferrite are denoted by D 1, D 2 and D 3, respectively. BCi B2 C ¼ Di 2 ; t > t0 ; ni1 ðt Þ < x < ni ðt Þ; i ¼ 1; 2; Bx Bt t > t0 ; n2 ðt Þ < x; i ¼ 3
ð6Þ
Ci ð x; t0 Þ ¼ f ð xÞ; x0i1 < x < x0i ; i ¼ 1; 2; x02 < x; i ¼ 3
ð7Þ
i Ci ðni0 ðt Þ; t Þ ¼ Cmin ; t > t0 ; i ¼ 1; 2
ð8Þ
C1 ð0; t Þ ¼ CS ; t > t0;
ð9Þ
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
iþ1 Ciþ1 ðniþ0 ðt Þ; t Þ ¼ Cmax ; t > ti ; i ¼ 1; 2
i Cmin Fi ðt Þ
ð10Þ
dni BCi ¼ Di ð x; t Þjx¼ni0 dt Bx BCiþ1 ð x; t Þjx¼niþ0 ; þ Diþ1 Bx
lim C3 ð x; t Þ ¼ C0 ; t > t0
ni ðt0 Þ ¼ x0i ; i ¼ 1; 2:
decreasing and reaches its maximum and minimum values at the beginning and at the end of the phase, respectively. 3.1. Simulation of the concentration profile in the quasisteady stage We propose
i ¼ 1; 2 xYþV
1461
Ci ð xÞ ¼ ai ðx˜ i xÞmi þ bi ðx˜ xÞ þ ci ; i ¼ 1; 2; 3
ð12Þ ð13Þ
Note that this model describes the three different stages of the process, characterized by the following facts: When t Z [t 0,t 1 we have F i (t) = f(n i (t)), i = 1, 2. For t Z [t 1,t 2) we have F 1(t) = C 2max, F 2(t) = f(n 2(t)) and finally, for t t 2 we have F i (t) = C i+1 max, i = 1, 2. To be consistent with the common notation, we define n 3 (t), t t 0 as the depth at which 3 : C3 ðn3 ðt Þ; t Þ ¼ 0:1Cmax
The growth of each layer is shown in Fig. 2b, for a time labeled t 1. In the schematic representation, the nitrogen concentration in gV layer has attained a value that corresponds to the equilibrium with austenite. The width of gVis x 1. The growth of this layer follows from the mass balance model; in contrast, the austenite width is x 2 x 1, and the maximum nitrogen concentration in austenite has not been attained. In the same schematic representation, it is shown that the nitrogen concentration in austenite has not yet attained equilibrium with ferrite, which is determined from the Fe – N phase diagram. Fig. 2c, labeled as time t 2, shows that the growth of nitride layers has attained the nitrogen concentration according to the Fe – N phase diagram. This is the equilibrium which corresponds to the nitrogen concentration considered in mass balance models and is taken from the Fe –N phase diagram [7,8]. Fig. 2d, labeled as t st, shows how the growth of the nitride layers follows from the mass balance model assumptions. The last step corresponds to a period previous to the layer growth ‘‘stabilization’’, where the layers and interfaces present the behavior of a moving boundary problem as in the standard mass balance model. This last step is finished after a large time t 4. The layer flow balance becomes negligible, which is justified from experimental essays and also analytically: pffi interfaces move with velocities proportional to 1= t , which becomes smaller for large values of time. Thus, the process reaches a quasi-stationary stage, where variation (flC/flt) is very small. Moreover, from the fact in every stage diffusion equation fulfills (flC i /flt ) = D i (fl2C i /flx 2) and the rate of change of the concentration is positive, then (fl2C i /flx 2) 0 and C i for every i is a convex function in x, which is also
ð14Þ
where x˜ i is the depth at which each layer reached the quasisteady state (experimental observation). m i (integers greater or equal than 2), a i , b i , c i , i = 1, 2, 3 are constants to be determined. Following the usual notation for layers, we define: x˜ i ¼ xcV ; x˜ 2 ¼ xc ; x˜ 3 ¼ xa : The form of (14) guarantees that the last two terms also satisfy the quasi-steady state equation (fl2C i /flx 2) = 0, while the first term should be such that m i (m i 1)a i (x˜ i x)m i 2 be very small for x in the i-th phase. Form (14) is inspired by Goodman’s method [16]. The problem is then to determine the relationships between m i , a i , b i , c i , i = 1, 2, 3 such that conditions for decreasing concentration and convexity at each phase, and a concentration jump and null net flow at the interfaces are fulfilled. More exactly, we write down: m 1 C1 ð xÞ ¼ a1 xcV x þ b1 xcV x þ Cmin n 2 C2 ð xÞ ¼ a2 xc x þ b2 xc x þ Cmin C3 ð xÞ ¼ a3 ðxa xÞk þ b3 ðxa xÞ
C min3
because guarantees
ð15Þ
could be considered null. Form (15)
1 2 C1 xcV ¼ Cmin ; C2 xc ¼ Cmin ; C3 ðxa Þ ¼ 0:
From the null net flow in the third interface, it follows that b 3 = 0, so (15) takes the form m 1 C1 ð xÞ ¼ a1 xcV x þ b1 xcV x þ Cmin n 2 C2 ð xÞ ¼ a2 xc x þ b2 xc x þ Cmin C3 ð xÞ ¼ a3 ðxa xÞk :
ð16Þ
In a straightforward fashion, we obtain a3 ¼ ð 1Þk
3 Cmax
xa xcV
k
b2 ¼ k
3 D3 Cmax D2 ðxa Þ xc
ð17Þ
xc xcV D3 3 1 2 2 n Cmax Cmin k C a2 ¼ xa xc D2 max xcV xc
ð18Þ
" # D2 ðn 1Þk D3 3 n 2 2 C þ C Cmin b1 ¼ D1 xa xc D2 max xcV xc max
ð19Þ
1462
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
(
"
ðn 1Þk D3 3 C CS ðxa xc Þ D2 max #) 2 n 2 Cmax : þ Cmin xcV xc
1 a1 ¼ m xcV
1 Cmin
D2 þ xcV D1
To reach the quasi-steady state, we should have B Ci ð xÞ 0 Bx2 2
for all x, i = 1, 2, 3 and large values of time. If we assume that the concentrations decrease following a convex profile as functions of depth, we immediately have that BCi ð xÞ B2 Ci ð xÞ V0; 0 Bx Bx2
for all x and i = 1, 2, 3. These two conditions mean that m1 BC1 ð xÞ ¼ ma1 xcV x b1 V0 Bð xÞ
BC3 ð xÞ ¼ ka3 ðxa xÞk1 V0 Bx m2 B2 C1 ð xÞ ¼ mðm 1Þa1 xcV x 0 Bx2 n2 B2 C2 ð xÞ ¼ nðn 1Þa2 xc x 0 2 Bx B2 C3 ð xÞ ¼ k ðk 1Þa3 ðxa xÞk2 0: Bx2
D3 V D2
2 Cmin
3 kCmax
D2 D3 ; D1 D2
2 2 ¼ a1 C1 ð2Þ Cˆ 1 ð2Þ þ a2 C1 ð4Þ Cˆ 1 ð4Þ þ a3 C1 ð6Þ Cˆ 1 ð6Þ 2
xa xc xc xcV
1 3 CS Cmin xc xcV D1 kCmax þ 2 2 2 2 D2 xcV ðn 1Þ Cmax Cmin Cmax Cmin D3 n : D2 n1
xcV ¼ 7 lm; xc ¼ 10 lm; xa ¼ 50 lm 1 2 2 CS ¼ 19:2; Cmin ¼ 19; Cmax ¼ 9:625; Cmin ¼ 6:5 3 3 Cmax ¼ 0:365; Cmin ¼ 0 D1 ¼ 1:0259 1013 m2 =s; D2 ¼ 2:186 1013 m2 =s D3 ¼ 1:987 1011 m2 =s
J
Then m, n, k 2, a i 0, i = 1, 2, 3. After some manipulations, it follows 2 Cmax
To find diffusion coefficients D i , i = 1, 2, 3 we use the approximate expression (15) of the quasi-steady problem and a nonlinear optimization algorithm to solve the leastsquare problem. Experimental measures of the quasi-steady depth of the layers and diffusion zone are taken as well as the experimental values of C S, C imax, C imin. All these values were obtained in the laboratory by the procedure described in the second section above. The values of the diffusion coefficients D i , i = 1, 2 were obtained from experimental measures in post-discharge nitriding by the mass balance approach, while D 3 was chosen from references.
From this data, theoretical expressions for concentration profiles at each layer and diffusion zone are computed, using expressions (16) and relations (17) – (20). Then a simulation process is performed, generating artificial data from the theoretical expressions of the profile (adding random errors) in order to test the feasibility and sensitivity of the solution of the inverse problems. Due to technical difficulties in obtaining real measurements, the artificial data Cˆ i ( y j ) in the first two layers are generated only at depths y j = 2, 4, 6, 8 Am, corresponding to those layers and at depths y j Am, j = 5, . . . N corresponding to the zone of diffusion. Afterwards, we use the generated data to solve the leastsquare problem of minimizing the function
n1 BC2 ð xÞ b2 V0 ¼ na2 xc x Bx
4. Numerical experiments
2 N 2 þ a4 C2 ð8Þ Cˆ 2 ð8Þ þ ~ aj C3 yj Cˆ 3 yj : j¼5
xc xcV xa xc
These relationships establish bounds for the diffusion coefficients D i , i = 1, 2, 3 under the assumption of boundedness in the second derivatives (fl2C i /flx 2), convexity and decreasing profile of C i (x). These bounds may be used to check whether the diffusion coefficients initially given are reliable. If so, we may use them to obtain corrections of the coefficients through an algorithm based on an inverse problem. Numerical experiments have been conducted to choose suitable values for m, n, k in (15).
Computing was performed with a PC at 900 MHz, using MATLAB regression program version 6.5. The results of the numerical experiments yield: – Estimates are not sensitive to the increase of the number of measurements in the diffusion zone in the case when the artificial data does not have random errors. – Estimates are very sensitive to the initial vector in the optimization process. It seems that an initial value of (D 3/D 2) near to the order of the actual values is the best option, while for (D 2/D 1) the initial value does not matter when random errors are not considered. – Estimates are also sensitive to measurements errors. A regularization process is needed to improve results. – More numerical experiments are needed to confirm these preliminary conclusions.
J.L. Bernal et al. / Surface & Coatings Technology 200 (2005) 1458 – 1463
5 . Conclusions & Based on the proposed moving boundary model with Stefan conditions, an algorithm which allows the estimation of approximate values for the diffusion coefficients in austenite and phase gV has been built. This algorithm assumes the diffusion coefficient of the ferrite to be known and also considers a gradual process of layer formation until solubility at each phase is reached, after the layer quasi-stabilization. & The optimization of the functional allows the analysis of whether numerical and experimental results are congruent. & The approach presented for obtaining diffusion coefficients through an inverse moving boundary problem with Stefan conditions may be generalized to the study of phenomena associated with mass transport and compact layer formation, e.g. cementation, boriding and oxidation processes. Acknowledgement The authors wish to acknowledge Dr. Juan Alfredo Gomez (UFRO, Chile) for his inspiring and fruitful collaboration on various parts of the work presented in this article.
References [1] A. Ricard, J.E. Oseguera, L. Falk, H. Michel, M. Gantois, IEEE Trans. Plasma Sci. 18 (6) (1990) 940. [2] Falk, J. Oseguera, A. Ricard, H. Michel, M. Gantois, Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 139 (1991) 132.
1463
[3] T. Belmonte, C. Jaoul, J.N. Borges, Surf. Coat. Technol. 188 – 189 (2004) 201. [4] A. Ricard, C. Joaul, F. Gaboriau, N. Gherardi, S. Villeger, Surf. Coat. Technol. 188 – 189 (2004) 287. [5] A. Medina-Flores, J. Oseguera, P. Santiago, J.A. Ascencio, Surf. Coat. Technol. 188 – 189 (2004) 7. [6] A. Medina-Flores, C. Arganis, P. Santiago, J. Oseguera, Surf. Coat. Technol. 188 – 189 (2004) 140. [7] U. Figueroa, J. Oseguera, P.S. Shabes-Retchkiman, Surf. Coat. Technol. 86 – 87 (1996) 728. [8] Ivan Campos, J. Oseguera, U. Figueroa, E. Mele´ndez, Surf. Coat. Technol. 102 (1998) 127. [9] M.A.J. Somers, Mittemeijer, Metall. Mater. Trans. 26A (1995) 57. [10] H. Du, J. Agren, Metall. Mater. Trans., A Phys. Metall. Mater. Sci. 27A (1996) 1073. [11] L. Torchane, P. Bilinger, J. Dulcy, M. Gantois, Metall. Mater. Trans. 27A (1996) 1823. [12] Lifang Xia, Mufu Yan, Acta Metall. Sin. (English Edition), Ser. B 2 (1) (1989) 18. [13] Mufu Yan, Jihon Yan, T. Bell, Model. Simul. Mater. Sci. Eng. 8 (2000) 491. [14] L.N. Tao, Acta Mech. 45, 49 – 64. [15] J.L. Bernal, O. Salas, U. Figueroa, J. Oseguera, Surf. Coat. Technol. 177 – 178 (2004) 665. [16] T.R. Goodman, Adv. Heat Transf. 1 (1964) 51. [17] J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1956. [18] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd edR, Clarendon Press, Oxford, 1965. [19] J.D. Fast, M.B. Verrijp, J. Iron Steel Inst. 176 (1954) 24. [20] P. Grievson, E.T. Turkdogan, Trans. Met. Soc. AIME 230 (1964) 1604. [21] A. Marciniak, Surf. Eng. 1,4 (1985) 283.