UDK 546.271:532.72 ISSN 1580-2949 Original scientific article/Izvirni znanstveni članek MTAEC9, 48(4)515(2014) ESTIMATION OF THE EFFECTIVE BORON-DIFFUSION COEFFICIENT IN THE Fe2B LAYERS GROWN ON GRAY CAST IRON DOLOČANJE EFEKTIVNEGA KOEFICIENTA DIFUZIJE BORA PRI RASTI PLASTI Fe2B NA SIVEM LITEM ŽELEZU Boudjema Bouarour1, Mourad Keddam1, Omar Allaoui2 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, Algiers, Algeria 2Laboratoire de Genie des Procedes, Universite Amar Tellidji de Laghouat, B.P. 37 G, 03000 Laghouat, Algeria keddam@yahoo.fr Prejem rokopisa - received: 2013-09-29; sprejem za objavo - accepted for publication: 2013-10-09 The present work evaluates the effective diffusion coefficient of boron in the Fe2B layers grown on a substrate from gray cast iron. The boride layers were generated using powder-pack boriding in the temperature range of 1173-1273 K for (2, 4, 6 and 8) h of treatment. First, the diffusion coefficient of boron in Fe2B (free of chemical stresses) was obtained through solving the mass-balance equation at the (Fe2B/substrate) interface using the experimental values of parabolic-growth constants taken from the literature. Second, a simple equation was derived to evaluate the effective diffusion coefficient of boron in Fe2B by considering the effect of chemical stresses. The estimated values of boron activation energies were 154.8 kJ/mol and 164.8 kJ/mol with and without the presence of chemical stresses. Furthermore, the calculated values of effective diffusion coefficients were found to be sensitive to the change in the boriding temperature and to the increase in the upper boron concentration in Fe2B. Keywords: boriding, chemical stresses, kinetics, Fick's law, effective diffusion coefficient, incubation time To delo ocenjuje efektivni koeficient difuzije bora v plasteh Fe2B, zraslih na podlagi iz sivega litega železa. Boridne plasti so nastale med boriranjem v prahu v škatli, v temperaturnem območju 1173-1273 K in trajanju (2, 4, 6 in 8) h. Najprej je bil določen koeficient difuzije bora v Fe2B (brez kemijskih napetosti) z rešitvijo enačbe za masno ravnotežje (Fe2B/podlaga) na stiku z uporabo eksperimentalnih vrednosti za konstanto parabolične rasti, dobljeno v literaturi. Nato je bila izpeljana enostavna enačba za oceno efektivnega koeficienta difuzije bora v Fe2B z upoštevanjem kemijskih napetosti. Določeni sta bili vrednosti aktivacijske energije bora 154,8 kJ/mol in 164,8 kJ/mol s kemijskimi napetostmi in brez njih. Ugotovljeno je bilo tudi, da so izračunane vrednosti efektivnega koeficienta difuzije občutljive za spremembo temperature boriranja in za naraščanje koncentracije bora v Fe2B. Ključne besede: boriranje, kemijske napetosti, kinetika, Fickov zakon, efektivni koeficient difuzije, čas inkubacije 1 INTRODUCTION iron substrate. In this context, some recent reference studies regarding the evaluation of the effective diffusion Boriding is a well-known thermochemical treatment coefficient of boron in Fe2B were reported for borided for creating boride layers with interesting properties by AISI 1018 and AISI 4140 steels and borided Armco saturating the surface layer of a material with boron. It is iron.8-10 used to improve the surface hardness, the frictional wear, Certain reference works11,12 report on the chemical the fatigue endurance and the corrosion resistance of fer- stresses resulting from the composition gradient similar rous and non-ferrous alloys.It can be performed between to those caused by the thermal stresses in an isotropic II23 K and I323 K to form iron borides on the material medium. Diffusion-induced stresses (or chemical stres- surfa1ce with the treatment times varying from 0.5 h to ses) are then built up due to the composition inhomo- l0 h1. Different boriding methods exist m practice. How- ge^eity during mass transfer. Diffusion-induced stresses ever, the powder-pack boriding has the advantages of also change the mechanical properties of metal systems during mass transfer. Larche and Cahn13-15 investigated simplicity, flexibility with respect to the powder composition, minimum equipment and cost-effectiveness.2-4 The boriding temperature, the time duration, the the stresses arising from the inhomogeneities of mate- boron potential of the medium and the chemical compo- rials. Consequently, chemical stresses enhance both the sition of the substrate are the key parameters controlling diffusion coefficient and the concentration.16 the morphology, the growth kinetics and the microstruc- The objective of this work was to evaluate the diffu- tural nature of boride layers.5-7 sivity of boron in Fe2B (with and without the presence of The present model considers the effect of chemical chemical stresses) in the temperature range of 1173- stresses when evaluating the effective diffusion coeffi- 1273 K by applying a kinetic model of a monolayer con- cient of boron in the Fe2B layers grown on a gray-cast- figuration of Fe2B grown on gray cast iron. 2 MATHEMATICAL MODEL A schematic non-linear concentration profile of boron in the Fe2B layer is depicted in Figure 1 where the boron potential allows the formation of Fe2B on the surface of the material. CuFe2B denotes the upper boron concentration in Fe2B, C Fe,2B (= 59.2 X 103 mol m 3) represents the lower boron concentration in Fe2B and toiT) is the boride incubation time depending on the boriding temperature. Cads is defined as the effective boron concentration17. Distance u is the layer depth at the (Fe2B/substrate) interface as a function of treatment time t. C0 is the boron solubility in the matrix. The upper boron concentration in Fe2B is between 59.2 x 103 mol m-3 and 60 x 103 mol m-3 since this phase exhibits a narrow composition range as pointed out by Massalski18. When building a mathematical model of the problem, the following assumptions are made: • The growth kinetics is controlled by the boron diffusion in the Fe2B layer • The growth of the Fe2B boride layer is a consequence of the boron diffusion perpendicular to the sample surface • The solid solution of boron in the matrix behaves ideally (i.e., the activity coefficient of boron in the solid solution is independent on the concentration) • The Fe2B iron boride nucleates after a certain incubation time • The boride layer is thin in comparison to the sample thickness • Local equilibrium is held at the (Fe2B/substrate) interface • A planar morphology is assumed for the (Fe2B/sub-strate) interface • The volume change during the phase transformation is neglected • A uniform temperature is assumed throughout the sample • No effect of the alloying elements on the boron diffusion is assumed • The pressure effect on the boron effective diffusion coefficient in Fe2B is ignored The initial condition of the diffusion problem is expressed with Equation (1): C f^B( x,0) = C, (1) The boundary conditions of the diffusion problem are given with Equations (2) and (3): C F^B {x[t = 10(T )] = 0,10( T )} = C for C ^Fe2B -up ^ 59.2 X 103 mol m-3 C F^B {x(t = t) = u, t}= C ,Fe2B low for Ca,s ^ 59.2 X 103 mol m 3 (2) (3) for 0 < x < u. The Fick's second law of diffusion19 relating to the change in the boron concentration throughout the Fe2B layer with time t and location x(t) is: dCFe2B(X, t) d t -= D dx ^ (4) where DjFe2B is the diffusion coefficient of boron in Fe2B free of chemical stresses and CF e9B(x,t) is the boron concentration. The boron concentration along the boride layer, which is the solution of the Fick's second law (Equation (4)) in a semi-infinite medium, is consistent with Figure 1. Its expression is given with Equation (5): C b(x, t)=C u:e2B+ C P®2B _c Fe. C low C up erf ^VD; • erf (5) 2 D Figure 1: Schematic non-linear concentration profile of boron in the Fe2B layer Slika 1: Shematični prikaz nelinearnega profila koncentracije bora skozi plast Fe2B for 0 < x < u. The continuity equation at the (Fe2B/substrate) interface is expressed with Equation (6): rar 1 dXl =_D.Fe=B _ dt _ F dx (6) with w C '^2") _ C [0.5X(Cu;e2VC low Boride layer thickness u follows the parabolic-growth law expressed with Equation (7), where k represents the parabolic-growth constant at the (Fe2B/substrate) interface: u = Ut _ 10( T) (7) where [t -t0(T)] is considered as the effective growth time of the Fe2B layer9'10,20-22. According to Brakman et al.23, boride incubation time t0(T) is decreased as the temperature goes up. The ß(T) parameter, which takes 1. Figure 2: Temperature dependence of the ß(T) parameter Slika 2: Temperaturna odvisnost parametra ß(T) into account the effect of the boride incubation time, is given with Equation (8): ß( T) = ^ 1- 10( T) (8) The ß(T) parameter can be approached with a linear relationship (Equation (9)), illustrated also in Figure 2 (with a correlation factor = 0.982): ß( T) = (5 • 7 x10 -4 T+0.24722) (9) By derivation of Equation (5) with respect to distance x(t) and Equation (7) with respect to time t, Equation (6) can be rewritten24 as follows: [0.5(C ;e2B +C iF:2B)-C 0 = = 2 D Fe2B (C Fe2B - C erf- kß( T) • exp ß' (T) k' I 4DBe2B ) ß( T) (10) 2. d: D To evaluate the diffusion coefficient of boron in Fe2B Be2B (free of chemical stresses), it is necessary to use the Newton-Raphson routine25 to solve the problem. For this purpose, a computer program was written in Matlab (version 6.5) to find the roots of Equation (10). By ignoring the pressure effect on diffusion, the effective diffusion coefficient of boron in Fe2B (see reference10 for more details) can be evaluated from Equation (11) as follows: D eff = D F 1+- 2Cf^b(x, t)V' E 9RT (1 -v) (11) where DBff and DB are the diffusion coefficients of boron in Fe2B with and without the chemical stresses, respectively. The origin of Equation (11) can be found elsewhere in13-15. Since the solid solution of boron in the matrix is ideal, the second term in Equation (11) is positive regardless of V. Partial molar volume V is taken to be equal to (1.01 x 10-5 m3mol-1).26 E = 290 GPa and v = 0.3 are the Young's modulus and Poisson's ratio of the Fe2B layer, respectively27 28. Taking the mean value of the boron concentration throughout the Fe2B layer10, i.e., C f.b(x, t) = [0.5 x (C Fpe2B + ClFew2B)], Equation (11) yields: D :ff = D Fe2B 1 + V ^(c ;e2B + C f:2B) e 9RT (1-v) (12) 3 RESULTS AND DISCUSSION To evaluate the diffusion coefficient of boron in Fe2B using Equation (10), the experimental results found in the reference work29 in terms of the parabolic growth constants at the (Fe2B/ substrate) interface and boride incubation times were used. The chemical composition (in mass fractions) of the gray cast iron (class 30, ASTM A48) to be borided is the following: (3.44-3.45 % C, 1.7-1.77 % Si, 0.5-0.6 % Mn, 0.2 % Cr, 0.45-0.5 % Cu). The boriding process was performed on the gray cast iron under an argon atmosphere in a conventional furnace at three temperatures (1173, 1223 and 1273) K and for variable times (2, 4, 6 and 8) h. The boriding medium is composed of boron carbide (B4C) as the boron source and KBF4 as the activator. Fifty measurements were taken on different cross-sections of the borided samples of the gray cast iron to estimate the thickness of the Fe2B layer. As the input data, the model uses the following parameters: the time, the temperature, the upper and lower boron concentrations in the Fe2B iron boride and the experimental values of the parabolic Figure 3: Evolution of the squared value of the Fe2B layer thickness versus the boriding time at different temperatures Slika 3: Razvoj kvadratne vrednosti debeline plasti Fe2B v odvisnosti od časa boriranja pri različnih temperaturah growth constants at the (Fe2B/substrate) interface. The effective diffusion coefficient of boron in Fe2B was estimated on the basis of Equation (12). Figure 3 shows the squared value of the experimental boride-layer thickness as a function of the boriding time, according to Equation (13): u' = k' t-t0(T) (13) The squared values of parabolic-growth constants (k2) are obtained from the slopes of the straight curves using the least-square method. The intercept with the abscissa determines the boride incubation time at each temperature. Table 1 gives the experimental values of parabolic-growth constants k in the temperature range of 1173-1273 K, along with the corresponding boride incubation times deduced from Figure 3. Table 1: Experimental values of parabolic-growth constants at the Fe2B/substrate interface with the corresponding boride incubation times Tabela 1: Eksperimentalne vrednosti konstant parabolične rasti na stiku Fe2B/podlaga z ustreznimi inkubacijskimi časi boriranja T/K Experimental parabolic-growth constant k/pm s-0 5 Incubation boride time (s) 1173 0.3150 3200 1223 0.4321 2428 1273 0.6122 1902 Table 2 lists the simulated values of and DBff obtained at each boriding temperature using Equations (10) and (12), for the upper value of the boron content in Fe2B 10 being equal to 59.80 x 103 mol m-3. Table 2: Computed values of the boron-diffusion coefficients for Fe2B and the effective diffusion coefficients of boron in Fe2B, for the upper boron content in the Fe2B phase equal to 59.80 x 103 mol m-3 Tabela 2: Izračunane vrednosti koeficientov difuzije bora v Fe2B in efektivni koeficienti difuzije bora v Fe2B za zgornjo vsebnost bora v Fe2B fazi 59,80 x 103 mol m-3 T/K DBFe2B/(m2 s-1) using Eq.(10) DFff/(m2 s-1) using Eq.(12) 1173 6.31 x 10-12 3.67 x 10-10 1223 11.881 x 10-12 6.64 x 10-1° 1273 23.86 x 10-12 12.82 x 10-10 When incorporating the effect of chemical stresses on the boron diffusion, the effective diffusion coefficient of boron in Fe2B is increased with the boriding temperature. So, the dependence between the diffusivity of boron in Fe2B and the boriding temperature can be expressed with the Arrhenius equation. The temperature dependence of the diffusivity of boron in Fe2B is then depicted in Figure 4. The activation energy of boron (with and without the presence of chemical stresses) can be easily obtained from Figure 4. As a result, the diffusion coefficient of boron in Fe2B in the temperature range of 1173-1273 K is given with: Figure Slika 4 4: Temperature dependence of the diffusivity of boron in Fe2B : Temperaturna odvisnost difuzivnosti bora v Fe2B DB" = 1.35 X10" -164.8kJ/mol 2 exp-RT-m2/s (14) The effective diffusion coefficient of boron in Fe2B is also determined as: -154.8kJ/mol DBff = 2.8 X10"3 exp- RT m2/s (15) where R is the universal gas constant (8.314 J/mol K) and T represents the absolute temperature. The boron activation energy obtained in this work, in the absence of chemical stresses, was compared with the values found in the literature29-34. Table 3 shows a comparison of the boron activation energies (in the absence of chemical stresses) obtained from different borided materials. The reported values in Table 3 differ from each other depending on different factors such as the chemical composition of the substrate, the boriding method and the kinetic approach used to estimate the boron activation energy. Table 3: Comparison of the boron activation energies (in the absence of chemical stresses) for borided ferrous alloys Tabela 3: Primerjava aktivacijskih energij bora (pri odsotnosti kemijskih napetosti) za primere boriranja zlitin železa Material Boron activation energy (kJ mol-1) Reference Armco iron 151 30 Armco iron 157 31 AlSl H13 186.2 32 AlSl 1045 169.6 33 Gray cast iron 177.4 29 Gray cast iron 175 34 Gray cast iron 164.8 Present work It is seen that the calculated value of the boron activation energy (154.8 kJ mol-1) under chemical stresses is lower than the one for 164.8 kJ mol-1 due to an enhancement of the boron diffusion. The temperature dependence of the computed values of the effective 'O 2- c I £ LU 1 - 1 1 1 1 1 1 1 1 ' 1 ' /(C) ß (C):60200 mol m / (B):59800 mol m"^ (A):59400 mol m"^ A^) - 1160 1180 1200 1220 1240 1260 1280 1300 Temperature (K) Figure 5: Temperature dependence of the computed values of the effective diffusion coefficient of boron in Fe2B for the increasing values of C ,F;2B Slika 5: Temperaturna odvisnost izra~unanih vrednosti efektivnega koeficienta difuzije bora v Fe2B pri nara{~ajo~ih vrednostih C,Fp2B diffusion coefficient of boron in Fe2B for the increasing values of the upper boron contents in Fe2B is shown in Figure 5. With the presence of chemical stress, the diffusion of boron atoms is accelerated with an increase in the boriding temperature since the diffusion process is a thermally activated phenomenon. At a given value of the upper boron content in Fe2B, the computed values of D^^ are affected by the change in the boriding temperature. The variation in the calculated values of the effective diffusion coefficient of boron in Fe2B as a function of the OT 6' 4- c _a> Ö £ o ^ 2-O OT 3 fc 73 a> > 0. ' 1 ' —1—1 ' 1 ' ' 1 ' 1273 K \ • 1223 K y • 1173 K V o £ LU 59000 59200 59400 59600 59800 60000 Upper boron content in Fe B (mol rh^) Figure 6: Variation in the calculated values of effective diffusion coefficients of boron in Fe2B as a function of the upper boron content in the same phase for different boriding temperatures Slika 6: Spreminjanje izra~unanih vrednosti efektivnega koeficienta difuzije bora v Fe2B v odvisnosti od zgornje vsebnosti bora v isti fazi pri razli~nih temperaturah boriranja upper boron content in the same phase (C,Fpe2B) is displayed in Figure 6 for different boriding temperatures. The effective diffusion coefficient of boron in Fe2B is decreased with an increase in the upper boron content in Fe2B. It can be explained with the saturation of the material surface with the active boron atoms for longer boriding times. It is concluded that the calculated values of the effective diffusion coefficients of boron in Fe2B vary notably with the upper boron content in Fe2B at a fixed boriding temperature. 4 CONCLUSION In the present work, the diffusion coefficient of boron in the Fe2B layers grown on gray cast iron was estimated through the mass-balance equation at the (Fe2B/sub-strate) interface under certain assumptions. The model included the effect of boride incubation times by forming the Fe2B layers. Afterwards, the boron effective diffusion coefficient in Fe2B was evaluated by applying a simple equation based on the elasticity theory. A lower value of the boron activation energy under chemical stresses was obtained (154.8 kJ mol-1) for an upper boron content in Fe2B equal to 59.80 x 103 mol m-3. 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