prediction of metal matrices behavior during thermochemical processing

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ScienceDirect Materials Today: Proceedings 19 (2019) 979–990

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BraMat 2019

Semi-empirical methods for estimation/prediction of metal matrices behavior during thermochemical processing Mihai Ovidiu Cojocarua, Leontin Nicolae Drugaa,b, Andrei Mihai Ghineaa,* a

Politehnica University of Bucharest, Splaiul Independentei nr.313, Bucharest, 060042, Romania b Uttis Industries SRL, Calea Bucuresti 20C, Vidra, Ilfov, 077185,Romania

Abstract Series of solutions for estimations/predictions of the behavior of metal matrices during thermochemical treatments are described on basis on minimal number of experimental results under relatively random chosen processing conditions. Three methods were considered: Popov, Kazeev and Baram. The Popov method is based on the solutions of the second differential equation of diffusion (Fick’s second law), obtained by analytical and criterial solving under third boundary conditions; in this way, the kinetic parameters D, h*, and k are estimated. The Kazeev method is based on the correct hypothesis that besides the atomic diffusion takes place also other phenomena/interactions in the metallic matrix. All these complex processes can be described by specific equations whose solutions allow to determine certain parameters which change continuously during the diffusion process. The Baram method considers the generalized equation of kinetics of heterogeneous chemical reactions. This was resulted by taking into account the law of mass action and the phenomenological dependence of change over time of the interphases separation surfaces positions in solid materials subjected to thermochemical processing. It was thus possible to describe the phenomena occurred during thermochemical processing. The algorithms for the utilization of these methods are exemplified by experimental data resulted from the gaseous nitriding of the pure technical iron (ARMCO). © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of the 11th International Conference on Materials Science & Engineering, BraMat 2019 Keywords: nitriding, diffusion equations solutions, diffusion kinetics parameters, heterogenous reactions kinetics, pure technical iron (ARMCO).

* Corresponding author. Tel.: +0 40 752 901 816. E-mail address: [email protected] 2214-7853 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of the 11th International Conference on Materials Science & Engineering, BraMat 2019

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1. Introduction There are multiple approaches of the diffusion process in the related literature, most of these starting from the solutions obtained by solving the differential equation of the diffusion [1, 2] under restrictive conditions strictly dependent on the thermochemical processing applied. The use of one or other solution has to carefully consider the particularities of the process and of the material subject to thermochemical processing. A more simplified approach which considers the necessity of estimation the overall growth kinetics of the layer in order to design a proper technological cycle, can start from minimum experimental/direct information related to the process kinetics, such as the interdependence equations - layer case depth-processing parameters can be customized for a specific material [3, 4]. 2. Materials and equipment used in research The purpose of the theoretical and experimental research was to check the possibility to use some methods mentioned in the related literature as particular tools able to predict by calculation the nitriding case depth/nitriding rate under certain processing conditions strictly specified. To determine a minimum number of preliminary data necessary for the theoretical estimations/verifications of the results, sample of 10x10x15 mm of pure technical iron have been nitrided in a nitriding pit furnace. The nitriding furnace is equipped with automatic temperature control system. The ammonia dissociation rate was determined conventionally by using a dissociation device and was maintained at about 40%. The total size of nitride case-depth was determined using a Reichert microscope. 3. Experimental results. Interpretation The predicting of time-temperature evolution of superficial layers growth kinetics of the metallic products under long time exposure in chemically active media is particularly interesting both theoretically and practically. A first solution would be the experimental programming resulting in regression equations by means of which the casedepth of the cemented layer can be estimated under specific conditions, namely the cementation rate. The experimental programing means the performing of a fixed number of experiments depending on the number of the independent variables related to the process considered and programmer adopted. Some solution on the estimation of growth kinetics of the cemented layers described in the literature are relatively simple and are based on a minimum number of information/experimental results. Starting from this minimal information, the explicating of series of equations meant to reach the goals targeted, becomes possible. Thus, Kazeev, considers that the time-temperature evolution of cemented layer case depth can be describe by means of a parabolic dependence as follows:

h   tm

(1)

where: h – is the case depth of the layer obtained by thermochemical processing (nitriding); β – is a complex parameter dependent on the process temperature (β=f(T)); has as unit cm.s-1 t - isothermal holding time m- kinetic parameter, temperature and time-dependent (when its value differs from m=0.5 the processes occuring simultanoeusly with the pure atomic diffusion are stronger). The explanation of the two parameters, β and m, is developed starting from the observation that m represents in fact the slope of the equation:

lg h  lg   m lg t

(Fig.1)

(2)

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where

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lg h  m . The values used as the calculation basis have been experimentally obtained and mentioned in lg t

table 1. Table 1. Nitrided layer total case-depth obtained on pure technical iron, nitrided in dissociated ammonia (α NH3=40%) at different times and temperatures. Nitrided layer case-depth, μm, after t hours of isothermal holding (experimental values) 4 hours 8 hours 12 hours

Nr.crt

T,oC

1

500

134

223

273

2

540

254

343

433

3

580

375

464

553

Fig. 1. The anamorphosed dependence of the equation kinetics (1) for the saturation by nitrogen diffussion of the pure technical iron. Note:The calculated values of the kinetic parameter m, for various temperatures are specified.

The value of the kinetic parameter m provides information about the process stage which develops at the slowest rate and which dictates the process overall kinetics. Thus, in the low temperature range, at 500oC, m=0.65→1; (1-m)=0.35<0.5→0, the process kinetics are governed by the processes occuring in the medium and at the interface medium-product (adsorption). At higher temperatures, T=580oC, where the parameter (1-m)=0.65 >0.5→1, the process kinetics are governed by the intensity of the diffusion processes in the metal matrix. The statistical processing of the results concerning the two kinetics parameters considered m şi β (



h tm

),

obtained for different temperatures and holding times led to the the following expressions of their temperature dependences (eq.3-4, Fig.2).

m  2.518  0.00375  T

(3)

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  1.03  0.00216  T

(4)

where T is the thermochemical processing temperature (nitriding), in degree Celsius. Therefore, the evolution in time of the total case depth of the nitrided layer obtained on pure technical iron, at different nitriding temperatures, can be described with the following relation:

h[ mm ]  ( 1.03  0.00216  T )  t ( 2.518  0.00375 T )

(5)

where t- is expressed in, and T in Celsius degree.

Fig. 2. Dependence of kinetic parameters, β and m, by temperature.

For the nitriding temperatures of 500 oC, 580 oC, 540 oC, the following simplified relations can be used:

h500  C  0.055  t 0.65

(6)

h540  C  0.128  t 0.48

(7)

h580  C  0.228  t 0.35

(8)

where the values of β parameter equal to 0.055;0.128, namely 0.228 represent the arithmetic means of the values of this parameter at different holding times (in the range 4÷8 hours) for the three temperatures considered. As conclusion, starting from the expression of the interdependence between the total case depth of the nitrided layer (eq.5) and the process time and temperature, the relation for the growth rate of the nitrided layer (ʋ) in the pure technical iron is :

v  (1.03  0.00216  T )  t (1.5180.00375T )

(9)

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Baram approaches the kinetics aspects of the thermochemical processing from the perspective of the heterogenous chemical reactions, considering the mass law and the phenomenological dependence of change over time of the inter-phases separation surfaces positions (eq.10).

K* 

(1   )V n S 0  t 1

m

dM

 [CV  (M  Mo)]

n

(10)

m0

where: S0- is the initial surface of the solid phase; M0 and M –mass of the solid phase at the initial time, respectively at time t of processing; V- volume of the phase which needs saturation; C- the limit concentration of the saturation element in solid phase; n- the order of the inter-phases reaction, depending on the type of cementing element; α- the kinetic constant, dependent on individual properties of reactant phases; Note. The significance of this element is similar to the element (1-m) used by Kazeev, indicating also in this case, the process stage that defines the layer generation and growth kinetics; K*- process rate constant In the case of an excess of the cementing element in the cementation medium, its concentration change in time can be neglected, therefore n=0, and by solving the integral of eq. (10) the following relation is obtained:

K* 

(1   )M S 0  t 1

(11)

where: M  M  M 0 is the mass variation of the solid phase (the sample subjected to thermochemical processing). Taking into consideration the proportionality:

h  K /  M

(12)

where, h is the case depth of the nitrided layer, the following relation is obtained:

K0 

(1   )h t 1

(13)

where :. K 0  K  K  S 0 *

/

The relationships (11-13) are useful for the calculation of the constants of the rates of the thermochemical processings and eq. 12 allows to determine the ratio between the case depth of the nitrided layer and the variation of the specific mass, size which is constant value for a certain thermochemical process [4;5]. The relation (13) developed in logarithmic coordinates, (lgh-lgt), having as reference the information contained in table 1, leads to straight lines with slopes with specific inclination towards the abscisa, which provide information about the value of the kinetic indicator (1- α), Fig.3.

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Fig. 3. The anamorphosed dependence of equation kinetics (13), for the saturation by nitrogen diffussion of the pure technical iron. Note:The calculated values of the kinetic parameter α, for various temperatures are specified.

For the calculation of the reaction rate constant, K0 (table 2) it was taken into account the sizes of the nitrided layer obtained on pure technical iron matrix, processed under different times and temperature (table 1; αNH3=ct=40%); the value of this allows the prediction by calculation of process kinetics. Table 2 Calculated values of the reaction rate constant, K0.

Time, hours

4 8 12

h(μm),at nitriding temperature, oC (according to.tab.1): 5000 540o 580o (1-α)=0.65 (1-α)=0.48 (1-α)=0.35 134 254 375 223 343 464 273 433 553

Ko,500grd

35.37 37.51 35.28

K o = 36.05

Ko,540grd

62.67 60.68 63.05

K o = 62.12

Ko,580grd

80.79 78.4 81.1

K o = 80.09

The results of the estimation of the reaction rate constant ( K o ) allow the calculation of the total case depth of the nitrided layer for the pure technical iron (eq.14) for certain processing conditions as well as the rate of germination and growth of nitrided layer (eq15).

hcalc 

v

K 0 1  t ( m) 1

K 0   t ( m / h) 1

(14)

(15)

Popov approaches differently the issue, his solution implies the use of graphical expressions of solutions of Fick’s differential equation (the second equation of diffusion), obtained by analytics and criterial solving in

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boundary conditions of third order (Θ=f(Fo;Ti))[5÷8]. Obs. Ti represents the expression of the Tihonov criterion, and Fo criterion Fourier. In the Popov’s approach, the time where the graphics describing the time-temperature evolution of the cemented layer case-depth (nitrided in the case) separates from the abscisa ( Fig.4), is particularly important. The algorithm necessary to achieve the goals - the explanation of the correlations between the total case-depth of the nitrided layer, namely the rate of germination and growth of the nitrided layer, depending on the nitriding time and temperature, according to Popov’s methodology, requires the fulfilment of the following steps:  determining the nitrogen relative concentration ,Θ , at a certain temperature;



C lim  C 0 C max  C 0

(16)

where :Clim - is the minimum solubility of nitrogen in ferrite,% Cmax- is the maximum solubility of nitrogen in ferrite at the thermochemical processing temperature,% C0 –is the initial nitrogen concentration in ferrite,%  determination of time (approximately) after which the time variation curve of the nitride layer case-depth separates the abscisa ( 0.5 hours, according to the data of fig.4);  starting from the hypotese that in certain nitriding conditions which are strictly specified, (temperature and medium) the statement

Ti1 = Ti x

t1 is valid, so Ti1  Ti x tx

t1 , results that the values corresponding to the tx

Tihonov criteria, respectively Ti2 ,Ti3.....Tix, can be calculated knowing the value of criterion Ti1=h’ D .t1 , (where

h* 

k represents a relative mass transfer coefficient, k- the cinetic rate of adsorbtion and D – D

diffusion coefficient) respectively t1; value Ti1=0.07 intersection of the curve

1 2 F0



x 2 D t

resulted from the diagram Θ=f(Ti;Fo)[6÷8], at the

 0 , with the horizontal related to the value Θ=0.077, t1=0.5h, representing the time

when the effects of nitriding become metallographically observable.  the values of the Tihonov criteria estimated in the previous stage for different isothermal holding times (Ti2 ,Ti3,....Tin), together with the value of the relative concentration criterion Θ, allow the estimation of values corresponding to the Biot criterion (Bi=h*‧x),Bi2,Bi3......Bin; in this purpose, the graphical expression of the differential diffusion equation Θ=f(Bi;Fo) is applied [6÷8]. Such calculated values of the Biot criterion allow the determinination of the relative mass transfer coefficients h’.  the value of the diffusion coefficient D of the nitrogen in the non-alloyed ferrite is determined: 2

1  Ti  D  [cm2 / s]  h *  3600  t1

(17)

 calculate the adsorption constant rate, k:

k  h * D[cm 2 / s ]

(18)

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Fig. 4. Dependence of all nitriding layer obtained in the case of iron technically pure, by nitriding parameters (values are those indicated in tab. 1) Note After t=0.5 hours, the layer becomes metallographic observable.

 estimation of the case-depth of the nitrided layer, using the information related to the value of the diffusion coefficients which were previously calculated:

h  a D t

(19)

where a- is a time-dependent correction coefficient; h-is the total case-depth of the nitrided layer-calculated value. Following the steps imposed by Popov’s algorithm result the following data of interest corresponding to the issue analyzed (table 3). Obs. X, cm, is the case-depth of the nitrided layer, experimentally determined, and equivalent of h (the calculated value of case-depth)  in the calculation of the diffusion coefficient have been considered the mean values of the size h*  the values of X, cm are those from table 1. Table 3. Summary of parameters values considered for the estimation of the sizes requested for the characterization of the nitriding gaseous process of the pure technical iron. T,oC 500

540

580

X,cm 0.0134 0.0223 0.0273 0.0254 0.0343 0.0433 0.0375 0.0464

Cmax/Θ/Ti1 0.05%N; Θ=0.08; Ti1=0.07 0.066%N Θ=0.06 Ti1=0.05 0.09%N Θ=0.044 Ti1=0.03

Ti 0.198 0.221 0.343 0.141 0.2 0.245 0.169 0.339

t,ore 4 8 12 4 8 12 4 8

Bi=h*.x 0.31 0.36 0.5 0.16 0.33 0.4 0.25 0.66

h*,cm-1 23.3 16.2 18.38 6.29 9.6 9.3 6.68 14.25

D,cm2/s

k,cm/s

7.3.10-9

140.8.10-9

3.86.10-8

32.38.10-8

7.78.10-7

1.058.10-5

By means of the calculated sizes (table 3) according to Popov’s algorithm, it is posible to calculate the correction coefficient from eq. (19), which is necessary to determine the relations that allow the prediction of the kinetics of the germination and growth of nitrided layer for pure technical iron undergoing the thermochemical processing. The correlations between these coefficients and the isothermal holding time at different temperatures, f(t), can be

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obtained by statistical processing of the information obtained for these correction coefficients obtained for the temperature and holding times of interest. In this way, resulted:

a (t ) 500  C  1 .22  0 .029  t

(20)

a (t ) 540  C  1 .077  0 .0031  t

(21)

a (t ) 580  C  0 .373  0 .0065  t

(22)

thus, the relations for the prediction of the kinetics of germination and growth of the nitrided layer, in the specific case of pure technical iron nitrided in ammonia, at a dissociation rate of 40%, will be:  at T= 500oC :

h500  C  a (t ) 500  C  2.62  10 5  t

a(t ) 500 C

(23)

2.62  10 5  t

(24)

h540  C  a (t ) 540  C  1.39  10 4  t

(25)

v500 C 

t

 at T= 5400C:

a (t ) 540 C

1.39  10 4  t

(26)

h580  C  a (t ) 580  C  2.8  10 3  t

(27)

v540 C 

t

 at T=580oC:

v580 C 

a(t ) 580 C t

2.8  10 3  t

(28)

Note: In the relations no. 20-28 the time is considered in hours because the value of diffusion coefficient is expressed in cm2/h, therefore the total case-depth of nitrided layer, is in cm, namely cm/h for the nitriding rates. The case-depths of the nitrided layers obtained in the pure technical iron nitrided in ammonia atmosphere (partially dissociated, α=40%) and the experimental nitriding rates have been compared with those resulted by using the solutions of the three methods (Kazeev, Baram and Popov) presented in Fig.5-6.

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Fig. 5. The total case-depth of the nitrided layer estimated by various methods 1-the experimental value; 2 a -estimation by Kazeev-complex relation; 2b- Kazeev simplified relation; 3-Baram; 4-Popov Note. From top to bottom, the times are: 4 hours 8 hours 12 hours; nitriding matrixpure technical iron (ARMCO).

It was found that there are differences between these which not exceed 8-10% (the highest resulted from the use of complex relations determined according to the Kazeev algorithm-eq. 5;9, namely being solutions which inmplies a relatively large number of statistical processings).

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Fig. 6. The growth rate of nitrided layer estimated by various methods 1 – experimental value; 2 a – estimation by Kazeev complex relationship; 2 b – Kazeev simplified relation; 3 – Baram; 4 – Popov Notes: From top to bottom, the times are: 4 hours 8 hours 12 hours; nitriding matrix –pure technical iron(ARMCO).

4.Conclusions The three methods, Kazeev, Baram and Popov described in the paper, can be used quite easily to explain the simple or complex dependencies between the case-depth of the nitrided layer, namely the nitriding rate and the thermochemical processing conditions. To obtain the equation that characterize the kinetics of any thermochemical

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process based on the solutions explained above, it is necesary to have series of experimental results which can be processed by a given algorithm, specific to each variant presented. In fact, these procedures can be applied to any type of thermochemical process. The results obtained through the verification of the three procedures for the case of gas nitriding of pure technical iron are obtained in the limits predicted, the maximum estimation errors being not higher than 8-10%. References [1] E.J.Mittemeijer; A.J. Somers, Thermochemical Surface Engineering of Steels-1-st Edition,Woodhead Publishing,sept.2014. [2] E.J.Mittemeier, M.A.Somers, Kinetics of thermochemical surface treatments, Thermochemical Surface Engineering of Steels,1-st Edition Woodhead Publishing,sept.2014. [3] Lahtin I.M, Difuzionniie osnovi protessa azotirovanii, Mitom ,nr.7,1995. [4] Baram I.I, Lahtin I.M, Kogan I.d, Kinetica protessov himico-termiceskoi obrabotki metallov i splavov, Mitom,nr.2,1979. [5] M.Cojocaru, Azotirovanie v electrostaticeskom pole, tezǎ de doctorat, Moskova, 1976. [6] Popov A A, Teoreticeskoe osnovâ himico-termiceskoi obrabotki stali, Sverdlovsk, 1962. [7] M.Cojocaru, M.Târcolea, Modelarea interacţiunilor fizico-chimice ale produselor metalice cu mediile, Matrix Rom, Bucureşti, 1998. [8] M.Cojocaru, Procese de transfer de masǎ, Matrix Rom, Bucureşti, 2004.