The paper deals with original methodology for characterization of chemical heterogeneity of welded joint by measurement
of concentration profi les of substitution elements in three straight lines in a plane perpendicular to the
weld interface. Three concentration profi les were evaluated by modelling of diff usion based on solution of the 2nd
Fick’s law. Primary goal of evaluation consists in characterization of chemical heterogeneity in the narrow area on
both sides of the welded joint on the basis of their mutual position. Experimental data are optimised by original
adaptation of the Levenberg-Marquardt’s algorithm of non-linear regression. The paper deals with diff usion of nickel
in the welded joint of two diff erent steels as an example of investigated elements.Web of Science52335835

Dobrovská, Jana

Kalousek, Jaroslav

Řeháčková, Lenka

The paper deals with original methodology for characterization of chemical heterogeneity of welded joint by measurement of concentration profiles of substitution elements in three straight lines in a plane perpendicular to the weld interface. Three concentration profiles were evaluated by modelling of diffusion based on solution of the 2nd Fick’s law. Primary goal of evaluation consists in characterization of chemical heterogeneity in the narrow area on both sides of the welded joint on the basis of their mutual position. Experimental data are optimised by original adaptation of the Levenberg-Marquardt’s algorithm of non-linear regression. The paper deals with diffusion of nickel in the welded joint of two different steels as an example of investigated elements

L. Řeháčková

J. Dobrovská

J. Kalousek

Directory of Open Access Journals

355 METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ, J. DOBROVSKÁ, J. KALOUSEKCHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO STEELS BY MODELLING OF DIFFUSION PROCESSESReceived – Prispjelo: 2012-09-12Accepted – Prihvaćeno: 2013-01-10Preliminary Note –Prethodno priopćenjeISSN 0543-5846METABK 52(3) 355-358 (2013)UDC – UDK 669.18:621.746.32:542.63:621.22.015.5:519.651=111L. Řeháčková, J. Dobrovská, J. Kalousek, VŠB-TU Ostrava, FMME, Ostrava, Czech RepublicThe paper deals with original methodology for characterization of chemical heterogeneity of welded joint by meas-urement of concentration profi  les of substitution elements in three straight lines in a plane perpendicular to the weld interface. Three concentration profi  les were evaluated by modelling of diff  usion based on solution of the 2nd Fick’s law. Primary goal of evaluation consists in characterization of chemical heterogeneity in the narrow area on both sides of the welded joint on the basis of their mutual position. Experimental data are optimised by original adaptation of the Levenberg-Marquardt’s algorithm of non-linear regression. The paper deals with diff  usion of nick-el in the welded joint of two diff  erent steels as an example of investigated elements. Key words: diff  usion, distribution of substitution elements, chemical heterogeneity, weld, steelINTRODUCTIONWelded joints of steel with increased contents of al-loying elements are almost always accompanied by higher chemical and structural heterogeneity that is par-ticularly characteristic for the transition zone of the welded joint. The accompanying redistribution of inter-stitial carbon strengthens the tendency of heterogeneity. Chemical and structural alterations in the transition zone caused by a redistribution of elements can also considerably determine strength and other mechanical properties of contacting metals. In technical practice adverse effects of chemical and structural heterogeneity manifest themselves in welded joints of heat-resistant steels that are subjected to a load which is both of high-temperature and long-duration. The margin of safety – or vice versa danger – certainly depends on time and temperature of annealing. Monographs [1] and [2] have introduced various models of diffusion solution in welded joints with ap-plication on interstitial elements, particularly carbon. Works [3 - 5] have applied these models on ternary and polycomponent systems of austenitic steels. The pilot study of the diffusion processes of substi-tution elements at the interface of two different steels from the point of chemical heterogeneity has been given in [6 - 8].The term “chemical heterogeneity” means the dif-ference in chemical composition in different places of the microstructure of an analysed system. In metallic materials, e.g. in steels, the chemical heterogeneity is mainly caused by the segregation behaviour of elements in the course of material solidiﬁ  cation. As both constitu-tive elements and impurities segregate into interden-dritic areas during steel solidiﬁ  cation, the difference in chemical composition in dendrites and in interdendritic areas can be relatively high in cast steels. The aim of the present work is primarily to show a possibility of an assessment of chemical heterogeneity of the weld joint on the basis of a difference in concen-tration proﬁ  les calculated from data sets measured along three lines perpendicular to the welded interface. This problem is illustrated on Ni redistribution in the welded joint of two different steels.    EXPERIMENT DESCRIPTIONA heat-resistant low-alloy CrNiMoV steel (P2) and low-alloy silicon steel (V) were used for the welded joint. Both steels are of ferrite structure. Table 1 gives the chemical composition of the steels. Table 1   Chemical composition of steels of welded joint / wt.% Steel C Mn Si P Cr Ni CuP2 0,16 0,48 0,28 0,009 2,20 1,29 0,06V 0,80 0,66 2,34 0,026 0,09 0,05 0,09Steel S Mo V Al FeP2 0,010 0,59 0,100 0,01 restV 0,033 0,004 0,007 0,15 restSamples in the form of small ﬂ  at cylinders with a diameter of 12 mm and a height of 4 mm were prepared from both steels. A polished metallographic section was prepared on the frontal surface of each cylinder. Sam-ples were welded on polished surfaces in diffusion pairs by electric shock under a protective argon stream. 356   METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ et al.: CHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO...Welded diffusion pairs were afterwards sealed into quartz tubes with titanium splinters and isothermally annealed in an electric resistance furnace in the tem-perature range from 500 to 1 000 °C with a grading of 50 °C with a different time of annealing [6]. Twelve samples were prepared altogether. After annealing, the welded diffusion pairs were removed from the quartz tubes, cut in half perpendicularly to the welded inter-face and polished; metallographic sections were pre-pared on the cut surfaces. Distribution of the selected elements was investi-gated by energy dispersion X-ray micro-analysis with use of the instrument JEOL – JXA 8600/KEVEX. This analysis was made in selected points, along three straight lines, perpendicularly to the weld interface, approx. 180 µm long with step of 3 µm. Concentration sets contain-ing 61 values of concentrations of the measured ele-ments (Al, Si, Ti, Cr, Mn, Fe, Ni, Mo) were obtained from one measured segment. Three such sets were ob-tained from each sample.THEORETICAL BASISSolution of the 2nd Fick’s law, used for this special case, results in the following relation [3 - 5]:  () ()  −− + =Dtx xerfc N N N t x N25 , 0 ,01 2 1  (1)Here N (x,t) corresponds to the concentrations at the distance x from the welded joint in the time t. Symbols N1, N2 correspond to initial concentrations in both areas of the welded joint, diffusivity D is equal for both sides of the weld for the element in question, x0 is a correction of the coordinate eliminating a possible inaccuracy of measurement of the interface between metals of the welded joint. Eq. (1) was used for constant temperature. Eq. (1) was derived from the fundamental solution, giv-en e.g. [9].  EVALUATION OF EXPERIMENTAL DATAThe objective of the data evaluation consists of the calculation of concentration proﬁ   les. The tool for achieving this objective is the original calculation opti-mization program, which determines free optimization parameters of the diffusion Eq. (1) i.e. D, x0, N1, N2. Calculation of concentration proﬁ  les and free parame-ters of Eq. (1) is based on non-linear regression. A pro-prietary original adaptation of Levenberg-Marquardt’ algorithm [10] was developed. Short description of the mentioned program was published [11]. Three sets of concentrations measured along three lines enable variant access of optimization processing:A) The complex of all concentration data measured along all three lines was used for input into computation program (i.e. 183 concentration values). B) Only the mean concentrations were used for in-put into computation program (i.e. 61 concentration values). Each such mean concentration was calculated from three concentrations measured along three lines at the same distance from the weld interface.  C) Each set of measured concentration data was used separately for input into computation program (i.e. 3 × 61 concentration values), (see Figure 1).D) From the output of method C) an arithmetic mean of parameters N1, N2, D and of their errors were deter-mined.E) Only the medians of measured concentrations were used for input into a computation program (i.e. 61 concen-tration values). Each such median was calculated from three concentrations measured along three lines at the same distance from the weld interface. This procedure is a certain form of robust regression; that is, it does not ex-clude boundary points, but gives lesser weight to them.Table 2 presents the results of evaluation of diffu-sion coefﬁ   cients according to the above mentioned methods. With regard to great errors of these coefﬁ  -cients the calculated values were rounded off. Due to the limited scope of the paper only results of one se-lected sample are presented. This selected sample most markedly encompasses solved problems, i.e. the inﬂ  u-ence of chemical heterogeneity on diffusion character-istics (particularly diffusivity) of an analysed element in a welded joint. DISCUSSIONA great many calculations of parameters for the ele-ments Cr [6, 14], Si [7], Mo and Ni were made by the Figure 1   Redistribution of nickel in 1st, 2nd and 3rd measured line, sample (700 °C, 40h)Table 2   Comparison of diff  usivities of sample (700 °C, 40 h) according to diff  erent methodsmethod 1015DNi  /cm2 s-1 Coeffi   cient of determinationA 900 ± 300 0,905B 900 ± 400 0,957C - line 1 200 ± 200 0,913C - line 2 500 ± 400 0,923C - line 3 3500 ± 1700 0,907D 1400 ± 300 0,914E 1000 ± 400 0,961357 METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ et al.: CHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO...authors by the methodological approach described above (Figure 1). They were successful even in case of large scatter of experimental data. Less favourable side of large scatters are the magnitude of parameters’ errors. The model used generally assumes diffusion in a ho-mogenous environment. This condition need not always be sufﬁ  ciently fulﬁ  lled. Various rates of diffusion in the mass of crystal and along grain boundaries can occur [12], forming of intermetallic compounds [13, 14], segre-gation in poly-component system, inﬂ  uencing of diffu-sion by phase transformations, by changing activities of diffusing components and by other situations of thermo-dynamic non-equilibrium. These phenomena may not be of random character and also may not have Gaussian dis-tribution. Until they are not sufﬁ  ciently quantiﬁ  ed and implemented by the system of physical-chemical laws into a calculation program, the statistic assessment of scatters remains an indirect quantitative indicator possi-ble secondary processes in the area of welded joint. The submitted model is based on use of the 2nd Fick’s law individually for each element, it neglects the inﬂ  u-ence of interdiffusion coefﬁ  cients, considered particu-larly in ternary systems, see e.g. [15]. Systematic error of simpliﬁ  ed method is also reﬂ  ected in errors of param-eters. The coefﬁ  cient of determination belongs to the statistical quantities appraising the ﬁ  tting of the opti-mised function to the measured points. It is advisable to pay enhanced attention to phenomena with irregular standard deviations of calculated parameters.In spite of unfavourable scatter of data in the case of nickel the developed methodology brings results. The ob-tained results show that diffusion characteristics of a sys-tem can signiﬁ  cantly differ depending on the selected locations for concentration data measuring. Thus, they can differ depending on the chemical heterogeneity of the system. The chemical heterogeneity is the reason for dif-ferent concentration proﬁ  les calculated in three investi-gated areas. The scatter of the measured concentrations (i.e. chemical heterogeneity) results in the difference in optimization parameters of the diffusion equation.Although the estimation of diffusivities is not the main aim of this work, the results of various methods (see the previous chapter) show relatively great differ-ences of quantities, included in Eq. (1), in case of the diffusivities see Table 2. The reason is very high chemi-cal and structural heterogeneity of the materials, form-ing the welded joint. Thus, if we measure the concentra-tion proﬁ  les perpendicular to welded joint in different places, we get a somewhat different concentration pro-ﬁ  les of the elements. Inﬂ  uenced diffusion layer is in close proximity to the welded joint. Diffusion charac-teristics calculation accuracy is inﬂ   uenced by the number of points measured in this area. For this reason, appear appropriate method A) (or B)), since it considers all points measured in the diffusion-affected zone.Because all the methods were performed by an iden-tical computation program, the results of all used meth-ods are mutually comparable. Seen from this aspect the different results from the different authors and laborato-ries are fully apprehensible. The value of the diffusion coefﬁ  cient of nickel in the pure α-Fe at 650 °C is 2.10-14 cm2/s [16]. Obviously the local conditions determine the parameters of the diffusion equation. The rates of diffusion differing in the three levels of welds’ depth may accelerate its unfavourable behaviour as far as the stability is concerned.From the engineering viewpoint it would be useful to conduct a similar evaluation for all analysed ele-ments, together with metallographic analysis and ther-modynamic calculations of possible chemical process-es, for complex comparison. Regrettably, such exten-sive appraisal is outside the frame of this paper.The authors are fully aware of the fact that the pre-sented model is simpliﬁ  ed: it assumes only homogenous molecular diffusion in one direction. This limitation does not take into account the above mentioned second-ary processes, which may run parallel or consequent with the basic process. In spite of these constraints the results obtained by the developed methodology are original and useful.        Knowledge and evaluation of heterogeneity of the welded joint is important in the ﬁ  rst place for technical practice. Generally, the result and methodology can serve also as an incentive for the next theoretical and experimental investigation of this topic.CONCLUSIONThe paper presents original results of methodology for experimental concentration data evaluation and for expression of chemical heterogeneity of a welded joint on the basis of diffusion characteristics.Adaptation of the Levenberg-Marquardt’s algorithm for the solution of optimisation of parameters of a diffu-sion equation is also original. On the basis of applica-tion of this algorithm the different values of optimisa-tion parameters of diffusion equation were determined for the concentration data sets measured in the individ-ual areas. The different values of optimisation parame-ters (i.e. diffusion characteristics), determined for the individual concentration proﬁ  les, indicate the chemical heterogeneity of material.Thus, in this work, diffusion is understood among others as the tool for expression of chemical heteroge-neity of the system. This approach in diffusion as the tool for expression of chemical heterogeneity is original and new.AcknowledgementsThis work was created within the frame of the project of the Ministry of Education, Youth and Sports of the Czech Republic (MŠMT ČR), reg. No. MSM6198910013.The authors express their acknowledgements to the Institute of Physics of Materials ASCR for sample prep-358   METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ et al.: CHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO...aration and to Doc. Ing. Antonín Rek, CSc. for per-formed concentration measurements of elements with use of the method of energy dispersion X-ray spectral micro-analysis.REFERENCES[1]   K. Stránský, Thermodynamics of quasistationary diffusion of carbon in steels and its application, (in Czech). Acade-mia Praha (1977).[2]   V. Pilous, K. Stránský, Structure stability of weld deposits and welded joints in energetic engineering, (in Czech). Academia Praha (1989).[3]   J. Kučera, B. Million, K. Stránský, Czech. J. Phys. B35 (1985), 1355.[4]   J. Kučera, et al., Czech. J. Phys. B36 (1986), 514.[5]  J. Kučera, et al., Kovove Mater. 34 (1996), 336.[6]   L. Řeháčková, et al. Mater. Sci. Forum 567-568 (2008), 305.[7]   L. Řeháčková, et al. Solid State Phenom 138 (2008), 193.[8]   L. Řeháčková, et al. Hutnické listy 7 (2008), 52.[9]   J. Crank, The Mathematics of Diffusion. Oxford Universi-ty Press, 2nd ed., London (1975).[10]   W. H. Press, et al., Numerical Recipes in C. Cambridge University Press, 2nd ed. (1992). [11]   L. Řeháčková, et al. Acta Metallurgica Slovaca 4 (2006), 388.[12]   J. Zhu, et al. Comp. Mater. Sci. 20 (2001), 37.[13]   G. Dimmler, et al. Mater. Charact. 512 (2003), 341.[14]   C. Schaffernak, et al. Calphad 25 (2001), 241-251.[15]   D. Zhang, et al. Mater. Sci. Eng. A, 447 ( 2006), 217. [16]   Smithells Metals Reference Book, 7th Ed., Eds. E. A. Branders and G. B. Brook, Butterworth-Heinemann (1992), 13- 20.Note:   The responsible translator for English language is the lecturer from VŠB-TU Ostrava, Czech Republic

Characterization of chemical heterogeneity of the welded joint of two steels by modelling of diffusion processes

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355METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ, J. DOBROVSKÁ, J. KALOUSEKCHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO STEELS BY MODELLING OF DIFFUSION PROCESSESReceived – Prispjelo: 2012-09-12Accepted – Prihvaćeno: 2013-01-10Preliminary Note –Prethodno priopćenjeISSN 0543-5846METABK 52(3) 355-358 (2013)UDC – UDK 669.18:621.746.32:542.63:621.22.015.5:519.651=111L. Řeháčková, J. Dobrovská, J. Kalousek, VŠB-TU Ostrava, FMME, Ostrava, Czech RepublicThe paper deals with original methodology for characterization of chemical heterogeneity of welded joint by meas-urement of concentration profi les of substitution elements in three straight lines in a plane perpendicular to the weld interface. Three concentration profi les were evaluated by modelling of diff usion based on solution of the 2nd Fick’s law. Primary goal of evaluation consists in characterization of chemical heterogeneity in the narrow area on both sides of the welded joint on the basis of their mutual position. Experimental data are optimised by original adaptation of the Levenberg-Marquardt’s algorithm of non-linear regression. The paper deals with diff usion of nick-el in the welded joint of two diff erent steels as an example of investigated elements. Key words: diff usion, distribution of substitution elements, chemical heterogeneity, weld, steelINTRODUCTIONWelded joints of steel with increased contents of al-loying elements are almost always accompanied by higher chemical and structural heterogeneity that is par-ticularly characteristic for the transition zone of the welded joint. The accompanying redistribution of inter-stitial carbon strengthens the tendency of heterogeneity. Chemical and structural alterations in the transition zone caused by a redistribution of elements can also considerably determine strength and other mechanical properties of contacting metals. In technical practice adverse effects of chemical and structural heterogeneity manifest themselves in welded joints of heat-resistant steels that are subjected to a load which is both of high-temperature and long-duration. The margin of safety – or vice versa danger – certainly depends on time and temperature of annealing. Monographs [1] and [2] have introduced various models of diffusion solution in welded joints with ap-plication on interstitial elements, particularly carbon. Works [3 - 5] have applied these models on ternary and polycomponent systems of austenitic steels. The pilot study of the diffusion processes of substi-tution elements at the interface of two different steels from the point of chemical heterogeneity has been given in [6 - 8].The term “chemical heterogeneity” means the dif-ference in chemical composition in different places of the microstructure of an analysed system. In metallic materials, e.g. in steels, the chemical heterogeneity is mainly caused by the segregation behaviour of elements in the course of material solidiﬁ cation. As both constitu-tive elements and impurities segregate into interden-dritic areas during steel solidiﬁ cation, the difference in chemical composition in dendrites and in interdendritic areas can be relatively high in cast steels. The aim of the present work is primarily to show a possibility of an assessment of chemical heterogeneity of the weld joint on the basis of a difference in concen-tration proﬁ les calculated from data sets measured along three lines perpendicular to the welded interface. This problem is illustrated on Ni redistribution in the welded joint of two different steels.    EXPERIMENT DESCRIPTIONA heat-resistant low-alloy CrNiMoV steel (P2) and low-alloy silicon steel (V) were used for the welded joint. Both steels are of ferrite structure. Table 1 gives the chemical composition of the steels. Table 1  Chemical composition of steels of welded joint / wt.% Steel C Mn Si P Cr Ni CuP2 0,16 0,48 0,28 0,009 2,20 1,29 0,06V 0,80 0,66 2,34 0,026 0,09 0,05 0,09Steel S Mo V Al FeP2 0,010 0,59 0,100 0,01 restV 0,033 0,004 0,007 0,15 restSamples in the form of small ﬂ at cylinders with a diameter of 12 mm and a height of 4 mm were prepared from both steels. A polished metallographic section was prepared on the frontal surface of each cylinder. Sam-ples were welded on polished surfaces in diffusion pairs by electric shock under a protective argon stream. 356  METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ et al.: CHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO...Welded diffusion pairs were afterwards sealed into quartz tubes with titanium splinters and isothermally annealed in an electric resistance furnace in the tem-perature range from 500 to 1 000 °C with a grading of 50 °C with a different time of annealing [6]. Twelve samples were prepared altogether. After annealing, the welded diffusion pairs were removed from the quartz tubes, cut in half perpendicularly to the welded inter-face and polished; metallographic sections were pre-pared on the cut surfaces. Distribution of the selected elements was investi-gated by energy dispersion X-ray micro-analysis with use of the instrument JEOL – JXA 8600/KEVEX. This analysis was made in selected points, along three straight lines, perpendicularly to the weld interface, approx. 180 µm long with step of 3 µm. Concentration sets contain-ing 61 values of concentrations of the measured ele-ments (Al, Si, Ti, Cr, Mn, Fe, Ni, Mo) were obtained from one measured segment. Three such sets were ob-tained from each sample.THEORETICAL BASISSolution of the 2nd Fick’s law, used for this special case, results in the following relation [3 - 5]: ( ) ( )  −−+=DtxxerfcNNNtxN25,0, 0121  (1)Here N (x,t) corresponds to the concentrations at the distance x from the welded joint in the time t. Symbols N1, N2 correspond to initial concentrations in both areas of the welded joint, diffusivity D is equal for both sides of the weld for the element in question, x0 is a correction of the coordinate eliminating a possible inaccuracy of measurement of the interface between metals of the welded joint. Eq. (1) was used for constant temperature. Eq. (1) was derived from the fundamental solution, giv-en e.g. [9].  EVALUATION OF EXPERIMENTAL DATAThe objective of the data evaluation consists of the calculation of concentration proﬁ les. The tool for achieving this objective is the original calculation opti-mization program, which determines free optimization parameters of the diffusion Eq. (1) i.e. D, x0, N1, N2. Calculation of concentration proﬁ les and free parame-ters of Eq. (1) is based on non-linear regression. A pro-prietary original adaptation of Levenberg-Marquardt’ algorithm [10] was developed. Short description of the mentioned program was published [11]. Three sets of concentrations measured along three lines enable variant access of optimization processing:A) The complex of all concentration data measured along all three lines was used for input into computation program (i.e. 183 concentration values). B) Only the mean concentrations were used for in-put into computation program (i.e. 61 concentration values). Each such mean concentration was calculated from three concentrations measured along three lines at the same distance from the weld interface.  C) Each set of measured concentration data was used separately for input into computation program (i.e. 3 × 61 concentration values), (see Figure 1).D) From the output of method C) an arithmetic mean of parameters N1, N2, D and of their errors were deter-mined.E) Only the medians of measured concentrations were used for input into a computation program (i.e. 61 concen-tration values). Each such median was calculated from three concentrations measured along three lines at the same distance from the weld interface. This procedure is a certain form of robust regression; that is, it does not ex-clude boundary points, but gives lesser weight to them.Table 2 presents the results of evaluation of diffu-sion coefﬁ cients according to the above mentioned methods. With regard to great errors of these coefﬁ -cients the calculated values were rounded off. Due to the limited scope of the paper only results of one se-lected sample are presented. This selected sample most markedly encompasses solved problems, i.e. the inﬂ u-ence of chemical heterogeneity on diffusion character-istics (particularly diffusivity) of an analysed element in a welded joint. DISCUSSIONA great many calculations of parameters for the ele-ments Cr [6, 14], Si [7], Mo and Ni were made by the Figure 1  Redistribution of nickel in 1st, 2nd and 3rd measured line, sample (700 °C, 40h)Table 2  Comparison of diff usivities of sample (700 °C, 40 h) according to diff erent methodsmethod 1015DNi  /cm2 s-1 Coeffi  cient of determinationA 900 ± 300 0,905B 900 ± 400 0,957C - line 1 200 ± 200 0,913C - line 2 500 ± 400 0,923C - line 3 3500 ± 1700 0,907D 1400 ± 300 0,914E 1000 ± 400 0,961357METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ et al.: CHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO...authors by the methodological approach described above (Figure 1). They were successful even in case of large scatter of experimental data. Less favourable side of large scatters are the magnitude of parameters’ errors. The model used generally assumes diffusion in a ho-mogenous environment. This condition need not always be sufﬁ ciently fulﬁ lled. Various rates of diffusion in the mass of crystal and along grain boundaries can occur [12], forming of intermetallic compounds [13, 14], segre-gation in poly-component system, inﬂ uencing of diffu-sion by phase transformations, by changing activities of diffusing components and by other situations of thermo-dynamic non-equilibrium. These phenomena may not be of random character and also may not have Gaussian dis-tribution. Until they are not sufﬁ ciently quantiﬁ ed and implemented by the system of physical-chemical laws into a calculation program, the statistic assessment of scatters remains an indirect quantitative indicator possi-ble secondary processes in the area of welded joint. The submitted model is based on use of the 2nd Fick’s law individually for each element, it neglects the inﬂ u-ence of interdiffusion coefﬁ cients, considered particu-larly in ternary systems, see e.g. [15]. Systematic error of simpliﬁ ed method is also reﬂ ected in errors of param-eters. The coefﬁ cient of determination belongs to the statistical quantities appraising the ﬁ tting of the opti-mised function to the measured points. It is advisable to pay enhanced attention to phenomena with irregular standard deviations of calculated parameters.In spite of unfavourable scatter of data in the case of nickel the developed methodology brings results. The ob-tained results show that diffusion characteristics of a sys-tem can signiﬁ cantly differ depending on the selected locations for concentration data measuring. Thus, they can differ depending on the chemical heterogeneity of the system. The chemical heterogeneity is the reason for dif-ferent concentration proﬁ les calculated in three investi-gated areas. The scatter of the measured concentrations (i.e. chemical heterogeneity) results in the difference in optimization parameters of the diffusion equation.Although the estimation of diffusivities is not the main aim of this work, the results of various methods (see the previous chapter) show relatively great differ-ences of quantities, included in Eq. (1), in case of the diffusivities see Table 2. The reason is very high chemi-cal and structural heterogeneity of the materials, form-ing the welded joint. Thus, if we measure the concentra-tion proﬁ les perpendicular to welded joint in different places, we get a somewhat different concentration pro-ﬁ les of the elements. Inﬂ uenced diffusion layer is in close proximity to the welded joint. Diffusion charac-teristics calculation accuracy is inﬂ uenced by the number of points measured in this area. For this reason, appear appropriate method A) (or B)), since it considers all points measured in the diffusion-affected zone.Because all the methods were performed by an iden-tical computation program, the results of all used meth-ods are mutually comparable. Seen from this aspect the different results from the different authors and laborato-ries are fully apprehensible. The value of the diffusion coefﬁ cient of nickel in the pure α-Fe at 650 °C is 2.10-14 cm2/s [16]. Obviously the local conditions determine the parameters of the diffusion equation. The rates of diffusion differing in the three levels of welds’ depth may accelerate its unfavourable behaviour as far as the stability is concerned.From the engineering viewpoint it would be useful to conduct a similar evaluation for all analysed ele-ments, together with metallographic analysis and ther-modynamic calculations of possible chemical process-es, for complex comparison. Regrettably, such exten-sive appraisal is outside the frame of this paper.The authors are fully aware of the fact that the pre-sented model is simpliﬁ ed: it assumes only homogenous molecular diffusion in one direction. This limitation does not take into account the above mentioned second-ary processes, which may run parallel or consequent with the basic process. In spite of these constraints the results obtained by the developed methodology are original and useful.        Knowledge and evaluation of heterogeneity of the welded joint is important in the ﬁ rst place for technical practice. Generally, the result and methodology can serve also as an incentive for the next theoretical and experimental investigation of this topic.CONCLUSIONThe paper presents original results of methodology for experimental concentration data evaluation and for expression of chemical heterogeneity of a welded joint on the basis of diffusion characteristics.Adaptation of the Levenberg-Marquardt’s algorithm for the solution of optimisation of parameters of a diffu-sion equation is also original. On the basis of applica-tion of this algorithm the different values of optimisa-tion parameters of diffusion equation were determined for the concentration data sets measured in the individ-ual areas. The different values of optimisation parame-ters (i.e. diffusion characteristics), determined for the individual concentration proﬁ les, indicate the chemical heterogeneity of material.Thus, in this work, diffusion is understood among others as the tool for expression of chemical heteroge-neity of the system. This approach in diffusion as the tool for expression of chemical heterogeneity is original and new.AcknowledgementsThis work was created within the frame of the project of the Ministry of Education, Youth and Sports of the Czech Republic (MŠMT ČR), reg. No. MSM6198910013.The authors express their acknowledgements to the Institute of Physics of Materials ASCR for sample prep-358  METALURGIJA 52 (2013) 3, 355-358L. ŘEHÁČKOVÁ et al.: CHARACTERIZATION OF CHEMICAL HETEROGENEITY OF THE WELDED JOINT OF TWO...aration and to Doc. Ing. Antonín Rek, CSc. for per-formed concentration measurements of elements with use of the method of energy dispersion X-ray spectral micro-analysis.REFERENCES[1]  K. Stránský, Thermodynamics of quasistationary diffusion of carbon in steels and its application, (in Czech). Acade-mia Praha (1977).[2]  V. Pilous, K. Stránský, Structure stability of weld deposits and welded joints in energetic engineering, (in Czech). Academia Praha (1989).[3]  J. Kučera, B. Million, K. Stránský, Czech. J. Phys. B35 (1985), 1355.[4]  J. Kučera, et al., Czech. J. Phys. B36 (1986), 514.[5] J. Kučera, et al., Kovove Mater. 34 (1996), 336.[6]  L. Řeháčková, et al. Mater. Sci. Forum 567-568 (2008), 305.[7]  L. Řeháčková, et al. Solid State Phenom 138 (2008), 193.[8]  L. Řeháčková, et al. Hutnické listy 7 (2008), 52.[9]  J. Crank, The Mathematics of Diffusion. Oxford Universi-ty Press, 2nd ed., London (1975).[10]  W. H. Press, et al., Numerical Recipes in C. Cambridge University Press, 2nd ed. (1992). [11]  L. Řeháčková, et al. Acta Metallurgica Slovaca 4 (2006), 388.[12]  J. Zhu, et al. Comp. Mater. Sci. 20 (2001), 37.[13]  G. Dimmler, et al. Mater. Charact. 512 (2003), 341.[14]  C. Schaffernak, et al. Calphad 25 (2001), 241-251.[15]  D. Zhang, et al. Mater. Sci. Eng. A, 447 ( 2006), 217. [16]  Smithells Metals Reference Book, 7th Ed., Eds. E. A. Branders and G. B. Brook, Butterworth-Heinemann (1992), 13- 20.Note:  The responsible translator for English language is the lecturer from VŠB-TU Ostrava, Czech Republic

Characterization of chemical heterogeneity of the welded joint of two steels by modelling of diffusion processes

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