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ИТЕИНОЕ1 ПРОИЗВОДСТВО
Model of precipitation kinetics of copper and manganese sulfides in ultra low-carbon and interstitial free steels was created on the basis of classical nucleation and growth theory. The model includes homogeneous and heterogeneous schemes for independent nucleation of manganese and copper sulfides and simultaneous growth/ \^hrinkage of inclusions. ^
M. H HONG, POSCO Department Korea,
K. Y. CHOI, Department of the Powder Metallurgy, Welding and Technology of Materials, BNTU,
E. I MARUKOVICH, Y. A. LEBEDINSKY, A. M. BRANOVITSKY, Institute of Technology of Metals of the
National Academy of Sciences of Belarus
MODELING OF PRECIPITATION KINETICS OF MANGANESE AND COPPER SULFIDES IN INTERSTITIAL FREE STEELS
Introduction
Low carbon steel production with copper sulfide precipitation strengthening is being now extensively investigated owing to its lower cost as compared to that with steelmaking using Nb and Ti precipitates. Copper is always available in steel scrap. In some cases steels with copper sulfide precipitates have even better mechanical properties than those with Nb/Ti precipitates. Currently such low-carbon steels with high mechanical properties are under active investigation. Precipitation hardening and strengthening have a great effect on mechanical properties of these steels.
Precipitation kinetic is modeled on the basis of classical nucleation and growth theory (CNGT). This method is most widely used for prediction of precipitation behavior. Its success is in its simplicity and a small number of physically transparent parameters. The correct definition of such parameters (by fitting or by atomic simulation and so on) enables obtaining the quantitative results for precipitation kinetic. The basic developments and applications of classical theory are presented in [1-25].
The different numerical schemes and details are presented in [11]. Application for copper precipitation in steel matrix is developed and good agreement with experimental data are presented in [13, 14].
In some cases precipitates such as nickel superalloys, copper HSLA steel, Al-Sc alloys as well as investigated inclusions with cuprum sulfide have low surface energy. Therefore, the coarsening induced by differences in interface surfaces fails to suppress continuous creation of new small subcritical-size inclusions. The majority of solute atoms belongs to low mobile clusters and do not contribute directly to coarsening process. Conventional models of nucle-
ation are valid for systems with high-surface energy and their application to systems with low values may result in erroneous interpretation of experimental data [18].
For many cases the modeling of homogeneous nucleation and growth by CNGT gives satisfactory agreement with experimental data. In some cases fitting of parameters is not used at all, in others one or two parameters are fitted. But as a rule even in the latter cases good agreement is observed for a great number of temperature regimes and initial concentrations of solutes as well as for multi-component and multiphase precipitation.
By now the physical nature of heterogeneous nucleation of copper sulfides is practically unknown. There are many hypotheses about heterogeneous nucleation sites for CuS including grain borders, matrix-MnS inclusion interfaces, dislocations, Cu or S mi-crosegregation and so on. In general, the method considered can be used for describing various models of nucleation, by means of simultaneous following both MnS and Cu2S nucleation. In this work the assumption is made that precipitations of copper and manganese sulfides are independent. Next we assumed that the diffusions of copper, manganese and sulfur atoms are independent as well.
The only one available work devoted to approximate modeling of copper sulfides precipitation kinetic is presented in [20]. We shall model precipitation using approximately the following composition: copper is 0,07 wt%, sulfur 0,04 wt% and manganese concentration 0,54 wt% for steels with hot and cold rolling.
The considerations accepted here to be used for simulation of precipitation kinetics on the basis of CNGT are as follows.
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1. Wholly diffusion-controlled precipitation is assumed for our case on the basis of spherical shapes of inclusions and reported cube-cube orientation between copper sulfide inclusions and matrix [20].
2. All heterogeneous nucleation sites have deformed lattice around nucleation sites and hence have higher mobility of Cu atoms as compared with that in ideal matrix. In this work the model of Cu diffusion near dislocations has been selected for nucleation and initial growth modeling. The characteristic feature of this model is that in the temperature range 200 to 500 °C the Cu diffusion coefficient may be a few orders of magnitude higher than this coefficient in ideal matrix.
3. The estimation of Gibbs free energy change for precipitation of copper sulfides is corrected in as compared with [20] on the basis of more correct equilibrium temperature estimation in [27], since thermodynamic modeling predicts unstable character of copper sulfides in Fe matrix, especially with adding of Mn solute.
MODEL OF PRECIPITATION Nucleation
CNGT is based on the changes in Gibbs energy AG associated with the formation of a precipitate in a supersaturated solid solution. In the case of spherical precipitates of radius R, AG(R) is given by:
AG(/?) = ^7ti?3Ag + 47ii?2a, (1)
where Ag is the chemical Gibbs energy density change and cr the specific interfacial energy.
According to CNGT the nucleation rate is given
by:
✓ . N
./ = NtoS Z exP
AG
' kT
exp
(2)
where Ntot is the overall number of potential nucleation sites; p* the frequency factor; Z the Zeldovich factor; AG* the activation energy barrier and finc the incubation time. They are calculated as:
* _ AnDcR*2 P ~ A '
a
Z =
va
2kR
o
2
AG* A*2a=167ia
AG0
2kTa4R*2 Dvaco
(3)
(4)
(5)
where vA is the volume of precipitate molecule, a the lattice parameter of matrix, D the diffusivity of solute atoms (Cu or Mn) and c the corresponding initial concentration:
(7)
(8)
(9)
R* II
Ro = 2 OVA kT '
= lll ( 2 ^ CCuCS
vCCu2S j
VA
(10)
in which it is assumed, that precipitate has stoichiometric composition Cu2S, and ccU2s is the product of solubility for equilibrium concentrations ce\
v 2
Sometime the critical radius is defined as
(11)
R = R +
kT 871a
taking into account the fact that a particle with a radius exactly equal to R* can not grow.
Gibbs energy
Expression (10) for chemical bulk energy may be estimated by two different ways. Experimental data concerning solubility product for Cu2S in silicon steel matrix are presented [20]:
log([%Cu]2 [%S]j = -44971 / T + 26,31. (12)
The following linear approximation is possible for small super saturations:
a nAT 1 1 e y m
(13)
where Q = 860700 J/mol-molecule of Cu2S and equilibrium temperature is estimated as Te = 1196 °C (derived from solubility product (12) for concentrations of S equal to 0,01 and Cu 0,07 mass%). AT is over-cooling below equilibrium temperature, Vm is molar volume of precipitates.
In [27] the thermodynamic calculation resulted in obtaining the dependence of driving force Ag for Cu2S on temperature. The equilibrium temperature is about 740 °C for 0,0015C-0,2Mn-0,06Si-0,01S-0,06P-0,02Nb-0,05Cu-0,035Al (wt %) steels assuming the absence of sulfur reactions with manganese and iron.
The equilibrium temperature in (13) may depend on initial solute concentration, phase transition pa-
t
v
j
j
\
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-4 (53), 2009 I
rameters and others. An insignificant error made in experimental measurement of copper solubility in matrix may actually lead to substantial deviation from correct equilibrium temperature. This inaccuracy is less important for estimation of phase transition enthalpy. On assumption that solubility product depends only on temperature as in (12), the Q value would depends only on enthalpy of phase transition [28], and Q can be estimated as in [20].
The equilibrium temperature for 0,05%Cu and 0,01%S is estimated as 740 °C according to [27].
It should be noted that this "combined" estimation ensures metastable behavior of copper sulfide in presence of manganese sulfide for temperatures usually used for annealing. Such approach makes reasonable evaluation of driving force possible. In other words, the accuracy of (13) when using more correct equilibrium temperature is higher than that with direct use of (12).
Then we can equalize Gibbs energies (10) and (13):
Q^ = RgT In
1 e
( ~2 ~ N
CCuCS v^Cu2S j
(14)
where c are the solute concentrations for which equilibrium temperature has been evaluated and Rg the gas constant. Then we have:
In cCu2s - + yy + In (ccuCs) • (15) & § e
As a result, (15) allows finding the corrected estimation of solubility product ccu2s f°r any temperature without direct using of approximation (12). For instance, at 650°C cCu2s = 1,57 • 10~15. Due to assumption, that such solubility product depends only on temperature, this value may be further used for another concentrations for calculation of Gibbs energy on the basis of (9) and (10).
Growth
The equilibrium concentrations c' of solutes in matrix at the matrix-precipitate boundary are increased owing to the curvature of surface between matrix and precipitate. It is assumed that the inclusion has spherical shape with current radius R. As to binary alloys new solute equilibrium concentration is defined by precipitate radius. With two-component precipitates the product of new equilibrium concentrations is defined by radius:
exp
R
2 / cs
CCu?S
(16)
crystal. As to frequently used Zener approximation growth/shrinkage of inclusion is determined by supersaturation and is given by:
v — -
dR DCu ecu - clcu Ds
C s
•ck
dt
R
a c{
Cu
"CCu
R acg-ck
(17)
where c£M and c£ are the fractions of Cu and S in precipitate, a the ratio of atomic volumes of this specie in matrix and precipitate and cCu and cs the current fractions of solutes in matrix. Two equations (17) give the second equation for definition of concentrations cqi and cs and are simultaneously used for growth speed determination.
The average precipitate radius is little dependent on critical radius for sufficiently long time of precipitation, but activation barrier, precipitation rate and inclusions density may considerably depend on critical radius.
Distinctive features of nucleation and growth on dislocations
The previous model describes homogeneous nucleation. As to heterogeneous nucleation some parameters in equations must de modified. The heterogeneous nucleation of Cu2S is expected to be mainly located on dislocations as soon as the density of dislocations rises to high values following hot and cold rolling.
As regards heterogeneous nucleation the interface energy is decreased by the value that depends on a particular nucleation type. In this work it is approximately assumed, that in accordance with [20] the value of a must be multiplied by a factor of 0,7.
The significant feature of growth on dislocations is rapid diffusion of copper atoms in a small zone around dislocation. In order to exactly consider this phenomenon the conception of depleted zone is introduced [2, 13]. The activation energy of pipe diffusion has been estimated as 0,73 of matrix diffusion, and around dislocation the diffusion coefficient Dd may be evaluated as [13]:
Dd - (DyDp)m,
(18)
The second equation for determination of ccu and cs can be derived from the fact, that diffusion flux of both species should lead to identical growth of
where Dp and Dv are the pipe diffusion and volume diffusions, respectively.
In this work simpler models without depleted zone calculation are used. Value Dd is used for nucleation and growth at initial stage. With increasing the inclusion radius the coefficient of diffusion decreases to Dv value by linear law. In such approach the mobility oi Cu atoms is sufficiently high to ensure nucleation and initial growth of precipitates even for relatively low temperatures of 300-500 °C. This mobility of coppei
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atoms at small distances around dislocation is comparable with that of sulfur atoms.
Numerical method
The solute balance equations are used for estimation of current concentrations of Cu and S i. e. cCu and cs. In this approximation it is assumed, that the solute fraction distribution is uniform in matrix. Then, if matrix volume change is neglected in precipitation process we have:
c = c - Vpnp
P p M Fe
P m MCu2S
(19)
where c is the initial solute molar fraction, Vp the volume fraction of precipitate, np the number of atoms of relevant specie in molecule of precipitate (for Cu np = 2, for S np is 1), pp and pm are the densities of precipitate and matrix and Mits molecular weights.
N-model [11] will be used for describing the evolution of precipitate size distribution. In this model the particle size distribution function (PSD) is dis-cretized into Nc number of size classes. The class is defined by radius i?z and the number of inclusions with radius size between - AR/2 and Rf + AR/2. The total number of particles is:
the average radius is:
I RiNi
R = -i-
N
the precipitate volume fraction is:
3 i
In case of using "Euler like" method for calculation the boundaries of classes are fixed and fluxes of particles between classes are estimated. The details are presented in [11].
The initial distribution for N{ is null or known (for example experimental data) distribution.
The adaptive step time At is used because the radius change during At time has not been above AR/2. This limitation eliminates the possibility to "jump" across one or more classes. The maximum speed v has classes with small 7?, according to (17).
New size classes are created for high R values during inclusion growth. And quite the contrary, if the number of particles in a class with small radius is less than 1 it will be excluded from calculation. This makes it possible to increase step time At and to calculate particle growth during long annealing periods.
Conclusions
Model for nucleation and growth the precipitation of manganese and copper sulfides inclusions are developed. New estimation of Gibbs free energy change for precipitation of copper sulfides is used. Model of nucleation and growth of copper sulfides precipitates on dislocations are used for heterogeneous nucleation. Numeric model, based on "Euler like" method, are developed for calculations.
References
1. K a m p m a n n R., Wagner R. Decomposition of Alloys: the early stages, ed. by P. Hassen, V. Gerold, R. Wagner and M. F. Ashby. Pergamon Press, Oxford, 1984.
2. Deschamps A., Brechet Y. Influence of Predeformation and Ageing of an Al-Zn-Mg alloy - II. Modeling of Precipitation Kinetics and Yield Stress A //Acta mater. 1999. Vol. 47. P. 293-305.
3. N i c o 1 a s M., Deschamps A. Characterization and modeling of precipitate evolution in an Al-Zn-Mg alloy during non-isothermal heat treatments //Acta mater. 2003. Vol. 51. P. 6077-6094.
4. Deschamps A., Solas D., Bréchet Y. Modeling of microstructure evolution and mechanical properties in age-hardening aluminum alloys // Proc. of Euromat 99, Munchen. 1999. Vol. 3.
5. Deschamps A., Militzer M., Poole W. J. Precipitation kinetics and strengthening of a Fe-0,8 wt% Cu alloy // ISIJ Int. 2001. Vol. 41. P. 196-205.
6. Deschamps A., Militzer M., Poole W. J. Comparison of precipitation kinetics and strengthening in an Fe-0,8%Cu alloy and a 0,8%Cu-containing low-carbon steel //ISIJ Int. 2003. Vol. 43. P. 1826-1832.
7. Perez M., Deschamps A. Microscopic modeling of simultaneous two-phase precipitation: application to carbide precipitation in low-carbon steels // Mater. Sci. Eng. 2003. Vol. A360. P. 214-219.
8. WerenskioldJ. C., Deschamps A., Brechet Y. Characterization and modeling of precipitation kinetics in an Al-Zn-Mg alloy // Mater. Sci. Eng. 2000. Vol. A293. P. 267-274.
9. K n i p 1 i n g K., D u n a n d D., S e i d m a n D. Nucleation and Precipitation Strengthening in Dilute Al-Ti and Al-Zr Alloys // Metall. mater. Trans. A. 2007. Vol. 38. P. 2552-2563.
10. P e r e z M., Courtois E., AcevedoD., Epicier T., Maugis P. Precipitation of niobium carbonitrides in ferrite: chemical composition measurements and thermodynamic modeling // Philosophical Magazine Letters. 2007. Vol. 87. P. 645 -656.
11. P e r e z M., D u m o n t M., Acevedo-Reyes D. Implementation of classical nucleation and growth theories for precipitation //Acta mater. 2008. Vol. 56. P. 2119-2132.
12 S o m m i t s c h C., Pô It P., Riif G., Mitsche S., Albu M. Interaction of RecrystaHization and Precipitation during hot Forming of Alloy 80A // Materials Science Forum. 2007. Vol. 539-543. P. 2988-2993.
/XTT^ FT (SMFJJ^frr.n //iq
-a (53), 2009/ HU
13. Y a n g J., Enomoto M. Numerical Simulaiton of Copper Precipitation during Aging in Deformed Fe-Cu Alloys // ISIJ Int. 2005. Vol. 45. P. 1335-1344.
14. H a s e g a w a H., N a k a j i m a K., M i z o g u c h i S. Effects of MnS on the Heterogeneous Nucleation of Cu precipitate in Fe-10 and 5 mass% Cu alloys // ISIJ Int. 2003. Vol. 43. P. 1021-1029.
15. C 1 o u e t E., N a s t a r M. Classical nucleation theory in ordering alloys precipitating with L12 structure //Phys. Rev. B. 2007. Vol. 75. P. 132102.
16.1 n d e n G. In Materials Issues for Generation IV Systems. 2008. Springer Netherlands.
17. M i 1 i t z e r M. Computer Simulation of Microstructure Evolution in low Carbon Sheet Steels // ISIJ Int. 2007. Vol. 47. P. 1-15.
18. T u r k i n A. A., B a k a i A. S. Modeling the Precipitation Kinetics in Systems with Strong Heterophase Fluctuations // Problems of Atomic Science and Technology. 2007. Vol. 3. P. 394-398.
19. Gottstein G., Schneider M., Lochte L. Simulation of Precipitation Processes in Commercial Aluminum Alloys// Proceedings of the 9th International Conference on Aluminium Alloys, 2004. Institute of Materials Engineering Australasia Ltd. 2004. P. 1116-1122.
20. L i u Z., K o b a y a s h i Y., N a g a i K. Crystallography and Precipitation Kinetics of Copper Sulfide in Strip Casting low Carbon Steel // ISIJ Int. 2004. Vol. 44. P. 1560-1567.
21. K h a n I. N., S t a r i n k M. I, Y a n J. L. A model for Precipitation kinetics and Strengthening in Al-Cu-Mg Alloys // Mater. Sci. Eng. A. 2008. Vol. 472. P. 66-74.
22. S t a r i n k M. J., G a o N., D a v i n L., Y a n J., C e r e z o A. Room temperature precipitation in quenched Al-Cu-Mg alloys: a model for the reaction kinetics and yield strength development // Phil. Mag. 2005. Vol. 85. P. 1395-1417.
23. S i m a r A. A multiscale multiphysics investigation of aluminum friction stir welds. Belgium, 2006.
24. Y a m a s a k i S., B h a d e s h i a H. K. D. H. Modelling and characterisation of Mo2C precipitation and cementite dissolution during tempering of Fe-C-Mo martensitic steel // Mater. Sci. Technol. 2003. Vol. 19. P. 723-732.
35. Y a m a s a k i S., B h a d e s h i a H. K. D. H. M4C3 precipitation in Fe-C-Mo-V steels and relationship to hydrogen trapping // Proc. R. Soc. 2006. Vol. 462. P, 2315-2330.
26. Y a n g and J., Enomoto M. Numerical Simulation of Copper Precipitation during Aging in Deformed Fe-Cu Alloys // ISIJ Int. 2005. Vol. 45. P. 1335-1344.
27. L e e B. - J., Sundman B., Kim S. I., Chin K. - G. Thermodynamic Calculations on the Stability of Cu2S in low Carbon Steels//ISIJ Int. 2007. Vol. 47. P. 163-171.
28. N i s h i z a w a T. Thermodynamics of Microstructure Control by Particle Dispersion // ISIJ Int. 2000. Vol. 40. P. 1269-1274.
