Theoretical and Experimental Nucleation and Growth of Precipitates in a Medium Carbon–Vanadium Steel

metals

Article Theoretical and Experimental Nucleation and Growth of Precipitates in a Medium Carbon–Vanadium Steel

Sebastián F. Medina 1,*, Inigo Ruiz-Bustinza 1, José Robla 1 and Jessica Calvo 2

1 National Centre for Metallurgical Research (CENIM-CSIC), Av. Gregorio del Amo 8, 28040 Madrid, Spain; [email protected] (I.R.-B.); [email protected] (J.R.) 2 Technical University of Catalonia (ETSEIB—UPC), Av. Diagonal 647, 08028 Barcelona, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-91-5538-900

Academic Editor: Hugo F. Lopez Received: 10 November 2016; Accepted: 30 January 2017; Published: 7 February 2017

Abstract: Using the general theory of nucleation, the nucleation period, critical radius, and growth of particles were determined for a medium carbon V-steel. Several parameters were calculated, which have allowed the plotting of nucleation critical time vs. temperature and precipitate critical radius vs. temperature. Meanwhile, an experimental study was performed and it was found that the growth of precipitates during precipitation obeys a quadratic growth equation and not a cubic coalescence equation. The experimentally determined growth rate coincides with the theoretically predicted growth rate.

Keywords: microalloyed steel; nucleation time; nucleus radius; precipitate growing

1. Introduction During the hot deformation of microalloyed steels an interaction takes place between the static recrystallization and precipitation of nanometric particles, which grow until reaching a certain size. The complexity of this phenomenon has been studied by many researchers and is inﬂuenced by all the external (temperature, strain, strain rate) and internal variables (chemical composition, austenite grain size) that participate in the microstructural evolution of hot strained austenite [1–8]. First of all, the precipitates nucleate and then increase in size by means of growth and coarsening or coalescence [9]. During coarsening, the largest particles grow at the expense of the smallest, and according to some authors this takes place due to the effect of accelerated diffusion of the solute along the dislocations, i.e., when the precipitation is strain-induced [10]. The present work basically addresses the precipitation in a vanadium microalloyed steel, distinguishing the growth and coarsening phases and determining which of these phenomena is really the most important for the increase in precipitate size. Calculations have been performed taking into account the general theory of nucleation [11,12]. Comparison of the calculated and experimentally obtained results helps to understand the precipitation phenomenon in its theoretical and experimental aspects, respectively.

2. Materials and Methods The steel was manufactured by the Electroslag Remelting Process and its composition is shown in Table1, which includes an indication of the γ→α transformation start temperature during cooling (Ar3), determined by dilatometry at a cooling rate of 0.2 K/s, which is the minimum temperature at which the different parameters related with austenite phase precipitation will be calculated.

Metals 2017, 7, 45; doi:10.3390/met7020045 www.mdpi.com/journal/metals Metals 2017, 7, 45 2 of 10

Table 1. Chemical composition (mass %) and transformation critical temperature (Ar3, 0.2 K/s).

◦ C Si Mn V N Al Ar3, C 0.33 0.22 1.24 0.076 0.0146 0.011 716

The specimens for torsion tests had a gauge length of 50 mm and a diameter of 6 mm. The austenitization temperature was 1200 ◦C for 10 min, and these conditions were sufﬁcient to dissolve vanadium carbonitrides. The temperature was then rapidly lowered to the testing temperature of 900 ◦C, where it was held for a few seconds to prevent precipitation taking place before the strain was applied. After deformation, the samples were held for times of 50 s, 250 s, and 800 s, respectively, and ﬁnally quenched by water stream. The parameters of torsion, torque, and number of revolutions, and the equivalent parameters, stress and strain, were related according to the Von Mises criterion. Accordingly, the applied strain was 0.35, and the strain rate was 3.63 s−1. A transmission electron microscopy (TEM) (CM20 TEM/STEM 200KV, Philips, Eindhoven, The Netherlands) study was performed and the carbon extraction replica technique was used. The distribution of precipitate sizes was always determined on a population of close to 200 precipitates. This technique is widely used, but there are other competitive techniques such as quantitative X-ray diffraction (QXRD) [13].

3. Results and Discussion

3.1. Theoretical Model and Calculation of Parameters The Gibbs energy for the formation of a spherical nucleus of carbonitride from the element in solution (V) is classically expressed as the sum of chemical free energy, interfacial free energy, and dislocation core energy, resulting in the following expression [14–16]:

16πγ3 γ ( ) = + µ 2 ∆G J 2 0.8 b (1) 3∆Gν ∆Gν

−2 where γ is the surface energy of the precipitate (0.5 Jm ), ∆Gν is the driving force for nucleation of precipitates, b is the Burgers vector of austenite, and µ is the shear modulus. For austenite, b = 2.59 × 10−10 m and γ = 4.5 × 104 MPa [9]. The equilibrium between the austenite matrix and the carbonitride VCyN1−y is described by the mass action law [15]:

ss ss ss −3 RgT XV XC XN ∆Gν J · m = − ln e + y ln e + (1 − y) ln e (2) Vm XV XC XN

ss e where Xi are the molar fractions in the solid solution of V,C, and N, respectively, Xi are the equilibrium fractions at the deformation temperature, Vm is the molar volume of precipitate species, Rg is the universal gas constant (8.3145 J·mol−1·K−1), and T is the deformation absolute temperature. The carbonitride is considered as an ideal mix of VC and VN, and the values of parameters e e e e (y; XV; XC; XC; XN) are determined using FactSage (Developed jointly between Thermfact/CRCT (Montreal, QC, Canada) and GTT-Technologies (Aachen, Germany)) [17,18]. The value of “y” in Equation (4) is the precipitated VC/VCN ratio, and “1 − y” is the VN/VCN ratio. On the other hand, 1.5 2 N0 = 0.5 ∆ρ is the number of nodes in the dislocation network, ∆ρ = (∆σ/0.2µb) is the variation in the density of dislocations associated with the recrystallization front movement in the deformed zone at the start of precipitation [9], and ∆σ is the difference between the ﬂow stress and yield stress at the deformation temperature. Metals 2017, 7, 45 3 of 10

The atomic impingement rate is given as [10]

4πR2D C β0 s−1 = c V V (3) a4 where DV is the bulk diffusivity of solute atoms (V) in the austenite, a is the lattice parameter of the precipitate, and CV is the initial concentration of vanadium in mol fraction. Here, the bulk diffusion coefficient (DV) is replaced by an effective diffusion coefficient, Deff, expressed as a weighted mean of the bulk diffusion (DV) and pipe diffusion (Dp) coefficients, and used in the description of the precipitate evolution [10,19]:

2 2 Deff = DpπRcoreρ + DV 1 − πRcoreρ (4) where Rcore is the radius of the dislocation core, taken to be equal to the Burgers vector, b. The Zeldovich factor Z takes into account that the nucleus is destabilised by thermal excitation compared to the inactivated state and is given as [20]

Vp r γ = at Z 2 (5) 2πRc kT

The ﬂow stress increment (∆σ) has been calculated using the model reported by Medina and Hernández [21], which facilitates the calculation of ﬂow stress. The dislocation density has been calculated at the nose temperature of the Ps curve corresponding to a strain of 0.35. When the austenite is not deformed the dislocation density is approximately 1012 m−2 [22]. The dislocation density corresponding to the curve nose will be given by 1012 + ∆ρ. The incubation time (τ) is given as follows [23,24]:

1 τ = (6) 2β0Z2

The critical radius for nucleation is determined from the driving force and is given as [10]

2γ Rc = − (7) ∆Gν

It is obvious that the nuclei that can grow have to be bigger than the critical nucleus, and in accordance with Dutta et al. [9] and Perez et al. [25] this value will be multiplied by 1.05. It must be noted that the factor 1.05 has little consequence on the overall precipitation kinetics. The integration of Zener’s equation for the growth rate will give the radius of the precipitates as a function of time [15]: SS i 2 2 XV − XV R = R0 + 2Deff ∆t (8) VFe − Xi Vp V where VFe is the molar volume of austenite, Vp is the molar volume of precipitate, R is the precipitate radius after a certain time ∆t, and R0 is the average critical radius of the precipitates that have nucleated during the incubation period, coinciding with 1.05R0. The FactSage software tool for the calculation of phase equilibria and thermodynamic properties makes it possible to predict the formation of simple precipitates (nitrides and carbides) and more complex precipitates (carbonitrides), and the results can be expressed as a weight fraction versus temperature [26]. The FSstel database containing data for solutions and compounds [26] was used. The calculations were carried out every 10 ◦C in the temperature range between 740 ◦C and 1250 ◦C with the search for transition temperatures [27]. Figure1 shows the fraction of AlN, VC, VN, and total VCN versus the temperature. In the model used (FactSage), the VCN fraction is the sum of VC and VN. Metals 2017, 7, 45 4 of 10 Metals 2017, 7, 45 4 of 10 MetalsMetals2017 2017, 7,, 457, 45 4 of4 10of 10

0.10 0.10 0.08 VC VC 0.08 VCVN 0.06 VN VCN 0.06 0.06 VCN AlN 0.04 AlN 0.04 0.02 0.02

Weight fraction (%)

Weight fraction (%) 0.00 Weight fraction (%) 0.00 0.00700 800 900 1000 1100 1200 700 800 900 1000 1100 1200 Temperature (ºC) Temperature (ºC) Figure 1. Equilibrium compounds predicted by FactSage. Figure 1. Equilibrium compounds predicted by FactSage. FigureFigure 1. Equilibrium1. Equilibrium compounds compounds predicted predicted by by FactSage. FactSage. The values calculated according to the above equations are shown in Figures 2–4. The incubation The values calculated according to the above equations are shown in Figures 2–4. The incubation timeThe (τ values) is shown calculated as a function according of the to the temperature above equations (Figure are 2). shown As the in effective Figures 2energy–4. The was incubation taken into timetime ((τ)) isis shownshown asas aa functionfunction ofof thethe temperaturetemperature (Figure(Figure 2).2). As the effective energy was taken into timeaccount (τ) is shownin Equation as a function (3) instead of theof only temperature the energy (Figure for bulk2). Asdiffusion, the effective the values energy of wasτ were taken lower. into The account in Equation (3) instead of only the energy for bulk diffusion, the values of τ were lower. The critical radius (Rc) for nucleation was calculated from Equation (7) as a function of τthe temperature accountcritical inradius Equation (Rc) for (3) nucleation instead of was only calculated the energy from for bulkEquation diffusion, (7) as thea function values of of thewere temperature lower. critical radius (Rc) for nucleation was calculated from Equation (7) as a function of the temperature The(Figure critical 3). radius (R ) for nucleation was calculated from Equation (7) as a function of the temperature (Figure(Figure 3).3). c (Figure3). 1200 1200 1100 1100 1000 1000 900 900 800

Temperature (ºC) 800

Temperature (ºC) Temperature (ºC) 700 700 1 10 100 1000 10000 1 10Incubation100 time1000 (s) 10000 Incubation time (s) Incubation time (s) Figure 2. Incubation time vs. temperature (curve Ps) for VCN precipitates. Figure 2. Incubation time vs. temperature (curve Ps) for VCN precipitates. FigureFigure 2. Incubation2. Incubation time time vs. vs temperature. temperature (curve (curve Ps )P fors) for VCN VCN precipitates. precipitates.

100 100

10 10

1 1

Critical radius (nm)

Critical radius (nm) 0.1 0.1700 800 900 1000 1100 700 800Temperature900 1000 (ºC) 1100 Temperature (ºC) Temperature (ºC) FigureFigure 3. Critical 3. Critical radius radius size size vs. vs temperature. temperature for for VCN VCN precipitates. precipitates. Figure 3. Critical radius size vs.. temperaturetemperature forfor VCNVCN precipitates.precipitates.

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8 975ºC 6 900ºC

Radius (nm) 2

0 0 25 50 75 100 125 150 175 200 t (s)

FigureFigure 4.4. PrecipitatePrecipitate sizesize (radius)(radius) growthgrowth vs.vs. timetime ((∆Δtt >> τ)) forfor VCNVCN precipitates.precipitates. EquationEquation (8).(8).

EquationEquation (8)(8) hashas beenbeen appliedapplied toto thethe precipitateprecipitate growth,growth, selectingselecting temperaturestemperatures ofof 975975 ◦°CC (curve(curve nosenose inin FigureFigure2 2)) and and 900 900 ◦°CC.. TheThe ∆Δtt timetime startsstarts toto bebe countedcounted afterafter thethe nucleationnucleation period,period, whichwhich waswas ◦ ◦ 1212 ss atat 975 °CC and 17 s s at at 900 900 °CC.. The The value value of of RR0 0waswas 0.8 0.8 nm nm and and 0.5 0.5 nm, nm, both both multiplied multiplied by by1.05, 1.05, at at975 975 °C◦ Cand and 900 900 °C◦,C, respectively. respectively. The The growth growth of of nuclei nuclei is is shown shown in in Figure 44.. TheThe calculationscalculations havehave beenbeen performedperformed usingusing parametersparameters extractedextracted fromfrom literatureliterature [[9,10,15,28,29]9,10,15,28,29] andand shownshown inin TableTable2 .2. The term ∆Gν Equation (2) is important in the calculations carried out, whose values at 975 °C and 900 °C were −1.347 × 109 J·mTable−3 and 2. −Parameters2.115 × 109 used J·m− for3, respectively. calculations. Dutta et al. [9] checked that the number of particles per unit volume decreased dramatically and this is not predictedParameter by a simple growth Symbol model. Such a reduction Valuein the density of particles Reference per unit volume requiresBurgers vector particle coarsening. Theb (m) coarsening of precipit2.59 ×ates10− by10 coalescence [ occurs9] once 4 precipitationShear is modulus complete, i.e., when theµ (MPa)plateau has ended and4.5 recrystall× 10 ization progresses[9] again. γ · −2 CoalescenceInterfacial can be energy explained by the modified(J m ) Lifshitz–Slyozov–0.5Wagner theory (MLSW) [10,15] [29,30], Lattice parameter (VCN) a (nm) 0.4118 [26] 1/3 which predicts that while the basic t kinetics2 − 1of the LSW theory− is4 maintained,(−264,000/RT) the coarsening rate Bulk diffusion of V DV (m ·s ) 0.28 × 10 e [27] 2 −1 −4 (−210,000/RT) increases asPipe the diffusion volume fraction increases,Dp (m even·s )at very small0.25 ×precipitated10 e volume fraction[26] values. 3 −1 −6 CoarseningMolar volume is given of VCN by the expression:VP (m · mol ) 10.65 × 10 [15] Molar volume of austenite V (m3·mol−1) 7.11 × 10−6 [15] Fe 2 8DVX i −10 Dislocation core radius R3core (m)3 Vγ p V 5.00 × 10 [10] R R0 Δ t (9) 9RTg ◦ The term ∆Gν Equation (2) is important in the calculations carried out, whose values at 975 C where Rg is the gas constant and the other parameters have already been explained. and 900 ◦C were −1.347 × 109 J·m−3 and −2.115 × 109 J·m−3, respectively. Dutta et al. [9] checked that the number of particles per unit volume decreased dramatically Table 2. Parameters used for calculations. and this is not predicted by a simple growth model. Such a reduction in the density of particles per unit volume requiresParameter particle coarsening.Symbol The coarsening of precipitatesValue by coalescenceReference occurs once precipitationBurgers is complete, vectori.e., when theb plateau (m) has ended and2.59 recrystallization× 10−10 progresses[9] again. Coalescence canShear be modulus explained by the modiﬁedμ (MPa) Lifshitz–Slyozov–Wagner 4.5 × 104 theory (MLSW) [29[9,]30 ], which predicts thatInterfacial while the energy basic t 1/3 kineticsγ of (J the·m− LSW2) theory is maintained,0.5 the coarsening[10,15 rate increases] as the volumeLattice fractionparameter increases, (VCN) even ata very (nm) small precipitated0.4118 volume fraction values.[26] CoarseningBulk diffusion is given of by V the expression:DV (m2·s−1) 0.28 × 10−4e(−264,000/RT) [27] Pipe diffusion Dp (m2·s−1) 0.25 × 10−4e(−210,000/RT) [26] 2 i 3 −1 D γV X −6 Molar volume of VCN VP3 (m ·mol3 8) V p V 10.65 × 10 [15] R = R0 + ∆t (9) Molar volume of austenite VFe (m3·mol−1) 9RgT 7.11 × 10−6 [15] Dislocation core radius Rcore (m) 5.00 × 10−10 [10] where Rg is the gas constant and the other parameters have already been explained. CoarseningCoarsening is thethe processprocess byby whichwhich thethe smallestsmallest precipitates dissolve to the proﬁtprofit of thethe largerlarger ones,ones, leadingleading toto aa distributiondistribution peakpeak shiftshift toto largerlarger sizes.sizes. ThisThis phenomenonphenomenon isis particularlyparticularly importantimportant whenwhen thethe systemsystem reachesreaches thethe equilibriumequilibrium precipitateprecipitate fraction.fraction. It is to bebe notednoted thatthat coarseningcoarsening cancan occuroccur atat everyevery stagestage ofof thethe precipitationprecipitation processprocess [[15].15].

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TheThe most most striking striking aspectaspect of of EquationEquation (9) (9) is is that that the the mean mean radius radius cubed cubed varies varies with with time, time, as as opposedopposed toto thethe squared radiusradius inin growthgrowth calculations.calculations. CoarseningCoarsening isis thusthus a much slower process than precipitateprecipitate growth,growth, asas isis reasonablereasonable givengiven thatthat thethe growthgrowth ofof oneone particleparticle onlyonly occursoccurs byby cannibalisticcannibalistic particleparticle growthgrowth [[22].22]. FollowingFollowing EquationEquation (9),(9), nucleusnucleus growthgrowth waswas calculatedcalculated forfor thethe temperaturestemperatures ofof 975975 ◦°CC and 900 ◦°CC.. Once again, DV has been replaced by Deff, which is one order of magnitude higher. The result is shown Once again, DV has been replaced by Deff, which is one order of magnitude higher. The result is shownin Figure in Figure5, where5, whereit can itbe can seen be that seen coarsening that coarsening does not does take not place take at place these at thesetemperatures temperatures and is therefore nil, unlike that shown in Figure 4. This is because the value of Deff, or DV, is very low at the and is therefore nil, unlike that shown in Figure4. This is because the value of Deff, or DV, is very lowmentioned at the mentioned temperatures temperatures and rises and as rises the as temperature the temperature increases. increases. In In other other words, words, at at higher higher temperaturestemperatures coarseningcoarsening by by coalescence coalescence would would also also probably probably take take place place during during precipitation precipitation and afterand thisafter has this ended. has ended.

1.6 975ºC 900ºC 1.2

0.8

R (nm) 0.4

0.0 0 50 100 150 200 250 t (s) FigureFigure 5.5. PrecipitatePrecipitate sizesize (radius)(radius) coarseningcoarsening vs.vs. timetime ((∆Δtt >> ττ).). Equation (9).

3.2.3.2. Evolution of Experimental PrecipitatePrecipitate SizeSize TheThe appliedapplied strainstrain waswas 0.35, which was insufficientinsufficient to promote dynamic recrystallizationrecrystallization [ 31]. nr AtAt highhigh temperaturestemperatures (above(aboveT Tnr), t thehe dislocation structure development governs the rates of recovery andand recrystallization,recrystallization, which affect thethe austeniteaustenite graingrain size,size, highhigh temperaturetemperature strength,strength, andand startstart ofof precipitationprecipitation [[3322].]. TheThe resolutionresolution ofof precipitatesprecipitates usingusing thethe carboncarbon extractionextraction replicareplica techniquetechnique byby TEMTEM isis shownshown inin FigureFigure6 6aa–c,–c, obtainedobtained onon aa specimen strained and and qu quenchedenched in in the the conditions conditions indicated indicated at at the the foot foot of ofthe the figure. figure. The The spectrum spectrum in Figure in Figure 6b6 showsb shows the the presence presence of ofV on V onthe the precipitate precipitate indicated indicated by bythe thered redarrow arrow in Figure in Figure 6a,6 a,and and the the lattice lattice parameter parameter determined determined from from Figure Figure 8c 8c reveals a fccfcc cubiccubic latticelattice withwith aa valuevalue ofof aa == 0.413 nm, which is identified,identified, in accordance withwith the referencereference valuevalue foundfound inin thethe literatureliterature [[22],22], asas aa vanadiumvanadium carbonitridecarbonitride (VCN).(VCN). TheThe twotwo unassignedunassigned peakspeaks are FeFe andand NiNi (6.4(6.4 keVkeV andand 7.47.4 keV,keV, respectively).respectively). TheyThey maymay bebe duedue toto gridgrid impuritiesimpurities oror residuesresidues fromfrom preparationpreparation inin thethe evaporator.evaporator. It is well knownknown that thethe coppercopper peakpeak alwaysalways appearsappears inin thethe carboncarbon extractionextraction replicareplica techniquetechnique andand isis duedue toto thethe coppercopper supportsupport grid.grid. OnOn thethe otherother hand,hand, thethe particleparticle sizesize distributiondistribution andand determinationdetermination ofof thethe averageaverage particleparticle havehave beenbeen obtainedobtained byby measuringmeasuring anan averageaverage numbernumber ofof 200200 particlesparticles onon thethe testedtested specimens.specimens. FigureFigure7 7 showsshows aa log-normallog-normal distributiondistribution thatthat correspondscorresponds toto aa holdingholding timetime ofof 5050 ss atat 900900 ◦°CC,, andand aa bimodalbimodal distributiondistribution correspondingcorresponding to to holding holding times times of of 250 250 s ands and 800 800 s, respectively,s, respectively, with with a weighted a weighted mean mean size _ __ (diametersize (diameterD) of 11.2D ) of nm, 11.2 19.2 nm, nm, 19.2 and nm 20.4, and nm, 20.4 respectively. nm, respectively. Other authors Other authors have found have similar found sizessimilar of V(C,N)sizes of precipitatesV(C,N) precipitates [33]. [33].

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(a) (b) (a) (b)

(c) (c) FigureFigure 6. TEM 6. TEM images images of steel of used. steel ( used.a) Image (a showing) Image precipitates showing precipitates for specimen for tested specimen at reheating tested at −1 temperatureFigurereheating 6. TEM temperatureof 1 2images00 °C /10 of of min steel 1200; Def. used.◦C/10 temp.= (a) min; Image 900 Def. °Cshowing; temp.=strain precipitates=900 0.35;◦C; strain strain for rate =specimen 0.35;= 3.63 strain s tested; holding rate at= reheating time 3.63 s=− 1; 250temperatureholding s; (b) EDAX time of= spectrum 2501200 s; °C (b/10) of EDAX minprecipitate; spectrumDef. temp.=; (c) Electron of 900 precipitate; °C diffraction; strain (c )= Electron 0.35; image. strain diffraction rate = 3.63 image. s−1; holding time = 250 s; (b) EDAX spectrum of precipitate; (c) Electron diffraction image. 50 =0.35 t=50 s; D=11.2nm 50 . =3.63 s-1 t=250 s; D=17.2nm 40 =0.35 t=50 s; D=11.2nm Temp.=900ºC t=800 s; D=20.4nm . -1 t=250 s; D=17.2nm 40 =3.63 s 30 Temp.=900ºC t=800 s; D=20.4nm 30 20 20 Frequency (%) 10 Frequency (%) 10 0 0 6 12 18 24 30 36 42 48 54 60 66 72 0 0 6 12 Precipitate18 24 30 size36 42 (nm)48 54 60 66 72 Precipitate size (nm) Figure 7. Size distribution of precipitates for specimen tested at reheating temperature of 1200 °C for 10Figure min, deformation 7. SizeSize distribution distribution temperature of of precipitates precipitates of 900 °C ,for and specimen holding timestested of at 50reheating s, 250 s, temperature and 800 s. of 1200 °C◦C for for 10 min, deformation temperature of 900 ◦°CC,, and holding times of 50 s, 250 s,s, and 800 s. If we accept that the precipitates grow once they are nucleated, then moment zero of growth coincides with the nucleation time (τ) and Δt will be the sample holding time after τ. Figure 8 shows IfIf we accept that the precipitates grow grow once once they they are nucleated, then then moment moment zero zero of of growth growth a representationcoincides with ofthe the nucleation average precipitatetime (τ) and growth Δt will at be a temperaturethe sample holding of 900 °Ctime, where after theτ. Figure size is 8 given shows coincides with the nucleation__ time (τ) and ∆t will be the sample holding time after τ. Figure8 shows a bya therepresentation mean radius of ( Rthe Daverage2 ). The precipitate holding times growth after at the a temperature nucleation time of 900 ◦were °C , 50where s, 250 the s, sizeand is800 given s, representation of the average__ precipitate growth at a temperature of 900 C, where the size is given by minusby the the mean value radius of 17 (sR corresponding D 2 ). The holding to the timescalculated after incubationthe nucleation time time at 900 were °C .50 At s, first 250 sights, and there 800 s, areminus seen tothe be value two growthof 17 s corresponding laws. Between tothe the first calculated two points incubation, the nucleus time grows at 900 notably °C . At first; between sight therethe are seen to be two growth laws. Between the first two points, the nucleus grows notably; between the

Metals 2017, 7, 45 8 of 10 last two, it barely grows. These two laws obey Equations (8) and (9), respectively, and show that the precipitates grow during precipitation but that coarsening by coalescence does not take place. As coalescence does not occur, it has already been seen that Equation (9) is a constant at 900 °C (Figure 5). If a horizontal line is drawn from the final point, with coordinates at 783 s; 10.2 nm, to its intersection with the quadratic curve, the intersection point is determined to be at coordinates 275 s; 10.2 nm, which means that precipitation has ended 275 s after the nucleation time (17 s). MetalsFrom2017, 7Figure, 45 8, in particular from the quadratic equation, it is possible to deduce the nucleus8 of 10 radius (R0). This indicates a size of 4.3 nm when Δt = 0, corresponding to the nucleus size at 900 °C , which is greater than _ the nucleus radius calculated at the theoretical curve nose, which was the mean radius (R = D/2). The holding times after the nucleation time were 50 s, 250 s, and 800 s, approximately 0.5 nm. It is necessary to highlight the conceptual difference between the theoretical minus the value of 17 s corresponding to the calculated incubation time at 900 ◦C. At first sight there and experimental incubation time curves, respectively. The calculated and experimental nucleation are seen to be two growth laws. Between the first two points, the nucleus grows notably; between times do not have the same meaning. The nucleation time calculated at any temperature means the the last two, it barely grows. These two laws obey Equations (8) and (9), respectively, and show that time necessary for the formation of a number N of nuclei, of size R0, whose growth continues when the precipitates grow during precipitation but that coarsening by coalescence does not take place. the precipitate radius reaches a size of 1.05R0. The experimental nucleation time refers to the time As coalescence does not occur, it has already been seen that Equation (9) is a constant at 900 ◦C necessary for the pinning forces to exceed the driving forces for recrystallization, temporarily (Figure5). If a horizontal line is drawn from the final point, with coordinates at 783 s; 10.2 nm, to its inhibiting the progress of recrystallization [34]. On the other hand, the carbon extraction replica intersection with the quadratic curve, the intersection point is determined to be at coordinates 275 s; technique does not easily detect precipitates with sizes of less than 1 nm. 10.2 nm, which means that precipitation has ended 275 s after the nucleation time (17 s).

16 14 12 (275s;10.2nm) 10 8 R(nm)=(802+0.320t)1/3 6 1/2 4 R(nm)=(18.4+0.304t) 2 Experimental R (nm) 0 0 200 400 600 800 1000 t (s)

FigureFigure 8. 8. ExperimentalExperimental growth growth and and coarsening coarsening of of precipitates. precipitates.

TheFrom results Figure confirm8, in particular that the from time the of 275 quadratic s deduced equation, for the it isend possible of precipitation, to deduce the to nucleuswhich the radius 17 s ◦ of(R the0). Thisnucleation indicates time a sizemust of be 4.3 added nm when, coincides∆t = 0, approximately corresponding with to the the nucleus experimentally size at 900 determinedC, which precipitationis greater than end the time nucleus [35]. radius calculated at the theoretical curve nose, which was approximately 0.5 nm.In other It is necessary words, to to explain highlight the the increase conceptual in the difference average precipitate between the size theoretical, it is sufficient and experimental to use the growthincubation equation time, curves, and coalescence respectively. can The be ignored. calculated On and the experimentalother hand, from nucleation the holding times time do not of have50 s tothe the same holding meaning. time of The 250 nucleations, the measured time precipitate calculated radius at any has temperature increased meansfrom 5.6 the nm time to 9.6 necessary nm, an increasefor the formation of 4 nm. If of the a number times of N 50 of nuclei,s and 2 of00 sizes areR 0inserted, whose in growth Figure continues 4, subtracting when from the precipitate both the calculatedradius reaches incubation a size oftime 1.05 (17R0 .s) The at 900 experimental °C , it is seen nucleation that the timenucleus refers radius to the increases time necessary from 2 fornm theto 4.6pinning nm. Therefore, forces to exceed the calculated the driving and forces experimental for recrystallization, growth rates temporarily are similar. inhibiting the progress of recrystallizationIt is thus deduced [34]. On from the otherboth hand,the experimental the carbon extraction and theoretical replica points technique of view does that not easilyprecipitate detect growthprecipitates between with the sizes start of less(Ps) thanand end 1 nm. of precipitation (Pf) can be predicted by the growth equation, as coarseningThe results is not conﬁrm taking that place, the so time it is of not 275 necessary s deduced to for include the end the of coalesc precipitation,ence equation. to which The the main 17 s differenceof the nucleation between time the mustcalculated be added, and experimental coincides approximately values is the with nucleus the experimentallysize. determined precipitation end time [35]. 4. ConclusionsIn other words, to explain the increase in the average precipitate size, it is sufﬁcient to use the growthIn conclusion,equation, and the coalescence calculated can mean be ignored. nucleus On radius the other washand, approximately from the holding 0.5 nm time and of 50 the s experimentalto the holding average time of radius 250 s, was the measured4.3 nm at the precipitate temperatu radiusre of has900 increased°C. This isfrom due to 5.6 the nm conceptual to 9.6 nm, differencean increase between of 4 nm. the If the theoretical times of 50 and s and experimental 200 s are inserted incubation in Figure time4 , curves,subtracting respectively. from both The the ◦ precipitatecalculated incubationsize growth time rate (17between s) at 900 the C,experimentally it is seen that determined the nucleus P radiuss and P increasesf practically from coincides 2 nm to with4.6 nm. the Therefore,calculatedthe growth calculated rate. Both and experimentalare expressed growth by a quadratic rates are growth similar. equation in such a way It is thus deduced from both the experimental and theoretical points of view that precipitate growth between the start (Ps) and end of precipitation (Pf) can be predicted by the growth equation, as coarsening is not taking place, so it is not necessary to include the coalescence equation. The main difference between the calculated and experimental values is the nucleus size. Metals 2017, 7, 45 9 of 10

4. Conclusions In conclusion, the calculated mean nucleus radius was approximately 0.5 nm and the experimental average radius was 4.3 nm at the temperature of 900 ◦C. This is due to the conceptual difference between the theoretical and experimental incubation time curves, respectively. The precipitate size growth rate between the experimentally determined Ps and Pf practically coincides with the calculated growth rate. Both are expressed by a quadratic growth equation in such a way that the coalescence of precipitates during precipitation is not taking place and can be ruled out. Since the effective diffusion coefﬁcient (Deff) parameter increases notably with the temperature, coarsening can take place at very high temperatures.

Acknowledgments: We acknowledge the financial support of the Spanish CICYT (Project MAT 2011-29039-C02-02). Author Contributions: Sebastián F. Medina conceived and designed the work and also performed and interpreted the results; Inigo Ruiz-Bustinza and José Robla made calculations and drew graphs; Jessica Calvo did thermodynamic calculations and drew the corresponding graphs. All contributed to the writing of the paper, especially S.F. Medina. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Gómez, M.; Medina, S.F.; Quispe, A.; Valles, P. Static recrystallization and induced precipitation in a low Nb microalloyed steel. ISIJ Int. 2002, 42, 423–431. [CrossRef] 2. Medina, S.F.; Quispe, A.; Gómez, M. Strain induced precipitation effect on austenite static recrystallization in microalloyed steels. Mater. Sci. Technol. 2003, 19, 99–108. [CrossRef] 3. Andrade, H.L.; Akben, M.G.; Jonas, J.J. Effect of molybdenum, niobium, and vanadium on static recovery and recrystallization and on solute strengthening in microalloyed steels. Metall. Trans. A 1983, 14, 1967–1977. [CrossRef] 4. Kwon, O. A technology for the prediction and control of microstructural changes and mechanical properties in steel. ISIJ Int. 1992, 32, 350–358. [CrossRef] 5. Luton, M.J.; Dorvel, R.; Petkovic, R.A. Interaction between deformation, recrystallization and precipitation in niobium steels. Metall. Trans. A 1980, 11, 411–420. [CrossRef] 6. Gómez, M.; Rancel, L.; Medina, S.F. Effects of aluminium and nitrogen on static recrystallization in V-microalloyed steels. Mater. Sci. Eng. A 2009, 506, 165–173. [CrossRef] 7. Gómez, M.; Medina, S.F. Role of microalloying elements on the microstructure of hot rolled steels. Int. J. Mater. Res. 2011, 102, 1197–1207. [CrossRef] 8. Kwon, O.; DeArdo, A. Interactions between recrystallization and precipitation in hot-deformed microalloyed steels. Acta Metall. Mater. 1990, 39, 529–538. [CrossRef] 9. Dutta, B.; Valdes, E.; Sellars, C.M. Mechanisms and kinetics of strain induced precipitation of Nb(C,N) in austenite. Acta Metall. Mater. 1992, 40, 653–662. [CrossRef] 10. Dutta, B.; Palmiere, E.J.; Sellars, C.M. Modelling the kinetics of strain induced precipitation in Nb microalloyed steels. Acta Mater. 2001, 49, 785–794. [CrossRef] 11. Russel, K.C. Nucleation in solids: The induction and steady state effects. Adv. Colloid. Interface Sci. 1980, 13, 205–318. [CrossRef] 12. Kampmann, R.; Wagner, R. A Comprehensive Treatment, Materials Science and Technology; Cahn, R.W., Ed.; VCH: Weinheim, Germany, 1991; pp. 213–303. 13. Wiskel, J.B.; Lu, J.; Omotoso, O.; Ivey, D.G.; Henein, H. Characterization of precipitates in a microalloyed steel using quantitative X-ray diffraction. Metals 2016, 6, 90. [CrossRef] 14. Fujita, N.; Bhadeshia, H.K.D.H. Modelling precipitation of niobium carbide in austenite: Multicomponent diffusion, capillarity and coarsening. Mater. Sci. Technol. 2001, 17, 403–408. [CrossRef] 15. Maugis, P.; Gouné, M. Kinetics of vanadium carbonitride precipitation in steel: A computer model. Acta Mater. 2005, 53, 3359–3367. [CrossRef] 16. Sun, W.P.; Militzer, M.; Bai, D.Q.; Jonas, J.J. Measurement and modelling of the effects of precipitation on recrystallization under multipass deformation conditions. Acta Metall. Mater. 1993, 41, 3595–3604. [CrossRef] Metals 2017, 7, 45 10 of 10

17. Salas-Reyes, A.E.; Mejía, I.; Bedolla-Jacuinde, A.; Boulaajaj, A.; Calvo, J.; Cabrera, J.M. Hot ductility behavior of high-Mn austenitic Fe-22Mn-1.5Al-1.5Si-0.45C TWIP steels microalloyed with Ti and V. Mater. Sci. Eng. A 2014, 611, 77–89. [CrossRef] 18. Mejía, I.; Salas-Reyes, A.E.; Bedolla-Jacuinde, A.; Calvo, J.; Cabrera, J.M. Effect of Nb and Mo on the hot ductility behavior of a high-manganese austenitic Fe-21Mn-1.3Al-1.5Si-0.5C TWIP steel. Mater. Sci. Eng. A 2014, 616, 229–239. [CrossRef] 19. Zurob, H.S.; Hutchinson, C.R.; Brechet, Y.; Purdy, G. Modeling recrystallization of microalloyed austenite: Effect of coupling recovery, precipitation and recrystallization. Acta Mater. 2002, 50, 3075–3092. [CrossRef] 20. Perrard, F.; Deschamps, A.; Maugis, P. Modelling the precipitation of NbC on dislocations in α-Fe. Acta Mater. 2007, 55, 1255–1266. [CrossRef] 21. Medina, S.F.; Hernández, C.A.; Ruiz, J. Modelling austenite flow curves in low alloy and microalloyed steels. Acta Mater. 1996, 44, 155–163. 22. Gladman, T. The Physical Metallurgy of Microalloyed Steels; The Institute of Materials: London, UK, 1997; pp. 28–56. 23. Perez, M.; Courtois, E.; Acevedo, D.; Epicier, T.; Maugis, P. Precipitation of niobium carbonitrides in ferrite: Chemical composition measurements and thermodynamic modelling. Phil. Mag. Lett. 2007, 87, 645–656. [CrossRef] 24. Radis, R.; Schlacher, C.; Kozeschnik, E.; Mayr, P.; Enzinger, N.; Schröttner, H.; Sommitsch, C. Loss of ductility caused by AlN precipitation in Hadfield steel. Metall. Mater. Trans. A 2012, 43, 1132–1139. [CrossRef] 25. Perez, M.; Deschamps, A. Microscopic modelling of simultaneous two phase precipitation: Application to carbide precipitation in low carbon steels. Mater. Sci. Eng. A 2003, 360, 214–219. [CrossRef] 26. FSstel Database. Available online: http//www.factsage.com (accessed on 4 December 2010). 27. Bale, C.W.; Bélisle, E.; Chartrand, P.; Degterov, S.A.; Eriksson, G.; Hack, K.; Jung, I.H.; Kang, Y.B.; Melancon, J.; Pelton, A.D.; et al. FactSagethermo-chemical software and databases—Recent developments. Calphad 2009, 33, 295–311. [CrossRef] 28. Mukherjee, M.; Prahl, U.; Bleck, W. Modelling the strain-induced precipitation kinetics of vanadium carbonitride during hot working of precipitation-hardened ferritic-pearlitic steels. Acta Mater. 2014, 71, 234–254. [CrossRef] 29. Oikawa, H. Lattice diffusion in iron-a review. Tetsu Hagane 1982, 68, 1489–1497. 30. Ardell, J. The effect of volume fraction on particle coarsening: Theoretical considerations. Acta Metall. 1972, 20, 61–71. [CrossRef] 31. Badjena, S.K. Dynamic recrystallization behavior of vanadium micro-alloyed forging medium carbon steel. ISIJ Int. 2014, 54, 650–656. [CrossRef] 32. Kostryzhev, A.G.; Mannan, P.; Marenych, O.O. High temperature dislocation structure and NbC precipitation in three Ni-Fe-Nb-C model alloys. J. Mater. Sci. 2015, 50, 7115–7125. [CrossRef] 33. Nafisi, S.; Amirkhiz, B.S.; Fazeli, F.; Arafin, M.; Glodowski, R.; Collins, L. Effect of vanadium addition on the strength of API X100 linepipe steel. ISIJ Int. 2016, 56, 154–160. [CrossRef] 34. Gómez, M.; Medina, S.F.; Valles, P. Determination of driving and pinning forces for static recrystallization during hot rolling of a Nb-microalloyed steel. ISIJ Int. 2005, 45, 1711–1720. [CrossRef] 35. Medina, S.F.; Quispe, A.; Gomez, M. New model for strain induced precipitation kinetics in microalloyed steels. Metall. Mater. Trans. A 2014, 45, 1524–1539. [CrossRef]

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