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
2
*
National Centre for Metallurgical Research (CENIM-CSIC), Av. Gregorio del Amo 8, 28040 Madrid, Spain;
[email protected] (I.R.-B.); [email protected] (J.R.)
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 Table 1, 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
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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]:
∆G (J) =
γ
16πγ3
+ 0.8µb2
2
∆Gν
3∆Gν
(1)
where γ is the surface energy of the precipitate (0.5 Jm−2 ), ∆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 VCy N1−y is described by the
mass action law [15]:
ss ss ss Rg T
XC
XN
XV
−3
∆Gν J · m
=−
ln
+ y ln
+ (1 − y) ln
(2)
e
e
e
Vm
XV
XC
XN
where Xiss are the molar fractions in the solid solution of V, C, and N, respectively, Xie are the equilibrium
fractions at the deformation temperature, V m 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 ; X e ; X e ; X e ) are determined using FactSage (Developed jointly between Thermfact/CRCT
(y; XV
C
C
N
(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,
N0 = 0.5 ∆ρ1.5 is the number of nodes in the dislocation network, ∆ρ = (∆σ/0.2µb)2 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
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The atomic impingement rate is given as [10]
4πR2c DV CV
β0 s−1 =
a4
(3)
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]:
Deff = Dp πR2core ρ + DV 1 − πR2core ρ
(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]
p
Z=
Vat
2πR2c
r
γ
kT
(5)
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]:
τ=
1
2β0 Z2
(6)
The critical radius for nucleation is determined from the driving force and is given as [10]
Rc = −
2γ
∆Gν
(7)
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]:
i
X SS − XV
∆t
(8)
R2 = R20 + 2Deff VV
i
Fe
Vp − XV
where VFe is the molar volume of austenite, V p 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]. Figure 1 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.
4 of 10
4 of
of 10
10
4 of4 10
Weight
fraction
(%)
Weight
fraction
(%)
Weight
fraction
(%)
Metals 2017, 7, 45
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2017,
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45
Metals
2017,
7, 45
0.10
0.10
0.10
0.08
0.08
0.08
0.06
0.06
0.06
0.04
0.04
0.04
0.02
0.02
0.02
0.00
0.00
0.00700
700
700
VC
VC
VC
VN
VN
VN
VCN
VCN
VCN
AlN
AlN
AlN
800 900 1000 1100 1200
800
900
1100
800 Temperature
900 1000
1000(ºC)
1100 1200
1200
Temperature
(º
C)
Temperature (ºC)
Figure 1. Equilibrium compounds predicted by FactSage.
Figure
1.
compounds
predicted
by FactSage.
Figure
1. Equilibrium
Equilibrium
compounds
predicted
FactSage.
Figure
1. Equilibrium
compounds
predicted
byby
FactSage.
Temperature
(º
C)
Temperature
(º
C)
Temperature
(ºC)
The values calculated according to the above equations are shown in Figures 2–4. The incubation
The
values
calculated
according
to
the
are
shown
in
2–4.
incubation
The
values
calculated
according
totemperature
the above
above equations
equations
areAs
shown
in Figures
Figures
2–4. The
The
incubation
time
(τ)
is
shown
as a function
of the
(Figure
2).
theineffective
energy
was
taken into
The
values
calculated
according
to the
above equations
are
shown
Figuresenergy
2–4. The
incubation
time
(τ)
is
shown
as
a
function
of
the
temperature
(Figure
2).
As
the
effective
was
taken
time
(τ)
is
shown
as
a
function
of
the
temperature
(Figure
2).
As
the
effective
energy
was
taken into
into
account
in
Equation
(3)
instead
of
only
the
energy
for
bulk
diffusion,
the
values
of
τ
were
lower.
The
time
(τ) is in
shown
as a function
of of
theonly
temperature
(Figure
2).diffusion,
As the effective
energy
was
taken
into
account
Equation
(3)
instead
the
energy
for
bulk
the
values
of
τ
were
lower.
account
in Equation
(3)nucleation
instead of was
only calculated
the energyfrom
for bulk
diffusion,
the
values
ofof
τ were
lower. The
The
critical
radius
(R
c) for
Equation
(7)
as
a
function
the
temperature
account
in
Equation
(3) instead of was
only the energyfrom
for bulk diffusion,
the values of
of τ were
lower.
critical
radius
(R
critical
radius
(Rcc)) for
for nucleation
nucleation was calculated
calculated from Equation
Equation (7)
(7) as
as aa function
function of the
the temperature
temperature
(Figure
3).
The
critical radius (Rc ) for nucleation was calculated from Equation (7) as a function of the temperature
(Figure
(Figure 3).
3).
(Figure 3).
1200
1200
1200
1100
1100
1100
1000
1000
1000
900
900
900
800
800
800
700
700
700 1
11
10
100
1000
10
100
(s)
10Incubation
100 time1000
1000
Incubation
Incubation time
time (s)
(s)
10000
10000
10000
Figure 2. Incubation time vs. temperature (curve Ps) for VCN precipitates.
Figure
2.
time
temperature
(curve
s) for
VCN
precipitates.
Figure
2. Incubation
time
vs.vs.
temperature
(curve
Ps )P
VCN
precipitates.
Figure
2. Incubation
Incubation
time
vs.
temperature
(curve
Pfor
s) for
VCN
precipitates.
Critical
radius
(nm)
Critical
radius
(nm)
Critical
radius
(nm)
100
100
100
10
10
10
1
11
0.1
0.1
0.1700
700
700
800
900
1000 1100
800
900
800Temperature
900 1000
1000
1100
(ºC) 1100
Temperature
(º
Temperature (ºC)
C)
Figure
3. Critical
radius
temperature
VCN
precipitates.
Figure
3. Critical
radius
sizesize
vs. vs.
temperature
for for
VCN
precipitates.
Figure
Figure 3.
3. Critical
Critical radius
radius size
size vs.
vs. temperature
temperature for
for VCN
VCN precipitates.
precipitates.
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2017, 7,
7, 45
45
55 of
of 10
10
10
Radius (nm)
8
975ºC
900ºC
6
4
2
0
0
25 50 75 100 125 150 175 200
t (s)
Figure
Figure4.
4. Precipitate
Precipitate size
size (radius)
(radius) growth
growthvs.
vs. time
time (∆t
(Δt >> τ)
τ) for
for VCN
VCN precipitates.
precipitates. Equation
Equation (8).
(8).
Equation (8)
(8) has
has been
been applied
applied to
to the
the precipitate
precipitate growth,
growth, selecting
selecting temperatures
temperatures of
of 975
975 ◦°C
(curve
Equation
C (curve
◦
nose in
in Figure
Figure 2)
2) and
and 900
900 °C.
The ∆t
Δt time
time starts
starts to
to be
be counted
counted after
after the
the nucleation
nucleation period,
period, which
which was
was
nose
C. The
◦ C.The
12 ss at
at 975 ◦°C
0.50.5
nm,
both
multiplied
byby
1.05,
at
12
C and 17
17 ss at
at 900
900 °C.
Thevalue
valueofofRR0 0was
was0.8
0.8nm
nmand
and
nm,
both
multiplied
1.05,
◦
◦
975
°C
and
900
°C,
respectively.
The
growth
of
nuclei
is
shown
in
Figure
4.
The
calculations
have
at 975 C and 900 C, respectively. The growth of nuclei is shown in
calculations have
been performed
performed using
using parameters
parameters extracted
extracted from
fromliterature
literature[9,10,15,28,29]
[9,10,15,28,29]and
andshown
shownin
inTable
Table2.2.
been
The term ∆Gν Equation (2) is important in the calculations carried out, whose values at 975 °C
−3 and
Table
2. −2.115
Parameters
calculations.
and 900 °C were −1.347 × 109 J·m
× 109used
J·m−3for
, respectively.
Dutta et al. [9] checked that the number of particles per unit volume decreased dramatically and
Parameterby a simple growthSymbol
Reference
this is not predicted
model. Such a reductionValue
in the density of particles
per unit
−
10
Burgers vector
b (m)
2.59 × 10 by coalescence [9]
volume requires
particle coarsening. The
coarsening of precipitates
occurs once
Shearismodulus
µ (MPa)
[9]
× 104
precipitation
complete, i.e., when the
plateau has ended and4.5recrystallization
progresses
again.
Interfacial energy
0.5
[10,15]
(J·m−2 )
Coalescence
can be explained by the γmodified
Lifshitz–Slyozov–Wagner
theory (MLSW)
[29,30],
Lattice parameter (VCN)
a (nm)
0.4118
[26]
1/3
which predicts
that while
theory is maintained, the coarsening
rate
2 −1of the LSW
Bulk diffusion
of V the basic t Dkinetics
[27]
0.28 × 10−4 e(−264,000/RT)
V (m ·s )
2
−
1
−
4
(
−
210,000/RT)
increases as
the
volume
fraction
increases,
even
at
very
small
precipitated
volume
fraction
values.
Pipe diffusion
[26]
Dp (m ·s )
0.25 × 10 e
Coarsening
is given
by the expression:
Molar volume
of VCN
[15]
V P (m3 ·mol−1 )
10.65 × 10−6
Molar volume of austenite
Dislocation core radius
V Fe (m3 ·mol−1 )
8DV γVp2 X Vi
R3core (m)
3
R  R0 
9 RgT
Δt
7.11 × 10−6
5.00 × 10−10
[15]
[10]
(9)
The term ∆Gν Equation (2) is important in the calculations carried out, whose values at 975 ◦ C
where R◦g 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
Parameter
Symbol
Value by coalescence
Reference
unit volume requires
particle coarsening.
The coarsening of precipitates
occurs once
−10
vectori.e., when the bplateau
(m)
2.59 recrystallization
× 10
[9]
precipitationBurgers
is complete,
has ended and
progresses
again.
Shear
μ (MPa)
4.5 × 104theory (MLSW) [29,30],
[9]
Coalescence can
bemodulus
explained by the modified
Lifshitz–Slyozov–Wagner
which
m−2LSW
)
0.5
predicts thatInterfacial
while theenergy
basic t1/3 kinetics γof(J·
the
theory is maintained,
the coarsening[10,15]
rate increases
Lattice fraction
parameter
(VCN) even at avery
(nm)small precipitated 0.4118
as the volume
increases,
volume fraction values.[26]
Bulk diffusion
DV (m2·s−1)
0.28 × 10−4e(−264,000/RT)
[27]
Coarsening
is given of
byVthe expression:
2
−1
−4
(−210,000/RT)
Pipe diffusion
Dp (m ·s )
0.25 × 10 e
[26]
2 Xi
−18D
−6
γV
V
Molar volume of VCN
VP3(m3·mol
)
10.65
×
10
[15]
p
V
R = R30 +−1
∆t
(9)
Molar volume of austenite
VFe (m3·mol
) 9Rg T
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.
Coarsening
Coarsening is
is the
the process
process by
by which
which the
the smallest
smallest precipitates
precipitates dissolve
dissolve to
to the
the profit
profit of
of the
the larger
larger
ones,
leading
to
a
distribution
peak
shift
to
larger
sizes.
This
phenomenon
is
particularly
important
ones, leading to a distribution peak shift to larger sizes. This phenomenon is particularly important
when
when the
the system
system reaches
reaches the
the equilibrium
equilibrium precipitate
precipitate fraction.
fraction. It
It is
is to
to be
be noted
noted that
that coarsening
coarsening can
can
occur
at
every
stage
of
the
precipitation
process
[15].
occur at every stage of the precipitation process [15].
Metals 2017, 7, 45
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The most
most striking
striking aspect
aspect of
of Equation
Equation (9)
(9) is
is that
that the
the mean
mean radius
radius cubed
cubed varies
varies with
with time,
time, as
as
The
opposed to
to the
the squared
squared radius
radius in
in growth
growth calculations.
calculations. Coarsening
Coarsening is
is thus
thus aa much
much slower
slower process
process than
than
opposed
precipitate
growth,
as
is
reasonable
given
that
the
growth
of
one
particle
only
occurs
by
cannibalistic
precipitate growth, as is reasonable given that the growth of one particle only occurs by cannibalistic
particle growth
growth [22].
[22].
particle
Following Equation
Equation (9),
(9), nucleus
nucleus growth
growth was
was calculated
calculated for
for the
the temperatures
temperatures of
of 975
975 ◦°C
and 900
900 ◦°C.
Following
C and
C.
Once
again,
D
V 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
in Figure
5, where
it can be
seenbethat
coarsening
does not
take
these
and is
shown
in Figure
5, where
it can
seen
that coarsening
does
notplace
takeat
place
attemperatures
these temperatures
therefore
nil, unlike
that shown
in Figure
This is
the valuethe
of value
Deff, orofDD
V, is very low at the
and
is therefore
nil, unlike
that shown
in 4.
Figure
4.because
This is because
eff , or DV , is very
mentioned
temperatures
and
rises
as
the
temperature
increases.
In
other
higher
low at the mentioned temperatures and rises as the temperature increases. In otherwords,
words,at
at higher
temperaturescoarsening
coarseningbybycoalescence
coalescence
would
also
probably
place
during
precipitation
and
temperatures
would
also
probably
taketake
place
during
precipitation
and after
after
this
has
ended.
this has ended.
1.6
975ºC
900ºC
R (nm)
1.2
0.8
0.4
0.0
0
50
100 150
t (s)
200
250
Figure5.5. Precipitate
Precipitatesize
size(radius)
(radius)coarsening
coarseningvs.
vs. time
time(∆t
(Δt>> τ).
τ). Equation (9).
Figure
3.2. Evolution
Evolution of
of Experimental
Experimental Precipitate
Precipitate Size
Size
3.2.
The applied
applied strain
strain was
was 0.35,
0.35, which
which was
was insufficient
insufficient to
to promote
promote dynamic
dynamic recrystallization
recrystallization [31].
The
[31].
At
high
temperatures
(above
T
nr
),
the
dislocation
structure
development
governs
the rates
rates of
of recovery
recovery
At high temperatures (above Tnr ), the dislocation structure development governs the
and recrystallization,
recrystallization, which
which affect
affect the
the austenite
austenite grain
grain size,
size, high
high temperature
temperature strength,
strength, and
and start
start of
of
and
precipitation
[32].
precipitation [32].
The resolution
resolution of
of precipitates
precipitates using
using the
the carbon
carbon extraction
extraction replica
replica technique
technique by
by TEM
TEM is
is shown
shown in
in
The
Figure
6a–c,
obtained
on
a
specimen
strained
and
quenched
in
the
conditions
indicated
at
the
foot
of
Figure 6a–c, obtained on a specimen strained and quenched in the conditions indicated at the foot
thethe
figure.
The
spectrum
in Figure
6b 6b
shows
thethe
presence
of Vofon
precipitate
indicated
by the
red
of
figure.
The
spectrum
in Figure
shows
presence
V the
on the
precipitate
indicated
by the
arrow
in Figure
6a,6a,
and
thethe
lattice
parameter
cubic lattice
lattice
red
arrow
in Figure
and
lattice
parameterdetermined
determinedfrom
fromFigure
Figure8c
8creveals
reveals aa fcc
fcc cubic
with
a
value
of
a
=
0.413
nm,
which
is
identified,
in
accordance
with
the
reference
value
found
in the
the
with a value of a = 0.413 nm, which is identified, in accordance with the reference value found in
literature
[22],
as
a
vanadium
carbonitride
(VCN).
The
two
unassigned
peaks
are
Fe
and
Ni
(6.4
keV
literature [22], as a vanadium carbonitride (VCN). The two unassigned peaks are Fe and Ni (6.4 keV
and 7.4
7.4 keV,
keV,respectively).
respectively). They
They may
may be
be due
due to
to grid
grid impurities
impurities or
or residues
residues from
from preparation
preparation in
in the
the
and
evaporator.
It
is
well
known
that
the
copper
peak
always
appears
in
the
carbon
extraction
replica
evaporator. It is well known that the copper peak always appears in the carbon extraction replica
technique and
and is
is due
due to
to the
the copper
copper support
support grid.
grid.
technique
On
the
other
hand,
the
particle
size
distribution
and determination
determination of
of the
the average
average particle
particle have
have
On the other hand, the particle size distribution and
been
obtained
by
measuring
an
average
number
of
200
particles
on
the
tested
specimens.
Figure
been obtained by measuring an average number of 200 particles on the tested specimens. Figure 77
shows aa log-normal
log-normal distribution
distribution that
that corresponds
corresponds to
to aa holding
holding time
time of
of 50
50 ss at
at 900
900 ◦°C,
and aa bimodal
bimodal
shows
C, and
distribution
corresponding
to
holding
times
of
250
s
and
800
s,
respectively,
with
a
weighted
distribution_ corresponding
to holding times of 250 s and 800 s, respectively, with a weighted meanmean
size
__
size
(diameter
)
of
11.2
nm,
19.2
nm,
and
20.4
nm, respectively.
Other authors
havesimilar
foundsizes
similar
D
(diameter D) of 11.2 nm, 19.2 nm,
and
20.4
nm,
respectively.
Other authors
have found
of
sizes
of
V(C,N)
precipitates
[33].
V(C,N) precipitates [33].
Metals 2017, 7, 45
7 of 10
Metals 2017, 7, 45
7 of 10
Metals 2017, 7, 45
7 of 10
(a)
(a)
(b)
(b)
(c)
(c)
Figure 6. TEM images of steel used. (a) Image showing precipitates for specimen tested at reheating
Figure 6. TEM images of steel used. (a) Image showing precipitates for specimen tested at
temperature
of 1200
°C/10
temp.=
900
°C;
strain precipitates
=900
0.35;
strain
rate
= 3.63strain
s−1tested
; holding
time
◦ C/10
◦ C;
Figure
6. TEM
images
of
steel
used.
(a)min;
Image
showing
for
specimen
reheating
temperature
ofmin;
1200Def.
Def.
temp.=
strain
=
0.35;
rateat=reheating
3.63 s=−1 ;
−1; holding time =
250
s;
(b)
EDAX
spectrum
of
precipitate;
(c)
Electron
diffraction
image.
temperature
of
1200
°C/10
min;
Def.
temp.=
900
°C;
strain
=
0.35;
strain
rate
=
3.63
s
holding time = 250 s; (b) EDAX spectrum of precipitate; (c) Electron diffraction image.
250 s; (b) EDAX spectrum of precipitate; (c) Electron diffraction image.
50
50
40
Frequency (%)
Frequency (%)
40
30
=0.35
.
=3.63 s-1
=0.35
Temp.=900º
C
.
=3.63 s-1
Temp.=900ºC
t=50 s; D=11.2nm
t=250 s; D=17.2nm
t=50 s; D=11.2nm
t=800 s; D=20.4nm
t=250 s; D=17.2nm
t=800 s; D=20.4nm
30
20
20
10
10
0
0 6 12 18 24 30 36 42 48 54 60 66 72
0
(nm)
0 6 12 Precipitate
18 24 30 size
36 42
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
temperature
of 900 °C,for
andspecimen
holding times
s, 250 s, temperature
and 800 s. of
◦ C for
7.
of
tested
at
reheating
Figure
7. Size
Size distribution
distribution
of precipitates
precipitates
for
specimen
testedof
at50
reheating
temperature
of 1200
1200 °C
for
10
of 900
900 ◦°C,
s, 250
250 s,
s, and
and 800
800 s.
s.
10 min,
min, deformation
deformation temperature
temperature of
C, and
and holding
holding times
times of
of 50
50 s,
If we accept that the precipitates grow once they are nucleated, then moment zero of growth
coincides
with
the nucleation
time (τ) andgrow
Δt will
be they
the sample
holding then
time after
τ. Figure
8 shows
If
we
accept
that the precipitates
once
are nucleated,
moment
zero of
growth
If we accept that the precipitates grow once they are nucleated, then moment zero of growth
a representation
of
the
average precipitate
growth
atbe
a temperature
of 900 °C,
where
the
size is8given
coincides
with
the
nucleation
time
(τ)
and
Δt
will
the
sample
holding
time
after
τ.
Figure
shows
coincides with the nucleation
time (τ) and ∆t will be the sample holding time after τ. Figure 8 shows a
__
representation
average
precipitate
growth
a temperature
of 900
°C, 50
where
the
given
bya the
mean radiusof( Rthe
holding times
afteratthe
nucleation time
were
s, 250
s, size
and is
800
s,
average
D
2 ). The
representation of the
precipitate growth at a temperature of 900 ◦ C, where the size is given by
__
by the
mean
radius
).
The
holding
times
after
the
nucleation
time
were
50
s,
250
s,
and
800
s,
minus
the
value
of 17 (sRcorresponding
to
the
calculated
incubation
time
at
900
°C.
At
first
sight
there
D2
are
seen to
two growth
laws. Between to
thethe
first
two points,
the nucleus
between
the
minus
thebevalue
of 17 s corresponding
calculated
incubation
timegrows
at 900notably;
°C. At first
sight there
are seen to be two growth laws. Between the first two points, the nucleus grows notably; between the
Experimental R (nm)
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).
MetalsFrom
2017, 7,Figure
45
8 of 10
8, in particular from the quadratic equation, it is possible to deduce the nucleus
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
(Figure 5). 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
10
8
6
4
2
0
(275s;10.2nm)
R(nm)=(802+0.320t)1/3
R(nm)=(18.4+0.304t)1/2
0
200
400 600
t (s)
800
1000
Figure8.8.Experimental
Experimentalgrowth
growthand
andcoarsening
coarseningof
ofprecipitates.
precipitates.
Figure
The
results
that thefrom
timethe
of 275
s deduced
for the
end
of precipitation,
to which
the
17 s
From
Figureconfirm
8, in particular
quadratic
equation,
it is
possible
to deduce the
nucleus
radius
◦
of
nucleation
time
must
coincides
thenucleus
experimentally
determined
(Rthe
indicates
a size
ofbe
4.3added,
nm when
∆t = 0,approximately
correspondingwith
to the
size at 900
C, which
0 ). This
precipitation
end
[35]. radius calculated at the theoretical curve nose, which was approximately
is greater than
thetime
nucleus
In other
words, totoexplain
thethe
increase
in thedifference
average precipitate
it is sufficient
to use the
0.5 nm.
It is necessary
highlight
conceptual
between thesize,
theoretical
and experimental
growth
equation,
and coalescence
canThe
be ignored.
Onand
theexperimental
other hand, from
the holding
of have
50 s
incubation
time curves,
respectively.
calculated
nucleation
times time
do not
to
the
holding
time ofThe
250 nucleation
s, the measured
increased means
from 5.6
nm
to 9.6
nm, an
the
same
meaning.
time precipitate
calculated radius
at any has
temperature
the
time
necessary
increase
of 4 nm. If
times of
s and 200
s areR0inserted
Figure
4, subtracting
both the
for the formation
of the
a number
N 50
of nuclei,
of size
, whose in
growth
continues
when from
the precipitate
calculated
incubation
time
(170 .s)The
at 900
°C, it is seen
that the time
nucleus
radius
from 2 for
nmthe
to
radius reaches
a size of
1.05R
experimental
nucleation
refers
to theincreases
time necessary
4.6
nm.
Therefore,
the
calculated
and
experimental
growth
rates
are
similar.
pinning forces to exceed the driving forces for recrystallization, temporarily inhibiting the progress of
It is thus deduced
both hand,
the experimental
and theoretical
of view
precipitate
recrystallization
[34]. Onfrom
the other
the carbon extraction
replica points
technique
does that
not easily
detect
growth
between
start
(Ps) than
and end
of precipitation (Pf) can be predicted by the growth equation,
precipitates
withthe
sizes
of less
1 nm.
as coarsening
is not
taking
place,
it is
include
theofcoalescence
equation.
The
main
The results
confirm
that
the so
time
ofnot
275necessary
s deducedtofor
the end
precipitation,
to which
the
17 s
difference
between
the must
calculated
and experimental
values is thewith
nucleus
size.
of the nucleation
time
be added,
coincides approximately
the experimentally
determined
precipitation end time [35].
4. Conclusions
In other words, to explain the increase in the average precipitate size, it is sufficient to use the
growth
equation, and
coalescence
be ignored.
the other
from the holding
time
of 50
In conclusion,
the
calculatedcan
mean
nucleusOn
radius
washand,
approximately
0.5 nm
and
thes
to the holding
time ofradius
250 s, was
the measured
precipitate
radius
5.6the
nmconceptual
to 9.6 nm,
experimental
average
4.3 nm at the
temperature
of has
900 increased
°C. This isfrom
due to
an
increase
of
4
nm.
If
the
times
of
50
s
and
200
s
are
inserted
in
Figure
4,
subtracting
from
bothThe
the
difference between the theoretical and experimental incubation time curves, respectively.
◦ C, it is seen that the nucleus radius increases from 2 nm to
calculated
incubation
time
(17
s)
at
900
precipitate size growth rate between the experimentally determined Ps and Pf practically coincides
4.6 nm.
calculated
and experimental
rates are growth
similar. equation in such a way
with
theTherefore,
calculatedthe
growth
rate. Both
are expressedgrowth
by a quadratic
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:
2011-29039-C02-02).
We acknowledge the financial support of the Spanish CICYT (Project MAT
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.
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