The derivations

The typical microstructure of interphase precipitation has been shown elsewhere in this thesis. The main characteristic features we are interested in are illustrated in Figure 5-1.

The following assumptions are made for the superledge model development:

a. Carbon enrichment in austenite during γ →α transformation is neglected since it is considered either large austenitic grains and/or low carbon content.

b. The carbides are spherical in shape.

c. The α ledge density is at a dynamic steady-state condition.

The superledge model can be subsequently divided into five steps: (1) α unit ledge forms on the interface which has already carbides on it; (2) α unit ledge moves laterally and is then pinned by the presence of carbide; (3) a new α unit ledge forms on the top of the base ledge, increasing the ledge height; (4) when the stacking of unit ledges into a superledge reaches a critical height, the superledge unpins from carbides and moves laterally, merging the neighboring superledges; (5) the _{α γ} interface

108

advances forward until another precipitation cycle occurs. These steps are schematically illustrated in Figure 5-2.

Table 5-1 The experimental data used for supporting the proposed numerical models

References Alloy in wt% Austenitization

at

Fe-0.05C-0.27V-0.0002N 1150 ^oC

810 ^oC 790 ^oC 760 ^oC 740 ^oC 720 ^oC

Figure 5-1 The characteristic features of interphase precipitation focused in the present superledge model.

λ

b

_p

v

_αγ

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b. The classical ledge mechanism of austenite-to-ferrite transformation

Let’s start from a α γ interface without any carbide. Using the interface mobility, M, the interface ledge velocity, ν, is given by

ν

= M G

D

(5-1)

where

D

G is the driving force for γ →α transformation.

The overall interface velocity of a ledged interface, ν_αγ^α , depends on the ledge spacing, b, and its height. Under a steady state assumption, the nucleation rate JB

and the ledge spacing are related:

τ

2

1

J_B = b (5-2)

where τ is the characteristic time for steady-state ledge-wise growth, which is defined as

τ ⁼ν

b (5-3)

Defining a as the ledge height, the conversion between overall interface velocity and ledge velocity can be formulated by

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ναγ ν

= =τ

α a a

b (5-4)

For a unit α ledge, the magnitude of a is taken as the lattice parameter of α.

Figure 5-2 The proposed sequence of the development of a superledge originated from the ferrite/austenite interface. (a) a unit ledge forms on an interface with carbides; (b) the first ledge moves laterally and is pinned by carbides, and then another unit ledge nucleates on the top of it; (c-d) the ledge continues to increase its height to unpin from carbides; (e) the superledge is able to move laterally again and then merge the neighboring superledges, and (f) the overall interface advances until carbides precipitate on the interface again, completing a cycle of interphase precipitation.

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c. The interaction of carbide with the growing ferrite phase

The competition between transformation driving force and carbide pinning force is the core of the present superledge model. In the case of interphase precipitation, the presence of carbides pins the lateral movement of ledges, leading ν to be temporarily zero. A superledge is then built up, and unpins from carbides as it reaches a critical height, h^*. The present model requires that the spacing between ledge nuclei, b, is much larger than the spacing between precipitates, b_p, and that the sheet spacing,

λ

, is larger than the critical ledge height, h^*

bp

b≥ (5-5)

*

λ

>h (5-6)

. Defining

τ

_λ as the time for the interface to advance a distance of

λ

, the overall velocity of an interface with IP is given by

λ

αγ τ

= λ

v (5-7)

τ

λ is equivalent to the time to complete one cycle of IP which can be further be divided into three sequences : (1) the waiting time to build a critical superledge,

τ

_w; (2) the time for unpinning superledge to merge neighboring superledges,

τ

_f , and (3) the time for the interface to move forward until next precipitation row is formed. It can be formulated as

112

λ τ

τw can be determined by considering repeated ledge nucleation on the top of the base ledge, JT, which is given by Using steady-state assumption and combining with equation (5-2), it yields

τ

This newly unit ledge then passes over on the top of base ledge with a distance of b_p. The required time, τ_α, can be deduced from steady-state assumptions

bp

τ_α = b τ (5-11)

The condition presented in equation (5-5) leads to

τ

_n

>> > τ τ

_α. It means that most of time is spent on the stage of α ledge nucleation, which reasonably leads the τ_w to be considered as the summation of the required time to reach h^*

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n

τ

f can be rationalized as follows. Describing the motion of a superledge of height h as resulting from the motion of successive unit ledges of height a allows to express the effective mobility of a ledge with height h, M , in terms of M the unit ledge mobility _h

hM

M_h = a (5-14)

In order to simplify the system, the superledge presenting on α γ interface are assumed to have the same height; therefore, the velocity of every superledge is the same.

When unpinning occurs (h=h^*), it leads to

* ^h *

v M G a M G

D h D

= = (5-15)

Then the flight time necessary to merge two unpinned superledges can be obtained from

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* 2

p f

v τ =l b⁻ (5-16)

where l is defined as the spacing of superledges. Using equations (5-4) and (5-15),

τ

_f , becomes cases of interests because of b>>b_p. Neglecting the flight time in the subsequent analysis, τ becomes _λ

Combining equations (5-7) and (5-18), the net global velocity of the ferrite/austenite interface

ν

_αγ can be obtained as

ν

αγ is independent of superledge spacing. Equation (5-19) also reveals the effect of carbide precipitation on the overall interface velocity. Compared it to equation (5-7), it shows that the overall interface velocity is slowed down by carbide precipitation.

We have now to derive equations to estimate the features of interests,

λ

, b_p, and h^*.

The ledge is pinned by carbides, when the transformation driving force,

D

Gis smaller than the pinning force, DG_ZP exerted by carbides. Then:

GZP G

D ≥D (5-20)

The pinning pressure is considered is a function of h: indeed the precipitates exert their pinning force at the root of the super ledge and this force is to be distributed on the whole height of the ledge. In that case:

o

where

σ

_α is the α γ interfacial energy. Note that this expression differs from the classical Zener pinning one because the precipitates are by construction lying of the interface while in the case of Zener they are randomly sampled by the interface.

To go further, we have to estimate in a simple way the characteristics of the precipitation. Representing a particle row with a slab of precipitate of thickness,

d

, and assuming a parabolic growth, we have

p

p

c Dt

d = c^α (5-23)

where c^α_p is the nominal solute concentration of carbide forming element, c_p is the solute concentration of precipitate, and D is the boundary diffusivity of carbide-forming element. Therefore, the particle radius can be obtained from

2 4 3

Setting 1λb²_pas the number of precipitates per unit volume, each precipitate having a

volume of 4 ³

3πR^unpin. The resulting volume fraction, f_p, is therefore given by 117

3

Assuming that all alloying elements are partitioned to precipitates, we obtain

p

p p

Fe

f V c

=V ^α (5-27)

where V_p and V are the molar volume of precipitation and iron matrix, respectively. _Fe Then, the radius of precipitates at the moment of unpinning can be expressed as

2 1/3

The ledge height is growing at a constant rate. At time t, the ledge height h is given by equation (5-13) which can be rewritten:

τ

Replacing equation (5-29) and into equation (5-21), the pinning force per unit surface GZP

At the time t =t^* which the superledge reaches its critical height and unpins from carbides, the driving force of transformation is equal to the pinning force due to carbides. Then:

G G_ZP =D

D (5-31)

The time t^* for ledge to reach its critical height, h^*, can be written by rearranging equation (5-29)

The only unknown in the equation is the sheet spacing. To derive the equation for

λ

, the nucleation conditions for the precipitates have to be analyzed. For sake of simplicity, one may assume that a sufficient nucleation rate for the formation of a new sheet of particles is attained in the limit where the solute concentration approaches some critical value, e.g. c_crit. Then, one may approximate

λ

as the diffusion distance during the time

τ

_λ, then:

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crit

p

c D

c ^λ

λ= _α τ (5-34)

By replacing equation (5-34) into equation (5-18) and using equations (5-5) and (5-33), the expression of sheet spacing now can be further simplified as

1/3 2/3

Subsequently, the interface velocity becomes:

2 1/2 2 1/4 3/4 1/2

Setting in as a first approximation, c_crit =c_p^α, the final forms of equations providing sheet spacing and overall interface velocity can be written as:

3/4 1/2

The particle spacing, b_p, is considered as the result from α and carbide nucleation competition. At the critical concentration under steady-state assumption, the particle nucleation rate, J_p, is obtained from equation (5-2)

120

2

With equations (5-1) and (5-4), b can be expressed as

3

such that the particle spacing in one sheet of IP is given by

( )

The remaining quantities to be determined to have a closed form model are the nucleation rate for the precipitates and for the ledges in the ferrite/austenite interface.

Now, we have successfully derived the equations for the characteristic features of interphase precipitation. Table 5-2 summarizes the most important equations in the present thesis.

Table 5-2 A summary of the equations used in the calculations

Descriptions Equations

Ferrite ledge nucleation rate

*

Particle spacing

( )

p

在文檔中 含釩鋼界面析出之顯微結構及模型之研究 (頁 135-149)