The models for interphase precipitation

The model proposed by Li and Todd (the L-T model) is based on the coupling between growth of carbide and ferrite. The precipitate sheet widths, precipitation repeat periods and boundary velocities are formulated. Their final outputs are a ratio of sheet spacing to carbide size.

In the L-T model, the term “pseudo-phase boundary” was introduced in order to apply Zener’s solution for plate precipitate growth and the following assumptions were made [58]:

a. the solute concentration is constant and equal to the average concentration of the sheet of interphase-precipitated carbides.

b. mass balance is applied at the interface

c. average solute concentration in ferrite at the interface boundary is constant d. The motion of interface does not influence the carbide nucleation.

e. The passage of ledge is accomplished by advancing small steps. The row of carbides and the layer of ferrite are stacked layer by layer.

Figure 2-13 highlights the sequences of interphase precipitation in L-T model. As the first row of precipitates formed, the pseudo-phase is generated just ahead of the

24

precipitates (Figure 2-13). The interphase-precipitated carbides are assumed to be small and uniformly distributed along the interface boundary, enabling the growth of precipitates in row to be approximately planar. The precipitates in the first row will grow with time, so does the austenite/ferrite interface boundary. Until the interface boundary moves away, accumulating sufficient solutes for further precipitation, the second cycle of interphase precipitation occurs. It should be kept in mind that the lateral growth of ledge is accomplished by small steps and the growth of carbide with interface motion is highlighted.

Figure 2-13 Schematic illustration showing the sequences of interphase precipitation in the L-T model. (a) the first row of precipitates nucleate with a size of y_p and generates the pseudo-phase boundary simultaneously; (b) the pseudo-phase boundary advances to the position where has sufficient solutes for further precipitation; the growth of the carbides in the first row is taken into considerations as well; and (c)

carbides continue to grow, from the size y^'_p to y^"_p [12]

Coupling with the experimental work from Honeycombe et al., it has found that the sheet spacing of interphase precipitation and the diffusivity of solute element can be expressed as:

' ( )1/2

c e V

y =K D (2-2)

25

where DV is the diffusivity of vanadium in ferrite. The proportional constant, Ke, is strongly dependent on compositions and austenitization temperatures.

Based on mass balance at the interface, the concentration in front of the advancing pseudo-phase has been expressed as

1 ( 2 )

1 ( 2)

m mo

mp mo

C C erf y Dt

C C erf s

− = −

− − (2-3)

where _{yp =s Dt}

D = solute diffusivity in ferrite t = time

C^m = solute concentration in ferrite at position y(t) C^mo =initial concentration of solute in ferrite

C^mp = concentration of solute in ferrite at the pseudo-phase/ferrite interface assuming to be zero in the present analysis

yp = forward growth distance of the pseudo-phase

Zener had provided an approximated value for s [58]:

s= 1Ω

− Ω (2-4)

and Li and Todd shows that the parameter Ω is expressed as [9]

26

'

After mathematically analysis, the concentration of solute in the pseudo-phase, C^p, and the sheet spacing are derived as follows:

'

where t_c^' is the interphase precipitate repeat period, and A is given by

2 A= 1− Ω

− Ω (2-8)

Then, the average velocity of interface can be obtained

' 1/ 2

In addition, the forward growth distance of the pseudo-phase at the instant when a new sheet of nuclei is formed is derived as

' '

p 1 c

y = Ω Dt

− Ω (2-10)

27

Finally, the equation for yc is shown as

For now, it has been shown that all the parameters of interests for interphase precipitation have been correlated with composition and position terms. Generally, all the quantities can be classified into two categories:

(a) Measured quantities:

sheet spacing ( )y^"_c , particle size ( )y^"_p , diffusivity (D), alloy concentration (C^mo), and activation energy ΔHV

(b) Calculated quantities:

sheet repeat period ( )t_c^' , solute concentration of pseudophase (C^p), average interphase velocity ( )V_b , and critical supersaturation (C^mc).

The final purpose of L-T model is to relate the measured quantities to the calculated ones. Because all the calculated quantities are the function of the ratio of sheet spacing to carbide width, y_c^' y^'_p , and it is independent of transformation temperatures, Li and Todd have suggested the first step is to measure the average interphase boundary velocity where sheet spacing is known. As the diffusivity of vanadium is calculated at a given temperature, the value of A can be obtained from

28

' determined. By measuring the interphase boundary velocity and sheet spacing, and with the knowledge of diffusivity of solutes, the value y^'_p can be obtained. Finally y^"_p is

In summary, the L-T model tries to relate all the interested parameters by using the ratio y_c^' y^'_p, which is simply obtained by measuring the interphase boundary velocity and sheet spacing. It firstly provided a systematically model describing interphase precipitation. It should be noted that the calculations in L-T model yield a ratio of

' '

c p

y y , instead of giving a direct value. The calculated results show great consistencies with the microscopic data in other published papers.

However, some of their propositions deviate from real physical situations and are doubted by Rios [10] and Liu [57]. In L-T model, the sheet structure of ferrite and carbide develop one after another. It implies that the ferrite ledge height is not necessarily equal to the sheet spacing of carbide. However, in fact, the ferrite ledges are observed to move as a “train”, instead of individual layer. Furthermore, L-T model

29

implies vanadium is enriched in austenite in front of the interface and re-distributed between ferrite and austenite during the motion of interface. Actually, the re-distribution of vanadium solute is expected to occur only at slow transformation rates (high transformation temperatures). In addition, vanadium diffusion in ferrite is about 3 times order of magnitude faster than in austenite; vanadium accumulated at the interface is expected to transfer into ferrite. Therefore, the enrichment in austenite is difficult to happen. These weaknesses have been discussed by Rios and Liu.

(2) The Model Proposed by P.R. Rios

Rios proposed an alternative to the L-T model. In that model, the vanadium volume diffusion is also considered but the ferrite ledge height is supposed to be in equal to the sheet spacing of interphase precipitation. The assumptions in Rios’ model are shown as follows:

a. The ledge velocity is constant

b. The carbides form as a continuous layer in the pseudo-phase region which is indicated as the shaded area in Figure 2-14.

Figure 2-14 Schematic illustration showing the Rios’ model. The pseudo-phase is indicated as a shaded area.

In the vanadium diffusion region, the pseudo-phase boundary changes linearly from the previous ledge riser to the latter one, with a final thickness, s_f.

30

c. The thickness of pseudo-phase increases linearly from the advancing riser to the trailing one and finally reaches its final thickness, s_f.

d. Based on stoichiometric composition, the carbon content of austenite remains constant, it means that the carbon rejected by ferrite growth goes into the carbides directly.

e. The vanadium is limited to diffuse into carbide between the region 0 < x < l and –h < y < 0. The growth of pseudo-phase is controlled by vanadium diffusion in ferrite.

f. The boundary conditions of vanadium concentration are: (1) at x = l along –y direction, Cv = Cvo

; (2) at ferrite/pseudo-phase boundary, Cv = 0; (3) Cv in austenite remains constant and equals to Cvo

. The three boundary conditions will be further used in the following calculations.

Using the imaginary coordinates in Figure 2-14, the vanadium concentration in ferrite can be approximated in a plane by

( , )

C x y

V

= + + A Bx Cy

(2-14)

The goal now turns into finding the exact values of constants A, B, and C. By applying the boundary conditions mentioned in assumption f. Two equations can be written as

0^h _V( , )

0 ( , 0)

V 0

C x dx

∫

^ =

 (2-16)

Furthermore, the average vanadium concentration in ferrite within the limited region is assumed to be equal to C_V^o/ 2 where the latter one is given by

Using the above Equations, the constants, A, B, and C are solved by the final expression of vanadium concentration in the limited diffusion region becomes

( , ) ( 1 )

From mass balance under steady-state consideration, the vanadium entering the ferrite through ledge riser is equal to that entering the growing pseudo-phase

0

The average pseudo-phase boundary velocity can be calculated by taking the flux of vanadium at position y = 0

32

0

where D is the vanadium diffusivity in ferrite, ^p

CV is vanadium composition in carbide,

p

kCV is the vanadium concentration in pseudo-phase, and k is a constant. Using equation (2-18), the average pseudo-phase boundary velocity becomes

p

fD

υ = − kh (2-21)

where f =C_V^o C_V^p , the volume fraction of carbides.

The ledge velocity is determined as follows. When the ledge riser moves from x = 0 to x = h (the ledge spacing has been shown to be equal to ledge height) at given time τ, it means that the pseudo-phase boundary moves its average position from y = -sf /2 to y

= -s_f. Therefore, the following relation between ledge and pseudo-phase velocities can be written as (the ledge and pseudo-phase velocities are assumed to be independent of time)

Assuming all the vanadium entering the ledge riser goes into pseudo-phase to form carbide, s kC_f( _V^p)=hC_V^o or f = (ksf)/h, a simplified expression for ledge velocity can be obtained by:

33

2D

υ = h (2-23)

Equation 2-23 shows that the ledge velocity is inversely proportional to ledge height but it does not demonstrate how the transformation temperatures and carbide volume fraction affect ledge height clearly. From equations 2-21 to 2-23, the three parameters, ks_f, h, and τ, can be related as

Equation 2-24 yields another relation describing the ledge height with K

D1/ 2

h K

= f (2-25)

The value of K is defined as a model parameter. Great efforts were made in Rios model to determine its value by using a phenomenological analysis from other researches.

Obviously, as we plotting the ledge height, h, against D^1/2/f, a straight line can be obtained. The slope of this line gives K. However, by using equation 2-25, the results have greater deviations with the experimental data which were obtained at higher transformation temperatures and lower alloy contents. Rios proposed that such deviations were associated with the ratio of total solutes, I_o, to the solutes which are

34

insoluble in austenite, Iin. In order to correct such a deviation, Rios introduced solubility product into K. Then, K is re-written as

( ^o)

Using the solubility product of VC in austenite, the Cs can be expressed as

VC

s o

V

C K

= C (2-28)

Then, equation 2-25 becomes

1/ 2

Replacing equation 2-29 into 2-23, the ledge velocity becomes

2 1/ 2

Equations 2-29 and 2-30 couple with the concept of super-saturation in term of carbon solubility ratio, Iin/Io, with the parameters of interests. By referring to equation 2-29, a greater ratio (I_o/I_in, super-saturation is low) will generate a wider sheet spacing of interphase precipitation. The physical interpretation is that as the degree of super-saturation is decreased, it would result in reducing the nucleation rate of carbide as well, forming a coarser and wider spaced carbide distribution. Of course, a smaller ratio of (Io/Iin) contributes to an opposite effect. The most advantageous part of equations 2-29 and 2-30 is that only one adjustable model parameter, K_s, is used for calculations. In addition, equation 2-29 has been established to be coupled with the relationships developed in the other work for calculating carbide size and volume fraction [59]. The sheet spacing calculated by equation 2-29 exhibits a better agreement (compared to equation 2-25 with the experimental data obtained by Batte and Honeycombe [29] and Balliger and Honeycombe [60].

The ledge velocity calculated by equation 2-30 represents the velocity of ledge riser during its migration. It indicates that the ledge velocity decreases with transformation temperatures, carbide volume fraction, and solubility of carbide.

However, compared to the average velocity calculated by Li and Todd, the results obtained by Rios present a smaller value, about half of that obtained by Li and Todd. As mentioned previously, the velocity calculated by Li and Todd is the overall interface velocity which advancing direction is normal to the sheet of carbides. Carbide precipitation on the terrace plane and the passage of ferrite ledge happen one after another and the ledge height does not have to be equal to sheet spacing of interphase precipitation. That is, the height of ferrite ledge is taken as an average value, so does the interface velocity. However, in Rios model, the velocity calculated by equation 2-30 is for ledge riser only. He proposed that the ledge mechanism of austenite-to-ferrite

36

transformation is accomplished by the passage of ferrite ledge which its height is equal to the sheet spacing of interphase precipitation, and the ledge riser advances with a constant velocity. It can be expected that the ledge height in Rios model is greater than that in L-T model and requires a longer time to develop, leading to a slower velocity than that calculated by Li and Todd.

(3) The Model Proposed by W.J. Liu

In addition to Rios, the propositions of L-T model were also challenged by Liu through different points of views. He pointed out that part of the assumptions made by Li and Todd were deviated from real thermodynamic and kinetic conditions. The arguments are summarized as follows:

a. In L-T model, vanadium solute is assumed to re-distribute with the advancing interface. Liu pointed out it is unreasonable. Because generally the activation energy of substitutional element is greater than that of interstitial element, the vanadium re-distribution is expected to operate at a higher transformation temperature. However, the experimental data coupled with the L-T model were obtained at the transformation temperatures around 700 ^oC. Liu argued that the vanadium atoms are difficult to be driven at these transformation temperatures.

b. Li and Todd proposed that the vanadium would accumulate in front of interface.

However, the diffusivity of vanadium in ferrite is three times order of magnitude faster than that in austenite. It means that the vanadium solute would diffuse into ferrite more rapidly. Accumulation of substitutional solute at the interface is almost impossible to happen

c. Carbide nucleation on the interface and its pinning effect were not involved in L-T 37

model. It implies that the precipitation of VC does not interfere with interface motion. However, it was recognized that interphase precipitation could effectively retard the interface motion and have influences on the growth kinetics of allotriomorphic ferrite. Liu had actually provided detailed derivations to against this point.

Based on these arguments, Liu proposed a computational model to modify the L-T model. The ferrite/austenite interface was supposed to be under para-equilibrium condition and a force balance between pinning force exerted from carbide, Fp, and transformation driving force, F_d, was supposed. The latter force affects directly interface velocity, which is dependent on the ease of carbon diffusion from the interface into austenite and gradually decreases because of carbon enrichment in austenite. The reduced velocity stimulates carbide precipitation, at the same time, increasing the pinning force, Fp. Until Fp = Fd, the interface is pinned by carbides and becomes stationary but the carbon continuously diffuses from the interface into austenite. This process will increase F_d again. On the other hand, the rate of increment of F_p will become slower because the solute concentration decreases as carbide precipitation. In the latter stage, F_d will take over F_d again, unpinning from the carbides. The evolutions of Fp and Fd with time are schematically illustrated in Figure 2-15. As a result, the first issue of Liu’s model is to determine the values of Fp and Fd, and this is obviously dependent on the carbon diffusion in austenite.

38

Figure 2-15 Schematic illustration showing the model proposed by Liu. The important time parameters are indicated in this figure. In his model, the process of interphase precipitation is a repeated force balance between the driving force, F_d, and the pinning force, F_p [57]

In the beginning of calculations, carbon diffusion in austenite should be first introduced because it determines the interface velocity and in calculating the sheet spacing of interphase precipitation. Using one-dimensional version of Fick’s second law and considering the composition-dependence of carbon diffusivity, the resulting differential equation is

2

where D_c is the diffusion coefficient of carbon in austenite, d_a is the austenite grain diameter, wf is the width of ferrite grain, x is the dimensional coordinate, c is the volume concentration of carbon in austenite, and

ν

is the interface velocity. With the boundary conditions

39

( ) respectively, and the subscript i stands for interface. The interface velocity can be obtained.

Then, the sheet spacing of interphase precipitation, λ , can be obtained by integrating the velocity from the t_unpin to t_repin.

Replacing the obtained interface velocity into the equation 2-31, the carbon concentration at the interface and composition-dependence carbon diffusivity in austenite can be derived as

[1 2 (1 0.088 exp(67.4 / ))]

where Yc is the site fraction of carbon in its corresponding sub-lattice, B is a carbon site

40

blocking parameter,

ψ

and

ψ

_M are the activity coefficients of carbon in austenite and of the activate complex of carbon, respectively.

The thermodynamic at the interface should be considered as well. Two free energy terms are introduced in Liu’s calculations, ΔGn and ΔG^*. From classical nucleation theory, the total free energy change accompanied with the formation of a carbide nucleus on the interface can be expressed as

( )

n chem

G V G G_e Sσ E_γα

D = D + D + − (2-36)

where V and S are the volume and surface area of the embryo, ΔGchem and ΔGε are the chemical driving force and coherency strain energy of forming an unit volume embryo, respectively, , σ is the embryo/matrix interface energy, and Eγα is the energy released from the interface nucleation site. The terms of V, Sσ, and Eγα are estimated by

3 represent as the interfacial energies at precipitate/ferrite, precipitate/austenite, and austenite/ferrite interface, respectively. From equations 2-36 and 2-37, the critical nucleation energy can be obtained. The detailed mathematical treatments will not be shown here but he final derivative results are presented as follows:

41

2

where

Ω

_MI is the molar volume of MI, log[M][I] is the solubility product of carbide in ferrite, y is the interstitial/substitution ratio in the MI phase, AFe, AM, and AI are atom weights of Fe, M, and I, respectively. The terms X_M^α and X_I^α are the mole fraction of M and I in ferrite.

µ

_α is the shear modulus of ferrite.

K K ,

_α, and

K

_MI are all elastic parameters, and D is the effective cubic dilatation. So far, the kinetics and ^* thermodynamics parameters have been constructed for further Fp and Fd calculations.

The nucleation rate of carbide is required to determine F_p. Ashby had shown that the maximum pinning pressure exerted on such an interface is expressed as

2 2

where s_h is the half-interparticle spacing which is calculated by

42

1 sh

= N (2-43)

where N is the precipitate density, being a function of nucleation rate, J, as.

0^t exp( )

N J dt

t τ

=

∫

^ − ^(2-44)

where

τ

is the incubation time of carbide. The integral time interval is from zero to the lifetime of the phase boundary,

t

_(see Figure 2-15), which can be determined by

t w

=ν

 (2-45)

where w is the thickness of an incoherent interface (or height of ledge on coherent interface) and v is the interface velocity in the direction normal to the interface which can be calculated by equation 2-32.

Using the classical nucleation theory, J and incubation time, τ, are written as

*

*

where Z is the Zeldovich non-equilibrium factor, β^* is the rate at which atoms are added to the critical nucleus, and Ns is the number of nucleation sites per unit area at the interface. The energy terms, ΔG^* and ΔGn, are the energy of forming a critical nucleus and an embryo that contains n atom pairs, respectively. The n^* is the atom pair number in the critical nucleus. In equation 2-49, r^* is the critical radius of the nucleus, D_Mⁱ is the diffusion coefficient of the metal element along the interface, Y_M is the site fraction of the metal solute in ferrite, and a_α is the lattice parameter of ferrite. The value of Ns is defined to be equal to number of lattice sites on

(

^{1 0 0}

)

ferrite plane

For the driving force, Fd, it is defined as the difference of iron and substitutional

For the driving force, Fd, it is defined as the difference of iron and substitutional

在文檔中 含釩鋼界面析出之顯微結構及模型之研究 (頁 51-85)