5. Application of the positions of nonequilibrium thermodynamics to the analysis of the nondiffusion transformation of austenite

Martensite is the basis of hardened steel, so studying the mechanism and kinetics of its transformation is still of extreme interest for the theory and practice of heat treatment.

In the works of G.V. Kurdyumov and coworkers, the martensitic transformation is considered as the usual phase transformation in a one-component system, further complicated by the influence of a strong interatomic interaction, which leads to the development of significant stresses in the martensite crystal and matrix [25].

In accordance with the alternative mechanism, the martensitic transformation takes place by means of an instantaneous shift of atomic planes that does not require thermal activation and is not associated with thermodynamic transformation stimuli [1], [26]. In this case, the stress initiating the transformation is believed to be the stresses arising from the sharp cooling of the sample (quenching) [26].

#### Figure 4.

The between interplate distance of perlite for a stationary growth process is

S0 <sup>¼</sup> <sup>2</sup>Хк � <sup>Δ</sup><sup>0</sup> <sup>¼</sup> <sup>2</sup>кХ<sup>k</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Using Eqs. (36) and (43), (44), we also find an improved expression for the

ΔT <sup>T</sup> <sup>1</sup> <sup>þ</sup>

We use the well-known dependence of the diffusion coefficient on tempera-

D = A exp.(�Q/RT),

After substituting the known values of the steel parameters and taking into account that to 2.0, we find the calculated dependence of the perlite growth rate on the supercooling value of the alloy (Figure 3). In this figure, the dependence of the perlite growth rate on the supercooling value, calculated according to Zener's for-

Dependence of the perlite growth rate on the supercooling value, calculated from formula (47) of the present

где Q is the activation energy, (Q ≈ 134 кJ/mol), and R is a constant (R = 8314 J/

The formula (45) is a more precise expression for determining the growth rate of perlite in the eutectoid transformation, than the expression obtained earlier by the

2Δl S0 � �

perlite growth rate for an isothermal transformation

mula (1) [22, 23], is given for comparison.

work (Vp1) and Zener's formula (1) [22, 23] (Vp2).

dX dt <sup>¼</sup> kDx S0

<sup>Δ</sup>lDx=α<sup>Т</sup> <sup>p</sup> : (44)

(45)

found from the formula:

Non-Equilibrium Particle Dynamics

authors of [20].

ture [17]:

(mol�К)).

Figure 3.

112

The calculated dependence of the between interplate distance of perlite on the magnitude of the supercooling of steel ( —experimental points from [24], p. 122, —calculated points).

Considering the martensitic transformation as a thermally activated process, B.Ya. Lyubov used the equations of normal transformation obtained on the basis of the positions of nonequilibrium thermodynamics to describe his kinetics [3].

Changes in a complex or composite system under constant external conditions can be described as the process of increasing entropy. The rate of increase of entropy σ can be represented as the sum of the flux products and the corresponding forces for all transfer substrates in an amount of N [7–10]:

$$\sigma = \frac{d\mathbf{S}}{dt} i r v v = \sum\_{k=1}^{N} f\_k \mathbf{X}\_k \ (\mathbf{k} = \mathbf{1}, \dots, \mathbf{N}), \tag{47}$$

In the general case, the flows can be represented in the form (1).

The irreversible change in the entropy dSirrev is equal to the sum of entropy changes in the system and the environment:

$$d\mathbf{S}\_{irrev} = d\mathbf{S} + d\mathbf{S}\_{\epsilon} \tag{48}$$

dF ¼ 3σdε þ φdn<sup>α</sup> (53)

J<sup>1</sup> ¼ L11Х<sup>1</sup> þ L12Х<sup>2</sup> (55) J<sup>2</sup> ¼ L21Х<sup>1</sup> þ L22Х2, (56)

(57)

RT (58)

: (60)

kT <sup>¼</sup> <sup>4</sup>:<sup>58</sup> � <sup>10</sup>�<sup>4</sup>ехрð Þ �252; <sup>000</sup>=RT (59)

dt : (54)

dεα

dt <sup>þ</sup> <sup>φ</sup>

Thus, in our system, in addition to the particle flow from the γ phase to the α-phase of J1 = dnα/dt, we will also take into account the change in the strain of the sample with time J2 = dε/dt. These flows are related to the driving forces by the Eq. (1).

If, as charges of the process of nondiffusion transformation of austenite, the two

where X1 = Δφ is the thermodynamic force for iron, the change in the chemical potential at the transition of particles from the γ-phase to the α-phase, and X2 = Δσ is the change in the internal stress during the transition from the γ-phase to the

dt <sup>¼</sup> L11 φγ–φα

In the normal kinetics of the phase transformation, the formation of the center (particle) of the α-phase occurs through separate (independent) acts of detachment of particles from the γ-phase and the attachment of atoms to the ferrite center. If we consider the process of formation of an α-phase close to the process of self-diffusion

L11 <sup>¼</sup> <sup>D</sup><sup>γ</sup>

The self-diffusion coefficient of iron is taken in the usual notation [17]:

where D0 is a multiplier and U is the activation energy of diffusion.

where σα is the stress in the α phase and σγ is the stress in the γ phase.

J2 <sup>¼</sup> <sup>d</sup>εα

where D<sup>γ</sup> is the self-diffusion coefficient of iron in the γ-phase (or the effective coefficient of self-diffusion in the γ-phase of the alloy),T is the transformation

dt <sup>¼</sup> L22 σγ–σα

Let σγ = 0. Let us take into account that for triaxial compression stretching [28]:

The system of Eqs. (55) and (56) describes the contribution of stresses and deformations to the nondiffusion transformation of austenite. However, we do not yet know the coefficients of the equations in it. We now find expressions for the coefficients of the system of Eqs. (55) and (56). The coefficient L11 characterizes the

J1 <sup>¼</sup> <sup>d</sup>n<sup>α</sup>

of iron in the γ-phase, then the coefficient L11 has the form [13]:

temperature, and R is the gas constant [27].

D<sup>γ</sup> ¼ D0e

�U

The coefficient L22 characterizes the direct relationship:

quantities are the concentration of α-phase particles and the strain value, then,

but the change in entropy:

DOI: http://dx.doi.org/10.5772/intechopen.83657

α-phase.

115

normal transformation:

dS

according to (1), the equations of motion take the form:

dt irrev ¼ �Т�<sup>1</sup> <sup>3</sup><sup>σ</sup> <sup>d</sup>εα

Using the Principles of Nonequilibrium Thermodynamics for the Analysis of Phase…

Under isothermal conditions, when the released heat is absorbed by the environment and the temperature remains constant:

$$dS\_e = -\,d\,Q/T,\,dQ = dU + PdV$$

$$dS\_{irrev} = dS - (dUa + PdV)T^{-1} = (TdS - dU - PdV)T^{-1}.\tag{49}$$

Since dU + PdV–TdS = dG, and if we take into account the low compressibility of bodies in the condensed state and relatively small pressures, then.

$$\frac{d\mathbf{S}}{dt}irrev = -T^{-1}\frac{d\mathbf{G}}{dt} \approx -T^{-1}\frac{d\mathbf{F}}{dt},\tag{50}$$

where F is the free energy of the system.

The change in free energy in a system with a variable number of particles and internal stresses can be represented in the form [3], p. 142:

$$dF = dF\varepsilon + dF n = \sigma\_{ik} d\varepsilon\_{ik} + \rho\_l d n\_l,\tag{51}$$

where dFε is the change in free energy in the system related to internal stresses, dFn is the change in the free energy in the system, determined by the variable number of particles of type l, σik is the stress tensor, εik is the strain tensor of the system, φ<sup>l</sup> is the chemical potential of the lth element of the system, and nl is the number of particles of the lst element of the system per unit volume, l = 1, N.

We now introduce some simplifying assumptions. First, for the nondiffusion transformation of austenite, only one kind of particles, the α-phase of iron nα, will be taken into account. Approximately, this is also true for alloys of iron with close elements (nickel, chromium, cobalt). Of course, φ is some effective (averaged) chemical potential of the atoms of the alloy.

Secondly, we assume that the deformation of the system is a triaxial compression-expansion, and in the expression for dFε, only the diagonal components of stress and strain tensors are taken into account:

$$
\sigma\_{ik} = \varepsilon\_{ik} = 0, i \neq k. \\
\sigma\_{i\bar{i}} = \sigma\_{\cdot} \ e\_{i\bar{i}} \text{--} \mathbf{\varepsilon} \tag{52}
$$

The change in internal energy can then be represented as:

Using the Principles of Nonequilibrium Thermodynamics for the Analysis of Phase… DOI: http://dx.doi.org/10.5772/intechopen.83657

$$dF = \mathcal{J}\sigma d\varepsilon + \varrho d n\_a \tag{53}$$

but the change in entropy:

Considering the martensitic transformation as a thermally activated process, B.Ya. Lyubov used the equations of normal transformation obtained on the basis of the positions of nonequilibrium thermodynamics to describe his kinetics [3].

Changes in a complex or composite system under constant external conditions

JkXk ð Þ k ¼ 1; ::; N , (47)

: (49)

dt , (50)

dSirrev ¼ dS þ dSe (48)

can be described as the process of increasing entropy. The rate of increase of entropy σ can be represented as the sum of the flux products and the corresponding

> N к¼1

The irreversible change in the entropy dSirrev is equal to the sum of entropy

Under isothermal conditions, when the released heat is absorbed by the envi-

dSirrev <sup>¼</sup> dS–ð Þ dUa <sup>þ</sup> PdV <sup>T</sup>�<sup>1</sup> <sup>¼</sup> ð Þ TdS–dU–PdV <sup>T</sup>�<sup>1</sup>

Since dU + PdV–TdS = dG, and if we take into account the low compressibility of

The change in free energy in a system with a variable number of particles and

where dFε is the change in free energy in the system related to internal stresses,

dFn is the change in the free energy in the system, determined by the variable number of particles of type l, σik is the stress tensor, εik is the strain tensor of the system, φ<sup>l</sup> is the chemical potential of the lth element of the system, and nl is the number of particles of the lst element of the system per unit volume, l = 1, N. We now introduce some simplifying assumptions. First, for the nondiffusion transformation of austenite, only one kind of particles, the α-phase of iron nα, will be taken into account. Approximately, this is also true for alloys of iron with close elements (nickel, chromium, cobalt). Of course, φ is some effective (averaged)

Secondly, we assume that the deformation of the system is a triaxial compression-expansion, and in the expression for dFε, only the diagonal compo-

dt <sup>≈</sup> � <sup>Т</sup>�<sup>1</sup> <sup>d</sup><sup>F</sup>

dF ¼ dFε þ dFn ¼ σikdεik þ φldnl, (51)

σik ¼ εik ¼ 0, i 6¼ k:σii <sup>¼</sup> σ, εii–ε (52)

dt irrev <sup>¼</sup> <sup>∑</sup>

dSe = � dQ/T, dQ = dU + PdV

bodies in the condensed state and relatively small pressures, then.

dt irrev ¼ �Т�<sup>1</sup> <sup>d</sup><sup>G</sup>

dS

internal stresses can be represented in the form [3], p. 142:

where F is the free energy of the system.

chemical potential of the atoms of the alloy.

114

nents of stress and strain tensors are taken into account:

The change in internal energy can then be represented as:

In the general case, the flows can be represented in the form (1).

forces for all transfer substrates in an amount of N [7–10]:

<sup>σ</sup> <sup>¼</sup> dS

changes in the system and the environment:

Non-Equilibrium Particle Dynamics

ronment and the temperature remains constant:

$$\frac{d\mathbf{S}}{dt}\_{irrev} = -T^{-1} \left( 3\sigma \frac{d\varepsilon\_a}{dt} + \rho \frac{d\varepsilon\_a}{dt} \right). \tag{54}$$

Thus, in our system, in addition to the particle flow from the γ phase to the α-phase of J1 = dnα/dt, we will also take into account the change in the strain of the sample with time J2 = dε/dt. These flows are related to the driving forces by the Eq. (1).

If, as charges of the process of nondiffusion transformation of austenite, the two quantities are the concentration of α-phase particles and the strain value, then, according to (1), the equations of motion take the form:

$$J\_1 = L\_{11}X\_1 + L\_{12}X\_2 \tag{55}$$

$$J\_2 = L\_{21}X\_1 + L\_{22}X\_2.\tag{56}$$

where X1 = Δφ is the thermodynamic force for iron, the change in the chemical potential at the transition of particles from the γ-phase to the α-phase, and X2 = Δσ is the change in the internal stress during the transition from the γ-phase to the α-phase.

The system of Eqs. (55) and (56) describes the contribution of stresses and deformations to the nondiffusion transformation of austenite. However, we do not yet know the coefficients of the equations in it. We now find expressions for the coefficients of the system of Eqs. (55) and (56). The coefficient L11 characterizes the normal transformation:

$$J\_1 = \frac{d n\_a}{dt} = L\_{11}(\rho\_\gamma - \rho\_a) \tag{57}$$

In the normal kinetics of the phase transformation, the formation of the center (particle) of the α-phase occurs through separate (independent) acts of detachment of particles from the γ-phase and the attachment of atoms to the ferrite center. If we consider the process of formation of an α-phase close to the process of self-diffusion of iron in the γ-phase, then the coefficient L11 has the form [13]:

$$L\_{11} = \frac{D\_{\gamma}}{RT} \tag{58}$$

where D<sup>γ</sup> is the self-diffusion coefficient of iron in the γ-phase (or the effective coefficient of self-diffusion in the γ-phase of the alloy),T is the transformation temperature, and R is the gas constant [27].

The self-diffusion coefficient of iron is taken in the usual notation [17]:

$$D\_{\gamma} = D\_0 e^{-\frac{U}{kT}} = 4.58 \cdot 10^{-4} \exp(-252,000/\text{RT})\tag{59}$$

where D0 is a multiplier and U is the activation energy of diffusion. The coefficient L22 characterizes the direct relationship:

$$J\_2 = \frac{d\varepsilon\_a}{dt} = L\_{22}(\sigma\_\gamma - \sigma\_a). \tag{60}$$

Let σγ = 0. Let us take into account that for triaxial compression stretching [28]: where σα is the stress in the α phase and σγ is the stress in the γ phase.

Let σγ = 0. Let us take into account that for triaxial compression-stretching [28]:

$$
\sigma\_a = \frac{\mathbf{E}}{\Im(\mathbf{1} \cdot \mathbf{2}\mu)} \frac{\Delta V}{V} = \frac{\mathbf{E}}{\mathbf{1} \cdot \mathbf{2}\mu} \varepsilon\_{a\nu} \tag{61}
$$

εα ¼ εγ þ Δεα ¼ εγ þ

Using the Principles of Nonequilibrium Thermodynamics for the Analysis of Phase…

ΔV<sup>γ</sup>!<sup>α</sup> V<sup>γ</sup>

 L<sup>22</sup>

Lt <sup>¼</sup> <sup>L</sup>11Δ<sup>φ</sup> <sup>þ</sup> <sup>L</sup>12σγ

L11Δφ þ L12σγ L v

the thermodynamic force Δφ, but also on the magnitude of the stresses in the

ation, characterized by <sup>Δ</sup>V<sup>γ</sup> ! <sup>α</sup> and the relative volume change <sup>Δ</sup>Vγ!<sup>α</sup>

<sup>Δ</sup>V<sup>γ</sup>!<sup>α</sup> <sup>¼</sup> <sup>0</sup>:<sup>268</sup> � <sup>1</sup>:62<sup>∗</sup>

εγ � αΔТ and the tensile stress σγ corresponding to this deformation:

volume is determined by the additional deformation: <sup>Δ</sup>Vγ!<sup>α</sup>

pressive stress arising in the α-phase has the value.

Then, substituting expression (53) into Eq. (51.1), we find:

dt <sup>¼</sup> L11Δ<sup>φ</sup> <sup>þ</sup> L12σγ � L12 L21Δ<sup>φ</sup> <sup>þ</sup> <sup>L</sup>22σγ

nα<sup>¼</sup>

sample as γ ! α-transformation:

DOI: http://dx.doi.org/10.5772/intechopen.83657

L11Δφ þ L12σγ <sup>1</sup> � <sup>e</sup>¼<sup>v</sup>

to obtain the kinetic equation for nα:

constructed model.

dn<sup>α</sup>

γ-phase.

the data of [3]:

117

Eq. (69) shows that the residual deformation of the α-phase after the transient process consists of the deformation of the austenite εγ and the additional deformation Δεα. This additional deformation determines the change in the volume of the

L21Δφð Þ <sup>1</sup>‐2<sup>μ</sup>

1 � e �v

> 1 � e �v

10�<sup>4</sup>Т,sm<sup>3</sup>

e�<sup>v</sup>

It can be concluded from expression (71) that the growth rate of α-phase particles depends on the stresses in the γ-phase. The greater the value of tensile stresses in the γ phase, the higher the growth rate of ferrite particles. The rate of growth of the α-phase particles at a constant temperature very rapidly (exponentially) decreases in time, determining the incompleteness of the transformation.

Integration of Eq. (71) with time-independent coefficients L11 and L12 allows us

In accordance with Eq. (72), the amount of α-phase formed depends not only on

6. Scheme of the nondiffusion transformation of austenite based on the

Before discussing the equations obtained, we introduce some more useful relations characterizing the γ ! α transformation. With the γ ! α transformation, the effective atomic volume of the iron lattice changes in the sample under consider-

We will assume that with the formation of the α-phase, the relative change in

When the alloy sample is cooled by ΔT, a deformation occurs in its surface layer:

σα <sup>¼</sup> <sup>E</sup> 1‐2μ <sup>L</sup>22<sup>Е</sup> : (69)

¼ 3nαΔεα: (70)

Lt <sup>¼</sup> L11Δ<sup>φ</sup> <sup>þ</sup> L12σγ�

Lt : (72)

<sup>V</sup><sup>γ</sup> . According to

=mol (73)

<sup>V</sup><sup>γ</sup> = 3Δεα, and the com-

Δεα: (74)

(71)

Lt :

where E is the modulus of elasticity of steel (�2.17�10<sup>5</sup> <sup>М</sup>Pa) and <sup>μ</sup> is the Poisson ratio (� 0.26).

Then, expression (60) can be transformed as follows:

$$\frac{d\varepsilon\_a}{dt} = L\_{22}\sigma\_a = L\_{22}\frac{\mathbf{E}}{\mathbf{1}\cdot\mathbf{2}\mu}\varepsilon\_a = \frac{\mathbf{v}}{L}\varepsilon\_a \tag{62}$$

where the following values are entered:

v is the propagation velocity of the microdeformation in sample (�1000 m/с) [3] and L is the characteristic distance over which the microdeformation of the shear is propagated (the size of the martensitic strips or plates). At the initial stage of the formation of the shear structure, it has a magnitude of the order of the diameter of the austenite grain (� 100 μm), and then decreases with decreasing temperature [1].

From Eq. (62), we find that the coefficient L22 is equal to:

$$L\_{22} = \frac{\mathbf{v}(\mathbf{1} \cdot \mathbf{2}\mu)}{LE}.\tag{63}$$

The cross-coefficients L12 = L21 for a nonequilibrium thermodynamic system are found with sufficient accuracy by the formulas proposed in [5]:

$$L\_{12} = \sqrt{L\_{11} L\_{22}} = \sqrt{\frac{D\_r}{RT} \frac{\mathbf{v}(\mathbf{1} \cdot \mathbf{2}\mu)}{LE}}\tag{64}$$

Thus, we obtained simple differential equations for a nonequilibrium thermodynamic system describing the nondiffusion transformation of austenite taking into account the influence of internal stresses.

Let us write the equations of motion of our system in the form:

$$\frac{d\mathfrak{n}\_a}{dt} = L\_{11}\Delta\varrho + L\_{12}\sigma\_\gamma - L\_{12}\sigma\_a. \tag{65}$$

$$\frac{d\varepsilon\_a}{dt} = L\_{21}\Delta\rho + L\_{22}\sigma\_\gamma - L\_{22}\sigma\_a. \tag{66}$$

We first transform Eq. (66) taking into account expression (62). We have:

$$\frac{d\varepsilon\_a}{dt} + \nu \varepsilon\_a/L = L\_{21} \Delta \rho + L\_{22} \sigma\_{\mathcal{V}} \tag{67}$$

where εα is the magnitude of deformations of the α-phase. The differential Eq. (66) with constant coefficients (temperature) has a solution:

$$
\varepsilon\_a = \frac{\left(\mathbf{L}\_{21}\Delta\rho + \mathbf{L}\_{22}\sigma\_\gamma\right)\left(\mathbf{1}\cdot\mathbf{2}\mu\right)}{L\_{22}E}\left(\mathbf{1} - e^{-\frac{\nu}{\hbar}t}\right). \tag{68}
$$

This kinetic equation describes the change in the magnitude of the deformation of the <sup>α</sup>-phase in time. At <sup>t</sup> <sup>=</sup> 0, εα <sup>=</sup> 0. When the time is counted, a fast (� <sup>10</sup>�<sup>6</sup> s) process of transition to deformation occurs:

Using the Principles of Nonequilibrium Thermodynamics for the Analysis of Phase… DOI: http://dx.doi.org/10.5772/intechopen.83657

$$
\varepsilon\_a = \varepsilon\_\gamma + \Delta \varepsilon\_a = \varepsilon\_\gamma + \frac{\mathbf{L}\_{21} \Delta \rho(\mathbf{1} \cdot \mathbf{2} \mu)}{L\_{22} E}. \tag{69}
$$

Eq. (69) shows that the residual deformation of the α-phase after the transient process consists of the deformation of the austenite εγ and the additional deformation Δεα. This additional deformation determines the change in the volume of the sample as γ ! α-transformation:

$$\frac{\Delta V\_{\gamma \to a}}{V\_{\gamma}} = \mathfrak{R}\_{a} \Delta \varepsilon\_{a}.\tag{70}$$

Then, substituting expression (53) into Eq. (51.1), we find:

$$\frac{d\mathfrak{n}\_a}{dt} = \mathcal{L}\_{11}\Delta\rho + \mathcal{L}\_{12}\sigma\_\gamma - \frac{\mathcal{L}\_{12}\left(\mathcal{L}\_{21}\Delta\rho + \mathcal{L}\_{22}\sigma\_\gamma\right)}{L\_{12}}\left(\mathbf{1} - e^{-\frac{\nu}{\mathcal{L}}t}\right) = \mathcal{L}\_{11}\Delta\rho + \mathcal{L}\_{12}\sigma\_\gamma - \mathcal{L}\_{12}\left(\mathbf{1} - e^{-\frac{\nu}{\mathcal{L}}t}\right),$$

$$\left(\mathcal{L}\_{11}\Delta\rho + \mathcal{L}\_{12}\sigma\_\gamma\right)\left(\mathbf{1} - e^{-\frac{\nu}{\mathcal{L}}t}\right) = \left(\mathcal{L}\_{11}\Delta\rho + \mathcal{L}\_{12}\sigma\_\gamma\right)e^{-\frac{\nu}{\mathcal{L}}t}.$$

It can be concluded from expression (71) that the growth rate of α-phase particles depends on the stresses in the γ-phase. The greater the value of tensile stresses in the γ phase, the higher the growth rate of ferrite particles. The rate of growth of the α-phase particles at a constant temperature very rapidly (exponentially) decreases in time, determining the incompleteness of the transformation.

Integration of Eq. (71) with time-independent coefficients L11 and L12 allows us to obtain the kinetic equation for nα:

$$m\_{a=} \frac{\left(L\_{11}\Delta\rho + L\_{12}\sigma\_{\gamma}\right)L}{v} \left(1 - e^{-\frac{\mu}{L}}\right). \tag{72}$$

In accordance with Eq. (72), the amount of α-phase formed depends not only on the thermodynamic force Δφ, but also on the magnitude of the stresses in the γ-phase.
