4. The nonequilibrium thermodynamics analysis of the eutectoid transformation

In [19], a generalization of the equations characterizing the growth of the pearlite colony is proposed, based on the application of nonequilibrium thermodynamic methods.

To this end, Eq. (19) from [20], which characterizes the growth rate of a perlite colony, is represented as:

$$\frac{d\mathbf{X}}{dt} = D\_x \left[ \left( \mathbf{C}\_{\Phi}^{'} - \mathbf{C}\_{\mathbf{u}}^{'} \right) / \left( \mathbf{C}\_{\Phi}^{'} - \mathbf{C}\_{\Phi} \right) + \left( \mathbf{C}\_{\Phi}^{'} - \mathbf{C}\_{\mathbf{u}}^{'} \right) / \left( \mathbf{C}\_{\mathbf{u}} - \mathbf{C}\_{\mathbf{u}}^{'} \right) \right] / \Delta = (D\_x / \Delta)(-\Delta \rho), \quad \text{(29)}$$

where Dx is the carbon diffusion coefficient in austenite along the x axis at a given temperature T, Δ is the thickness of a layer of austenite with different concentration of carbon, C'<sup>Φ</sup> and С'<sup>y</sup> is the carbon concentration in the austenite near the ferrite and cementite plates, respectively, at a temperature T (Figure 2), С<sup>y</sup> is the carbon content in cementite (�6.67%), C<sup>Φ</sup> is the carbon content in the ferrite at a given temperature T, and �Δφ is the thermodynamic force of perlite lateral growth. It is determined by the carbon concentrations in ferrite and cementite and has a dimensionless value.

### Figure 2. Carbon distribution in the austenite-perlite system [20].

The second equation characterizing our system—the heat balance Eqs. (23)–(25) from [20]–is written in the form:

$$\text{Cyl}T/dt = \text{a } \Delta T - (q\gamma/\Delta) \,\text{dX}/dt\tag{30}$$

J<sup>2</sup> ¼ ð Þ qγ=Δ ðð Þ� Dх=Δ ð Þþ Δφ L12ð Þ �ΔТ=Т Þ � αТð Þ �ΔТ=Т

Using for the kinetic coefficients, the Onsager reciprocity relations Lik = Lki [9],

L12 <sup>¼</sup> L21 <sup>¼</sup> <sup>q</sup>γDх=Δ<sup>2</sup>

γ2

In accordance with (36), the perlite growth rate is affected not only by the concentration thermodynamic force, but also by the temperature difference between the sample and the environment. Let us further consider the phase transformation of austenite under special conditions of steady growth of the pearlite colony, when it can be assumed that ΔT ≈ сonst, dТ/dt ≈ 0. In this case, Eq. (37)

> γ2 Dх=Δ<sup>3</sup>

By analogy with the previously obtained solutions [21], we introduce the fol-

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi кqγDx=αТ

where Δ<sup>l</sup> ¼ qγ=k is the characteristic parameter of the system: (41)

For Δ<sup>l</sup> = 0, as expected, the solution of Eq. (31) Δ = Δ0. We obtain a well-known solution for the pearlite transformation of austenite [20]. In the real domain, there is one solution of Eq. (44). For small Δl (<0.5), the root Xk is in the region close to 1 (Xk ! D0), with increasing Δ<sup>l</sup> (in units of D0), the root value increases. For large

> Хk≈ ffiffiffiffiffiffiffiffiffiffiffiffiffi Δl=Δ<sup>0</sup>

γ2 Dх=Δ<sup>3</sup>

L21 <sup>¼</sup> <sup>q</sup>γDх=Δ<sup>2</sup> (33)

, (34)

<sup>D</sup>х=Δ<sup>3</sup> � <sup>α</sup>Т: (35)

–α<sup>Т</sup> � � ð Þ �ΔТ=<sup>Т</sup> : (37)

–α<sup>Т</sup> � � ð Þ¼ �ΔТ=<sup>Т</sup> <sup>0</sup>: (38)

Δφ ¼ кΔТ=Т, (39)

; (40)

Δ<sup>l</sup> ¼ 0 (42)

p<sup>3</sup> : (43)

ð Þ �ΔТ=Т (36)

(32)

<sup>¼</sup> <sup>q</sup>γDх=Δ<sup>2</sup> � �ð Þþ �Δ<sup>φ</sup> <sup>q</sup>γL12=Δ—α<sup>Т</sup> � � ð Þ �ΔТ=<sup>Т</sup>

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

L22 <sup>¼</sup> <sup>q</sup><sup>2</sup>

The system of Eqs. (31) and (32) takes the form:

<sup>J</sup><sup>2</sup> <sup>¼</sup> <sup>q</sup>γDх=Δ<sup>2</sup> � �ð Þþ �Δ<sup>φ</sup> q2

where k is the proportionality coefficient.

Eq. (38) can now be represented in the form:

values of Δl, the root of Xk is approximately equal to

For small ΔT, we can write approximately, as was done in [20]:

Δ<sup>0</sup> ¼

<sup>Δ</sup><sup>3</sup> � <sup>Δ</sup><sup>0</sup> 2 Δ � Δ<sup>0</sup> 2

J1 <sup>¼</sup> ð Þ� <sup>D</sup>х=<sup>Δ</sup> ð Þþ <sup>Δ</sup><sup>φ</sup> <sup>q</sup>γDх=Δ<sup>2</sup>

<sup>J</sup><sup>2</sup> <sup>¼</sup> <sup>q</sup>γDх=Δ<sup>2</sup> � �ð Þþ �Δ<sup>φ</sup> <sup>q</sup><sup>2</sup>

Relating Eqs. (3) and (32) to each other, we obtain:

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

we find that

whereas

takes the following form:

lowing notation:

111

where α is the heat transfer coefficient, C is the specific heat, ΔТ is the temperature difference between the sample (T) and the cooling medium, q is the specific amount of heat released during the formation of perlite, and γ is the density of steel.

If two quantities are used as charges for the eutectoid transformation of austenite-the temperature of the sample T and the thickness of the plates of perlite X, then, according to (4), the Onsager motion equations must have symmetric forms (2) and (3),

where J<sup>1</sup> = � dХ/dt is the flow of the pearlitic layer (with increasing absolute value of the thermodynamic growth force of perlite, the flow increases in absolute value), and J<sup>2</sup> = � CγdТ/dt is the heat flow in the sample (with a drop in sample temperature, the flow is positive), Х<sup>1</sup> = (�Δφ), Х<sup>2</sup> = (�ΔТ/Т) is the thermodynamic forces of perlite growth and temperature [14].

In order for Eq. (29) to correspond to Eq. (2), it must contain an additional term L12 (�ΔТ/Т); with the value of the coefficient L11 = (Dх/Δ):

$$J\_1 = (D\_x/\Delta)(-\Delta\rho) + L\_{12}(-\Delta T/T) \tag{31}$$

where L12 is a cross ratio whose value is not yet known. Thus, we introduce (we assume) an additional dependence of the growth rate of the perlite layer not only on the carbon concentrations in the phases but also on the temperature.

Substituting expression (34) into the energy balance Eq. (33), we find the expression for the heat flow J2:

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

$$\begin{split} J\_2 &= (q\gamma/\Delta)((D\_\mathbf{x}/\Delta)(-\Delta\varrho) + L\_{12}(-\Delta T/T)) - aT(-\Delta T/\mathbf{T}) \\ &= \left(q\gamma D\_\mathbf{x}/\Delta^2\right)(-\Delta\varrho) + \left(q\gamma L\_{12}/\Delta - aT\right)(-\Delta T/\mathbf{T}) \end{split} \tag{32}$$

Relating Eqs. (3) and (32) to each other, we obtain:

$$L\_{21} = q \gamma \mathcal{D}\_x / \Delta^2 \tag{33}$$

Using for the kinetic coefficients, the Onsager reciprocity relations Lik = Lki [9], we find that

$$L\_{12} = L\_{21} = q\eta D\_x / \Delta^2,\tag{34}$$

whereas

The second equation characterizing our system—the heat balance Eqs. (23)–(25)

where α is the heat transfer coefficient, C is the specific heat, ΔТ is the temperature difference between the sample (T) and the cooling medium, q is the specific amount of heat released during the formation of perlite, and γ is the density of steel. If two quantities are used as charges for the eutectoid transformation of austenite-the temperature of the sample T and the thickness of the plates of perlite X, then, according to (4), the Onsager motion equations must have symmetric

where J<sup>1</sup> = � dХ/dt is the flow of the pearlitic layer (with increasing absolute value of the thermodynamic growth force of perlite, the flow increases in absolute value), and J<sup>2</sup> = � CγdТ/dt is the heat flow in the sample (with a drop in sample temperature, the flow is positive), Х<sup>1</sup> = (�Δφ), Х<sup>2</sup> = (�ΔТ/Т) is the thermodynamic

In order for Eq. (29) to correspond to Eq. (2), it must contain an additional term

where L12 is a cross ratio whose value is not yet known. Thus, we introduce (we assume) an additional dependence of the growth rate of the perlite layer not only on

Substituting expression (34) into the energy balance Eq. (33), we find the

CγdТ=dt ¼ α ΔТ � ð Þ qγ=Δ dХ=dt (30)

J1 ¼ ð Þ� Dх=Δ ð Þþ Δφ L12ð Þ �ΔТ=Т (31)

from [20]–is written in the form:

Non-Equilibrium Particle Dynamics

Carbon distribution in the austenite-perlite system [20].

forces of perlite growth and temperature [14].

expression for the heat flow J2:

110

L12 (�ΔТ/Т); with the value of the coefficient L11 = (Dх/Δ):

the carbon concentrations in the phases but also on the temperature.

forms (2) and (3),

Figure 2.

$$L\_{22} = q^2 \gamma^2 D\_x / \Delta^3 - \mathfrak{a}T. \tag{35}$$

The system of Eqs. (31) and (32) takes the form:

$$J\_1 = (D\_x/\Delta)(-\Delta \rho) + q\gamma D\_x/\Delta^2(-\Delta T/T) \tag{36}$$

$$J\_2 = \left( q\gamma \mathcal{D}\_x / \Delta^2 \right) (-\Delta \rho) + \left( q^2 \gamma^2 \mathcal{D}\_x / \Delta^3 - aT \right) (-\Delta T / \mathcal{T}).\tag{37}$$

In accordance with (36), the perlite growth rate is affected not only by the concentration thermodynamic force, but also by the temperature difference between the sample and the environment. Let us further consider the phase transformation of austenite under special conditions of steady growth of the pearlite colony, when it can be assumed that ΔT ≈ сonst, dТ/dt ≈ 0. In this case, Eq. (37) takes the following form:

$$J\_2 = \left( q\gamma \mathcal{D}\_x / \Delta^2 \right) (-\Delta \rho) + \left( q^2 \gamma^2 \mathcal{D}\_x / \Delta^3 - aT \right) \left( -\Delta T / \mathcal{T} \right) = \mathbf{0}.\tag{38}$$

For small ΔT, we can write approximately, as was done in [20]:

$$
\Delta \rho = \kappa \Delta T / T,\tag{39}
$$

where k is the proportionality coefficient.

By analogy with the previously obtained solutions [21], we introduce the following notation:

$$
\Delta\_0 = \sqrt{\kappa q \chi D\_\mathbf{x} / \alpha T}; \tag{40}
$$

where Δ<sup>l</sup> ¼ qγ=k is the characteristic parameter of the system: (41)

Eq. (38) can now be represented in the form:

$$
\Delta^3 - \Delta\_0^2 \Delta - \Delta\_0^2 \Delta\_l = \mathbf{0} \tag{42}
$$

For Δ<sup>l</sup> = 0, as expected, the solution of Eq. (31) Δ = Δ0. We obtain a well-known solution for the pearlite transformation of austenite [20]. In the real domain, there is one solution of Eq. (44). For small Δl (<0.5), the root Xk is in the region close to 1 (Xk ! D0), with increasing Δ<sup>l</sup> (in units of D0), the root value increases. For large values of Δl, the root of Xk is approximately equal to

$$Xk \approx \sqrt[3]{\Delta l/\Delta\_0}.\tag{43}$$

The between interplate distance of perlite for a stationary growth process is found from the formula:

$$\mathbf{S}\_0 = \mathbf{2}\mathbf{X}\boldsymbol{\kappa} \times \boldsymbol{\Delta}\_0 = \mathbf{2}\boldsymbol{\kappa}\mathbf{X}k\sqrt{\boldsymbol{\Delta}l\mathbf{D}\_\mathbf{x}/aT}.\tag{44}$$

According to the constructed model, the perlite growth rate in the direction of the X axis has a maximum value at supercooling ΔТ = 140.0°С. The perlite growth rate calculated according to Zener's formula has a theoretical maximum value at overcooling ΔТ = 96.0°С. Consequently, the theoretical expressions (31) and (32) make it possible to describe with greater accuracy the maximum and the course of the experimental curve for the perlite formation rate presented in [3, 24] for high-

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

The expression for perlite growth rate obtained in this section has a significant value at supercooling of 300–400°С, thereby determining the possibility of perlite formation in this temperature range. Indeed, the formation of perlite in carbon

The calculated dependence of the between interplate distance of perlite by formula (46) on the magnitude of the supercooling of steel is shown in Figure 4.

A fairly good agreement of the calculated dependence with the results of the latest experiments is observed, which indicates the adequacy of the proposed

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

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

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

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).

takes place by means of an instantaneous shift of atomic planes that does

steels in the temperature range 375–325°С was revealed in [24].

The same figure shows the experimental points from [24].

purity eutectoid steel.

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

model.

treatment.

(quenching) [26].

Figure 4.

113

Using Eqs. (36) and (43), (44), we also find an improved expression for the perlite growth rate for an isothermal transformation

$$\frac{d\mathbf{X}}{dt} = \frac{kD\mathbf{x}}{\mathbf{S}\_0} \frac{\Delta T}{T} \left(\mathbf{1} + \frac{2\Delta l}{\mathbf{S}\_0}\right) \tag{45}$$

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 authors of [20].

We use the well-known dependence of the diffusion coefficient on temperature [17]:

$$D = A \exp.(-Q/RT),$$

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

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

#### Figure 3.

Dependence of the perlite growth rate on the supercooling value, calculated from formula (47) of the present work (Vp1) and Zener's formula (1) [22, 23] (Vp2).

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

According to the constructed model, the perlite growth rate in the direction of the X axis has a maximum value at supercooling ΔТ = 140.0°С. The perlite growth rate calculated according to Zener's formula has a theoretical maximum value at overcooling ΔТ = 96.0°С. Consequently, the theoretical expressions (31) and (32) make it possible to describe with greater accuracy the maximum and the course of the experimental curve for the perlite formation rate presented in [3, 24] for highpurity eutectoid steel.

The expression for perlite growth rate obtained in this section has a significant value at supercooling of 300–400°С, thereby determining the possibility of perlite formation in this temperature range. Indeed, the formation of perlite in carbon steels in the temperature range 375–325°С was revealed in [24].

The calculated dependence of the between interplate distance of perlite by formula (46) on the magnitude of the supercooling of steel is shown in Figure 4. The same figure shows the experimental points from [24].

A fairly good agreement of the calculated dependence with the results of the latest experiments is observed, which indicates the adequacy of the proposed model.
