2. Formation of carbides in chrome steel during tempering

Consider the process of separation of carbides in a low-carbon steel system of iron-carbon-chromium with 0.15% carbon and about 5% chromium at 600°C. In this model system, there are two phases—the doped α-phase (F) and carbides (K), in which carbon, iron, chromium, and vacancies flows (Figure 1). As charges, we will use four quantities—the concentrations of carbon, iron, chromium, and vacancies. The flow of vacancies in the carbide phase will be assumed to be equal to the flow of vacancies in the ferrite.

In the absence of a change in the volume of the system, for flows in the doped α phase, condition [13] is fulfilled:

tempering, graphitization of undoped cementite, separation of carbides in alloyed

When the rate of transformation of austenite is determined by the rate at which the interface separates, differing only in its crystalline structure, the transformation is called nondiffusion [1]. Kinetically, the normal polymorphic and martensitic transformations of austenite are distinguished. When the temperature of the normal transformation decreases, its velocity first increases and then decreases. The kinetics of the martensitic transformation is characterized by a very high rate of growth of individual crystals and the maximum space velocity at the initial moment

In addition to martensite, at least two other structural components are known, which are formed with a shear ("martensitic") morphology of crystal formation ferrite side-plates and acicular ferrite. They can also be attributed, with some simplifying assumptions, to the products of the nondiffusion transformation of austenite. In addition, in some alloys martensitic and normal transformations occur at the same temperature [1]. The consistent theory of nondiffusion transformations should explain this phenomenon. Thus, the theoretical description of the processes of phase transformations in iron-carbon alloys is a complex and urgent task of

Nonequilibrium thermodynamics provides the necessary apparatus for analyz-

where Ji are flows, Xk are the thermodynamic forces, Lik = Lki are the Onsager kinetic coefficients [9], and i, k are the charge numbers (transfer substrates).

The main driving forces of phase transformations in nonequilibrium thermodynamics are gradients of the chemical potentials of their components [6–9]. When discontinuous systems are considered, the finite differences of chemical potentials (�Δμi,) as the transition from a metastable state to a stable state are used as thermodynamic forces [10, 11]. Equations of nonequilibrium thermodynamics were first used in the physics of metals to describe the process of graphitization of

As is known, unalloyed cementite in iron-carbon alloys at normal pressure is a metastable phase, its activity in phases with it in equilibrium exceeds the solubility of graphite, a stable phase [11]. Therefore, at a sufficiently high temperature, graphitization of such alloys takes place, that is, phase transition from metastable to stable equilibrium. Despite the seeming simplicity of this process, its theoretical

If two values are used as charges of the graphitization process-carbon and iron concentrations, then, according to (1), the equations of motion take the form:

where J1 is the carbon flow characterizing the rate of the graphitization process, J2 is the flow of iron, and X1 = (�ΔμFe) and X2 = (�ΔμC) are the thermodynamic forces of iron and carbon. The potential drop has a "þ" sign as it increases, and the flow is directed toward a decrease in the potential, so the expressions for the forces

LikXkð Þ i ¼ 1; ::; N , (1)

J<sup>1</sup> ¼ L11Х<sup>1</sup> þ L12Х<sup>2</sup> (2) J<sup>2</sup> ¼ L21Х<sup>1</sup> þ L22Х2, (3)

ing the processes of phase transformations in iron-carbon alloys [7–9]. In the general case, the thermodynamic equations of motion have the form [7]:

> Ji ¼ ∑ N к¼1

steels, and others [4–6].

Non-Equilibrium Particle Dynamics

modern metal physics.

nonalloyed iron-carbon alloys [6, 11].

description is a complex task.

contain the sign "�."

104

of transformation under isothermal conditions.

$$J\_{\rm Fe} + J\_{\rm Gr} + J\_v = \mathbf{0},\tag{5}$$

Let us find the expressions for the cross coefficients, which make it possible to obtain a nontrivial solution of the system of Eqs. (9)–(11). From the first Eq. (9), we

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

For independent variations δμ<sup>С</sup> and δμСr, the linear system of Eqs. (13) and (14)

and the sign before the root is selected based on the sign (direction) of the flows under consideration (see Figure 1). The considered procedure of variation allows us to find cross rates in the Onsager equations after the direct kinetic coefficients are calculated. In this case, the established connection (15) is satisfied for systems not

Let us find the values of the thermodynamic forces and kinetic coefficients for the steel of the Fe-C-Cr system with 0.15% C at 600°C. We will assume that in a solid α-solution, there is chromium with a concentration of СCr = 0.05 and a carbon with a concentration of С<sup>C</sup> = 0.007, an iron concentration of СFe = 0.943. In cementite-type carbide, chromium is found with a mass fraction of �20% (with a concentration of СCr = 0.2) and carbon with a carbon concentration of 0.25, an iron

It is known from the experimental data that carbon is removed very rapidly (approximately 1 minute) from the α-solution of alloyed steel at a temperature of 550–650°C and, consequently, the formation of carbide inclusions is primarily due

The thermodynamic force for carbon can be calculated from the formula [11]:

С аα С

ln ð Þ¼ аС=аС<sup>0</sup> βi Ni, (17)

, (16)

С is the

�Δμ<sup>С</sup> ¼ �RT ln аК

<sup>С</sup> is the thermodynamic activity of carbon in α-solution, а<sup>К</sup>

thermodynamic activity of carbon in cementite, R is the universal gas constant, and

The change in the thermodynamic activity of carbon in the alloy upon doping

where βi is the coefficient of the element's influence on the thermodynamic activity of carbon in the alloy, Ni is the content of the element in the alloy in atomic fractions, and аС<sup>0</sup> is the thermodynamic activity of carbon for the alloy in the

with component i can be found by the method of [15, 16] from the equation:

is compatible if the coefficients of δμ<sup>С</sup> and δμС<sup>r</sup> are equal to 0, from which we

δμFe ¼ �ð Þ L12=L<sup>11</sup> δμ<sup>С</sup> � ð Þ L13=L<sup>11</sup> δμCr: (12)

<sup>12</sup>=L<sup>11</sup> � �δμ<sup>С</sup> <sup>þ</sup> ð Þ <sup>L</sup><sup>23</sup> � <sup>L</sup>12L13=L<sup>11</sup> δμCr <sup>¼</sup> <sup>0</sup>, (13)

<sup>13</sup>=L<sup>11</sup> � �δμС<sup>r</sup> <sup>¼</sup> <sup>0</sup>, (14)

Lii � Lkk <sup>p</sup> , <sup>i</sup>, <sup>k</sup> <sup>¼</sup> <sup>1</sup>…<sup>3</sup> (15)

establish a connection between the variations of forces:

Substituting (12) into Eqs. (10) and (11), we find

<sup>J</sup>С<sup>r</sup> <sup>¼</sup> ð Þ <sup>L</sup><sup>32</sup> � <sup>L</sup>13L12=L<sup>11</sup> δμ<sup>С</sup> <sup>þ</sup> <sup>L</sup><sup>33</sup> � <sup>L</sup><sup>2</sup>

immediately find the relation between Onsager's kinetic coefficients:

very far from equilibrium and for the real system is approximate.

Lik <sup>¼</sup> Lki ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3. Calculation of thermodynamic forces and kinetic coefficients

<sup>J</sup><sup>С</sup> <sup>¼</sup> <sup>L</sup><sup>22</sup> � <sup>L</sup><sup>2</sup>

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

concentration in the carbide C'Fe = 0.55.

to carbon diffusion [14].

T is the temperature of the alloy.

where а<sup>α</sup>

standard state.

107

so one of the threads (in our case—Jv) is a dependent quantity. According to (1), the thermodynamic equations for flows in the carbide phase take the form:

$$J\_{\rm Fe} = -L\_{11} \Delta \mu\_{\rm Fe} - L\_{12} \Delta \mu\_{\rm c} - L\_{13} \Delta \mu\_{\rm Cr} \tag{6}$$

$$J\_{\mathcal{C}} = -L\_{21} \Delta \mu\_{\text{Fe}} - L\_{22} \Delta \mu\_{\text{c}} - L\_{23} \Delta \mu\_{\text{Gr}} \tag{7}$$

$$J\_{Cr} = -L\_{31} \Delta \mu\_{Fe} - L\_{32} \Delta \mu\_c - L\_{33} \Delta \mu\_{Cr} \tag{8}$$

where JFe, JС, and JCr are the flows of iron, carbon, and chromium, respectively.

Based on the general principles of nonequilibrium thermodynamics, we can find the values of the thermodynamic forces �ΔμFe, ΔμCr, and �ΔμС, as well as the values of the kinetic coefficients L12, L13, and L23, as it was previously performed in [5] for a system with two flows. In the conditions of complete equilibrium, ΔμFe = 0, Δμ<sup>С</sup> = 0, and ΔμCr = 0. However, for a linear thermodynamic system, there is also the possibility of dynamic equilibrium, in which all the flows are 0, but some thermodynamic forces in the system are not equal to zero (there are their variations) [5, 7].

Let us consider this possibility for a triple thermodynamic system. From Eqs. (6)–(8), it follows that near equilibrium, in the presence of variations of thermodynamic forces, the following conditions must be fulfilled:

$$J\_{Fe} = \mathbf{0} \Rightarrow L\_{11} \delta \mu\_{Fe} + L\_{12} \delta \mu\_C + L\_{13} \delta \mu\_{Cr} = \mathbf{0},\tag{9}$$

$$J\_C = \mathbf{0} \Rightarrow L\_{21}\delta\mu\_{\rm Fe} + L\_{22}\delta\mu\_C + L\_{13}\delta\mu\_{\rm Cr} = \mathbf{0},\tag{10}$$

$$J\_{Cr} = \mathbf{0} \Rightarrow L\_{31} \delta \mu\_{Fe} + L\_{32} \delta \mu\_C + L\_{33} \delta \mu\_{Cr} = \mathbf{0},\tag{11}$$

where the index δμ denotes the coordinated variations of the thermodynamic forces that ensure the dynamic equilibrium of the system. It follows from the system of Eqs. (9)–(11) that the expressions for the flows of iron, chromium, and carbon are connected: the cross rates L12, L13, and L<sup>23</sup> in expressions for the flows must have values such that the determinant of the matrix A composed of the coefficients of this system was equal to 0. In this case, the values of the flows of iron and chromium can significantly increase due to cross-kinetic coefficients in comparison with the independent diffusion of these elements [7, 16].

Figure 1. Scheme of the process of carbides formation in chromium steel.

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

Let us find the expressions for the cross coefficients, which make it possible to obtain a nontrivial solution of the system of Eqs. (9)–(11). From the first Eq. (9), we establish a connection between the variations of forces:

$$
\delta\mu\_{\rm Fe} = -(L\_{12}/L\_{11})\delta\mu\_{\rm C} - (L\_{13}/L\_{11})\delta\mu\_{\rm Cr}.\tag{12}
$$

Substituting (12) into Eqs. (10) and (11), we find

JFe þ JCr þ Jv ¼ 0, (5)

JFe ¼ �L11ΔμFe � L12Δμ<sup>c</sup> � L13ΔμCr (6) J<sup>С</sup> ¼ �L21ΔμFe � L22Δμ<sup>c</sup> � L23ΔμCr (7) JCr ¼ �L31ΔμFe � L32Δμ<sup>c</sup> � L33ΔμCr, (8)

JFe ¼ 0 ) L11δμFe þ L12δμ<sup>С</sup> þ L13δμCr ¼ 0, (9) J<sup>С</sup> ¼ 0 ) L21δμFe þ L22δμ<sup>С</sup> þ L13δμCr ¼ 0, (10) J<sup>С</sup><sup>r</sup> ¼ 0 ) L31δμFe þ L32δμ<sup>C</sup> þ L33δμ<sup>С</sup><sup>r</sup> ¼ 0, (11)

so one of the threads (in our case—Jv) is a dependent quantity. According to (1),

where JFe, JС, and JCr are the flows of iron, carbon, and chromium, respectively. Based on the general principles of nonequilibrium thermodynamics, we can find

the values of the thermodynamic forces �ΔμFe, ΔμCr, and �ΔμС, as well as the values of the kinetic coefficients L12, L13, and L23, as it was previously performed in [5] for a system with two flows. In the conditions of complete equilibrium, ΔμFe = 0, Δμ<sup>С</sup> = 0, and ΔμCr = 0. However, for a linear thermodynamic system, there is also the possibility of dynamic equilibrium, in which all the flows are 0, but some thermodynamic forces in the system are not equal to zero (there are their varia-

Let us consider this possibility for a triple thermodynamic system. From Eqs. (6)–(8), it follows that near equilibrium, in the presence of variations of

where the index δμ denotes the coordinated variations of the thermodynamic forces that ensure the dynamic equilibrium of the system. It follows from the system of Eqs. (9)–(11) that the expressions for the flows of iron, chromium, and carbon are connected: the cross rates L12, L13, and L<sup>23</sup> in expressions for the flows must have values such that the determinant of the matrix A composed of the coefficients of this system was equal to 0. In this case, the values of the flows of iron and chromium can significantly increase due to cross-kinetic coefficients in com-

thermodynamic forces, the following conditions must be fulfilled:

parison with the independent diffusion of these elements [7, 16].

Scheme of the process of carbides formation in chromium steel.

tions) [5, 7].

Non-Equilibrium Particle Dynamics

Figure 1.

106

the thermodynamic equations for flows in the carbide phase take the form:

$$J\_C = \left(L\_{22} - L\_{12}^2 / L\_{11}\right) \delta \mu\_C + (L\_{23} - L\_{12} L\_{13} / L\_{11}) \delta \mu\_{Gr} = 0,\tag{13}$$

$$J\_{Gr} = (L\_{32} - L\_{13}L\_{12}/L\_{11})\delta\mu\_C + \left(L\_{33} - L\_{13}^2/L\_{11}\right)\delta\mu\_{Gr} = 0,\tag{14}$$

For independent variations δμ<sup>С</sup> and δμСr, the linear system of Eqs. (13) and (14) is compatible if the coefficients of δμ<sup>С</sup> and δμС<sup>r</sup> are equal to 0, from which we immediately find the relation between Onsager's kinetic coefficients:

$$L\_{ik} = L\_{ki} = \pm \sqrt{L\_{ii} \times L\_{kk}}, \text{ i. } k = 1...3 \tag{15}$$

and the sign before the root is selected based on the sign (direction) of the flows under consideration (see Figure 1). The considered procedure of variation allows us to find cross rates in the Onsager equations after the direct kinetic coefficients are calculated. In this case, the established connection (15) is satisfied for systems not very far from equilibrium and for the real system is approximate.
