3. Calculation of thermodynamic forces and kinetic coefficients

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 concentration in the carbide C'Fe = 0.55.

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 to carbon diffusion [14].

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

$$-\Delta\mu\_{\mathcal{C}} = -RT\ln\frac{a\_{\mathcal{C}}^{K}}{a\_{\mathcal{C}}^{a}},\tag{16}$$

where а<sup>α</sup> <sup>С</sup> is the thermodynamic activity of carbon in α-solution, а<sup>К</sup> С is the thermodynamic activity of carbon in cementite, R is the universal gas constant, and T is the temperature of the alloy.

The change in the thermodynamic activity of carbon in the alloy upon doping with component i can be found by the method of [15, 16] from the equation:

$$\ln\left(a\_C/a\_{C0}\right) = \beta i \text{ Ni},\tag{17}$$

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 standard state.

We will assume that for our steel in the standard state а<sup>α</sup> <sup>С</sup><sup>0</sup> =аК <sup>С</sup><sup>0</sup> = аС0, i.e., unalloyed cementite in steel with 0.15%, C is stable and in equilibrium with the solid solution at a tempering temperature of 600°C [13]. Using this condition and Eqs. (16) and (17), we find:

$$\ln \left( a^{K}{}\_{C} / a^{a}{}\_{C} \right) = \beta^{K}{}\_{Cr} N^{K}{}\_{Cr} - \beta^{a}{}\_{Cr} N^{a}{}\_{Cr} \tag{18}$$

Dα

At a temperature of 600° C:

Fe <sup>≈</sup> 3.03�10�<sup>19</sup> cm2

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

0.95 � <sup>10</sup>�26, L13 <sup>=</sup> �0.611 � <sup>10</sup>�24, and

D1 = D<sup>α</sup>

for our system:

calculations show:

JV <sup>=</sup> 6.07 � <sup>10</sup>�<sup>15</sup> cm<sup>2</sup>

transformation

colony, is represented as:

has a dimensionless value.

ф–С<sup>0</sup> ц � �<sup>=</sup> <sup>С</sup><sup>0</sup>

methods.

dX

109

dt <sup>¼</sup> <sup>D</sup><sup>х</sup> <sup>С</sup><sup>0</sup>

/s.

10�<sup>21</sup> cm2

<sup>С</sup><sup>r</sup> <sup>¼</sup> <sup>3</sup>, 05 exp �<sup>358000</sup>

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

/s, D2 = D<sup>α</sup>

<sup>L</sup><sup>11</sup> <sup>=</sup> 0.394 � <sup>10</sup>�22, L22 <sup>=</sup> 0.984 � <sup>10</sup>�13, L12 <sup>=</sup> �1.97 � <sup>10</sup>�17, L33 <sup>=</sup>

RT � �см<sup>2</sup>

Using relations (23)–(25) and (15), we find the values of the kinetic coefficients

L23 <sup>=</sup> 0.306 � <sup>10</sup>�19. Consequently, the system of Eqs. (6)–(8) takes the form:

JFe <sup>¼</sup> <sup>0</sup>:<sup>394</sup> � <sup>10</sup>�<sup>22</sup> �ΔμFe ð Þ� <sup>1</sup>:<sup>97</sup> � <sup>10</sup>�<sup>17</sup> �Δμ<sup>С</sup> ð Þ� <sup>0</sup>:<sup>611</sup> � <sup>10</sup>�<sup>24</sup> �Δμс<sup>r</sup> ð Þ,

<sup>J</sup><sup>С</sup> ¼ �1:<sup>97</sup> � <sup>10</sup>�<sup>17</sup> �ΔμFe ð Þþ <sup>0</sup>:<sup>984</sup> � <sup>10</sup>�<sup>13</sup> �Δμ<sup>С</sup> ð Þþ <sup>0</sup>:<sup>306</sup> � <sup>10</sup>�<sup>19</sup> �ΔμCr ð Þ,

<sup>J</sup><sup>С</sup><sup>r</sup> ¼ �0:<sup>611</sup> � <sup>10</sup>�<sup>24</sup> �ΔμFe ð Þþ <sup>0</sup>:<sup>306</sup> � <sup>10</sup>�<sup>19</sup>ð Þþ �Δμ<sup>С</sup> <sup>0</sup>:<sup>95</sup> � <sup>10</sup>�<sup>26</sup> �ΔμCr ð Þ: (28)

It was established in [18] that during the tempering period, a certain amount of nanoparticles of special chromium carbide with a size of �100 nm can be formed in

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

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

h i � � <sup>=</sup><sup>Δ</sup> <sup>¼</sup> ð Þ� <sup>D</sup>х=<sup>Δ</sup> ð Þ <sup>Δ</sup><sup>φ</sup> , (29)

ц

ф–С<sup>0</sup> ц � �<sup>=</sup> <sup>С</sup>ц–С<sup>0</sup>

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

It follows from Eqs. (26)–(28) that the values of iron and chromium fluxes increase substantially due to the cross-ratios L12 and L32 of a significant thermodynamic force (�ΔμC). The value of the carbon flux having a positive sign is determined mainly by the intrinsic coefficient L22. The thermodynamic forces of iron and chromium make an insignificant contribution to the fluxes, because of the small value of the kinetic coefficients and their influence can be neglected. Then, as direct

JFe <sup>=</sup> � 6.08 � <sup>10</sup>�15, JC <sup>=</sup> 3.04 � <sup>10</sup>�11, JCr <sup>=</sup> 0.94 � <sup>10</sup>�17, and

4. The nonequilibrium thermodynamics analysis of the eutectoid

/s.

<sup>ф</sup>–С<sup>ф</sup> � � <sup>þ</sup> <sup>С</sup><sup>0</sup>

the steel, which were detected experimentally.

<sup>C</sup> ≈ 1.02 10�<sup>6</sup> cm2

=сек: (25)

/s, and D3 = D<sup>α</sup>

Cr ≈ 1.38

(26)

(27)

The value of βi is calculated through the interfacial distribution coefficient of the alloying element Ki = Ni (K)/Ni (α) and the atomic fraction of carbon in the alloy Nc [15, 16]:

$$\beta \dot{\imath} = -\frac{(\text{Ki} - \mathbf{1}) + (\text{Nc}(K) - \text{KiNc}(a))}{(\text{Ki} - \mathbf{1})\text{Nc} + (\text{Nc}(K) - \text{KiNc}(a))}.\tag{19}$$

With a slight error for low-alloyed alloys, we can take Nc (K) = 0.25, Nc (α) ≈ 0.001—the carbon content in the undoped phases of steel, taken from the Fe-C state diagram.

Using the coefficient of chromium distribution between the α-phase and the carbide KCr, equal to 4, we find the equations for calculating the coefficients of influence βCr:

$$
\beta\_{Cr} = -3.246/(3,0\text{Nc} + 0.246),
$$

whence β<sup>α</sup> <sup>С</sup><sup>r</sup> <sup>=</sup> �12.16 and <sup>β</sup><sup>К</sup> <sup>С</sup><sup>r</sup> = �3.26. Then from expressions (16)–(18), one can find the values.

$$\ln \left( a^{\mathcal{K}}\_{\mathcal{C}} / a^{\mathcal{a}}\_{\mathcal{C}} \right) = -0.6085 + 0.652 = -0.0425 \text{ and } -\Delta\_{\mu\mathcal{C}} = 308.47 \text{ Joule.} \tag{21}$$

The work done in the diffusion of carbon from the α-phase to cementite is positive. For the diffusion of iron, it is not possible to calculate the difference of thermodynamic potentials, since the coefficient of iron activity in carbide is unknown. However, from the experimental data and the thermodynamics of the process, it is known that diffusion of carbon is the leading one, the diffusion of chromium accompanies the diffusion of carbon, and the diffusion of iron is forced, since it is directed toward increasing the concentration of iron.

With this in mind, we find the values of the kinetic coefficients Lii in the Onsager equations.

As is known [8, 13], the kinetic coefficients Lii are related to the diffusion coefficients Di by the relation:

$$L\_{ii} = C\_i D\_i / RT,\tag{22}$$

where C1 is the concentration of iron in the alloy (0.943), C2 is the concentration of carbon in the alloy (0.007), and C3 is the concentration of chromium in the alloy (0.05).

Dependences of the diffusion coefficients of chromium and carbon in doped chromium ferrite on the temperature have the form [14, 17]:

$$D\_C^a = 0, \mathbf{1}77 \exp\left[\frac{-88230}{RT}\right] c\omega^2 / ce\kappa,\tag{23}$$

$$D\_{Fe}^{a} = 2,910^{-4} \exp\left[\frac{-251000}{RT}\right] c\omega^2 / ce\kappa,\tag{24}$$

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

$$D\_{Cr}^{a} = 3,05\exp\left[\frac{-358000}{RT}\right]c\omega^2/ce\kappa.\tag{25}$$

At a temperature of 600° C:

We will assume that for our steel in the standard state а<sup>α</sup>

<sup>С</sup>=а<sup>α</sup> С <sup>¼</sup> <sup>β</sup><sup>К</sup>

ln а<sup>К</sup>

<sup>С</sup><sup>r</sup> <sup>=</sup> �12.16 and <sup>β</sup><sup>К</sup>

Eqs. (16) and (17), we find:

Non-Equilibrium Particle Dynamics

Nc [15, 16]:

Fe-C state diagram.

influence βCr:

whence β<sup>α</sup>

<sup>С</sup>=а<sup>α</sup> С

Onsager equations.

(0.05).

108

coefficients Di by the relation:

ln а<sup>К</sup>

unalloyed cementite in steel with 0.15%, C is stable and in equilibrium with the solid solution at a tempering temperature of 600°C [13]. Using this condition and

Cr N<sup>К</sup>

The value of βi is calculated through the interfacial distribution coefficient of the alloying element Ki = Ni (K)/Ni (α) and the atomic fraction of carbon in the alloy

<sup>β</sup><sup>i</sup> ¼ � ð Þþ Ki � <sup>1</sup> ð Þ Ncð Þ� <sup>К</sup> KiNcð Þ <sup>α</sup>

Nc (α) ≈ 0.001—the carbon content in the undoped phases of steel, taken from the

Using the coefficient of chromium distribution between the α-phase and the carbide KCr, equal to 4, we find the equations for calculating the coefficients of

βС<sup>r</sup> = �3.246/(3,0Nc + 0.246),

¼ �0:<sup>6085</sup> <sup>þ</sup> <sup>0</sup>:<sup>652</sup> ¼ �0:0425 and � <sup>Δ</sup>μ<sup>С</sup> <sup>¼</sup> <sup>308</sup>:47 Joule: (21)

The work done in the diffusion of carbon from the α-phase to cementite is positive. For the diffusion of iron, it is not possible to calculate the difference of thermodynamic potentials, since the coefficient of iron activity in carbide is unknown. However, from the experimental data and the thermodynamics of the process, it is known that diffusion of carbon is the leading one, the diffusion of chromium accompanies the diffusion of carbon, and the diffusion of iron is forced,

With this in mind, we find the values of the kinetic coefficients Lii in the

As is known [8, 13], the kinetic coefficients Lii are related to the diffusion

where C1 is the concentration of iron in the alloy (0.943), C2 is the concentration of carbon in the alloy (0.007), and C3 is the concentration of chromium in the alloy

> RT

RT 

см<sup>2</sup>

см<sup>2</sup>

Dependences of the diffusion coefficients of chromium and carbon in doped

<sup>С</sup> <sup>¼</sup> <sup>0</sup>, 177 exp �<sup>88230</sup>

Fe <sup>¼</sup> <sup>2</sup>, <sup>910</sup>�<sup>4</sup> exp �<sup>251000</sup>

<sup>С</sup><sup>r</sup> = �3.26.

Then from expressions (16)–(18), one can find the values.

since it is directed toward increasing the concentration of iron.

chromium ferrite on the temperature have the form [14, 17]:

Dα

Dα

With a slight error for low-alloyed alloys, we can take Nc (K) = 0.25,

Cr � <sup>β</sup><sup>α</sup>

Cr N<sup>α</sup>

ð Þ Ki � <sup>1</sup> Nc <sup>þ</sup> ð Þ Ncð Þ� <sup>К</sup> KiNcð Þ <sup>α</sup> : (19)

Lii ¼ СiDi=RT, (22)

=сек, (23)

=сек, (24)

<sup>С</sup><sup>0</sup> =аК

<sup>С</sup><sup>0</sup> = аС0, i.e.,

Cr (18)

D1 = D<sup>α</sup> Fe <sup>≈</sup> 3.03�10�<sup>19</sup> cm2 /s, D2 = D<sup>α</sup> <sup>C</sup> ≈ 1.02 10�<sup>6</sup> cm2 /s, and D3 = D<sup>α</sup> Cr ≈ 1.38 10�<sup>21</sup> cm2 /s.

Using relations (23)–(25) and (15), we find the values of the kinetic coefficients for our system:

<sup>L</sup><sup>11</sup> <sup>=</sup> 0.394 � <sup>10</sup>�22, L22 <sup>=</sup> 0.984 � <sup>10</sup>�13, L12 <sup>=</sup> �1.97 � <sup>10</sup>�17, L33 <sup>=</sup> 0.95 � <sup>10</sup>�26, L13 <sup>=</sup> �0.611 � <sup>10</sup>�24, and

L23 <sup>=</sup> 0.306 � <sup>10</sup>�19. Consequently, the system of Eqs. (6)–(8) takes the form:

$$J\_{Fe} = 0.394 \times 10^{-22} (-\Delta\mu\_{Fe}) - 1.97 \times 10^{-17} (-\Delta\mu\_C) - 0.611 \times 10^{-24} (-\Delta\mu\_{cr}),\tag{26}$$

$$J\_C = -1.97 \times 10^{-17} (-\Delta\mu\_{Fe}) + 0.984 \times 10^{-13} (-\Delta\mu\_C) + 0.306 \times 10^{-19} (-\Delta\mu\_{Gr}),\tag{27}$$

$$J\_{Cr} = -0.611 \times 10^{-24} (-\Delta\mu\_{Fe}) + 0.306 \times 10^{-19} (-\Delta\mu\_C) + 0.95 \times 10^{-26} (-\Delta\mu\_{Cr}).\tag{28}$$

It follows from Eqs. (26)–(28) that the values of iron and chromium fluxes increase substantially due to the cross-ratios L12 and L32 of a significant thermodynamic force (�ΔμC). The value of the carbon flux having a positive sign is determined mainly by the intrinsic coefficient L22. The thermodynamic forces of iron and chromium make an insignificant contribution to the fluxes, because of the small value of the kinetic coefficients and their influence can be neglected. Then, as direct calculations show:

$$J\_{Fe} = -6.08 \times 10^{-15}, J\_C = 3.04 \times 10^{-11}, J\_{Cr} = 0.94 \times 10^{-17}, \text{and} \\ J\_V = 6.07 \times 10^{-15} \text{ cm}^2/\text{s}. \\ \text{I}\_{N\_{2,2}} = 6.07 \times 10^{-15} \text{ cm}^2/\text{s}. \\ \text{I}\_{N\_{2,2}} = 1.01 \times 10^{-17}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and} \\ J\_{N\_{2,2}} = 1.01 \times 10^{-15}, \text{and}$$

It was established in [18] that during the tempering period, a certain amount of nanoparticles of special chromium carbide with a size of �100 nm can be formed in the steel, which were detected experimentally.
