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Application of diffusion and transition state theories on the carburizing of steel AISI 316 by annealing in uranium carbide powder

The steel specimens were tempered in contact with uranium carbide powder by sodium bonding at 500, 600, 700, and 800 °C for 1000h. Carburizing zone of the specimens was determined by measuring of microhardness which is taken as a kinetic variable instead of the corresponding carbon content. Arrhenius equation was determined for the diffusion of carbon atoms in the steel by using the solution of Fick's second law. Temperature dependency of the activation enthalpy, Gibbs energy, and entropy was calculated from the transition state theory by an assumption that the carburizing occurs over an activated complex. Kinetic and thermodynamic for formation an activated complex were discussed depending on the obtained numerical values.


Introduction
Fuels and their cladding are the main materials for nuclear reactors. Plutonium-uranium oxide, (Pu x U 1-x )O 2 , or plutonium-uranium carbide, (Pu x U 1-x )C, has been used to be nuclear fuel. Carbide fuel is obtained by the vacuum carbothermic reaction from oxide fuel [1]. Cladding for the fuel tips are made of usually austenitic stainless steels due to their excellent mechanical and physicochemical properties [2,3]. These properties such as corrosion resistance, hardness, ductility, elongation at rupture, wear resistance, fracture strength, fatigue strength, tensile strength, and yield strength change by the physicochemical interactions with the fuels at the working temperature of the reactors [4,5].
Chemical compatibility between carbide fuel and cladding is more than those oxide one. Namely, chemical effect of an oxide fuel and its fission products on cladding is a great extent, whereas a carbide fuel limited [6]. Nevertheless, preparation and using a carbide fuel are much more difficult than those oxide one.
Diffusion of carbon atoms into a steel from a carbonaceous medium in solid, liquid or gas phase is known as carburizing. Carbon content decreased from the surface to interior of the steel. The carburized zone and interior core in the steel remain integrally bonded. The effect of carburizing in several carbonaceous mediums on the mechanical and physicochemical properties of steels have been investigated [7,8,9,10].
Carburizing process have been clearly examined with out-of-pin annealing test. This test realized between cladding materials and fuels as well as their fission product is very important to prevent the most dangerous accidents would be arisen from the core of nuclear reactors. Since they have similar chemical properties, a uranium carbide powder has been used instead of the real carbide fuel by the test [11,12]. Distribution and depth of the carburized zone in the steel have been determined by the measurement of carbon content or microhardness [13,14,15,16]. Since the hardness changes depending on the carbon content in the steel, there is a direct correlation between the microhardness and carbon content profiles which was experimentally proved [12,13,14,15]. Microscopy also has been used to examine the carburizing [17,18,19]. Carbon content or microhardness profiles change depending on the annealing temperature and its time as well as the rate of the cooling process [20,21,22]. Metal-carbon nanostructures such as Cr 23 C 6 were found on the contact surface and carburized zone of the steel [18]. Several attempts have been made to mathematical evaluate of the profiles [23,24]. However, kinetic and thermodynamic explanations have not been adequate to understand the theoretical nature of the carburizing [20,21,22]. So, the objective and the novelty of the present study respectively are to examine the high temperature carburizing of a steel in contact with a uranium carbide powder starting from microhardness profiles and firstly using the combination of diffusion and transition state theories for the kinetic calculations.

Materials
Austenitic stainless steel AISI 316 and a uranium carbide powder were used as a cladding material and carbon medium, respectively [6].

Experimental
Cladding specimens were die-pressed from the steel received in sheetform with the thickness of 1mm. They were annealed in vacuum at 1050 C for 30 min to remove fin-grained surface zone by the preparation.
Carburizing of the specimens was realized by the out-of-pile annealing tests. Because of the uranium carbide powder is greatly pyrophoric, the preparation procedure was carried out under a pure argon atmosphere in a glove box.
Special prepared four steel capsules each having four parallel strip specimens were filled with the uranium carbide powder and then adding a sufficient amount liquid sodium tightly enclosed. The steel capsules taken out of the glove box were encapsulated again by evacuating silica capsules. Each capsule was heated in a muffle oven at various temperatures of 500, 600, 700, and 800 C for 1000h. The capsules stand to cooling in the oven up to room temperature and then the carburized specimens were taken out by cut of them. After completion of the elementary treatments the microhardness gradients from the contact surface to the interior of the specimens were measured at room temperature with a Leitz Durimet machine [6].

Kinetics for carburizing
Carburized zone of the steel specimens annealed at the different temperatures is seen from the microhardness profiles which are given in Fig. 1a, b, c, and d. These profiles revealed that the microhardness at the contact surface (h s ) and maximum depth (x max ) for the carburized zone greatly change depending on the annealing temperature. The h s and x max values which are taken from the profiles and given in Table 1.
Because the hardening is due to the diffusion of carbon atoms in the steel the microhardness would be taken as a kinetic variable instead of the corresponding carbon content. The shape of the microhardness pro- files appeared to verify the Fick's diffusion laws. Thus, the equation for the microhardness profiles and mean square depth, <x 2 > are respectively derived from the approximate solution of Fick's second law in the following forms [22,23,24]: where h s is the maximum microhardness, h o ¼ 220 HV is the minimum microhardness for the uncarburized steel, h is the microhardness at any depth in the carburized zone, H is defined to be relative microhardness, D is the diffusion coefficient which depends only on temperature, x is any depth of the carburized zone and t ¼ 1000h is the constant annealing time. These relationships are respectively known as parabolic and linear law equation for diffusion. The lnH against x 2 plots which are straight lines as seen in Fig. 2 indicated that the microhardness profiles obey Eq. (1). Diffusion coefficients were obtained from the slope of the straight line and listed in Table 1. The plot of the lnD against 1/T is a straight line as seen in Fig. 3 obeys the Arrhenius equation in the following form, where D 0 is called to be a frequency factor or pre-exponential constant, E # is the energy of activation, T is the absolute carburizing temperature, and R ¼ 8.314 J mol À1 K À1 is the gas constant. The values of E # ¼ 34910 J mol À1 and D 0 ¼ 6.66 Â 10 À14 m 2 s À1 were respectively calculated from the slope and intercept of the straight line in Fig. 3. Accordingly, temperature dependency of the diffusion coefficient can be written in exponential form given as follows: The root mean square depth, <x 2 > 1/2 value of the each temperature was calculated from Eq. (2) and listed Table 1. Their temperature dependence is represented in Fig. 4. The < x 2 > 1/2 is much lower than the corresponding x max . The functional increase of the x max and <x 2 > 1/2 show that the carburized zone extended to inside steel by increasing the annealing temperature.
Several equations to get along with each others in the experimental errors reported by many authors for the carburizing of similar alloys. For example, theoretical carbon content profiles for the carburizing of a steel at a temperature for different times were obtained by using a mathematical model derived from the solution of Fick's second law and compered experimental ones [21]. Carbon content as well as microhardness profiles which are determined depending on the temperature at a constant time or vice versa have been used to evaluate the kinetic parameters as mentioned above [18,20]. The carburized layer thickness, x, has been used instead of the root mean square distance, <x 2 > 1/2 , to calculate the diffusion coefficient from Eq. (2) in several studies [23]. Also, layer thickness is directly used in Arrhenius equation instead of the diffusion coefficient [18,20,25].

Activation thermodynamics for the carburizing
Transition state theory (TST) was applied to the carburizing by supposing the Cr 23 C 6 nanostructure acts as an activated complex (AC). According to TST a chemical equilibrium exists between the Cr 23 C 6 and their atoms in the following schematic form: The forward decomposition of the activated complex given in Eq. (5) causes the diffusion of carbon atoms into the steels. In contrary, during the backward decomposition forms uranium carbide powder again. The forward and backward rates depend on temperature and its change with time. The forward rates would be taken as the carburizing rate.
All of the thermodynamic relations are valid for the equilibrium between reactant and activated complex not fully carburized. Eyring equation which is derived from the TST for this equilibrium would be written in the following form [22]: where k B ¼ 1.38 Â 10 À23 J K À1 is the Boltzmann constant, h ¼ 6.62 Â 10 À34 J s is the Planck constant, and K # is the equilibrium constant for the formation reaction of the activated complex. The K # value of the each temperature is calculated and listed in Table 1.
The temperature dependence of the activation enthalpy (ΔH # ) can be evaluated by inserting Arrhenius (dlnD/dT ¼ E # /RT 2 ) and van't Hoff (dlnK # /dT ¼ ΔH # /RT 2 ) relations in the temperature derivative of the logarithmic Eyring equation in the following form: According to this equation ΔH # >0 between 500-800 C and the Table 1 The maximum microhardness (h s ), maximum distance from the contact surface (x max ), diffusion coefficient (D), root mean square distance (<x 2 > 1/2 ), and the equilibrium constant (K # ) for the carburizing at different temperatures.  formation of the AC is endothermic. Change in heat capacity (ΔC p # ) by the formation of the AC is found by using Kirchhoff's equation as follows: This negative value indicated that the total vibration modes of atoms decreased during the formation of the activated complex.
Variation of the lnK # with the carburizing temperature between 500 and 800 C was calculated from the integration of the van't Hoff relation in the following form: where -54 is an integral constant that is the arithmetic mean of the values calculated for each temperature by using the corresponding K # values. Variation of the Gibbs free energy (ΔG # ) and entropy (ΔS # ) with the temperature were respectively calculated using the basic thermodynamic equations in the following forms: ΔS # ¼ À dΔG # =dT ¼ À440:6 À 8:314 lnT As a result, the positive value of ΔG # and the negative value of ΔS # increase with the increasing of the carburizing temperature between 500 and 800 C. However, the spontaneous diffusion of carbon atoms into steel reveals that the total Gibbs energy (ΔG) could be negative by the carburizing.
The positive value of the ΔG # indicated that the activated complex does not form spontaneously. But it is carried out mandatory by given continuous energy from surrounding as similar as the formation of the ozone layer in the atmosphere. Also, the negative value of ΔC p # and ΔS # is verified the formation of Cr 23 C 6 nanostructure as an activated complex [7]. Because of decrease in the number of vibration modes and disorder during the formation an activated complex by the bonding of many atoms causes the lowering in the heat capacity and entropy, respectively. The h s against T plot has a maximum as seen in Fig. 5. The change in the kinetic parameters in an opposite way before and after 600 C would be the reason of the maximum. This maximum would be originated from the increasing of the forward decomposition rate of the activated complex before 600 C and after the backward rate.

Conclusions
Fick's second law and transition state theory were firstly applied to calculate kinetic parameters for high temperature carburizing of a steel in a uranium carbide powder. Microhardness profile for the carburized zones was used in the calculations instead of the carbon content profile. Fig. 4. The plot of the maximum distance (x max ) and root mean square distance (<x 2 > 1/2 ) from the contact surface against the carburizing temperature (T).  The kinetic and thermodynamic results indicated that the transfer of carbon atom from a solid phase to the steel would be over an activated complex. The proposed calculation method would be used also to examine the carburizing of the other alloys and pure metals by tempering in contact with an inorganic and organic carbon medium.