Abstract
In this paper, the reactive diffusion in Ni80C20r-Ti ternary system is discussed at 1173K. The diffusion couple was prepared and annealed for 100 h. The two intermetallic phases and two two-phase zones occurred, namely: Ti2Ni, TiNi, TiNi+Cr, and TiNi3+Cr. Based on the experimental results (molar fractions, thicknesses of the intermetallic phases), the intrinsic diffusion coefficients of all components in each phase were numerically approximated. The Wagner method was used in the pure intermetallic phases. In the two phase zones the approximation was based on the generalized Darken and Wagner methods. The presented methods allowed for determination of the effective diffusion coefficients in each presented phase.
Keywords: Reaction diffusion; Equilibrium; Ni80Cr20-Ti system
Apstrakt
U ovom radu je razmatrana reaktivna difuzija u Ni80C20r-Ti trojnom sistemu na temperaturi od 1173K. Difuzioni par je pripremljen, a zatim je narednih 100 časova podvrgnut žarenju. Pojavile su se dve intermetalične faze, kao i dve dvofazne zone Ti2Ni, TiNi, TiNi+Cr i TiNi3+Cr. Na osnovu rezultata eksperimenata (molarna frakcija i debljina intermetaličnih faza), približno su numerički određeni koeficijenti unutrašnje difuzije za sve komponente u svakoj fazi. Vagnerova metoda je korišćena kod čistih intermetaličnih faza. Kod dvofaznih zona su za aproksimaciju korišćene uopštene Darkenova I Vagnerova metoda. Predstavljene metode su omogućile određivanje efikasnih koeficijenata difuzije za svaku prikazanu fazu.
Ključne reči: Reakciona difuzija; Ravnoteža; Ni80Cr20-Ti sistem
1. Introduction
Diffusion coupling technique is one of the most often used methods of examination of the diffusion phenomena. This method can be used in studying the phase relations (phase diagrams determination) in multicomponent systems and is based on the assumption of local equilibrium in the diffusion zone. The experimental results, obtained on annealed diffusion couple in the form of the concentration profile, help in determination of the diffusion path, which is a mapping of the stationary concentrations onto the isothermal section of the equilibrium phase diagram [1]. The diffusion path is a common way of presenting the sequence of the layers growth and their compositions on the phase diagram. The experimental determination of the diffusion (reaction) path during the diffusion coupling experiments allows for determination of many phenomena, such as the order of the product layers, their morphology and compositions, however, the information about their spatial distribution is lost. The diffusion path connects initial compositions of the diffusion couple and can go across the single-, two-, and three-phase fields (in ternary systems). It starts at the composition of one alloy and ends at the other [1]. Kirkaldy and Brown [2], in one of their seventeen theorems, show the possibility of mapping the concentration profiles (diffusion path) onto the ternary isotherm.
One of the most important kinetic data in high temperature alloys characterization is the diffusion coefficient. The diffusion coefficient describes the speed of atom interaction - entropy and enthalpy in the alloy. There are some experimental methods allowing for determination of these material constants, however, they are limited to the binary [3] or ternary alloys [4,5]. Such methods are based on insertion of the markers between the alloys that create the diffusion couple. Then, based on the position of the markers after the annealing, the diffusion coefficient can be determined [6]. Such experimental approach is time- and cost-consuming, and that is why the numerical methods are presently widely used for determination of the diffusivity data. One of the most known methods of this kind, based on the experimental results after annealing of the binary diffusion couple, is Boltzmann-Matano analysis [7,8], further generalized by Sauer-Freise by introduction of different molar volumes [9]. However, such approach is limited to the binary one-phase systems. There were some attempts to modify the Boltzmann-Matano method into ternary systems [10], but these modifications allow for determination of the intrinsic diffusion coefficients only in multicomponent one-phase systems. None of the above mentioned approaches could determine the diffusion coefficient in multi-phase alloys, which is essential in many applications as the diffusion coefficients help to understand the mechanical properties of heat-treated materials, i.e. stress induced diffusion [11] or phase transformation [12].
There were many theoretical as well as experimental studies describing the phase diagrams for the Ni-Cr-Ti system [13-15]. Knowledge of phase equilibrium in such a system is of great importance as addition of Ti into Ni-Cr alloys improves their high-temperature corrosion resistance and mechanical properties [16-17]. However, those properties depend strongly on the order and morphology of the intermetallic phases that form in the interdiffusion zone which in turn depend on initial alloy compositions.
In this paper, the annealing process of the Ni80Cr20-Ti diffusion couple at 1173K for 100h was conducted. The alloy composition was chosen to check what intermetallic phases will form (one and two phase regions) during annealing under these conditions - such data is lacking in the literature. The experimental results will be used to verify if it is possible to determine the diffusion coefficients of particular elements (Ni, Cr, and Ti) in each phase of the multicomponent ternary systemby the proposed approach.The procedure of determination of diffusion coefficients is based on the Wagner approximation [19] and generalization of the Darken method into the two-phase systems [20]. Ultimately, the diffusion coefficients in the whole system were approximated.
2. Experimental
In this paper, two materials were connected to create a diffusion couple, namely Ni80Cr20 (Ni - 80 at. %, Cr - 20 at. %) alloy and Ti (99.9% purity from Sigma Aldrich supplier). Before diffusion annealing, the samples were cut to the rectangular shapes of the dimensions 20 x 10 x 1 mm. The surface of the samples were further ground and polished up till 1 μm grain size of SiO2 particles in a polishing suspension to obtain mirror-like surface finishing. The diffusion couple was inserted into the molybdenum holder (Figure 1) and placed into the glass tube of the Carbolite STF 16/450 tube furnace. Initially the holder with the samples was flushed with high purity argon (5.0 purity) in the cold zone of the furnace to extract the ambient atmosphere. After the flushing time, the holder was moved to the hot zone of the furnace. The proper heat treatment experiment was performed at 1173K for 100h also in an argon atmosphere to prevent the oxidation of the diffusion couple. After annealing and standard preparation of metallographic cross-sections, the diffusion couple was examined using scanning electron microscopy (SEM) Hitachi S3400 N coupled with energy dispersive spectrometer (EDS). Images were taken in back-scattered electron (BSE) mode, which enabled the increase of the contrast between the phases. Concentration profiles of diffusion couple's cross-section were measured using EDS and the qualitative analysis of the elements that occurred in the diffusion zone was made using EDX (energy dispersive X-Ray spectroscopy). Thicknesses of identified phases were determined by means of NIS-Elements software based on SEM/BSE images.
3. Results and discussion 3.1. Experimental results
The SEM/BSE image of diffusion zone that was created between pure titanium and Ni80Cr20 alloy after annealing at 1173 K for 100 hours is presented in Figure 2. The SEM/EDX analysis was performed (Figure 3, Table 1) and enabled the identification of the phases that were distinguished and named in Figure 2. During the diffusion, the Ti2Ni and TiNi phases (points 2, 3 and 4, 6 in Figure 3, Table 1, respectively) were generated as well as two-phase zones: TiNi+Cr and TiNi+Cr and TiNi3+Cr. The formation of such phases wasvisible in the cross-section: TiNi created a brighter matrix (point 8 in Figure 3, Table 1) within which a darker chromium precipitates (point 7 in Figure 3, Table 1) were present. Similarly, dark chromium spots (point 10 in Figure 3, Table 1) were distinguishable from a brighter TiNi3 matrix (point 9 in Figure 3, Table 1).The SEM/BSE image revealed also, that there were many pores and cracks visible within the Ti2Ni phase (Figure 2). It is probably an effect of the cross-section preparation, mainly grinding steps - TiNi2 was brittle and therefore, it was very likely that some of it broke away while preparation.
The concentration profile of the elements as a function of distance, following the direction of the line-scan marked by a white arrow in Figure 2, is shown in Figure 4. The concentration profile may be also drawn as a ternary diagram where concentration of chromium, nickel, and titanium are x, y, and z-axis, respectively. Diffusion path, obtained by placing the mentioned ternary form of concentration profile onto isothermal section of the equilibrium phase diagram of Ni-Cr-Ti system at 1123K is presented in Figure 5. Based on such a diagram, an occurrence of TiNi2 and TiNi phases, that were observed on SEM/BSE images (Figure 2 and 3a), may be confirmed. Moreover, as it is visible in Figure 5, diffusion path crossed the conodes in two points, which resulted in generation of two two-phase regions, namely TiNi+Cr and TiNi3+Cr - such observation is also in a good agreement with metallographic observations (Figure 2 and 3b).
All the presented results allow for determination of the phases which were created after annealing of the diffusion couple at 1173K for 100 h. Summarizing, diffusion path between pure Ti and Ni80Cr20 alloy in the given conditions went through the following phases: Ti → TiNi2 → TiNi → TiNi + Cr → TiNi3+Cr → Ni80Cr20.
The phases mentioned above possessed different thickness. It was clearly visible that TiNi intermetallic phase was characterized by greater thickness than TiNi2 intermetallic phase. Also, the two-phase zone TiNi + Cr was substantially thicker than TiNi3 + Cr zone. The exact values of a thickness ?xj of the distinguished phases - measured based on Figure 2, are collected in Table 2. The molar fractions of the elements in the phases as well as molar volume of the phases are also presented in Table 2.
3.2. Diffusion coefficients approximation
Determination of diffusion path, sequence of growth, and thickness of appearing phases in ternary systems during diffusion process is still an unsolved problem. One of the greatest obstacles that restrain the progress in understanding of diffusion phenomena in such systems is lacking of kinetic data, namely diffusion coefficients of the elements in particular phases and systems under specific temperature conditions. The experimental diffusion path observed after annealing of the Ni80Cr20-Ti diffusion couple for 100h at 1173K (Figure 5) revealed which phases and regions were formed in the diffusion zone. Therefore, the diffusion coefficients ( D i j) of the i-th element (i= Ni, Cr, Ti) in each phase (j) may be calculated - initially only the diffusion coefficients for the high purity Ti ( [21]) were known (Table 3).
3.2.1. Wagner's diffusion coefficients
One of presently available ways of calculation of diffusion coefficients in multiphase, multicomponent systems was proposed by Wagner. The formula for calculation of intrinsic diffusion coefficients had been presented as follows [12]:
where: N±i is the molar fraction of the element ion the left hand-side (-) and right-hand side (+) of the diffusion couple, Nf - molar fraction of the i-th element in the /J-phase^and V m j - thickness and molar volume of the j-th phase (j = /? or v where /? -the phase of interest and v- currently calculated phase), respectively, t - time of diffusion annealing.
The Wagner diffusion coefficients rely strongly on the neighboring phases, namely on concentration of particular elements and thicknesses of the phases. The above mentioned experimental data (phase thickness and molar fraction of elements, time of annealing) as well as molar volume of all the phases have to be known to apply Equation 1. The required data for calculation of diffusion coefficients for Ni80Cr20-Ti diffusion couple after annealing at 1173K for 100 hours are presented in Table 3. Results of calculations of intrinsic diffusion coefficients by Wagner method (Equation 1) are gathered in Table 4.
3.2.2. Determination of diffusion coefficients in the two-phase region
In the previous section of the paper, the two-phase regions, namely TiNi+Cr and TiNi3+Cr were treated as regular intermetallic phases, thus the Wagner formula could be applied for calculation of diffusion coefficients. However, the Wagner formula enables for calculation of diffusion coefficients only in the phases that appear in the diffusion zone while annealing - it is impossible to calculate diffusion coefficients in the phases that are missed by diffusion path. Therefore, even if the diffusion coefficient in the TiNi3+Cr has been calculated, the diffusion coefficient of the TiNi3 phase is still missing.
Diffusion coefficients, that are presented in Table 4 for these phases are effective diffusivities that are influenced by diffusion coefficients of phases that appear in the two-phase regions, molar fraction of the phases and the gradient of component's concentration over displacement. Therefore, for the two-phase region TiNi3+Cr, the equation describing the flux of the i-th component may be formulated as follows:
where: f(TiNi3) and f(Cr)are the molar fractions of the TiNi3 and Cr phases in the two-region zone and Di (TiNi3) and Di(Cr) are diffusion coefficients of i-th component in the TiNi3 and Cr phase, respectively.
A problem arises in determination of the gradient of component's concentration over displacement, as only one point is visible on diffusion path between the TiNi3 and Cr phases. Nevertheless, from the properties of derivative, the above mentioned gradient may be rewritten in the following form:
where N i is the average molar fraction in the two-phase region.
For the NiCrTi system, when the conodes between the TiNi3 and Cr phases are almost parallel and restrain a thin two-phase region, the first derivative may be simplified to:
Thus, Equation 2 may be rewritten in the following form:
and the equation for effective diffusion coefficient can be approximated as:
Finally, the diffusion coefficients in TiNi3 can be calculated as:
Molar fractions of components in particular phases that appear in Equation 7 have been depicted on schematic representation of Ni80Cr20-Ti diffusion path after annealing at 1173K in Figure 6.
Calculation of diffusion coefficient in the TiNi3 phase by Equation 7 is possible, when diffusion coefficients in the Cr phase are known. To calculate their values the analogous approximation can be used in the TiNi+Cr two-phase zone, as diffusion coefficients in the TiNi phase had been previously calculated by the Wagner method (Equation 1, Table 4). Thus, the Cr intrinsic diffusion coefficient and the impurities diffusion coefficients in Cr can be approximated as:
Results of calculation of Di(Cr) by Equation 8 are used for calculation of Di(TiNi3) by Equation 7. The required data for calculation of above mentioned diffusion coefficients in the two-phase zone after annealing at 1173K for 100 hours are presented in Table 5. Results of calculations are gathered in Table 5.
3.2.3. Diffusion coefficient of pure components inNi80Cr20 alloy
Lack of diffusion coefficients in the literature is not the only obstacle that inhibits description of diffusion phenomena. Problem of determination of ternary interdiffusion coefficients may be solved based on thermodynamic models, which present interdiffusion coefficients depending on binary and ternary interaction parameters. Therefore, for ternary system, one obtains main and cross interdiffusion coefficients for dependent components. The Wagner method enables for calculation of intrinsic diffusion coefficients of particular elements in particular phases, thus it is an effective diffusivity. Therefore, it is impossible to directly compare the values of diffusion coefficients obtained in such different ways.
So even though the Ni-Cr-Ti ternary system, especially in its fcc phase, was already compressively studied by Huang et al. [15] it is impossible to use the Huang's method for calculation of the diffusion coefficients of particular elements in the alloy what is the subject of this study as the Huang's method is based on thermodynamical data (namely gradient of chemical potential), which is lacking in the literature.
Nevertheless, these diffusion coefficients, namely diffusion coefficients of elements in Ni80Cr20 alloy may be estimated thanks to their correspondence to atomic mobility:
where R - gas constant and T - absolute temperature. Atomic mobility Mimay be further defined as:
where Fi is a property that depends on the composition and temperature. It may be calculated thanks to Redlich-Kister polynomials:
where F pi denotes the value of Fi for i-th element in p-th phase,rFp,qi and sFp,q,vi are parameters of interaction for binary and ternary systems, respectively. In case of calculation of diffusion coefficients of pure components in Ni80Cr20 system, Equation 11 may be simplified only to the first addend. Values of Fi were calculated by equations given by Zhu et al. [22], which are presented in Table 7.
Results of calculations of parameter F and atomic mobilitiesMi of pure components along with molar fractions with these components for Ni80Cr20 alloyat 1173K are presented in Table 8.
Based on the data collected in Table 8, it is finally possible to calculate diffusion coefficients of Ti, Ni, and Cr in Ni80Cr20 alloy at 1173K by Equation 9. Results are presented in Table 9.
4. Conclusions
In the present paper, an attempt to determine the diffusion coefficients in Ni80Cr20-Ti had been made based on the experimental results of annealing of such a diffusion couple at 1173K for 100h. Diffusion coefficients were calculated by various possible methods, mainly by the Wagner formula (TiNi2, TiNi, TiNi+Cr, TiNi3+Cr), separation of effective diffusivity in the two-phase region (TiNi3), and thermodynamic data (Ni80Cr20). Summary of the obtained results is presented in Table 10.
It was shown, that a single diffusion experiment enabled the determination of all the diffusion coefficients in the ternary Ni80Cr20 system at 1173K, however, three different methods of calculations had to be used, as each of them had its own limitations. An experimental study had to be performed, because the kinetic information, namely the thickness of the phases that occurred in the diffusion zone, was needed for implementation of the Wagner formula. However, once calculated, the results, summarized in Table 10, may be further applied in determination of the phases and their thicknesses that appear while annealing of the ternary Ti-Ni-Cr diffusion couple at 1173 K.
Acknowledgments
This work has been supported by the National Science Centre (NCN) in Poland, decision number 2014/15/B/ST8/00120. The authors declare no conflict of interest.
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Abstract
In this paper, the reactive diffusion in Ni80C20r-Ti ternary system is discussed at 1173K. The diffusion couple was prepared and annealed for 100 h. The two intermetallic phases and two two-phase zones occurred, namely: Ti2Ni, TiNi, TiNi+Cr, and TiNi3+Cr. Based on the experimental results (molar fractions, thicknesses of the intermetallic phases), the intrinsic diffusion coefficients of all components in each phase were numerically approximated. The Wagner method was used in the pure intermetallic phases. In the two phase zones the approximation was based on the generalized Darken and Wagner methods. The presented methods allowed for determination of the effective diffusion coefficients in each presented phase.