Kinetics and Mechanical Characterization of Hard Layers Obtained by Boron Diffusion in 80/20 Nickel–Chromium Alloy

: This study examines the formation of hard layers containing Ni-B and Cr-B on the surface of 80/20 nickel–chromium alloy. The work evaluates the mechanical properties of the boride layers using instrumented nanoindentation. In addition, the growth kinetics of the coatings were assessed by applying a kinetic model that relates the layer thickness with the experimental parameters of temperature and treatment time. First, the boride layers were achieved using the powder-pack boriding process in a conventional furnace. The treatment time was set at 2, 4, and 6 h at temperatures of 900, 950, and 975 ◦ C, respectively. The microstructure of the layers was analyzed by X-ray diffraction. The thickness of the layers showed a closed correlation with the experimental parameters of time and temperature, and was established between 38.97 and 156.49 µ m for 2 h to 900 ◦ C and for 6 h to 975 ◦ C, respectively. The hardness and Young’s modulus values agree with those presented in the literature for boriding nickel alloys, being in the range of 1.3 GPa on average and 240 to 270 GPa, respectively. The resulting layers exhibited a characteristic diffusion zone where the hardness values decrease gradually without the typical high hardness gradient observed on borided steels.


Introduction
The nickel-chromium system reveals that chromium is quite soluble in nickel. It has a maximum soluble rate of 47% at the eutectic temperature, decreasing to nearly 30% at room temperature [1]. This characteristic enables nickel-chromium alloys to be chemically stable at high temperatures and suitable for working at high temperatures. Nickel-chromium alloys show corrosion resistance that could be due to the passive layer on the system that is mainly composed of chromium oxy/hydroxide-like stainless steel [2]. This metal is usually used as a thermal element in furnace manufacturing due to its superb stability under high temperatures. Similarly, these non-ferrous alloys have a large variety of uses, such as in: aircraft gas turbines, steam turbines, medical applications, and equipment parts in the chemical industry [3]. These alloys have become so important in the industry that some of them have commercial names, such as Inconel ® and Hastelloy ® . Nevertheless, due to their high nickel content, nickel-chromium (Ni-Cr) alloys show poor wear resistance, which is their main disadvantage and the reason for their limited use [4]. Specifically, 80/20 nickel-chromium alloy is well known for its excellent mechanical properties and corrosion resistance [5,6]. The 80/20 nickel-chromium alloy is frequently used for wrought and cast parts for high-temperature applications, as it has better oxidation and hot corrosion resistance than other alloys [7]. In that sense, several researchers have studied how to The boriding process was carried out at three different temperatures (900, 950, and 975 • C) for 2, 4, and 6 h each, under atmospheric air conditions. After boriding, the samples were cooled to room temperature inside the furnace to reduce the risk of thermal crashes and the subsequent fracture of the layers [14,15,18]. Standard metallographic techniques were used to prepare the samples for microscopic examination. The thickness of the boride layers was measured by optical examination with a GX-51 optical microscope (Olympus, Center Valley, PA, USA). At least 50 measurements were performed to establish a mean value of the layer thickness.

Kinetics of Growth
It is well known that boron diffusion controls the kinetics of the growth of boride layers, so the development of layers occurs as a consequence of boron diffusion in a perpendicular direction to the surface of the samples [18][19][20][21][22][23]. Additionally, the boron concentration in the boride phases is established by the second Fick's law (Equation (1)): The second Fick's law predicts how a specimen's concentration changes as a function of time due to its diffusion in a specific medium. The second Fick's law is a partial differential equation expressing the mass's conservation during the diffusion process.
The deduction of the second Fick's law is always complex. Nevertheless, a particular solution can be expressed as: Then, extracting the layer thickness from the equation, it can be expressed as: It is essential to point out that Equation (3) considers a particular case where the boron concentration profile on the boride layer is of a linear function, as shown in Figure 1. where is the boron concentration on the surface of the layer (mol m −2 ), ( , ) rep the boron concentration at a distance ( ) in time ( ) and 0 (mol m −2 ) denotes the concentration in the substrate. On the other hand, considering that, for distance any time of treatment  where C s is the boron concentration on the surface of the layer (mol m −2 ), C (x,t) represents the boron concentration at a distance (x) in time (t) and C 0 (mol m −2 ) denotes the boron concentration in the substrate. On the other hand, considering that, for distance (x) at any time of treatment (t), the relationship between the boron concentration and the diffusion coefficient tends to be constant (as shown in Figure 1), Equation (3) can be rewritten as: Equation (4) indicates that the boride layer's growth obeys a parabolic law [23,24] where (x) is the thickness of the boride layers (m), (K) represents the constant of parabolic growth (m 2 /s), and (t) stands for the treatment time (s).
(K) can be estimated from the slope of the graph layer thickness squared (x 2 ) versus treatment time. The relationship between the constant of parabolic growth (K), temperature (T), and activation energy (Q) can be expressed by an Arrhenius-type equation as follows: where (K 0 ) is called the pre-exponential constant, (Q) denotes the activation energy required to make the reaction occur (J mol −1 ), and (T) refers to the absolute temperature (Kelvin). In addition, (R) is the constant of ideal gases (8.3144 J mol −1 K −1 ). The activation energy can be estimated by plotting Equation (5) in a logarithm form as follows:

Characterization
The thicknesses of the boride layers were measured using the methodology described in Figure 2. At least 100 measurements were realized in 10 different zones of the boride samples.  The hardness and Young's modulus of the boride layers were evaluated by instrumented indentation with a nanohardness tester (TTX-NHT, CSM Instruments, Needham, MA, USA) using a Berkovich indenter, according to the methodology established by Oliver and Pharr [21]. The hardness profiles were realized at each 25 µm from the surface to the substrate, 10 indentations with a constant indentation load of 250 mN each The instrumented indentation technique is based on the curve load deformation generated during the test (Figure 3). The hardness and Young's modulus of the boride layers were evaluated by instrumented indentation with a nanohardness tester (TTX-NHT, CSM Instruments, Needham, MA, USA) using a Berkovich indenter, according to the methodology established by Oliver and Pharr [21]. The hardness profiles were realized at each 25 µm from the surface to the substrate, 10 indentations with a constant indentation load of 250 mN each The instrumented indentation technique is based on the curve load deformation generated during the test (Figure 3). mented indentation with a nanohardness tester (TTX-NHT, CSM Instruments, Needham, MA, USA) using a Berkovich indenter, according to the methodology established by Oliver and Pharr [21]. The hardness profiles were realized at each 25 µm from the surface to the substrate, 10 indentations with a constant indentation load of 250 mN each The instrumented indentation technique is based on the curve load deformation generated during the test (Figure 3).

Figure 3.
Schematic indentation curve obtained using a Berkovich nanoindenter: a is the curve of application of load F; b is the curve of removal of load F; c is the tangent to curve b at Fmax; d is the dwell period to Fmax, F is the test load; Fmax is the maximal test load; hp is the permanent indentation depth; hr is the tangent indentation depth; hc is the contact depth of the indenter with the sample at Fmax; hmax is the maximum indentation depth; S is the contact stiffness; and ε is a geometric constant related to the shape of the indenter, according to the methodology established by Oliver and Pharr [21].
In the nanohardness test, the hardness values were obtained by measuring the depth of the indentation where the indenter is in contact with the material. This is because, at this level of load, the material experiences an elastic recovery, so the permanent indentation print does not indicate the real penetration of the indenter. Figure 4 shows a schematic of the measurement of the contact depth during the indentation test. . Schematic indentation curve obtained using a Berkovich nanoindenter: a is the curve of application of load F; b is the curve of removal of load F; c is the tangent to curve b at F max ; d is the dwell period to F max , F is the test load; F max is the maximal test load; h p is the permanent indentation depth; h r is the tangent indentation depth; h c is the contact depth of the indenter with the sample at F max ; h max is the maximum indentation depth; S is the contact stiffness; and ε is a geometric constant related to the shape of the indenter, according to the methodology established by Oliver and Pharr [21].
In the nanohardness test, the hardness values were obtained by measuring the depth of the indentation where the indenter is in contact with the material. This is because, at this level of load, the material experiences an elastic recovery, so the permanent indentation print does not indicate the real penetration of the indenter. Figure 4 shows a schematic of the measurement of the contact depth during the indentation test. The resulting load-displacement response typically shows an elastic-plastic loading followed by elastic unloading (see Figure 3). The elastic equations of contact are then used in conjunction with the unloading data to determine the Young's modulus and hardness of the specimen material as follows: where ( ) is the hardness of the specimen, max refers to the maximum applied load, stands for the contact area at peak load (24.49 ℎ 2 ), ℎ denotes the experimentally meas- The resulting load-displacement response typically shows an elastic-plastic loading followed by elastic unloading (see Figure 3).
The elastic equations of contact are then used in conjunction with the unloading data to determine the Young's modulus and hardness of the specimen material as follows: where (H) is the hardness of the specimen, F max refers to the maximum applied load, A c stands for the contact area at peak load (24.49 h 2 c ), h c denotes the experimentally measured contact indentation depth, 24.49 is a constant related to the geometry of the indenter, E is the Young's modulus, υ s represents the Poisson's ratio of the sample (0.3), υ i stands for the Poisson's ratio of the indenter (0.07 for diamond), E i denotes the Young's modulus of the indenter (1141 GPa), E r represents a reduced modulus of the indentation contact, and S is the stiffness of the sample [25].

Microstructure
SEM examination (JSM-6360LV, JEOL, JEOL Ltd., Akishima, Japan) of a cross-section of the borided samples revealed the presence of three zones of interest ( Figure 5). The outermost is assumed to be a layer containing Ni-B and Cr-B compounds with flat morphology, similar to borided stainless steel [15,21]. The second is a large diffusion zone and the substrate, which is not affected by the diffusion process.  As shown in Figure 5, the thickness of the layers depends not only on the temperature, but also on the treatment time (see Table 2). The layers with the lowest thickness are those exposed to the lowest temperature for the shortest time (900 °C and 2 h). Therefore, once the temperature necessary for boron mobility is reached, the layer starts to grow and continues growing during the process [24]. This behavior confirms the assumption that boriding is a thermally activated process.
The flat morphology of the boride layers (similar to those obtained in stainless steel) can be explained because of the high Cr content in the alloy. It has been shown that the high contents of Cr and Ni in the alloys exposed to boriding tend to act as a diffusion barrier, limiting the growth of the layers and consuming high amounts of energy during the process [15]. As shown in Figure 5, the thickness of the layers depends not only on the temperature, but also on the treatment time (see Table 2). The layers with the lowest thickness are those exposed to the lowest temperature for the shortest time (900 • C and 2 h). Therefore, once the temperature necessary for boron mobility is reached, the layer starts to grow and continues growing during the process [24]. This behavior confirms the assumption that boriding is a thermally activated process. The flat morphology of the boride layers (similar to those obtained in stainless steel) can be explained because of the high Cr content in the alloy. It has been shown that the high contents of Cr and Ni in the alloys exposed to boriding tend to act as a diffusion barrier, limiting the growth of the layers and consuming high amounts of energy during the process [15].
On the other hand, the thickness of the layers obtained on the alloys containing Ni-Cr will be lower than those obtained in low-alloyed steel under the same treatment conditions. Similar results were reported by Campos et al. [26], even when they used a different technique for the boriding process to accelerate the growth of the layers.
A dendritic phase can be observed in the microstructure (Figure 5e), which is assumed to be chromium borides (CrB and Cr 2 B), while the brighter phase corresponds to NiB, Ni 2 B, and Ni 4 B 3 [23]. Interesting results were obtained from the elementary analysis applied to the dendritic and to the brighter phases (Figure 5e). The boron content in the dendritic phase was near to 50% (atomic content), which means that the main structure in this phase is CrB with a 50/50% atomic composition. The low content of Ni indicates that Ni was displaced with the Cr-B structures. Additionally, when the brighter phases were analyzed, the boron content matched better with the Ni 3 B structure, and the Cr content was reduced due to the Ni-B compound formation.
The XRD analysis corroborated the composition of the boride layers, as shown in Figure 6. this phase is CrB with a 50/50% atomic composition. The low content of Ni indicates tha Ni was displaced with the Cr-B structures. Additionally, when the brighter phases wer analyzed, the boron content matched better with the Ni3B structure, and the Cr conten was reduced due to the Ni-B compound formation.
The XRD analysis corroborated the composition of the boride layers, as shown i Figure 6.  The XRD analysis (Figure 6a) revealed the presence of Ni-B and Cr-B characteristic peaks. Indications of NiB, Ni 4 B 3 , Ni 2 B, Ni 3 B, CrB, and Cr 2 B were evidenced. Interestingly, even though the alloy contains only 0.5% by weight of iron (Table 1), some indications of iron boride type Fe 2 B can be observed in the XRD analysis. The presence of iron borides in the boride layer indicates the affinity of iron to boron. According to the XRD analysis (Figure 6b), the layer is mainly compounded by Ni 4 B 3 and Ni 3 B. The results indicate how the boron concentration decreases through the boride layer until it reaches the diffusion zone and finally approaches zero. This behavior confirms the assumption that the concentration profile on the boride layer is of a linear function (Figure 1). The formation of this type of boron compound enhances the surface properties of the treated material, such as the hardness, Young's modulus, corrosion resistance, and wear resistance [8,10,23]. A probable explanation for the improvement in the mechanical properties could be the combination of Ni-B and Cr-B, which have a hardness of approximately 1300 to 2400 HV [23].

Kinetics of Growth
The mean values of the layer thickness are depicted in Table 2 and shown in Figure 7.

Kinetics of Growth
The mean values of the layer thickness are depicted in Table 2 and shown in Figure  7.  The results show how the thickness of the layer evolves as a function of the temperature and treatment time. The most relevant parameter is the temperature because the layer thickness increases as the temperature increases; for example, from 38.97 ± 2.8 µm for 2 h and 900 °C to 99.68 ± 08.8 µm for 2 h and 975 °C. This behavior confirms the affinity of nickel to boron since it is possible to obtain boride layers even faster than in low carbon steels [24]. On the other hand, it is clear that the layer thickness evolves as a function of the treatment time. However, at low treatment time and temperature, the layer's thickness is low due to the layer's growth requiring a certain time to start. This time necessary for The results show how the thickness of the layer evolves as a function of the temperature and treatment time. The most relevant parameter is the temperature because the layer thickness increases as the temperature increases; for example, from 38.97 ± 2.8 µm for 2 h and 900 • C to 99.68 ± 08.8 µm for 2 h and 975 • C. This behavior confirms the affinity of nickel to boron since it is possible to obtain boride layers even faster than in low carbon steels [24]. On the other hand, it is clear that the layer thickness evolves as a function of the treatment time. However, at low treatment time and temperature, the layer's thickness is low due to the layer's growth requiring a certain time to start. This time necessary for initiating the layer's growth is known as incubation time, and its effect is more evident at low temperatures (see Table 2).
The values of the parabolic growth constant (K) were estimated using the slope of the curves ( Figure 7); their values are summarized in Table 3. According to the values shown in Table 3, it is feasible to consider that the values of K are correct due to the excellent correlation of the points to a straight line. The results indicate that the boriding process of the 80/20 nickel-chromium alloy is a controlled process, where the growth of the layers is directly dependent on the treatment parameters, such as temperature and time.
The values of the parabolic growth constant were concordant with those reported by Campos et al. [26], even when they used an electrochemical method to accelerate the process.
Once the parabolic growth constant for the borided 80/20 nickel-chromium alloy was estimated, it was possible to assess the activation energy (Q) necessary for boron mobility during the boriding process. Therefore, the activation energy was calculated by plotting the Arrhenius equation in logarithmic form (Figure 8), and estimated to be 145.9 kJmol −1 .
are correct due to the excellent correlation of the points to a straight line. The results indicate that the boriding process of the 80/20 nickel-chromium alloy is a controlled process, where the growth of the layers is directly dependent on the treatment parameters, such as temperature and time.
The values of the parabolic growth constant were concordant with those reported by Campos et al. [26], even when they used an electrochemical method to accelerate the process.
Once the parabolic growth constant for the borided 80/20 nickel-chromium alloy was estimated, it was possible to assess the activation energy ( ) necessary for boron mobility during the boriding process. Therefore, the activation energy was calculated by plotting the Arrhenius equation in logarithmic form (Figure 8), and estimated to be 145.9 kJmol −1 . This result was compared with those reported in the literature (Table 4) for nickel alloys, such as Ni3Al, and borided steels [23,[27][28][29]. This result was compared with those reported in the literature (Table 4) for nickel alloys, such as Ni 3 Al, and borided steels [23,[27][28][29]. Table 4. Different activation energy values are achieved for different materials exposed to the boriding process.

Material
Activation energy (KJ/mol) Reference The 80/20 nickel-chromium alloy (present work) requires the lowest activation energy to diffuse boron and to form compounds such as Ni-B and Cr-B. This means that it is possible to enhance the surface properties of 80/20 nickel-chromium alloy at a relatively low cost.
The pre-exponential constant (K 0 ) was estimated through the intersection of the ordinated axis, and was evaluated as 1.31 × 10 −6 m 2 s −1 .
The values of (K 0 ) and (Q) determined from the experimental results can be used to propose a particular solution to the diffusion process applied to 80/20 nickel-chromium alloy, so Equation (10) can be rewritten as: where K is the growth rate, and T represents the treatment's absolute temperature (Kelvin). By analyzing Equations (4) and (10), it is possible to develop a practical formula for estimating the layer thickness under pre-determined treatment conditions. Equation (4) can be rewritten as: According to the results presented in Table 5, the data calculated show specific errors, especially at a short treatment time. However, the errors decrease as the temperature and treatment time increase. This behavior can be explained because the boriding process depends on the treatment parameters. Therefore, as the time and temperature increase, the process becomes more stable, and the model becomes more efficient. One way to diminish the error could be to add one more time and temperature condition, and eliminate the first condition (2 h and 900 • C). Finally, once stabilized, the model (Equation (11)) calculates the first condition data and compares it with the experimental data.

Mechanical Characterization
The hardness and Young's modulus were estimated by instrumented indentation. The indentations were performed perpendicularly to the diffusion surface. Only the indentations with suitable geometry were considered for measurement.
Once the boride layers were observed ( Figure 5), it was decided to make the indentations starting at 25 µm from the surface and every 25 µm until reaching the substrate. The above because from 25 µm, there is a consolidated phase of borides, in all treatment conditions. The hardness behavior of the boride layers is shown in Figure 9. As can be observed, the hardness values of the layers tended to increase as the temperature and treatment time increased. This behavior can be attributed to the enrichment of the layers with boron as the process evolves. The results show that the layers are more compact and the compounds formed tend to be saturated until the equilibrium is reached. The hardness profiles presented in Figure 9 show a gradually reduction in hardness, and this behavior indicates that the hardness of the boride layers is highly dependent on the boron concentration. Thus, according to the model presented in Figure 1, the boron concentration in the boride layer is higher on the surface and decreases gradually to the substrate. The results match well with the layer's thicknesses presented in Table 2, where the slope of the hardness gradient is more pronounced for the samples exposed to 900 • C than for those exposed to 975 • C, indicating a gradient of concentration that decreases from the surface to the substrate.
The highest hardness value was 1360 ± 70 HV near to the surface of the sample exposed to 975 • C for 6 h. Compared with the hardness of the 80/20 nickel-chromium alloy (270 ± 12 HV) achieved directly from the measured values, the increase in the hardness values at the surface of the 80/20 nickel-chromium alloy indicates an excellent improvement in its mechanical surface.
The behavior of the Young's modulus as a function of the different treatment conditions is shown in Figure 10.
dicates that the hardness of the boride layers is highly dependent on the boron concentration. Thus, according to the model presented in Figure 1, the boron concentration in the boride layer is higher on the surface and decreases gradually to the substrate. The results match well with the layer's thicknesses presented in Table 2, where the slope of the hardness gradient is more pronounced for the samples exposed to 900 °C than for those exposed to 975 °C, indicating a gradient of concentration that decreases from the surface to the substrate.

Figure 9.
Hardness behavior for 900 °C, 950 °C and 975 °C (a-c) respectively. (d) represents the zones where the hardness profiles were measured (Sample exposed to 975 °C for 6 h). Figure 9. Hardness behavior for 900 • C, 950 • C and 975 • C (a-c) respectively. (d) represents the zones where the hardness profiles were measured (Sample exposed to 975 • C for 6 h).
Coatings 2022, 12, x FOR PEER REVIEW 11 of 15 The highest hardness value was 1360 ± 70 HV near to the surface of the sample exposed to 975 °C for 6 h. Compared with the hardness of the 80/20 nickel-chromium alloy (270 ± 12 HV) achieved directly from the measured values, the increase in the hardness values at the surface of the 80/20 nickel-chromium alloy indicates an excellent improvement in its mechanical surface.
The behavior of the Young's modulus as a function of the different treatment conditions is shown in Figure 10. Distance from the surface (µm) 2h 4h 6h Figure 10. The behavior of the Young's modulus for the samples exposed to 975 • C for 2, 4, and 6 h.