Surface Growth of Boronize Coatings Studied with Mathematical Models of Diffusion

Abstract

The following investigation focused on examining the kinetics of Fe₂B coating formation on the surface of ASTM A681 steel during the powder-pack boronizing process. The study measured Fe₂B coating thicknesses at various temperatures and exposure times to confirm the diffusion-controlled growth mechanism during boronizing. Five distinct mathematical models were devised to determine the boron diffusion coefficients in Fe₂B coatings. Understanding the growth kinetics of boronize coatings is imperative as it facilitates the optimization and automation of industrial processes. This ensures the efficient and consistent production of boronize coatings on cutting tools, such as drills and milling cutters, due to their high hardness and wear resistance. The value of the activation energy estimated with five mathematical diffusion models for the Fe₂B coating was 209.8 kJ∙mol⁻¹. The X-ray diffraction technique was used to identify the presence of the iron boronize phase. Tribological studies were also performed to evaluate the coefficient of friction (COF) of the boronized (0.256) and untreated (0.781) samples, having a 300% positive effect of the boronize coating on wear resistance. Finally, the models were empirically validated for two supplementary treatment conditions for 1223 K for 3 h and 1273 K for 1.5 h, where the percentage error for both conditions was estimated to be approximately 2.5%.

1. Introduction

Various surface treatments can augment steel’s wear and corrosion resistance. These methods encompass nitriding, boronizing, and aluminizing, as well as physical vapor deposition (PVD) and chemical vapor deposition (CVD). Nevertheless, thermochemical processes are favored due to the superior adhesion properties of the resultant coatings under elevated temperatures. Boronizing is a widely employed thermochemical process, and it is distinguished among these methodologies. This technique engenders the development of a durable surface comprising single- or multi-phase boronizes, achieved through the diffusion of element B onto the surface of the base material. The resultant boronize coatings exhibit exceptional hardness and strength, which are challenging to attain through conventional methods like carburizing, nitriding, and carbonitriding. Consequently, the tribological performance and service life of steels are significantly improved.

The process can be used on ferrous and non-ferrous materials, typically at temperatures between 973 and 1273 K (1300 and 1830 °F) for 2 to 10 h. By doing this, a metal boronize coating of approximately 20–300 µm thickness can be formed. The metal boronize coating that results from boronizing exhibits excellent characteristics such as high hardness, reasonable wear and corrosion resistance, and moderate resistance to high-temperature oxidation. However, it is essential to note that boronizing cannot be applied to some metals and alloys, such as aluminium and magnesium, due to their low melting points. Moreover, copper alloy is incompatible with boron atoms [1]. Notably, a singular Fe₂B coating in the industrial sector is typically favored over a dual-phase coating consisting of both FeB and Fe₂B due to the potential risk of crack propagation along the (FeB/Fe₂B) interface [2]. Solid boronizing has emerged as the most extensively utilized method in the industry for this purpose, owing to its cost-effectiveness, ease of handling, and the capability to modify the chemical composition of the powders as per the requirements.

The academic literature contains numerous documented diffusion models focusing on the kinetics of Fe₂B coating formation on various substrates. These simulation techniques have real-world applications that aim to enhance the surface properties of treated steels by optimizing the thickness of the boronize coating. For example, Ortiz-Domínguez et al. [3] utilized two distinct methods to evaluate the growth kinetics of Fe₂B coatings on the surface of AISI 1018 steel. The first approach was founded on a diffusion model that employed the mass balance equation at the Fe₂B/substrate interface. The second approach involved a dimensional analysis in constructing an equation that could accurately describe the thickness of the boronize coating as a function of process parameters, including treatment time and boronizing temperature. Additionally, a series of experiments were conducted at different temperatures, with an exposure time of 5 h, to facilitate a comparison between the predicted values and the experimental coating thicknesses. Arslan-Kaba et al. [4] conducted a study where they utilized a boronizing process known as pulsed current integrated cathodic reduction/thermal diffusion (PC/CRTD-Bor) on low-carbon steels. The aim was to create Fe₂B-type monocoatings in a shorter period. The study utilized a constant duty cycle of 1/4 and various current densities (50, 200, and 700 mA∙cm⁻²) at a treatment temperature of 1223 K for exposure times of 5, 10, and 30 min. An average diffusion coefficient (ADC) model was utilized to evaluate the boron diffusion coefficients in FeB and Fe₂B coatings, an average diffusion coefficient (ADC) model was used, and the calculated coating thicknesses were found to agree with the experimental data. Bouarour et al. [5] studied the kinetics of Fe₂B boronize coatings formed on the surface of SAE 1020 steel by considering four mathematical mass transfer models. They also considered the boronize incubation period for the Fe₂B coating in the mathematical models. The activation energies of boronize evaluated on SAE 1020 steel were relatively equal and approximately equal to 183 kJ mol⁻¹.

The investigation of boronize needle orientation during the forming of Fe₂B coatings on pure Armco iron was carried out by Ramdan et al. [6] through 2D multi-phase field (MPF) simulations. The study aimed to determine the growth behaviour of texture in these boronizes parallel to the substrate surface. The study’s main finding was observing texture growth in these boronizes, which occurs perpendicular to the substrate surface. Notably, the growth behavior was found to be independent of the initial size of the boronize needle. The study conducted by Campos et al. [7] is significant as it provides a comprehensive analysis of the paste-boronizing process of AISI 1045 steel. Fuzzy logic Mamdani and Takagi-Sugeno’s mathematical techniques to simulate the process are particularly interesting due to their ability to handle imprecise or uncertain input data. The formation of Fe₂B coatings on the steel’s surface, which is a desirable outcome in the paste-boronizing process, was successfully achieved by varying the thickness of the boron paste. The reported mean errors of 2.61% and 3.62% for the Mamdani and Takagi-Sugeno techniques indicate that the simulation results are highly accurate and reliable. The study’s findings are of practical importance to researchers and practitioners in materials science and engineering as they can optimize the paste boronizing process of AISI 1045 steel.

In a previous study, Campos et al. [8] examined the growth interface between the Fe₂B/substrate while accounting for mass conservation. However, the boronize incubation times for pure Armco iron paste boronizing were not taken into consideration. The authors proposed investigating the growth kinetics of boronize coatings by exploring the relationship between the molar volume of the Fe₂B phase and the substrate.

Nait Abdellah et al. [9] proposed two mathematical mass transfer models to estimate the coefficients of boronize in the Fe₂B coating formed on the surface of ASTM A36 steel in the temperature range from 1173 to 1273 K with treatment times of 2 h, 4 h, 6 h, and 8 h. They also considered the boronize incubation time required to form the primitive coating, which is constant and independent of the boronizing conditions. The first mathematical model was derived from the mass balance equation at the growth interface (Fe₂B/substrate), while the second one employed the integral method. The calculated values of boron activation energies for ASTM A36 steel turned out to be very close to each other for both approaches (161.65 kJ·mol⁻¹ and 160.96 y kJ·mol⁻¹).

This research aimed to study the growth kinetics of the boronized coatings (Fe₂B) formed on the surface of ASTM A681 steel through five mathematical mass transfer models. Likewise, the models were experimentally validated for two extra boronizing conditions (1223 K for 3 h and 1273 K for 1.5 h). The boronizing treatment was performed using the traditional powder-pack technique. For this reason, four different temperatures and four different durations were selected for the treatment. Furthermore, a powder mixture containing 33.5 wt.% B₄C, 5.4 wt.% KBF₄, and 61.1 wt.% SiC was used to treat ASTM A681 steel. In this context, the SEM method was used for microstructural characterization, and the XRD and EDS methods were used for chemical characterization of the boronize coatings formed due to the boronizing treatment. The investigation examined the influence of the boronizing treatment on the coefficient of friction (COF) value after the wear test and compared it with the unboronized sample.

2. Materials and Methods

2.1. Properties

During the boronizing process, boron atoms diffuse and adsorb onto the metal lattice of the component’s surface, forming an interstitial boron compound that manifests as a single-phase or polyphase boronize coating. The boronize coating’s morphology, growth, and phase composition are intricately linked to the elements present in the substrate materials and their interactions with the boron atoms. As demonstrated in

Table 1. Exhibits significant data on the microhardness values and the constitution of boronize coatings formed on diverse substrates, obtained after the boronizing process.

Substrate	Type of Boronize Coating	Hardness of Boronize Coating (HV)
Fe	FeB	1900–2100
	Fe2B	1800–2000
Co	CoB	1850
	Co2B	1550
Co-27·5Cr	CoB	2200
	Co2B	1550
Ni	Ni4B3	1600
	Ni2B	1500
	Ni3B	900
W	W2B	2700
	WB	2700
	W2B5	2700
Nb	Nb2B4	2600–3000
	NbB4	2600–3000
Zr	Zr2B	2300–2600
	ZrB2	2300–2600
Ta	Ta2B	3200–3500
Ta2B	TiB2	2500
	TiB	2500
Ti-6Al-4V	TiB2	3000
	TiB	3000
Mo	Mo2B	2400–2700
	Mo2B5	2400–2700
Re	ReB	2700–2900
	Re2B	2700–2900

, the boronize coating’s characteristics depend on the substrate materials’ constituents [10].

A very interesting and much more complete work is the one presented by Akopov et al. [11], which reviews the advances in the crystal chemistry of the different boronizes in the last 60 years and compiles data on the crystal structure, material properties, and synthetic routes of metal boronizes. Likewise, metal boronizes are chosen because of their rich crystalline chemistry, which gives rise to several properties that offer many application possibilities. The formation of a boronize coating on iron and steel can result in a single-phase or two-phase composition, which is determined by the Fe-B phase diagram (refer to Figure 1) [12].

The following technical information concerns the morphological characteristics of the boronize coating. It has been observed that forming a single-phase coating results in the creation of Fe₂B, while a two-phase coating featuring an outer FeB phase and an inner Fe₂B phase is formed. The morphology of the boronize coating is characterized by a sawtooth structure, illustrated in Figure 2. This structure is vital in enhancing the mechanical adhesion at Fe₂B/substrate interfaces, as documented in reference [13].

It is essential to note that the development of boronize coatings on the substrate surface is subject to a certain period of nucleation or incubation time, as referenced in scholarly literature [14]. During this time, the initial spreading of the coating is limited to specific locations on the substrate surface, where it merges to form a skinny initial coating of iron boronize. At this stage, the initial coating is randomly oriented [15]. However, as the coating grows, it is strongly favored along the preferential direction [002] for FeB and Fe₂B [15]. It is important to note that when undergoing boronizing treatment, the formation of the Fe₂B phase is preferred over the FeB phase due to the latter being more brittle [2]. The differences in thermal expansion coefficients between the FeB and Fe₂B phases and the base material can create a complex stress state, leading to tensile residual stresses in the outer FeB coating and compressive residual stresses in the Fe₂B phase [16]. However, the formation of both phases can result in cracks at the FeB/Fe₂B interface of the two-phase coating, which may cause spalling and even separation of the coating under mechanical stress or thermal/mechanical shock [2]. Fortunately, the annealing process can help reduce the occurrence of the FeB phase after boronizing treatment, thus mitigating the risk of crack formation [17,18].

The data presented in Figure 2 indicate that the boron concentration at the phase surface is denoted as $C_{up}^{F{e}_{2}B}$ (=9 wt.%B), while the boron concentration at the Fe₂B/substrate growth interface is represented as $C_{low}^{F{e}_{2}B}$ (=8.83 wt.%B). Additionally, the boron solubility limit within the substrate is symbolized as $C_0$ (=35 × 10⁻⁴ wt.%B), though it cannot be considered due to its magnitude [12]. It is essential to note that the information presented in Figure 2 is critical for understanding the boron concentration at the phase surface and the Fe₂B/substrate growth interface. Therefore, the data can provide valuable insights into the growth process of Fe₂B on the substrate.

The unique characteristics of the Fe₂B phase are notable [19]:

• The composition contains approximately 9.0 wt.% boron.
• The crystal structure of Fe₂B is identified as body-centered tetragonal and is characterized by axial lengths a = b = 0.5078 × 10⁻⁹ m and c = 0.4249 × 10⁻⁹ m (as exemplified in Figure 3).
• The density of Fe₂B is 7.43 × 10⁻³ kg/cm³.
• Microhardness ranges from approximately 18–20 × 10⁹ Pa.
• The Young’s modulus range of Fe₂B is between 285–295 × 10⁹ Pa.
• The thermal expansion coefficient is measured at 7.65 × 10⁻⁶/K within the 473–873 K range and 9.2 × 10⁻⁶/K within the 373–1073 K range.

2.2. Substrate and Boronizing Process

ASTM A681 is a high-speed tool steel widely acclaimed for its remarkable wear resistance, strength, and toughness. This type of steel is commonly employed in manufacturing cutting tools, dies, and various machinery where its exceptional properties can be highly advantageous. The notable characteristics of this steel make it an ideal choice for applications with high-stress conditions. The material’s high wear resistance maintains its cutting edge even under harsh conditions. Its superior strength and toughness also enable it to withstand the high forces and pressures commonly encountered in industrial applications. The samples to be boronized were cubic and had 10 mm sides. They were sectioned from a square bar using a Buehler-IsoMet^TM 1000 precision gravity-fed bench (Lake Bluff, IL, USA). The steel samples were roughened with silicon carbide (SiC) paper grits before undergoing the thermochemical process. The grits ranged from 80 to 2500 and were applied using an EcoMetTM 30 single-hand grinding machine (Lake Bluff, IL, USA). The samples were then cleaned for 20 min in a SONOREX SUPER RK 52 high-performance ultrasonic cleaner (Heinrichstraße, Berlín, Germany). After washing, the samples were placed inside a cylindrical AISI 316L stainless steel container and embedded in a boron-rich powder mixture, as shown in Figure 4.

The chemical mixture has the following weight percentage composition: 33.5% boron carbide (B₄C) as the main boronize, 5.4% potassium fluoroborate (KBF₄) as the catalyst, and 61.1% silicon carbide (SiC) as the diluent. This specific composition of the boronize powder facilitates the formation of a Fe₂B-type boronize monocoating on the treated surface, as cited in [20]. Chemical reactions that form the Fe₂B coatings are shown in Figure 5, as referenced by [21].

During the thermochemical treatment process, the surface of the substrate material, ASTM A681 steel, undergoes ionization through the reaction of BF₂ gas, resulting in the formation of B²⁺ and [BF]⁺ ions. The diffusion of boron (B) atoms further ensues into the interior of the substrate. A TEFIC 1473 K brand furnace (Weiyang District, Shaanxi, China) was employed to execute this process. Different treatment times and temperatures were utilized, while argon gas protected the samples from atmospheric pollution, oxygen, and hydrogen. Argon gas is chemically inert, tasteless, and odorless, making it an ideal choice. The AISI 316L steel container was heated, with the samples embedded in the boron-rich mixture.

According to the phase diagram, the boronizing treatment temperatures (refer to Figure 1) ranged from 1123 K to 1273 K, while the treatment duration ranged from 2 to 8 h. After completion of the boronizing process, the container was removed from the oven and allowed to cool to room temperature (296 K). The sample was then removed from the container and encapsulated in resin using a Buehler SimpliMet 4000 mounting press (Lake Bluff, IL, USA). This encapsulation was carried out to facilitate sample holding during manual polishing. It was chosen based on the shape of the part and the type of analysis to be performed, including edge analysis as part of a surface treatment. The metallographic samples underwent a rigorous two-step polishing process. Firstly, the samples were polished with a Buehler MicroCloth™ (Hong Kong, SAR, China), a soft, versatile, synthetic rayon cloth with a long pile and magnetic backing. The cloth was used with Al₂O₃ aluminium oxide abrasive (3 and 1 × 10⁻⁶ m) to remove any scratches that may have been produced during the roughing operation. Secondly, with each size, the samples were subjected to diamond suspension polishing (0.25 and 0.05 × 10⁻⁶ m, Buehler) for 20 min. This allowed for the creation of a specular surface necessary for accurate analysis. Additionally, the samples were cleaned after each step using an ultrasonic bath of acetone for thirty seconds to ensure the highest possible quality. A reagent was employed in immersion with 4% volume Nital to reveal and analyze the samples’ microstructure. The thickness of the boronized coatings was observed using a 4K optical microscope from the VHX-7000 series (Higashi-Yodogawa-ku, Osaka, Japan). The metallographic procedure, including the observation of phases, grain boundaries, impurities, and deformation zones, is presented in Figure 6.

The thicknesses of boronize (Fe₂B) coatings were estimated using the specialized software Image-Pro Plus 6.3 (Rockville, MD, USA). An automated process was utilized for this purpose. The growth kinetics study involved six samples, and a replicate was made for each one. To determine the average values of Fe₂B coating thickness, one hundred measurements were taken on each of the thirty-two cross sections selected. This approach enabled us to obtain accurate and reliable data. Figure 7 shows a graphical illustration of the selected cross-sections. The phase was examined using the grazing angle X-ray diffraction technique. The peaks of the X-ray diffraction spectrum were analyzed through the use of Match! crystallographic software version 3.15 (Karlsruhe, Baden-Württemberg, Germany). The scanning range was from 25 to 85, and a CoKα-type diffraction radiation with a wavelength of λ = 0.18 × 10⁻⁹ m was used. The Intel Equinox 2000 diffractometer (Waltham, MA, USA) was utilized for this purpose.

3. Mathematical Models of Mass Transfer

Correctly understanding growth kinetics optimizes process parameters to reduce production time and costs. Minimizing the processing time required to achieve the desired properties can result in significant savings in energy and resources. Five mathematical models were formulated to study the growth kinetics of boronize (Fe₂B) coatings deposited on the surface of ASTM A681 steel via the solid boronizing process. The initial proposed model pertains to the linear mass transfer model, encompassing the mass balance equation at the Fe₂B/Fe growth interface $(\mathsf{\Delta }C){\left.(dx/d{t}_u)\right|}_{x=u}=J_{in}^{F{e}_{2}B}{\left.(x)\right|}_{x=u}-J_{out}^{Fe}{\left.(x)\right|}_{x=u+du}$ and the solution of Fick’s second law without time dependence $C_{F{e}_{2}B}(x)=C_{1}x+C_2$ (steady state). The second proposition involves utilizing the mass transfer model called the error function. This model considers the mass balance equation at the Fe₂B/Fe growth interface, incorporating time dependence, the solution of the second Fick’s law $C_{F{e}_{2}B}(x,t_u)=A+Berf(x/2{(D_{F{e}_{2}B}{t}_u)}^{1/2})$, and the parabolic growth equation of the boronize (Fe₂B) coatings $u^2=k{t}_u$. For the third model, the integral method of Goodman [22,23] was used, where the mass balance equation at the Fe₂B/Fe growth interface with time dependence was considered $(\mathsf{\Delta }C){\left.(\partial C_{F{e}_{2}B}(x,t_u)/\partial t_u/\partial C_{F{e}_{2}B}(x,t_u)/\partial x\right|}_{x=u}=J_{in}^{F{e}_{2}B}{\left.(x,t_u)\right|}_{x=u}-J_{out}^{Fe}{\left.(x,t_u)\right|}_{x=u+du}$, a parabolic profile with time dependence $C_{F{e}_{2}B}(x,t_u)=b{(u-x)}^2+a(u-x)+c$ and the parabolic growth equation. Only the mass balance equation at the Fe₂B/Fe growth interface without time dependence $(\mathsf{\Delta }C){\left.(dx/d{t}_u)\right|}_{x=u}=-D_{F{e}_{2}B}(d{C}_{F{e}_{2}B}(x)/d{t}_u){\left.(d{t}_u/dx)\right|}_{x=u}$ and the chain rule were considered for the optional fourth mass transfer diffusion model. Finally, the fifth model is based on an inverse diffusion problem approach, where the concentration profile is considered known experimental data $C_{F{e}_{2}B}(x,t_u)$, which allows the determination of the diffusion coefficient as a function of the concentration $D_{F{e}_{2}B}(C_{F{e}_{2}B}(x,t_u))$. It is based on Fick’s second law but with the consideration that the boron diffusion coefficient is a function of the concentration profile $\partial C_{F{e}_{2}B}(x,t_u)/\partial t_u=\partial ((D_{F{e}_{2}B}(C_{F{e}_{2}B}(x,t_u)))\partial C_{F{e}_{2}B}(x,t_u)/\partial t_u)/\partial x$ and is combined with the similarity variable $x/2t_u^{1/2}$. Figure 8 displays the boronize concentration profile for the Fe₂B monocoating in a graphical format. The $C_{ads}$ term mentioned in Figure 8 refers to the adsorbed content of active boron on the substrate surface [24].

3.1. Linear Mass Transfer Model

The boron concentrations in Figure 8 ($C_{up}^{F{e}_{2}B}$, $C_{low}^{F{e}_{2}B}$, and $C_{low}^{F{e}_{2}B}$) remain consistent and are determined by the phase diagram, as depicted in Figure 1. Under these specific conditions, the conservation of mass equation can be derived at the interface separating the Fe₂B phase and the substrate. Notably, the forward velocity of the interface between the Fe₂B/substrate phases is directly proportional to the difference between the inflow and outflow rates (as demonstrated in Figure 8).

$$\begin{array}{cc}\hfill \left(\frac{C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0}{2}\right){\left.\left(\frac{dx}{d{t}_u}\right)\right|}_{x=u}& ={\left.J_{in}^{F{e}_{2}B}(x)\right|}_{x=u}-{\left.J_{out}^{Fe}(x)\right|}_{x=u+du}\hfill \\ & =-D_{F{e}_{2}B}{\left.\left(\frac{d{C}_{F{e}_{2}B}\left(x\right)}{dx}\right)\right|}_{x=u}-\left(-D_{Fe}{\left.\left(\frac{d{C}_{Fe}\left(x\right)}{dx}\right)\right|}_{x=u+du}\right).\hfill \end{array}$$

(1)

The formation time of the Fe₂B phase $t_u=(t-t_0^{F{e}_{2}B})$ (s) [25], which denotes the treatment time $t$ (s) and $t_0^{F{e}_{2}B}$ is equivalent to the characteristic time of Fe₂B iron boronize [14], has an essential role in material engineering. Additionally, the incoming flux in the Fe₂B phase is $J_{in}^{F{e}_{2}B}(x)$ (atoms/m²∙s), and $J_{out}^{Fe}(x)$ corresponds to the outgoing flux in the substrate (atoms/m²∙s). The diffusion coefficient in the Fe₂B phase ($D_{F{e}_{2}B}$) and the diffusion coefficient in the Fe phase ($D_{Fe}$) are significant parameters that affect the diffusion process. Notably, the diffusion coefficients in Equation (1) are concentration-independent. Therefore, there are direct methods to address the problem of various conditions at the boundary. In particular, there is no boron flux from the Fe₂B surface coating to the substrate $J_{out}^{Fe}{\left.(x)\right|}_{x=u+du}=0$, as indicated in Figure 8, which illustrates the boron concentration profile for this system $C_{F{e}_{2}B}(x)$. The thickness of the boronize coating is specified as $u$ (m). Lastly, the boundary conditions for no time dependence are as follows:

$$C_{F{e}_{2}B}\left(x=u_0\approx 0\right)=C_{up}^{F{e}_{2}B}=9\hspace{0.33em}\mathrm{w}\mathrm{t}.\%,\hspace{0.33em}\mathrm{for}\hspace{0.33em}{C}_{ads}^B>9\hspace{0.33em}\mathrm{w}\mathrm{t}.\%, \mathrm{and} t_u>0,$$

(2)

$$C_{F{e}_{2}B}\left(x=u\right)=C_{low}^{F{e}_{2}B}=8.83\hspace{0.33em}\mathrm{w}\mathrm{t}.\%,\hspace{0.33em}\mathrm{for}\hspace{0.33em}{C}_{ads}^B<8.83\hspace{0.33em}\mathrm{w}\mathrm{t}.\%, \mathrm{and} t_u>0.$$

(3)

The parameter ‘$u_0$’ denotes a fundamental Fe₂B (m) coating that terminates at the point of emergence of the first iron boronizes following a characteristic duration specific to boronizes. Fick’s second law, which incorporates time dependence, is expressed as follows:

$$\frac{\partial C_{F{e}_{2}B}\left(x,t_u\right)}{\partial t_u}=D_{F{e}_{2}B}\frac{{\partial }^2{C}_{F{e}_{2}B}\left(x,t_u\right)}{\partial x^2}.$$

(4)

Equation (4) can be rewritten without time dependence as follows:

$$\frac{d^2{C}_{F{e}_{2}B}\left(x\right)}{d{x}^2}=0.$$

(5)

Solving Equation (5) and considering Equations (2) and (3), we arrive at the following:

$$C_{F{e}_{2}B}\left(x\right)=\frac{C_{low}^{F{e}_{2}B}-C_{up}^{F{e}_{2}B}}{u}x+C_{up}^{F{e}_{2}B}.$$

(6)

Substituting Equation (6) into Equation (1) and considering that $C_{Fe}(x)=cte.$, we obtain the following:

$$u^2=4D_{F{e}_{2}B}\left(\frac{C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}}{C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}}\right)\left(t-t_0^{F{e}_{2}B}\right)=4D_{F{e}_{2}B}{\epsilon }_{F{e}_{2}B}^2{t}_u.$$

(7)

In Equation (7), it is observed that the coatings obey the law of parabolic growth with

$${\epsilon }_{F{e}_{2}B}^2=C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}=9.5\times {10}^{-3},$$

(8)

where ${\epsilon }_{F{e}_{2}B}^2$ is defined as a concentration-dependent adimensional constant.

3.2. Mass Transfer Model of the Error Function

The solution to Fick’s second law, which incorporates time dependence, is typically expressed in terms of the error function. The error function is a mathematical construct extensively used in various academic disciplines and industries, such as finance, engineering, and physics. Specifically, the error function is widely employed in statistical mechanics, quantum mechanics, and probability theory, among other fields. Equation (4) is commonly used to express Fick’s second law regarding the error function. This efficient approach enables accurate modeling of diverse phenomena using a single mathematical construct. As such, the error function is a valuable tool in many research areas, enabling precise calculations and predictions.

$$C_{F{e}_{2}B}\left(x,t_u\right)=A+Berf\left(\frac{x}{2\sqrt{D_{F{e}_{2}B}{t}_u}}\right).$$

(9)

The boundary conditions for $C_{F{e}_{2}B}(x,t_u)$ with time dependence are:

$$C_{F{e}_{2}B}\left(x=u_0,t_u=t_0^{F{e}_{2}B}\right)=C_{up}^{F{e}_{2}B}=9\hspace{0.33em}\mathrm{w}\mathrm{t}.\%,\hspace{0.33em}\mathrm{for}\hspace{0.33em}{C}_{ads}^B>9\hspace{0.33em}\mathrm{w}\mathrm{t}.\%, \mathrm{and} t_u>0.$$

(10)

$$C_{F{e}_{2}B}\left(x=u,t_u=t_u\right)=C_{low}^{F{e}_{2}B}=8.83\hspace{0.33em}\mathrm{w}\mathrm{t}.\%,\hspace{0.33em}\mathrm{for}\hspace{0.33em}{C}_{ads}^B<8.83\hspace{0.33em}\mathrm{w}\mathrm{t}.\%, \mathrm{and} t_u>0.$$

(11)

To determine constants A and B for Equation (9), the boundary conditions given in Equations (10) and (11) must be considered. Based on these conditions, the values of A and B can be derived accordingly. We must adhere to this process to obtain accurate and reliable results.

$$C_{F{e}_{2}B}\left(x,t_u\right)=C_{up}^{F{e}_{2}B}+\frac{C_{low}^{F{e}_{2}B}-C_{up}^{F{e}_{2}B}}{erf\left(\frac{u}{2\sqrt{D_{F{e}_{2}B}{t}_u}}\right)}erf\left(\frac{x}{2\sqrt{D_{F{e}_{2}B}{t}_u}}\right).$$

(12)

Conversely, the equation governing the balance of matter at the interface of coating/substrate growth can be expressed as follows. This equation plays a significant role in understanding the fundamental processes involved in thin coating growth.

$$\begin{array}{cc}\hfill \left(\frac{C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0}{2}\right){\left.\left(\frac{dx}{d{t}_u}\right)\right|}_{x=u}& =J_{F{e}_{2}B}{\left.(x,t_u)\right|}_{x=u}-J_{Fe}{\left.(x,t_u+d{t}_u)\right|}_{x=u+du}\hfill \\ & =-D_{F{e}_{2}B}{\left.\left(\frac{\partial C_{F{e}_{2}B}\left(x,t_u\right)}{\partial x}\right)\right|}_{x=u}-\left(-D_{Fe}{\left.\left(\frac{\partial C_{Fe}\left(x,t_u+d{t}_u\right)}{\partial x}\right)\right|}_{x=u+du}\right).\hfill \end{array}$$

(13)

The transfer of boron from the Fe₂B surface coating to the substrate at $J_{out}^{Fe}{\left.(x,t_u+d{t}_u)\right|}_{x=u+du}=0$ has yet to be considered. This study assumes that the parabolic growth law applies to the boron coatings, similar to the linear model. We obtained the following result by substituting Equation (12) into Equation (13).

$$\left(C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0/2\right){\epsilon }_{F{e}_{2}B}=-\left(C_{low}^{F{e}_{2}B}-C_{up}^{F{e}_{2}B}\right)exp\left(-{\epsilon }_{F{e}_{2}B}^2\right)/\sqrt{\pi }erf\left({\epsilon }_{F{e}_{2}B}\right).$$

(14)

Equation (14)’s numerical values can be obtained through the implementation of the Newton–Raphson numerical method [26]. This method is frequently employed to determine the roots of an equation and can converge rapidly.

$${\epsilon }_{F{e}_{2}B}^2=9.6\times {10}^{-3}.$$

(15)

3.3. Goodman Integral Mass Transfer Model

Fe₂B coating growth was studied using Goodman’s integral method [22,23]. Specifically, a semi-infinite solid with a parabolic profile was preliminarily defined for this purpose. Implementation of the method allowed for an in-depth examination of the growth process. The results obtained using this method are expected to provide valuable insights into the fundamental mechanisms underlying the growth of Fe₂B coatings.

$$C_{F{e}_{2}B}\left(x,t_u\right)=b{\left(u-x\right)}^2+a\left(u-x\right)+c.$$

(16)

Reference [22] indicates that parameters a and b in Equation (16) are time-dependent functions. Upon closer examination, it is observed that Equation can be rewritten in a different form by applying the chain rule on the left-hand side. Parameters a and b are noted as functions that vary with time.

$$\left(C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}/2\right)\left(\frac{{\left.\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial t_u}\right|}_{x=u}}{{\left.\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial x}\right|}_{x=u}}\right)=-D_{F{e}_{2}B}{\left.\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial x}\right|}_{x=u}.$$

(17)

Combining Equations (4) and (17), the following is obtained.

$$\left(C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}/2\right)\left(\frac{{\left.D_{F{e}_{2}B}\frac{\partial }{\partial x}\left(\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial x}\right)\right|}_{x=u}}{{\left.\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial x}\right|}_{x=u}}\right)=-D_{F{e}_{2}B}{\left.\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial x}\right|}_{x=u}.$$

(18)

Replacing Equation (16) with Equation (18), the following occurs:

$$a^2=\left(C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0\right)b.$$

(19)

Similarly, by replacing Equation (16) with Equation (13), we obtain the following:

$$\left(C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0/2\right){\left.\frac{dx}{d{t}_u}\right|}_{x=u}=D_{F{e}_{2}B}\hspace{0.17em}a.$$

(20)

The concentration profile described in Equation (16) is defined within a specific range ($0\le x\le u$). Substitution of the boundary condition stipulated in Equation (11) into Equation (16) may result in a resultant output.

$$c=C_{low}^{F{e}_{2}B}.$$

(21)

With the result obtained from Equation (21), the concentration profile given in Equation (16) can be rewritten as follows:

$$C_{F{e}_{2}B}\left(x,t_u\right)=b{\left(u-x\right)}^2+a\left(u-x\right)+C_{low}^{F{e}_{2}B}.$$

(22)

Applying the boundary condition given by Equations (10)–(22) produces the following:

$$b=\left(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}-au\right)/u^2.$$

(23)

Combining Equation (19) with Equation (23), we obtain the following:

$$a=\frac{\left(C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}\right)\left(\sqrt{1+4\left(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0\right)}-1\right)}{2u}.$$

(24)

Substituting Equations (7) and (24) in Equation (20), we find the following:

$${\epsilon }_{F{e}_{2}B}^2=\frac{\left(\sqrt{1+4\left(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0\right)}-1\right)}{2}\approx 9.4\times {10}^{-3}.$$

(25)

3.4. Optional Mass Transfer Diffusion Model

This optional diffusion model postulates an examination of the mass balance equation at the Fe₂B/substrate growth interface without considering the shape of the boron concentration profile along the Fe₂B coating. Equation (1) can be rephrased by applying the chain rule on the right-hand side of equality.

$$\left(\frac{C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0}{2}\right){\left.\left(\frac{dx}{d{t}_u}\right)\right|}_{x=u}=-D_{F{e}_{2}B}\left(\frac{d{C}_{F{e}_{2}B}\left(x\right)}{d{t}_u}\right){\left.\left(\frac{d{t}_u}{dx}\right)\right|}_{x=u}.$$

(26)

The term $d{C}_{F{e}_{2}B}(x)/d{t}_u$ appearing on the right-hand side of Equation (26) is non-zero when considering the differential of the concentration profile, which is expressed as follows:

$$d{C}_{F{e}_{2}B}\left(x\right)=\frac{\partial C_{F{e}_{2}B}\left(x\right)}{\partial x}dx.$$

(27)

Deriving both sides with respect to $d{t}_u$, we obtain the following:

$$\frac{d{C}_{F{e}_{2}B}\left(x\right)}{d{t}_u}=\frac{\partial C_{F{e}_{2}B}\left(x\right)}{\partial x}\frac{dx}{d{t}_u}.$$

(28)

It is observed that the right-hand side of Equation (28) does not cancel, so $d{C}_{F{e}_{2}B}(x)/d{t}_u$ does not cancel either. Equation (26) can also take the following form:

$$\left(\frac{C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0}{2}\right){\left(\frac{dx}{d{t}_u}\right)}^2{\left.d{t}_u\right|}_{x=u}=-D_{F{e}_{2}B}{\left.d{C}_{F{e}_{2}B}\left(x\right)\right|}_{x=u}.$$

(29)

Substituting Equation (7) on the left side of Equation (29), we obtain the following:

$$\left(\frac{C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0}{2}\right){\epsilon }_{F{e}_{2}B}^2\frac{d{t}_u}{t_u}=-{\left.d{C}_{F{e}_{2}B}\left(x\right)\right|}_{x=u}.$$

(30)

Integrating both sides of the equal of Equation (30) with the corresponding limits, we obtain the following:

$$\left(\frac{C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0}{2}\right){\epsilon }_{F{e}_{2}B}^2{\displaystyle \underset{t_u=t_0^{F{e}_{2}B}}{\overset{t_u=t_u}{\int }}\frac{d{t}_u}{t_u}}=-{\displaystyle \underset{C_{F{e}_{2}B}(x=u_0)=C_{up}^{F{e}_{2}B}}{\overset{C_{F{e}_{2}B}(x=u)=C_{low}^{F{e}_{2}B}}{\int }}d}{\left.C_{F{e}_{2}B}\left(x\right)\right|}_{x=u}.$$

(31)

The left side of Equation (31) reveals a characteristic initial time for forming the first iron boronize crystals $t_0^{F{e}_{2}B}$. In other words, the coating does not attain instantaneous growth at $t_u=0$. Equation (31) further elucidates that the formation of crystals necessitates a specific time interval. This phenomenon highlights the intricate process involved in the growth of the coating and the formation of crystals.

$${\epsilon }_{F{e}_{2}B}^2=2\left(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0\right)/ln(t/t_0^{F{e}_{2}B}).$$

(32)

In the case of ASTM A681 steel, experimental data on Fe₂B coating thickness indicate that the characteristic incubation time of iron boronizes is roughly $t_0^{F{e}_{2}B}$ = 1941.9569 s. As a result, Equation (32) can be approximated accordingly.

$${\epsilon }_{F{e}_{2}B}^2=9.7\times {10}^{-3}.$$

(33)

3.5. Boltzmann–Matano Inverse Diffusion Model

The mathematical models discussed previously are limited to scenarios where the diffusion coefficient, denoted by $D_{F{e}_{2}B}$, remains constant. However, the diffusion coefficient is subject to variability in practical diffusion experiments. To address this, researchers utilize an inverse diffusion problem approach, where the concentration field, denoted by $C_{F{e}_{2}B}(x,t_u)$, serves as a known experimental datum. This approach aims to determine the diffusion coefficient as a function of concentration, a technique widely documented in the literature [27]. Additionally, the nonlinear diffusion equation expresses the second Fick’s law for a linear time-dependent flow with a varying diffusion coefficient, denoted by $D_{F{e}_{2}B}(C_{F{e}_{2}B}(x,t_u))$.

$$\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial t_u}=\frac{\partial }{\partial }\left((D_{F{e}_{2}B}(C_{F{e}_{2}B}(x,t_u)))\frac{\partial C_{F{e}_{2}B}(x,t_u)}{\partial t_u}\right).$$

(34)

In certain boundary conditions, Boltzmann demonstrated that the proportionate diffusion coefficient, $D_{F{e}_{2}B}(C_{F{e}_{2}B}(x,t_u))$, is exclusively a function of concentration. Moreover, $C_{F{e}_{2}B}(x,t_u)$ can be represented as a single variable called the similarity variable ($x/2t_u^{1/2}$). This permits Equation (34) to be reduced to an ordinary differential equation by introducing a new variable, $\eta$. The latter variable is expressed as a function of the following:

$$\eta =x/2t_u^{1/2}.$$

(35)

Conditions for a semi-infinite medium:

$$C_{F{e}_{2}B}\left(x,t_u>0\right)=C_0=35\times {10}^{-4}\hspace{0.33em}\mathrm{w}\mathrm{t}.\%,\hspace{0.33em}\mathrm{for}\hspace{0.33em}x=0,$$

(36)

$$C_{F{e}_{2}B}\left(x,t_u=0\right)=0,\hspace{0.33em}\mathrm{for}\hspace{0.33em}x>0,.$$

(37)

With the similarity variable, the conditions of Equations (36) and (37) are transformed into the following:

$$C_{F{e}_{2}B}\left(\eta \right)=C_0=35\times {10}^{-4}\hspace{0.33em}\mathrm{w}\mathrm{t}.\%,\hspace{0.33em}\mathrm{for}\hspace{0.33em}\eta =0,$$

(38)

$$C_{F{e}_{2}B}\left(\eta \right)=0,\hspace{0.33em}\mathrm{for}\hspace{0.33em}\eta =\infty .$$

(39)

With this new variable, we obtain the following:

$$\frac{\partial C_{F{e}_{2}B}\left(x,t_u\right)}{\partial x}=\frac{\partial C_{F{e}_{2}B}\left(\eta \right)}{\partial \eta }\frac{\partial \eta }{\partial x}=\frac{1}{2t_u^{1/2}}\frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta },$$

(40)

$$\frac{\partial C_{F{e}_{2}B}\left(x,t_u\right)}{\partial t_u}=\frac{\partial C_{F{e}_{2}B}\left(\eta \right)}{\partial \eta }\frac{\partial \eta }{\partial t_u}=-\frac{x}{4t_u^{3/2}}\frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta },$$

(41)

$$\begin{array}{cc}\hfill \frac{\partial }{\partial x}\left(D_{F{e}_{2}B}\left(C_{F{e}_{2}B}\left(x,t_u\right)\right)\frac{\partial C_{F{e}_{2}B}\left(x,t_u\right)}{\partial x}\right)& =\frac{\partial }{\partial x}\left(D_{F{e}_{2}B}\left(C_{F{e}_{2}B}\left(x,t_u\right)\right)\frac{1}{2t_u^{1/2}}\frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta }\right)\hfill \\ & =\frac{d}{4t_{u}d\eta }\left(D_{F{e}_{2}B}\left(C_{F{e}_{2}B}\left(\eta \right)\right)\frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta }\right).\hfill \end{array}$$

(42)

Substituting Equations (41) and (42) into Equation (34), obtains the following:

$$-2\eta \frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta }=\frac{d}{d\eta }\left(D_{F{e}_{2}B}\left(C_{F{e}_{2}B}\left(\eta \right)\right)\frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta }\right).$$

(43)

Since Equation (43) contains only total differentials, we can cancel $1/d\eta$ from each side and integrate between $C_{F{e}_{2}B}=0$ and $C_{F{e}_{2}B}=C_{F{e}_{2}B}^{\prime }$, where $C_{F{e}_{2}B}^{\prime }$ is a concentration in the range $0<C_{F{e}_{2}B}^{\prime }<C_0.$

$$-2{\displaystyle \underset{C_{F{e}_{2}B}=0}{\overset{C_{F{e}_{2}B}=C_{F{e}_{2}B}^{\prime }}{\int }}\eta d{C}_{F{e}_{2}B}\left(\eta \right)}={\displaystyle \underset{C_{F{e}_{2}B}=0}{\overset{C_{F{e}_{2}B}=C_{F{e}_{2}B}^{\prime }}{\int }}d\left(D_{F{e}_{2}B}\left(C_{F{e}_{2}B}\left(\eta \right)\right)\frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta }\right)}=D_{F{e}_{2}B}\left(C_{F{e}_{2}B}\left(\eta \right)\right){\left.\frac{d{C}_{F{e}_{2}B}\left(\eta \right)}{d\eta }\right|}_0^{C_{F{e}_{2}B}^{\prime }}.$$

(44)

Since $D_{F{e}_{2}B}(C_{F{e}_{2}B}(\eta ))d{C}_{F{e}_{2}B}(\eta )/d\eta =0$, then $C_{F{e}_{2}B}=0$. Finally, rewriting Equation (42), we obtain the following:

$$D_{F{e}_{2}B}\left(C_{F{e}_{2}B}^{\prime }\right)=-2\frac{d\left(x/2t_u^{1/2}\right)}{d{C}_{F{e}_{2}B}\left(x,t_u\right)}{\displaystyle \underset{0}{\overset{C_{F{e}_{2}B}^{\prime }}{\int }}\left(x/2t_u^{1/2}\right)d{C}_{F{e}_{2}B}\left(x,t_u\right)}=-\frac{1}{2t_u}\frac{dx}{d{C}_{F{e}_{2}B}\left(x,t_u\right)}{\displaystyle \underset{0}{\overset{C_{F{e}_{2}B}^{\prime }}{\int }}xd{C}_{F{e}_{2}B}\left(x,t_u\right)}.$$

(45)

The diffusion coefficient of the dissolved component is represented by $D_{F{e}_{2}B}(C_{F{e}_{2}B}^{\prime })$. In contrast, $dx/d{C}_{F{e}_{2}B}(x,t_u)$ denotes the inverse of the dissolved component $C_{F{e}_{2}B}(x,t_u)$ concentration profile derivative concerning the diffusion path in the x phase. By utilizing the Ratajski proposal [28], we may rewrite the integral of Equation (45) as follows:

$$D_{F{e}_{2}B}\left(C_{F{e}_{2}B}^{\prime }\right)=\frac{1}{2t_u}\frac{dx}{d{C}_{F{e}_{2}B}\left(x,t_u\right)}\left({\displaystyle \underset{0}{\overset{x_i}{\int }}{C}_{F{e}_{2}B}^{\prime }\left(x,t_u\right)dx+{\displaystyle \underset{x_i}{\overset{\infty }{\int }}{C}_{F{e}_{2}B}\left(x,t_u\right)dx}}\right).$$

(46)

With

$$\begin{array}{cc}\hfill {\displaystyle \underset{0}{\overset{x_i}{\int }}{C}_{F{e}_{2}B}^{\prime }\left(x,t_u\right)dx+{\displaystyle \underset{x_i}{\overset{\infty }{\int }}{C}_{F{e}_{2}B}\left(x,t_u\right)dx}}& ={\displaystyle \sum _{j=1}^{i-1}{\overline{c}}_i}\mathsf{\Delta }{x}_j+{\overline{c}}_i\frac{\mathsf{\Delta }{x}_i}{2}+\frac{{\overline{c}}_i+c_{i+1}}{2}\frac{\mathsf{\Delta }{x}_i}{2}+{\displaystyle \sum _{j=i+1}^n{\overline{c}}_j}\mathsf{\Delta }{x}_j,\hfill \\ & ={\displaystyle \sum _{j=1}^{i-1}{\overline{c}}_i}\mathsf{\Delta }{x}_j+\frac{1}{4}\mathsf{\Delta }{x}_i\left(3{\overline{c}}_i+c_{i+1}\right)+{\displaystyle \sum _{j=i+1}^n{\overline{c}}_j}\mathsf{\Delta }{x}_j.\hfill \end{array}$$

(47)

Substituting Equation (47) into Equation (46), we obtain the following:

$$D_{F{e}_{2}B}\left(C_{F{e}_{2}B}^{\prime }\right)=\frac{1}{2t_u\mathsf{\Delta }{c}_i}{\displaystyle \sum _{j=1}^n{\delta }_{ij}}\mathsf{\Delta }{x}_i\mathsf{\Delta }{x}_j.$$

(48)

$\mathsf{\Delta }{c}_i$ represents the difference of concentrations of the diffusion element of the i-th phase. On the other hand, the following can be obtained:

$${\delta }_{ij}=\left\{\begin{array}{cc}{\overline{c}}_i& i>j\\ (3{\overline{c}}_i+c_{i,\hspace{0.33em}i+1})/4,& j=1\\ {\overline{c}}_j& i<j\end{array}\right\}.$$

(49)

The following Equation is presented to transform Equation (48) for a single-phase Fe₂B boron coating: ${\overline{c}}_i$ represents the average value of boron concentration in the i-th phase, while $c_{i,i+1}$ represents the lower limit of boron in the i-th phase, and $\mathsf{\Delta }{x}_i$ represents the thickness of the boronize coating of the i-th phase, as determined by Equation (7). It is important to note that formal terminology is necessary for clear and concise communication in academic settings.

$$D_{F{e}_{2}B}=\frac{1}{2t_u\mathsf{\Delta }{c}_1}{\delta }_{11}{\left(\mathsf{\Delta }{x}_1\right)}^2=\frac{1}{2t_u\mathsf{\Delta }{c}_1}{\delta }_{11}{u}^2=\frac{2{\delta }_{11}}{4t_u\mathsf{\Delta }{c}_1}{u}^2.$$

(50)

Equation (50) represents the diffusion coefficient of the Fe₂B phase; the parameters on which it depends are expressed as follows: $\mathsf{\Delta }{c}_1=C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}=0.17\hspace{0.33em}\mathrm{w}\mathrm{t}.\%$, ${\delta }_{11}=(3{\overline{c}}_1+{\overline{c}}_{12})/4=8.893\hspace{0.33em}\mathrm{w}\mathrm{t}.\%$, ${\overline{c}}_1=(C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B})/2=8.915\hspace{0.33em}\mathrm{w}\mathrm{t}.\%$, and ${\overline{c}}_{12}=C_{low}^{F{e}_{2}B}=8.83\hspace{0.33em}\mathrm{w}\mathrm{t}.\%$, and $u^2=4D_{F{e}_{2}B}{\epsilon }_{F{e}_{2}B}^2{t}_u$. Thus, from Equation (50), we obtain the following:

$${\epsilon }_{F{e}_{2}B}^2=9.6\times {10}^{-3}.$$

(51)

4. Results

4.1. SEM Cross-Sections of the Boronize Coatings

Figure 9 depicts the cross-sectional views of boronize coatings formed on the surface of an ASTM A681 steel, subjected to a treatment temperature of 1273 K over varying periods ranging from 2 to 8 h. The morphology of the boronizes is characterized by an acicular shape, which indicates perpendicular growth relative to the material’s surface. This observation suggests that the boronizes formed through an epitaxial mechanism, in which the development of the crystals is aligned with the underlying crystal structure of the steel. The results of this study have important implications for the design of high-performance materials and coatings, as well as for the development of novel processing techniques that leverage the unique properties of boronize coatings.

The obtained boronize coating thickness ranged from 58 ± 9.61 µm after two hours to 96 ± 15.9 µm after 8 h at a constant temperature of 1273 K. A research study by VillaVelazquez-Mendoza [29] revealed that the process temperature had a significant influence, approximately 67%, compared to the treatment time, approximately 16%, on the kinetics of boronized coatings in AISI 1018 steel, according to ANOVA analysis.

Further, Figure 10 illustrates the energy-dispersive spectroscopy analyses performed on the boronized coatings’ cross-sections generated at 1273 K for 8 h at two distinct locations. The results of an EDS analysis of the surface, as presented in Figure 10a, indicate the presence of several elements, including boron, carbon, manganese, silicon, and iron. The formation of the Fe₂B phase under boronizing conditions involves a chemical reaction between boron and the base element, iron. However, due to its characteristic low-intensity peak, EDS analysis cannot reliably identify the boron element. Notably, the distinctive boron peak is absent near the interface in Figure 10b, while other components, such as carbon and silicon, which are insoluble in iron boronize, migrate towards the substrate.

4.2. XRD Analysis

Figure 11 presents the X-ray diffraction patterns of the diffraction peaks obtained from the analyzed surface of ASTM A681 steel boronized after 2 and 8 h at 1273 K for both treatment times. In both images, distinct peaks of the Fe₂B phase are evident, and their diffracted intensities are highly similar.

The X-ray diffractogram in Figure 11 provides evidence of the presence of Fe₂B coatings in ASTM A681 steel. The most prominent peak in the diffractogram corresponds to the crystallographic plane (002). At the same time, the observed variation in diffracted intensities is attributed to the growth of the coatings with a well-defined texture along the most favorable crystallographic direction [001]. This texture minimizes the growth-induced stress [30]. The literature suggests that the phase composition of boronizing coatings on iron and Armco steels depends on both the amount of boron present in the solid boronizing powder mixture and the type of boronizing agent used in other boronizing methods [15].

4.3. Coefficient of Friction

Figure 12 shows the variation of the coefficient of friction concerning sliding distance on the surface of untreated and boron-treated samples at 1273 K for 8 h. The tribological behavior of the two surfaces under dry conditions was evaluated by pressing a diamond indenter on the tested surfaces. Results indicate a significant difference in the coefficient of friction between the two surfaces, owing to the formation of a hard boronize coating on the boronized sample’s surface. The average coefficient of friction on the boronized surface ranged from 0.485 to 0.256, while that of the untreated substrate was recorded between 0.646 and 0.781. The study further compared the results with previously published literature, and the findings agreed with the data reported by other authors, regardless of the bidding conditions. The survey by Türkmen et al. [31] investigated the evolution of the coefficient of friction (COF) versus sliding distance in the case of steel specimens with and without boronizing treatment.

4.4. Evaluation of the Activation Energy of Boron in Diiron Boronize for ASTM A681 Steel

The present study aimed to calculate boron diffusion coefficients using five different mathematical models, namely Equations (8), (15), (25), (33), and (51), along with the parabolic growth law represented by Equation (7). The calculation process required the determination of experimental parabolic growth constants, as provided in

Table 2. Experimental kinetic constants at the growth interface (Fe₂B coating/substrate) with the presence of incubation times.

Experimental Temperature T (K)	Experimental Kinetic Constant $4{\mathit{\epsilon }}_{\mathit{F}{\mathit{e}}_2\mathit{B}}^2{\mathit{D}}_{\mathit{F}{\mathit{e}}_2\mathit{B}}$ (m2⸱s−1)	Incubation Time ${\mathit{t}}_0^{\mathit{F}{\mathit{e}}_2\mathit{B}}$ (s)
1123	1.4661 × 10−13	1941.9976
1173	3.9790 × 10−13	1941.9725
1223	9.2007 × 10−13	1941.9906
1273	2.0991 × 10−12	1941.9869

. The coefficients were computed based on the slopes of the curves that depict the relationship between coating thickness squared and exposure time (refer to Figure 13). Notably, the incubation periods remained consistently similar across the chosen temperature range.

Figure 14 shows the relationship between the thickness of the iron boronize (u) Fe₂B coatings with treatment time and temperature. The boronize coating thickness increases with treatment time and temperature [32].

Five distinct models were utilized to evaluate the diffusion coefficients of boron in the Fe₂B phase, with a maximal boron concentration of 9 wt.%. The results were fitted with the Arrhenius relation [32], as depicted in Figure 15.

Table 3. Evaluated values of boron diffusivities in Fe₂B coating using five diffusion models.

Pre-Exponential Factor D0 (m2·s−1)	Activation Energy Q (kJ·mol−1)	Diffusion Model
2.257 × 10−2	209.867	Linear
2.233 × 10−2		Error function
2.280 × 10−2		Goodman integral
2.210 × 10−2		Optional diffusion
2.233 × 10−2		Boltzmann-Matano inverse diffusion

presents the grouping of the activation energies of boron in the Fe₂B phase and their corresponding pre-exponential factors for ASTM A681 steel based on five distinct models. The energies in question were derived from the slopes of $ln{D}_{low}^{F{e}_{2}B}=f(1/T)$, as per the Arrhenius relation [32].

Table 4. Comparison of boron activation energy values from this work with literature data.

Steel	Activation Energy Q (kJ·mol−1)	Method	References
Hardox-450	157.9	Empirical growth law	[32]
ASTM A36	161.0	Diffusion model	[9]
AISI 1018	91.20–155.2	Empirical growth law	[33]
Nimonic 80A-Alloy	190.93	Diffusion model	[34]
X65Cr14 stainless steel	206.53	Empirical growth law	[35]
AISI P20	200.0	Empirical growth law	[36]
SAE 1020	183.15	Empirical growth law	[31]
ASTM A681	209.867	Diffusion model	This study
		Diffusion model
		Diffusion model
		Diffusion model
		Diffusion model

compares the values of boron activation energies estimated in the present work with those reported for some boronizing steels [9,31,32,33,34,35,36].

Two supplementary boronizing conditions were employed to verify the validity of five models experimentally. The scanning electron microscope (SEM) cross-sectional images in Figure 16 illustrate the findings. The boronize incubation period required to generate the Fe₂B coating on ASTM A681 steel was determined to be 1941.9569 s through analysis of the experimental data fit in Figure 15. Equations (8), (15), (25), (33), and (51), in conjunction with the parabolic growth law (as outlined in Equation (7)), were utilized to calculate the estimated Fe₂B coating thickness under conditions of 1223 K for 3 h and 1273 K for 1.5 h.

Table 5. Comparing the predicted values of the thickness of the diiron boronize coating with the experimental value obtained at 1273 K for 1.5 h.

Boronizing Condition	Empirical Coating Thickness (µm)	Thickness of Simulated Coating (µm)	Models
1273 K For 1.5 h	83.33 ± 10.56	85.155	Linear
		85.602	Error function
		84.705	Goodman integral
		86.046	Optional diffusion
		85.602	Boltzmann-Matano inverse diffusion

and

Table 6. Comparing the predicted values of the thickness of the diiron boronize coating with the experimental value obtained at 1223 K for 3 h.

Boronizing Condition	Empirical Coating Thickness (µm)	Thickness of Simulated Coating (µm)	Models
1223 K For 3 h	88.88 ± 11.48	90.871	Linear
		91.348	Error function
		90.392	Goodman integral
		91.823	Optional diffusion
		91.348	Boltzmann-Matano inverse diffusion

provide the results of these estimations.

The study revealed that the estimated values of boronize coating thickness, as determined by Equations (8), (15), (25), (33), and (51), are consistent with the experimental outcomes presented in Table 5 and Table 6, for a Fe₂B phase surface boron content of 9 wt.%. Therefore, in industrial applications of this steel type, knowledge of the variables that govern the boronizing process is critical in obtaining the optimal Fe₂B coating thickness. The thickness of the boronize coating plays a crucial role in protecting against wear and tear, with thin coatings considered effective against adhesive wear and thicker coatings intended to combat abrasive wear. The optimal thickness of the boronize coating varies according to the steel type used, ranging from 50 to 250 µm for low-carbon and low-alloy steels and from 25 to 76 µm for high-alloy steels [37].

5. Discussion

5.1. Growth Kinetics of the Boronized Coatings

In this section, we discuss the differences between the five mass transfer models that have been used to determine the growth kinetics of the boron coatings formed on the surface of ASTM A681 steel; according to the results obtained, it can be noted that the estimated values with the five mathematical models of the boron activation energy (Q = 209.867 kJ·mol⁻¹) for ASTM A681 steel is the same value. Likewise, the calculated values for the pre-exponential factors of the linear model (D₀ = 2.257 × 10⁻² m²/s), error function (D₀ = 2.233 × 10⁻² m²/s), Goodman integral (D₀ = 2.280 × 10⁻² m²/s), optional diffusion (D₀ = 2.210 × 10⁻² m²/s), and Boltzmann–Matano inverse diffusion (D₀ = 2.233 × 10⁻² m²/s) show that there is minimal variation. To determine how this coincidence is possible, we analyze the model of the error function (see Equation (14)), developing in a Taylor series to the parameters $exp(-{\epsilon }_{F{e}_{2}B}^2)$ and $erf({\epsilon }_{F{e}_{2}B})$ considering only the first order; we obtain the following: $exp(-{\epsilon }_{F{e}_{2}B}^2)\approx 1$ and $erf({\epsilon }_{F{e}_{2}B})\approx 2{\epsilon }_{F{e}_{2}B}/\sqrt{\pi }$, in such a way that Equation (14) is transformed into the following: ${\epsilon }_{F{e}_{2}B}^2=C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}$, which coincides with Equation (8). Continuing with the analysis of Goodman’s integral model of mass transfer (see Equation (25)), developing in a Taylor series to the term $\sqrt{1+4(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0)}$, considering the first order, the following is obtained: $\sqrt{1+4(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0}\approx 1+2(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B}-2C_0)$, so Equation (25) is expressed as follows: ${\epsilon }_{F{e}_{2}B}^2=C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}$, as in the previous case, and coincides with the expression given in Equation (8). For the case of the optional mass transfer diffusion model (see Equation (32)), in the term $ln(t_u/t_0^{F{e}_{2}B})$ with $t_u=t-t_0^{F{e}_{2}B}$ and $t_0^{F{e}_{2}B}$= 1941.9569 s, for t = 2 h, 4 h, 6 h, and 8 h considered, it can be approximated to be as follows: $ln(t_u/t_0^{F{e}_{2}B})$ ≈ 1.95, so Equation (32) can be expressed as follows: ${\epsilon }_{F{e}_{2}B}^2=1.025(C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B})$. It is also observed that it approximates the expression given in Equation (8). Finally, in the Boltzmann–Matano inverse diffusion model, the term ${\epsilon }_{F{e}_{2}B}^2=C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/(3(C_{up}^{F{e}_{2}B}+C_{low}^{F{e}_{2}B})/2+C_{low}^{F{e}_{2}B})/2$ is equivalent to the term $C_{up}^{F{e}_{2}B}-C_{low}^{F{e}_{2}B}/C_{low}^{F{e}_{2}B}-2C_0+C_{up}^{F{e}_{2}B}$. Even though diffusion models are different, we have an equivalence between the five mathematical mass transfer models for the growth kinetics of the boronized coatings. We also determined a common adimensional constant (${\epsilon }_{F{e}_{2}B}^2$) for the five models. Because of its simplicity, the linear model is the most suitable for studying growth kinetics; the study of the growth kinetics of boronized coatings is essential in the field of metallurgy and materials engineering, especially in applications where the wear resistance, hardness, and other mechanical properties of metallic materials are sought to be improved. Understanding the growth kinetics of boronize coatings is essential to optimize and automate surface treatment processes, control the quality of the formed coatings, and design materials with specific properties for industrial applications requiring wear and corrosion resistance.

5.2. Microstructural Characterization of the Boronize Coating (SEM)

The images presented in Figure 9 illustrate the vertical sections of boronized samples obtained through the implementation of scanning electron microscopy (SEM). These visual representations unveil an internal structure characterized by two distinct regions: a surface boronize coating and an underlying matrix region. Upon thoroughly examining the formation of boronize coatings in all samples, it was observed that the coatings exhibited continuous coverage on the material surface with consistent and uniform thickness. Figure 9 depicts the single-phase sawtooth morphology (Fe₂B) of boronized coatings. Within the tetragonal lattice of Fe₂B, the highest atomic density of boron aligns along the crystallographic direction [001], which is perpendicular to the sample surface. This orientation facilitates boron diffusion within the lattice, establishing it as the favored growth direction for Fe₂B crystals. Consequently, the crystals manifest a sawtooth morphology, notably prominent in iron boronize coatings. The development of this morphology is contingent upon various factors, including the presence of alloying elements in the base material, their respective concentrations, process temperature, and duration. Typically, boronized coatings formed during the boronizing processes of unalloyed and low-alloy steels exhibit a sawtooth morphology. This particular structural configuration significantly enhances the interconnection and adhesion between the substrate material and the boronized coating. Upon careful examination of the microstructure photographs of all samples in this study and considering the circumstances, it can be deduced that the boronize coatings are exclusively composed of the Fe₂B phase. The XRD analysis presented in Section 4.2 further substantiates this deduction. The determination of the average thickness of boronized coatings on boronized samples involved the use of optical microscope software (Image-Pro Plus 6.3). To ensure precise data in assessing the boronize coating’s thickness, it is vital to conduct a comprehensive examination encompassing a significant number of measurements from various segments of the coating.

5.3. Chemical Characterization (XRD and EDS)

X-ray diffraction analysis was performed on boronized samples to determine the phase or phases forming the boronize coating. Whether the boronize coating has a single-phase or multi-phase structure can affect the tribological, mechanical, and adhesion properties of the boronized material. The XRD patterns obtained from the analysis of two samples boronized at a temperature of 1273 K for two distinct time durations of 2 h and 8 h are depicted in Figure 11. Based on the analysis results, it was determined that the boronize coating had a single-phase (Fe₂B) structure. XRD analysis of the relevant samples used a broader range of diffraction angles. The XRD analysis data confirm the idea of a single-phase coating structure formation, as expressed in the microstructural examination results in Section 4.2. The vertical cross-sectional SEM image of the boronized sample and EDS analysis results performed in different regions are shown in Figure 10. The EDS analyses aimed to determine the possible presence of elements such as Fe and B in the boronize coating and their concentrations. Elemental analysis was performed on a total of two regions. Upon evaluation of the results obtained from regions 1 and 2 of the sample (at 1273 K for 8 h), the presence of the B element was determined. It is well established that the boronize coating formed through boronizing unalloyed or low-alloyed iron-based materials may contain a Fe₂B or Fe₂B+FeB phase. The Fe-B phase diagram shows that the Fe2B and FeB phases consist of approximately 9 wt% and 16.4 wt% B, respectively. The amount of element B in the boronize coatings was determined to be at least 6.03% and 8.15% by weight. Comparing these values with the theoretical weight per cent of element B in the Fe₂B phase, it is clear that they are relatively close. It should be noted that the EDS method may not accurately detect elements with low atomic numbers, particularly those below eight. As a result, it was concluded that the Fe₂B coating was formed on the surface of ASTM A681 steel by boronizing processes since the determined values were below the amount of boron in the FeB structure and closer to the amount of boron in the Fe₂B structure.

5.4. Pin-on-Disc Test (COF)

The graphs depicted in Figure 12 present the variation in COF (coefficient of friction) observed during the wear test of the samples. The COF values were derived from the friction forces recorded about the sliding distance, calculated under steady-state wear conditions (200–800 m). Upon analyzing the initial COF values obtained after the wear test, it is evident that the initial friction coefficient of the unboronized sample is approximately three times higher than that of the boronized sample. The low coefficient of friction (COF) exhibited by boronized steels can be ascribed to the interaction between iron (Fe) and boron (B) within the boronize coating and the oxygen present in the surrounding atmosphere [38]. This observed decrease in friction is postulated to be a result of forming an oxide film. In the study, it was found that the initial COF values of boronized sample were lower than those of unboronized sample.

6. Conclusions

This study focused on the study of the growth kinetics through five mathematical models of mass transfer of boronize coatings formed on the surface of ASTM A681 steel commonly used in the manufacture of cutting tools, dies, and various machinery, where their exceptional properties can be very advantageous, using the powder-pack boronizing process. The process involved the use of a mixture of powders containing 33.5% by weight of B₄C (primary boron source), 5.4% by weight of KBF₄ (activator), and 61.1% by weight of SiC (diluent). The boronizing processes were carried out at temperatures between 1123 and 1273 K, with 2, 4, 6, and 8 h of treatment. The main results are listed below:

• As a result of the boronizing processes, the formation of a single-phase boronize coating (Fe₂B) with sawtooth morphology, homogeneously distributed on the surface of the substrate material, was observed. The highest boronize coating thickness was obtained in the boronized sample at 1273 K for 8 h.
• The results indicated that the thickness of the Fe₂B boronize coating increased proportionally to the treatment time and temperature.
• The growth kinetics of the boronize coating were observed to follow the parabolic rate law ($u^2=4D_{F{e}_{2}B}{\epsilon }_{F{e}_{2}B}^2{t}_u$).
• The activation energy for the growth of the boronized coating was determined to be 209.867 kJ·mol⁻¹ for all five mathematical mass transfer models.
• According to the results presented in Figure 12, the values of the coefficients of friction (COF) obtained after the wear test show that the coefficient of friction of the unboronized sample was approximately three times higher than that of the boronized sample.
• The growth kinetics of boronized coatings were investigated by proposing five mathematical mass transfer models: linear, error function, Goodman integral, optional diffusion, and inverse Boltzmann–Matano diffusion. The results revealed a dimensionless constant (${\epsilon }_{F{e}_{2}B}^2$) shared by all models, concluding that the models are equivalent.
• The confirmation of the presence of the Fe₂B phase was accomplished through the use of X-ray diffraction (1273 K for 8 h).
• Two extra processing parameters (1223 K for 3 h and 1273 K for 1.5 h) were employed to validate the five mathematical models of mass transfer empirically. According to Table 5 and Table 6, the predicted coatings’ thicknesses were in line with the experiments.

Consequently, the growth kinetics of the boronized coatings can be studied from the linear model due to its simplicity and equivalence with the other models. The practical importance of the growth kinetics of the boronize coatings is fundamental to understanding and controlling the boronizing process because the growth kinetics influences the thickness of the boronize coating, which in turn affects the final properties of the material, such as wear resistance, corrosion resistance, and surface hardness. The boronizing process has experienced significant advances in recent decades and is expected to continue to develop in several directions; one of them is to understand the growth kinetics with other modeling alternatives, such as artificial neural networks, because it is essential to optimize the boronizing process to maximize efficiency and minimize costs, and thus ensure consistent production. Finally, the proposed mathematical models of mass transfer allow estimating the thickness of the boronize coatings on the surface of the material, which favors the control of factors such as temperature, time, and boriding medium to achieve the desired properties in the surface coating. For example, in the aerospace industry, it is essential to control the thickness of the boronized coating, because wear and corrosion resistance are critical. Components such as turbine shafts, bearings, and structural parts are hardened through boronizing treatment.