4. Results of Theoretical Simulation and Discussion
The experimental profiles of the solute atoms across the bimetallic interfaces have been fitted by the procedure that is reported in the previous
Section 2.2. Equation (1) characterizes the element transport controlled by solid state diffusion according to the Fick’s second law, so it expresses the theoretical concentration profiles. Assuming that the diffusivity does not change with concentration, it is possible to calculate the
D∙t parameter in Equation (1) that gives a good fit of the EDS experimental profiles, as shown in
Figure 15 for Ni diffusion at the interface ASTM A515 Gr.60/AISI 304 L.
To perform the fitting, preliminarily the starting concentration in the two diffusion regions, which correspond to the cladding alloy and the base steel, must be fixed according to the reference values in
Table 1 (10.07 and 0 wt. %, respectively), for both the experimental profile obtained by EDS and the theoretical one to be drawn by Equation (1). In the first case, this means imposing the average values of the concentrations measured experimentally in arbitrary units (A.U.), in the portions of the two regions that are not affected by the diffusion process, equal to the reference values. In the second case, it consists in imposing the reference values to the two plateaus on the theoretical profile, that is
cA = 10.07 and
cB = 0 wt. % in Equation (1).
By means of this preliminary setting, it is possible to couple the experimental curve, and the theoretical one that will fit the former, even though they are expressed in different measure units (A.U. and wt. %, respectively).
The theoretical fitting has been optimized by minimizing the sum of the squared distances between the experimental curve (
Figure 4a, and
Figure 15 blue line) and the theoretical curve (
Figure 15 red line), as calculated by means of Equation (1) within the interval outside the two plateaus of starting concentrations, obtaining a value of parameter
D∙t = 4.24 × 10
−12 m
2. The fitting has been performed by the evolutionary solving tool that was implemented in Excel, while assuming the sum of the squared distances between the experimental and theoretical curves as the fitness function of the algorithm, and the parameter
D∙t as the variable of the optimization problem. The algorithm showed a uniform convergence to stable solution (convergence parameter 0.0001), also as the main genetic parameters change (population size 100 ÷ 150, mutation rate 0.2 ÷ 0.5, variable seeding factor), and allowed to obtain a fitting solution with a coefficient of determination
R2 = 0.956. Furthermore, the solution has been validated by the Pandat
TM diffusion simulator module (PanDiffusion), which provided a diffusion curve (
Figure 15 yellow line) that was substantially superimposable on the fitted theoretical one.
As stated in advance (
Section 2.2.), in general terms, the parameter
D∙t expresses the shape factor of the diffusion profile, since it characterizes the trend of concentration in the diffusion transient and its slope across the Matano interface and, therefore, the diffusion depth, which determines the width of bonding in the direction orthogonal to the interface. Particularly, the value of the parameter
D∙t previously obtained, allows for analytically modeling the diffusion of Ni experimentally detected at the interface, so as to investigate the possible combination of temperature and time of the rolling process, which are not always made known by the producers, and the effect of their variations and instabilities on cladding.
By means of the fitting value of parameter
D∙t, setting some values of time (
t), the corresponding values of diffusivity (
D) can be fixed, and while using Equation (2) with
Do = 3.00 × 10
−4 m
2/s and
Q = 314 kJ/mol for Ni diffusion in γ-Fe [
30], the corresponding equivalent temperatures (
Teq.) can be calculated, to be considered as constant during the diffusion transient.
Table 4 gives the resulting couples of
Teq. and
t for Ni diffusion at the interface ASTM A515 Gr.60/AISI 304 L, together with values of the diffusion coefficient D.
For each duration of the diffusion transient, the sensitivity of the diffusion phenomena to temperature has been investigated.
Figure 16 shows the dependence of the
D∙t parameter on temperature with regards to the four diffusion times set in
Table 4, and highlights how the shorter is the duration of the diffusion transient, and the lower is the value of
D∙t, the higher is the robustness of the diffusion processes and cladding features to temperature fluctuations. This is due to the different slopes of the curves: the higher is the slope, the higher the effect of temperature variations on the
D∙t parameter, which is on the shape of diffusion curve, on diffusion depth, and subsequently on the width and properties of the bonding at the interface.
With regard to the interface ASTM A515 Gr.60/AISI 304 L examined here, the sensitivity of Ni diffusion curve and depth to the temperature variations can be quantified by the values of the slopes at the points corresponding to
D∙t = 4.24 × 10
−12; these values, as calculated by the derivative of
D∙t with respect to temperature at
T =
Teq. for the four diffusion times, are reported in the same
Figure 16.
This theoretical approach to interface characterization allows for assessing the robustness of its main properties to temperature fluctuations, which can occur during rolling processes.
Being the diffusion process, the more influenced by temperature variation, the longer the duration of the diffusion transient, in order to emphasize the effects of process temperature fluctuations,
t = 3600 s has been assumed.
Figure 17 shows some diffusion profiles that are simulated for different values of temperature fluctuations around the equivalent value (1438 K).
Due to these fluctuations, the value of diffusivity D changes, and the same does the parameter D∙t (being t fixed), with the consequent variation of the slope and depth of the diffusion transient, and therefore of the width of the zone involved in Ni diffusion, which deviates significantly from the experimental value of 10 μm when temperature changes starting from 50 K, in particular for positive fluctuations.
As highlighted before, an interface bonding effective in terms of mechanical stability is generally attributed to a sufficient alloying elements diffusion, being the interfacial shear strength of clad plates as stronger as thicker is the diffusion distance. Consequently, while positive fluctuations have a beneficial effect from the point of view of bonding strength, negative fluctuations can significantly reduce it.
With regard to the effect of the diffusion time, process settings that provide longer transient durations results are more effective for high bonding strength, since, on the contrary, their reduction tends to compress the diffusion depth. This conflicts with the beneficial effect that shorter diffusion transients have in terms of the robustness of the diffusion processes to temperature fluctuations, which can make the bonding process more uniform.
Furthermore, too deep carbon diffusion might lead to detrimental effects that are caused by the formation of decarburized and carburized layers. In this case, the main problem is to avoid carbide precipitations at the grain boundaries. Equation (1) has been used to theoretically simulate carbon diffusion from base steel to cladding austenitic steel, but, while taking into account that C diffusion in the γ-range is strongly concentration-dependent, its diffusivity D cannot be assumed constant varying the concentration. Therefore, the relation for D as a function of temperature and molar fraction that was proposed by Agren [
31] has been used, assuming
T =
Teq. and iterating the calculation of
c(
x) by Equation (1), so to take into account the effect of carbon content variation on its diffusivity along the diffusion direction. In this way, concentration profiles that have been simulated at various temperatures and diffusion times have been obtained, such as those shown in the
Figure 18.
Actually, these profiles are characterized by high penetration depths (even at temperature lower than 100 degrees of the equivalent one), but luckily the time that is required to carbide precipitation grows considerably with a decrease of carbon content. Referring to the diagram TTS (Time Temperature Sensitisation) shown in [
32], the time of 1 h, which is required for sensitisation at 600 °C when C = 0.042%, becomes equal to 10 h when C = 0.03%. Obviously, this time is too long for carbide precipitation to occur during the cooling stage of plates after hot rolling. This explains why the hardening effect due to carbide precipitation was detected up to a depth of about 450 μm (
Figure 3).
In any case, as for the diffusion of Ni, also for that of C, it occurs that the decrease in the diffusion transient duration results in a compression of the diffusion depth, as shown in
Figure 18, where this effect is highlighted for the two extreme diffusion condition (
t = 3600 s and
Teq. = 1438 K, solid lines;
t = 1800 s and
Teq. = 1477 K, dashed lines), and the corresponding temperature fluctuations more severe (±100 K).
The previous observations, as a whole, allow for completing the interpretation of the parameter D∙t. As a concentration profile shape factor, it determines the diffusion depth and the bonding width; moreover, it governs the relationship between diffusion time and temperature, which, on the basis of the previous evidences, must be appropriately set to obtain a well-balanced compromise between bonding strength and uniformity, and metallurgical stability. Therefore, it is possible to conclude that D∙t acts as a diffusion bonding efficiency parameter.
The experimental profiles of the solute atoms across the bimetallic interfaces have been fitted and validated for the other two interfaces analysed, ASTM A283/Alloy 59 and AISI 1010/Monel 400, according the same procedure of
Section 2.2. Additionally, in these two other cases, as for ASTM A515 Gr.60/AISI 304 L interface, before to perform the fitting, the starting concentration in the diffusion regions, which correspond to the cladding alloy and the base steel, have been set to the reference values of the diffusing element (reported in
Table 2 and
Table 3), to preliminarily couple the experimental profiles and the theoretical ones that will fit them. By fitting the EDS profiles for Ni diffusion (
Figure 11a and
Figure 14a), the values of 3.88 × 10
−12 and 9.08 × 10
−12 m
2, respectively, have been obtained for the parameter
D∙t. Setting the four values of time (
t) used before, the corresponding values of diffusivity (
D) have been fixed, and using Equation (2) the corresponding equivalent temperatures (
Teq.) have been calculated.
For the ASTM A283/Alloy 59 interface the same value of
Do and
Q for Ni diffusion in γ-Fe introduced in the previous analysis of the ASTM A515 Gr.60/AISI 304 L interface, have been used.
Table 5 illustrates the resulting couples of
Teq. and
t.
Interface ASTM A283/Alloy 59 presents substantial similarity with the diffusion conditions of Ni across the interface ASTM A515 Gr.60/AISI 304 L, with the exception of the different content of Ni in the cladding alloy. Therefore, the analyses of diffusion profiles simulated for different values of temperature fluctuations around the equivalent values provide results that are quite similar to those represented in
Figure 17 for the first interface, showing that: low duration of the diffusion transient increases the robustness of the diffusion process to temperature fluctuations; conversely, a high duration of the transient extends the diffusion depth, which is a beneficial condition for mechanical strengthening of the interface.
For the AISI 1010/Monel 400 interface, lower cladding temperatures have to be set, because of the melting temperature of Monel 400 (a value of solidus temperature just below 1600 K corresponds to the Ni-Cu alloy composition reported in
Table 3 [
33]). Hence, in this case, the diffusion of Ni in the α-Fe field has been considered, and the values
Do = 1.30 × 10
−4 m
2/s and
Q = 234.5 kJ/mol [
34] have been used in Equation (2).
Table 6 provides the resulting couples of
Teq. and t for Ni diffusion.
Figure 19 shows the diffusion profiles that were simulated for different values of temperature fluctuations around the equivalent value (1144 K) for
t = 3600 s. This simulation differs from the previous ones, in that it presents a change in the properties of the diffusion region, which arises due to positive high temperature fluctuations: for increases in
Teq. that are greater than +30 K, the diffusion of Ni leaves the α-Fe field to enter the γ-Fe field, being 1183 K the transition temperature between these two fields commonly assumed for diffusion in iron matrix [
35]; the result is the downfall of the diffusion mechanism for fluctuations of +50 K and +100 K, being well represented by the dashed curves in the figure, which would be a particularly insidious phenomenon if arisen during the cladding process.
Contrary to the first interface, in the last two cases analysed, the diffusion of carbon does not represent a critical issue, due to the negligible content that characterizes this element in the base material for the interface ASTM A283/Alloy 59 (
Table 2), and the absence of chromium in the cladding material for the interface AISI 1010/Monel 400 (
Table 3).