3. Results and Discussion
The process of cataphoresis varnishing can be viewed from two angles. The first is the electro-osmotic theory of cataphoresis. It is assumed that an electric bi-layer emerges at the interface between the solid and liquid phase. A part of this double layer is deposited as a liquid coating on top of the solid phase and the other part is scattered in the adherent liquid layer. As long as the solid phase can move freely in the liquid, the tangential component of the electrical force sets the suspended particles into motion. Cataphoresis varnishing uses the principle of cathodic organic coating creation based on epoxy or acrylic cataphoresis materials. Water-soluble cationic coatings with very low organic solvent content contain particles of varnish in the form of polymer cations. Thus, if an electric field is created in this system, with the solid phase particles scattered in the liquid phase, the particles begin to move in the direction of the electric field under the influence of the electric force. A direct current between the coated part, which is the cathode in the cataphoresis varnishing process, and an anodic counter electrode (anode) creates an electric field that becomes the carrier of the polymer cations that travel towards the cathode. In the course of the reactions with hydroxyl ions resulting from the breakdown of water, the solubility is suppressed on the cathode, and the organic coating deposition process is activated on the surface of the cathode. The second view of the cataphoresis varnishing process is the ionic theory of cataphoresis. In this theory, suspended particles are considered to be high-molecular-weight electrolyte molecules. These molecules then disassociate into high-power ions and associated electrolytic ions, which carry the same amount of electrical charge, but of the opposite polarity. The moving particles of the solid phase, which are suspended in the liquid phase under the influence of the electric field, are seen as electrolytic ions in electrolyte solutions. The electric charges of the ions are affected by an electric force in the electric field, the magnitude of which is determined by the product of the magnitude of the electric charge and the magnitude of the electric field. This force accelerates the ion, which is, at the same time, hampered by the movement of the frictional force emerging in the liquid environment. The ion is, at this time, considered to be a sphere with a radius corresponding to its resistance in the given environment, defined by the Stokes equation of resistance of a sphere in a liquid. The friction force is proportional to the velocity of the ion. Upon the introduction of the electrical charge, a steady state occurs. The mobility of the ions is directly proportional to the power and inversely proportional to the radius of the ion [
28,
29].
Since the implemented methodology of the experimental verification (Design of Experiments) represented a statistical approach, the subsequent analysis of the experimentally obtained data was, too, carried out using mathematical–statistical procedures. The initial analysis of the applied model pointed to the fact that the proportion of the variability of the measured thickness of the cataphoresis coating was 86.70225% and the adjusted index of determination, determining the degree of explanation of the data variability by the model, was 85.8775%. The average thickness of the cataphoresis layer formed, covering all the individual trial runs, was th = 24.464 ± 3.677 μm.
The table of the variance analysis (
Table 2), as a basic requirement for the correctness of the regression model, enabled us to conclude that the variability caused by random errors was significantly lower than the variability of the measured values explained by the model, and the value of the achieved significance level (
p) indicated the adequacy of the model used, based on the Fisher–Snedecor test criterion. Another view of this analysis is through assessing the adequacy of the model itself and is based on the very essence of the variance analysis. For testing the null (
H0) statistical hypothesis, which followed from the nature of the test and said that none of the effects (factors) used in the model effected a significant change on the examined variable, it followed that the achieved level of significance (
p) was less than the selected level of significance
α = 0.05, and it could be concluded that we did not have enough evidence to accept
H0 and we could say that the model was significant [
30].
The applied model was further tested in the so-called insufficient model adaptation error test (
Table 3), where we tested the scatter of the residues and scatter of the data measured within the groups; thus, we tested the premise of whether the regression model adequately described the observed dependence. Based on the error test of insufficient model adaptation, due to the achieved significance level of 0.1853, a zero statistical hypothesis could be accepted at the selected significance level of α = 5% and it could be said that the scatter of the residues was less than or equal to the scatter within the groups and, therefore, the model could be considered sufficient.
Based on the above assumptions and their fulfillment (
Table 2 and
Table 3), the following table (
Table 4) presents an estimate of the regression model parameters with testing the significance of the individual effects and their combination at the significance level α = 0.05.
The results shown in
Table 4 thus enabled the building of a predictive mathematical-statistical model at a coded scale:
Since the DoE methodology worked with a code scale, in order to ensure the numerical and statistical correctness of the results, it was necessary to convert Equation (3) to the scale of the original variables, the natural scale. Considering that the code scale represented the DoE standardization of the variable input factors, it was necessary to use the following equation to convert to the natural scale:
where
x(
i) represents the original basic variable,
i = 1, 2, …,
n is the number of basic factors,
xmax is the maximum value of the original variable
x(
i), and
xmin is the minimum value of the original variable
x(
i).
Thus, when using the regression model (3) in the code scale, taking into account the conversion Equation (4) for the individual variable input factors and subsequent adjustment, it was possible to make a notation of a prediction equation for the thickness of the cataphoresis layer in the form of:
The analysis of
Table 4 showed that the largest share in explaining the variability in the parameter under study, the thickness of the cataphoresis layer per absolute element of the model (intercept), which was involved in changing the thickness value of the layer, was that of 57.387%. From a methodological point of view, the absolute element of the model was characterized by “neglected” influences, which we kept at a constant level in the experiment (especially the chemical characteristics of the cataphoresis electrolyte, current density, and anode-to-cathode ratio), or we did not consider them. If we neglected this model element and subsequently analyzed only the basic variable input factors, we would have come to the conclusion (
Table 4) that the most significant factor that affected the thickness of the layer formed was the voltage (
U). It accounted for 33.82% of the change in the thickness of the layer. The second most significant element of the model (5) was the deposition time in the cataphoresis varnishing process (
tKTL), accounting for 28.67% of the change in the thickness. At the same time, the interaction of the voltage and deposition time accounted for 20.25% of the change in the thickness of the layer formed. The nonlinear model elements (5), namely, the voltage squared and the deposition time squared, accounted for 5.94% and 11.32%, respectively, of the change in the thickness of the created layer. As seen in
Table 4 and model (5), it was clear that the processes of the surface treatment of the metals, including the cataphoresis varnishing, were best described by non-linear models with a significant influence and mutual interaction of the individual factors. The model (5) also needed to be expanded and modified by the influence of the chemical factors acting in the process of the cataphoresis varnishing. The model (5) represented a steppingstone to a comprehensive analysis of the cataphoresis varnishing process using a statistical approach. The statistical approach was chosen because the studied layer parameters were understood as random variables in the mathematical sense [
31].
The plotted thickness of the cataphoresis layer formed during the respective deposition times in the cataphoresis coating process under various voltages is shown in
Figure 2.
The graph shows that, by increasing the deposition time of the varnishing medium, the thickness of the layer formed was reduced under different varnishing voltages. A varnishing voltage of 200 V and a deposition time of the varnishing medium of 3 min affected the layer thickness the most. The thickness of the layer increased during the 3 min deposition period, after which, the thickness of the layer decreased. This was due to a low electric current, which caused the coagulation of the paint on the surface where it stopped; thus, the coated part became electrically non-conductive. With an increased varnishing medium deposition time, the value of the electric current decreased due to an increase in the thickness of the deposited layer th, and the value of the current decreased to zero.
Under a 220 V voltage, the thickness of the coating increased for 5.5 min when the coating also reached its maximum thickness. After this time, the thickness of the coating decreased. Increasing the varnishing voltage to 240 V meant increasing the deposition time of the varnishing medium up to 8 min, without significantly affecting the thickness of the layer. A further increase in the varnishing voltage resulted in an accelerated layer formation and a similar change was observed when the varnishing voltage was increased to 260 V and 280 V, when the thickness of the layer reached its maximum values throughout the deposition time of the varnishing medium t in the electrolyte. This phenomenon could be attributed to the color deposition technology that ran in the following sequence: water electrolysis, ion migration (electrophoresis), the coagulation of the polymer on the cathode, and water ejection via osmotic pressure. This phenomenon, as such, could be explained by Ohm’s law, which says an electric current of a constant voltage is created between a cathode and anode, which can be described by the equation U = R*I. Cataphoresis varnishing works on the principle of creating cathodic organic coatings based on epoxy materials. Cationic coatings soluble in water contain a small number of organic solvents and, at the same time, particles in the form of polymer cations.
Once the coating was deposited on the cathode in the process of cataphoresis, the resistance reached its maximum values. The layer ceased to be conductive and became insoluble in water again. This deposited layer needed to be subsequently cured in a reaction with another polymer. At this stage, hydroxyl groups along the molecular chain in the cationic resin were applied, which reacted with isocyanates (they were equally present in the resin) to form urethane compounds.
The function gradient (5), i.e., the direction of the steepest addition to the layer thickness under the input parameters examined, namely the voltage (
U) and the deposition time in the course of the cataphoresis varnishing (
tKTL), is defined by the vector:
The relationship (6) thus defines the direction, depending on the input variables, in which the function (5) grew the fastest, that is, the direction where the thickness of the forming layer reached its maximum in the shortest possible time.
In terms of the cataphoresis layer formation, based on the mathematical–statistical models (5) and (6), it is possible to define the layer formation rate as the first derivative of the function (5), according to the deposition time (
tKTL):
Thus, Equation (7) represents the direction of the steepest rise in the function (6) in the direction of the voltage. Thus, from Equation (7), it follows that the rate of formation of the cataphoresis layer under the given experimental verification conditions (
Table 1) was a function of the voltage and deposition time in the process of the cataphoresis varnishing. In accordance with theoretical knowledge and on the basis of Equation (7), we can conclude that, by increasing the voltage, the rate of deposition of the cataphoresis layer also increased, and, on the other hand, by increasing the deposition time, this rate in the cataphoresis varnishing process decreased. The decrease in the rate of the formation of the cataphoresis layer and the influence of the deposition time depended on the electrical properties of the forming layer. Considering the fact that the layer formed during cataphoresis was electrically non-conductive, its electrical resistance must have inevitably increased with an increase in its thickness; therefore, the rate of its formation must have decreased. However, if we wanted to ensure a constant rate of formation of the layer throughout the entire deposition period in the cataphoresis varnishing process, we would have to increase the voltage in proportion, as per Equation (7). The rate of the formation of the layer is a fairly important indicator of the cataphoresis varnishing process. However, there are two opposing requirements of the rate of the layer formation. On the one hand, there is a requirement to achieve the highest possible rate of layer formation, thereby reducing the time required for the cataphoresis varnishing to run its course, which results in an increased economic efficiency of the process itself. The counter requirement stems from the process of layer formation in relation to its quality. If the rate of the layer formation is too high, the hydrogen that emerges on the surface of the treated structural part does not have enough time to "escape" the surface, and the resulting layer “traps” it in the surface of the part. However, in the process of polymerization, this trapped hydrogen creates defects in the layer in the form of craters. Further research is needed to determine the optimal value for the rate of deposition, taking into account the basic requirements above [
32].
Equation (7) is a statistical equation and, therefore, within, it holds only the input variables’ intervals and the factors used (
Table 1). Its extrapolation beyond these factor values intervals may lead to incorrect results and conclusions.
The second partial part of the analysis, shown in
Table 1, was devoted to the evaluation of the thickness of the cataphoresis layer in relation to the polymerization time (
tpol), using three different times as part of the experiment plan, namely 15 min, 20 min, and 25 min, respectively, for each combination. The basic graphical representation of the influence of the polymerization time on the thickness of the layer created in individual combinations of the input factors (
U,
tKTL) is shown in
Figure 3.
Figure 3 makes it evident that the polymerization time affected the resulting thickness of the cataphoresis layer in a relatively random manner. However, the polymerization process itself showed that, depending on the type of cataphoresis paint used, 10% and 20% of it was lost in the polymerization process. This was because the polymerization process did not directly participate in the formation of the cataphoresis layer, but affected its resulting properties. The average thickness of the cataphoresis layer formed after the polymerization at a constant temperature of 200 °C and a polymerization time of 15 min was 23.709 ± 0.192 μm. Here, it is necessary to say that in this analysis, all the values of the measured thickness were used, including repetitions of individual measurements (7565 measurements). For a polymerization time of 20 min at a constant temperature of 200 °C, the average thickness value of the formed layer was 24.349 ± 0.267 μm, and for a polymerization time of 25 min, the average thickness of the layer was 24.937 ± 0.289 μm.
Thus, the average difference in the thickness of the cataphoresis layer between the individual polymerization times was 0.613 ± 0.103 μm between the polymerization times of 20 min and 15 min, 0.619 ± 0.102 μm between the polymerization times of 25 min and 20 min, and finally, 1.220 ± 0.102 μm between the polymerization times of 25 min and 15 min (
Figure 4).
A graphic representation of the model verification (5) under the practical conditions of the production process is shown in
Figure 5. The verification was carried out at
UKTL = 240 V on the same samples, listed in the Material Selection and Technological process of production section, and under the same conditions as the main experiment.
As part of the analysis of the modelled thickness values of the cataphoresis layer created and the values obtained from repeated measurements of the verification experiment, we came to conclusion that the average deviation in all the measurements carried out was 0.632 µm (2.14%), while the lowest negative deviation in the calculated thickness of the layer measured and the lowest deviation in the model (5) was at the level of −1.801 µm (8.690%), and the maximum positive value of the examined difference was at the level of +2.306 µm (8.407%). At the same time, based on the Shapiro–Wilks test, it can be said that the residues showed a normal Gaussian distribution (
p = 0.233) at the selected level of significance, which indicated that the model (5) also met the last condition for the regression triplet analysis and could be considered correct. A graphical representation of the differences between the measured and modeled values of the thickness of the created layer is shown in
Figure 6.
The morphology of the surface was also of significant importance from the point of view of the quality of the created cataphoretic layer. The change in the morphology and structure of the surface of the created layer depended primarily on the conditions of the process of creating the layer, that is, on the operation of the cataphoretic painting itself.
Figure 7 shows the surfaces of the layers created at voltages of
UKTL = 220 V, 260 V, and 280 V with deposition times of
tKTL = 7.5 min and 6.0 min at a constant polymerization temperature of 200 °C, but with a different polymerization times: 15 min (a), 20 min (b), and 25 min (c). It is clear from the mentioned morphologies that the tension in the process of creating the cataphoretic layer had a significant influence. At a voltage of 220 V, in all cases (a, b, c), a relatively smooth surface without a distinct structure was scanned. At a voltage of 260 V, morphological changes began to appear in the form of a slightly distinct structuring of the surface, but in the presence of a significant defect in the form of craters. These craters could be attributed to the process of cataphoretic painting in the form of the binding of the hydrogen on the surface of the painted sample with its binding by the created layer and subsequent “explosion” in the polymerization process. However, this defect was no longer observed when using a voltage of 280 V, but the surface of the created layer already had a pronounced wrinkled structure. In general, it can therefore be said that, by increasing the tension in the process of creating a layer, the morphology of the created layer deteriorated and the created layer acquired a significantly wrinkled structure.
The confirmation of the above conclusions was carried out using additional experiments at a voltage of 400 V and deposition times of 5.0 min and 4.0 min, while the morphology of the surface of the cataphoretic layer is shown in
Figure 8. It is obvious that the surface morphology of the formed layer at a high voltage was significantly structured with very pronounced wrinkling; however, with a deposition time of 5.0 min, the surface was significantly more heterogeneous than that in the case of a deposition time of 4.0 min. Thus, in addition to the applied voltage, the deposition time of the KTL process also had an effect on the surface morphology of the created cataphoretic layer and, with an increase in the deposition time, a more pronounced heterogeneity of the surface occurred.
An important consequence of the defined predictive dependence (5) was a determination of the optimal values of the analyzed input variables (
U, tKTL). Due to the technological requirements placed on the thickness of the layer under formation, it was advisable to look for the maximum regression function (5). The general optimization problem was to select n decision variables
x1;
x2; …;
xn from a given implemented area, in such a way as to optimize (minimize or maximize) the purpose function:
The optimization problem was a non-linear programming problem (
NLP) if the purpose function was nonlinear or the implemented area was defined by nonlinear constraints. Then, the maximization of the general nonlinear programming is defined in the form of:
for restrictions:
where each of the constraints g
1 through
gm is defined. A special case is linear programming. The obvious relation for this case is:
and
Non-negative variables constraints can be included simply by attaching additional constraints:
In some cases, these constraints are considered explicit, as is any other issue in the delimited areas. In other cases, it is appropriate to consider them implicit if the non-negative constraints are manipulated, as is the case with simplex methods.
To simplify the proposition, let
x denote the vector of the control variables
x1,
x2, …,
xn, which represents
x = (
x1,
x2, …,
xn). The problem is more aptly written in the form:
according to the:
As in solving the tasks of linear programming, there are no restrictions on these formulations. When maximizing the f(x) functions and, of course, also when minimizing its f(x), the conditions of equality h(x) = b can be written as two separate conditions of inequality, h(x) ≤ b and −h(x) ≤ −b.
To optimize the thickness of the created cataphoresis layer, a regression model (5) was applied as a functional function and the interior point method was used for the nonlinear optimization. The ranges of the intervals of the variable input factors used were the basic constraints (
Table 1), which are defined as follows:
A MATLAB software product Optimization Toolbox was used to implement the optimization of the thickness of the cataphoresis layer created. The task of the nonlinear optimization, in our case, was to find the maximum of the problem, which is defined as:
where
x,
b,
beq,
lb, and
ub are vectors,
A and
Aeq are matrices,
c(
x) and
ceq(
x) are vector functions, and
f(
x) is a scalar function. The course of the optimization process itself, as an output from the optimization program, is shown in
Figure 9.
The result of the non-linear optimization process of the cataphoresis varnishing, considering only two variable technological factors, different voltages and deposition times in the process of the cataphoresis varnishing, was the determination of the maximum thickness of the purpose-built regression function (5). The maximum of the purpose-built function, while respecting the constraints given by Equation (16), was
thmax = 26.114 µm under the following technological conditions:
U = 240 V and
tKTL = 6.0 min. Therefore, in order to create the thickest layer possible, it was necessary to set these basic factors at a defined level [
33].
However, we must also define the limitations of the conducted experimental research. The conclusions of the submitted study are valid only in the range of the experimental conditions listed in
Table 1, which resulted from the applied statistical approach. A further limitation is imposed by the other relevant input conditions in the processes of degreasing, activation, and phosphating. Therefore, it will be necessary to expand the model (5) by including these impacts, thus defining the complex technological dependence of the process factors on the layer forming.