2.1. Adhesion Results of the Coating/Steel Systems
The changes of adhesion of EV coating/steel and EGF coating/steel under two environments are shown in
Figure 1. The scatter plots represent the experimental data and the lines represent the fitting results of grey models, which will be discussed in the following sections. The average values of dry adhesion of EV coating and EGF coating are 13.14 MPa and 12.43 MPa, respectively, and the latter is a little lower, which may due to the addition of glass flake. However, the dry adhesion will change to be wet adhesion after immersion in the water. Referring to
Figure 1, the declines in wet adhesion of the coatings under different pressures can be found during the entire test period. For the EV coating, wet adhesion under two pressure conditions decreased dramatically during the initial period; in particular, the degradation under hydrostatic pressure was greater than that under ordinary pressure. When the immersion time was 192 h, the wet adhesion under ordinary pressure and under hydrostatic pressure declined to a very low level, 2.15 MPa and 1.42 MPa, respectively. Several visible corrosion spots appeared on the substrate of the sample under ordinary pressure (
Figure 2a). However, there mainly were obvious blisters for the sample under hydrostatic pressure (
Figure 2c) at the same time, indicating that such poor wet adhesion could not ensure the protective performance of the organic coating. For the purpose of comparison, the macro morphology of samples under ordinary pressure after immersion 720 h is shown in
Figure 2b, and a large amount of macroscopic corrosion products could be observed. However, blistering was the main cause for coating degradation rather than corrosion under hydrostatic pressure (
Figure 2d). Microscopic corrosion morphologies showed that the corrosion was cystiform under ordinary pressure (inset in
Figure 2b), while, under hydrostatic pressure, corrosion was observed in areas under the blisters (inset in
Figure 2d). Consequently, different corrosion states were obtained under different pressure conditions.
The variations on EGF coating/steel followed a similar changing trend. The wet adhesion under hydrostatic pressure degraded greater than that under ordinary pressure. After serving 192 h, the values under ordinary pressure and under hydrostatic pressure declined to 3.21 MPa and 3.09 MPa, respectively. Macro morphology in
Figure 3a,c (serving 192 h) also indicated that the coating degradation under hydrostatic pressure was more severe for identical duration, with visible blisters and corrosion pitting. Finally, the failure states under ordinary pressure and hydrostatic pressure were mainly related to corrosion products (
Figure 3b) and blisters (
Figure 3d), respectively.
From the analysis above, the coating under hydrostatic pressure had a rapid loss of wet adhesion. The failure state was mainly related to the blisters, and dramatic adhesion degradation occurred before obvious corrosion of the substrate, which is different from that under ordinary pressure. In addition, EGF coating had a better protective performance than EV coating in the same condition, which was attributed to the barrier properties of glass flakes [
9]. The changes in glass transition temperatures of the coatings have proved this (
Table 1). In addition, comparisons of the fracture area of coatings after pull off tests were made to support the results above. In
Figure 4, both coatings before immersion had a mixed adhesive/cohesive fracture form. In other words, the adhesion of coating/steel interface in some areas is even stronger than the coating structure, so that the coating body first fractured when a pull off test was performed, implying that both coatings have a very high dry adhesion. Then, the fracture form of the EV coating turned into a complete adhesive fracture (48 h,
Figure 5a) followed by the failure fracture (192 h,
Figure 5b) under ordinary pressure. By contrast, the EV coating under hydrostatic pressure had bigger exposed areas at each time period (
Figure 5c,d). The EGF coating developed from mixed fracture (48 h,
Figure 6a) to complete adhesive fracture (192 h,
Figure 6b), and the coating under hydrostatic pressure had bigger exposed areas (
Figure 6c,d). The photos of fracture surface suggest that the coatings under hydrostatic pressure have a faster interface delamination, and the glass flakes can allay the reduction of wet adhesion.
From the results above, the failure mechanisms of the coating can be demonstrated. For the coating under ordinary pressure, inevitable permeation of the water gradually causes coating degradation and substrate corrosion. However, the failure behaviour of coating is mainly related to blister rather than corrosion under high hydrostatic pressure. This is because the water diffusion is accelerated by high hydrostatic pressure. More water is accumulated at the coating/steel interface, resulting in the presence of many small blisters. Therefore, the adhesion decreases rapidly. As the immersion continues, the blisters would connect to each other. A water film with corrosion products forms on the interface (
Figure 3d), at which time the coating fails eventually.
2.2. Establishment of the GM (1, 1) Models by GST
In GST, the white system is the system in which the information is completely defined. The black system is the system in which the information is undefined. The grey system is the system in which some information is defined and some undefined. The grey prediction method of GST is making the most of the available information, to predict the future trends of an uncertain system from only a limited amount of data [
21]. The GM (
α,
β) model is the core of grey prediction, where
α is the order of the differential equation and
β is the number of the variables.
The GM (1, 1) model, which is composed of a differential equation of first order with one variable, is the most widely used one among different grey prediction models due to its computational efficiency [
31]. In this work, GM (1, 1) was used to build the mathematical models on wet adhesion and conduct a further analysis of lifetime prediction of the coatings. In order to assess the precisions of the models, the first 12 groups of the original data were used to make the establishment, and the last four groups were stayed to examine the accuracy of the models. Taking the EV coating/steel system under 0.1 MPa, for example, the establishment of GM (1, 1) is shown as follows.
Defining a sequence
that denotes the wet adhesion of a coating/steel system at different immersion times, and the initial sequence is:
The corresponding time sequence is:
where
is the wet adhesion (non-negative value),
t is the immersion time, and
n is the sample size of the data. Therefore, the sequence of the EV coating/steel system under 0.1 MPa is:
The establishment of the corresponding grey model requires initial data with equal time intervals, and its specific time sequence is:
The generation treatment of initial data, such as Accumulating Generation Operator (AGO) or Inverse AGO, is the premise of the establishment of the grey model, which can smooth the randomness and strengthen the regularity of the sequence. Here, the grey sequence generation is performed by AGO, and the following monotonically increasing sequence
is obtained:
where
The generated mean sequence
of
is defined as:
where
The AGO sequence
and the generated mean sequence
of the EV coating/steel system under 0.1 MPa are calculated according to Equations (4) and (6), respectively. Then, the next step involves the building and solving of grey differential equation. GM (1, 1) is the first order differential equation model, and the form of equation is:
The least square estimate sequence of the grey differential equation is defined as follows [
31]:
Then, the GM (1, 1) whitening differential equation of
is therefore:
By solving Equations (10)–(12) based on the data above, the parameters a and u of the EV coating/steel system under 0.1 MPa were 0.0804 and 7.5778, respectively.
According to Equation (9), the solution of
at time
k is:
In the above,
p denotes the predicted value. Then, to obtain the predicted value of the primitive data at time
k, the Inverse Accumulating Generation Operator (IAGO) is used to establish the following GM (1, 1):
for
Replacing the parameter
k in Equation (14) with
t, the following equation can then be obtained:
where
t1 is the initial time of the time sequence
t, and
N is the interval of the arithmetic series. Equations (14) and (16) are grey models GM (1, 1) based on GST with different expressions for the wet adhesion of the organic coating/steel system. The same procedures were adopted for EV coating under 3.5 MPa and EGF coating under two pressures. Finally, the parameter values and the grey model GM (1, 1) formulas by Equation (16) are summarized in
Table 2.
Figure 1 shows the fitting results of the established grey models (the values of dry adhesion have been ruled out). It can be seen that the calculated values of GM (1, 1) models are in good agreement with the experimental data, and the fitted curves reflect the changing trend of wet adhesion with immersion time very well.