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Finite element analysis of the water diffusion behaviour in pigmented epoxy coatings under alternating hydrostatic pressure
Progress in Organic Coatings 123 (2018) 168–175
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Finite element analysis of the water diffusion behaviour in pigmented epoxy coatings under alternating hydrostatic pressure
T
⁎
Rui Liua,b, Li Liua,c, , Fandi Mengc, Wenliang Tiana, Ying Liua, Ying Lia, Fuhui Wangc a Corrosion and Protection Division, Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Wencui Road 62, Shenyang 110016, China b School of Materials Science and Engineering, University of Science and Technology of China, Hefei 230000, China c Key Laboratory for Anisotropy and Texture of Materials (MoE), School of Materials Science and Engineering, Northeastern University, NO. 3-11, Wenhua Road, Heping District, Shenyang 110819, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Epoxy coatings Water diffusion Alternating hydrostatic pressure Finite element analysis
The water diffusion behaviour in pigmented epoxy coatings under alternating hydrostatic pressure (AHP) has been calculated by the finite element method. The water diffusion in the coating under AHP follows the nonFickian diffusion, which is caused by cracks and superposed by the occurrences of many independent Fickian diffusion processes. The cracks largely contribute to the water absorption of the coating. If the occurrence time of the cracks is measured by the experiment, the lifetime of the coating under AHP can be quickly determined.
1. Introduction Epoxy resin based organic coatings have been widely and effectively used in marine environments to protect structure materials from corrosion [1–3]. Unfortunately, organic coatings are confronted with the tough challenges and dramatic deterioration served in deep-ocean environments because of significant water uptake. The diffusion of water marks the start of coating failure and it has been demonstrated to induce swelling [4], cracking [5] and so on. In order to solve these problems, a lot of researchers have studied the behaviour of water diffusion in organic coatings. For Fickian diffusion, many researchers discussed the free volume in a polymer which contributed to water diffusion in coatings [6–8]. Wong and Broutman [9] draw attention to the water diffusion mechanism in epoxy resins and proposed a model where epoxy network was consisted of two regions in which water molecules possessed different mobilities by the differences of free volume. However, the non-Fickian diffusion is more common in the water diffusion process of coatings, because of the swelling and cracking of the coatings [10–12]. Van Westing et al. [13] proposed the swelling coefficient to adjust the ideal Fickian relation for water-epoxy system. After that Nguyen et al. [10] and Zhu et al. [14] studied the transition from Fickian to non-Fickian behaviour during water uptake. They suggested that the diffusion was rate-determined first in the ideal Fickian sorption then the amorphisation crystallisation of the polymer
started and water transport processes got disordered inducing nonFickian sorption. Based on the studies of Berens [15] and Van Westing et al. [13], Reichinger et al. [16] pointed out the substrate-dependent water transport with a focus on the polymer-pigment coating by quantifying the transport rates and found that the diffusion rate of the pigmented coating on steel was five times slower than the clear coating. In deep-ocean environments, one of the principal determinants of the coating failure is the high hydrostatic pressure (HP) [17–20]. The coupled effects of water diffusion and stress lead to the premature failure of the coatings [12]. In recent years, a series of studies on water diffusion of epoxy coatings under HP and alternating hydrostatic pressure (AHP) have been performed [21–23]. Liu et al. [21] have studied the water diffusion behaviour of the epoxy coating under HP and indicated that its ability of water absorption is enhanced by high HP. While under AHP, the water diffusion becomes more complex and is totally different, because the cracks in the coating contribute largely to the water absorption [24]. Meanwhile, AHP also makes the water diffusion coefficient of the coating change frequently [25]. Therefore, it is difficult to describe the water diffusion process under AHP by a simple equation originating from the theories proposed by Berens and Van Westing [13,15]. A two-Fickian-stage absorption curve decomposed into two simultaneous or successive Fickian sub-curves is able to provide useful insights into the mechanism of water diffusion within organic coatings [26] and it was used in this study. Due to the complex structure inside epoxy coatings and variable
⁎ Corresponding author at: Corrosion and Protection Division, Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Wencui Road 62, Shenyang, 110016, China. Fax: +86-24-2392-5323; Tel: +86-24-8108-3918. E-mail address: [email protected] (L. Liu).
https://doi.org/10.1016/j.porgcoat.2018.07.011 Received 28 December 2017; Received in revised form 22 June 2018; Accepted 5 July 2018 Available online 20 July 2018 0300-9440/ © 2018 Published by Elsevier B.V.
Progress in Organic Coatings 123 (2018) 168–175
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HPs, it is helpful to solve the water diffusion process in the organic coating under AHP by means of the finite element method (FEM). Numerical investigations have been done with the FEM to study Fickian and non-Fickian process in composites [27–29]. Gagani and Echtermeyer [30] studied the effect of cracks on water diffusion of the glass fiber/epoxy composite by FEM and found that the cracks on the external layer of the sample increase the rate of water uptake, while cracks in the internal layer do not influence diffusivity. However, the cracks in their simulations were caused by tensile stress, which is different from HP in deep-sea environments. The nature of water diffusion under AHP is still ambiguous, because the high HP limits the in-situ observation of the changes of coating morphology and the in-situ measurement of the water uptake. In this study, a detailed description on the non-Fickian behaviour of water diffusion in epoxy coatings under AHP was made by FEM according to Wong’s model and Coniglio’s idea. Based on the model, the lifetime of a coating under AHP can be quickly determined. 2. Experiment 2.1. Sample preparation Fig. 1. Schematic diagram of the automatic deep ocean simulation device: 1) nitrogen cylinder, 2) valve, 3) solid reference electrode, 4) thermocouple, 5) working electrode, 6) counter electrode, 7) pressure controller, 8) automatic elevator, 9) circulation line, 10) temperature controller, 11) temperature measuring and 12) autoclave.
The epoxy glass flake (EG) coating consisted of E-44 epoxy resin (biphenol A, Wuxi Resin Factory, China) as the binder, polyamide (TY650, Tianjin Yanhai Chemical Co., Ltd., China) as the hardening agent, dimethylbenzene as the solvent, and glass flake (produced by Wen’an Huaxing Glass Flake Factory, China) as a single component pigmented (thickness: 2∼5 μm, diameter: 30∼40 μm). They were mixed in a weight proportion of 1:0.8:0.3:0.3 for stoichiometric reaction, and then stirred using a commercial magnetic stirrer machine (Thermostat magnetic stirrer hotplate 85-2 of Guohua Electric Appliance Co., Ltd.) for 2 h to mix sufficiently. Finally, the coating was left for 0.5 h to make them partially cure before painting. The free film sample was made by brushing the coating on a silica gel plate. After being cured in an oven at 40 °C for 4 h and 60 °C for 2 h, the film was peeled off from the plate and cut into the dimension of 20 mm by 20 mm by 0.2 mm for gravimetric experiment. Subsequently, the film was cured at 60 °C for 20 h and room temperature (25 °C, 30% RH) for 7 days. The coating thickness was measured by a portable electronic gauge (PosiTector 6000 of Defelsko) and the average thickness of 200 ± 10 μm was obtained according to ISO 2808-2007 [31]. Prior to gravimetric tests, samples were stored in a desiccator to keep dry and avoid any change in properties due to adsorption of moisture from the atmosphere.
for each test and the average of the water absorption was calculated according to the following equation [32]:
Qt =
mt − m 0 × 100% m0
(1)
where mt is the mass of free film at time t, m0 is the initial mass before immersion and Qt is the water absorption (mass%) at time t. Different models have been developed to describe the water diffusion process inside materials [33–36]. If it follows an ideal Fickian process, water absorption data for a plane sheet geometry is represented by the equation [2,37,38]:
Mt 8 =1− 2 M∞ π
∞
∑ n=0
1 −(2n + 1)2Dπ 2 ⎤ exp ⎡ t ⎥ ⎢ (2n + 1)2 l2 ⎦ ⎣
(2)
where Mt is the mass of absorbed water, which equals to mt -m0, at time t, M∞ is the mass of water absorbed at saturation, l is the thickness of the free film, D is the diffusion coefficient which is considered to be constant during the immersion time. For a free film, when Mt/ M∞ < 0.6, the water uptake is proportional to the square root of immersion, and Eq. (2) can be approximated as the following equation [10,39,40].
2.2. Experimental setup and gravimetric tests The immersion experiments were separated in two parts according to the different experimental environments (HP and AHP). For the HP environments, the experiments under 0.1 MPa were conducted in atmospheric pressure. While for the HP of 3.5 MPa and AHP, the experiments were carried out in a specially designed deep ocean simulation system, shown in Fig. 1. The high HP was obtained by pumping high purity nitrogen into the autoclave, which was controlled by a pressure valve. AHP was obtained by the instant alternation between HP at 3.5 MPa (equal to 350 m depth HP in the ocean) and atmospheric pressure at 0.1 MPa. Each alternating pressure cycle (12 h) consisted of high static pressure for 6 h and atmosphere pressure for 6 h, and started from 0.1 MPa. The testing solution was de-ionized water and the temperature kept 25 ± 1 °C. Free films were immersed under the HP of 0.1 MPa and 3.5 MPa, and the AHP of 3.5 MPa, respectively. After removal from the test solution, each sample was dried quickly with a filter paper. Then, the mass gain was measured using a Sartorius MC5 microbalance (1 μg resolution) at different immersion time. Three replications were needed
Mt 4 D = M∞ l π
t
(3)
3. Water absorption modelling In this work, COMSOL multiphysics was applied to establish a model to satisfy the experiment results of the water diffusion in the epoxy coating under AHP. 3.1. Water absorption modelling under HP Due to the variation of the diffusion coefficient and the occurrence of cracks, the model of the water absorption under AHP is difficult to set up directly. Fortunately, the fact that water diffusion under HP follows Fick’s law has been confirmed [22]. Thus, the water absorption of the EG coating under different HPs was firstly calculated by Eq. (3), 169
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Table 1 The gravimetric test results of the average water absorption under different HPs (%). t (h)
0
2
4
8
10
12
24
36
48
72
96
120
168
240
360
0.1 (MPa) 3.5 (MPa)
0 0
0.21 0.19
0.37 0.51
0.49 0.76
0.55 0.89
0.61 0.99
0.84 1.34
0.97 1.54
1.13 1.78
1.36 1.80
1.52 1.82
1.67 1.82
1.80 1.83
1.78 1.72
1.81 1.84
HP of 3.5 MPa. It can be speculated that there must be other factors affected by AHP that increase the water absorption in addition to HP. Some researchers have found that some cracks occurred inside the epoxy coatings under AHP [24], and the occurrence of the cracks can improve the water absorption. Therefore, a model with easy and hard diffusion zone plus a cracking zone to represent the EG coating under AHP was established. We assumed that the volume of the coating after addition of cracks under AHP was the same as the volume of the coating without cracks under HP. The size of the coating with a crack and the size of the crack were set to 235.5 μm by 235.5 μm by 200 μm and 70.3 μm by 213.7 μm by 200 μm respectively, according to the values of the saturated water absorption under HP and AHP (Tables 1 and 2). Under AHP, the cracks were located at the coating/pigment interface and on the coating surface. However, because of a large number of the coating/pigment interfaces in the coating [24], the inner crack fraction is more than the surface crack fraction. Therefore, the cracking zone was assumed to be separated in half into two parts for the most severe scenario: a) the surface cracking zone, where the cracks occur near the surface of the coating, and b) the inner cracking zone, where the cracks occur at the interface between the coating and pigments. For the surface cracking zone, the water enters the cracks from the outside of the coating. The water diffusion coefficient here is obviously larger than that in the coating. We assumed that the diffusion coefficient in the surface cracking zone was 1 × 10−9 cm2/s. Whereas, for the inner cracking zone, the water enters the cracks from the coating, and the diffusion rate of the inner cracking zone is controlled by the water diffusion in the coating [30]. We assumed that the water diffusion coefficient in the inner cracking zone was equal to the diffusion coefficient in the easy diffusion zone of the coating. Distinguished from the model under HP, the pressure changes periodically with AHP. In accordance with the immersion experiment under AHP, each simulated AHP cycle (12 h) consisted of the static pressure of 3.5 MPa and the atmosphere pressure of 0.1 MPa. The AHP cycle started from 0.1 MPa for 6 h, then instantly increased to 3.5 MPa for remaining 6 h. Due to the variation of the pressure, the diffusion coefficient of the coating is not constant any more, which has been confirmed by the fact that the diffusion coefficient immediately changes when HP increases [25]. Moreover, the volume ratio of the easy diffusion zone versus the hard diffusion zone, the initial conditions and the boundary conditions under the AHP of 3.5 MPa were the same as under the HP of 3.5 MPa. The following assumptions were made for the simulation under AHP, and the schematic diagram of water absorption under AHP is shown in Fig. 2.
according to the experimental results (Table 1). In Table 1, the water absorption values of the EG coating first increase obviously and then approach a steady state. The water absorption under 0.1 MPa and 3.5 MPa reaches the saturated value of 1.79% and 1.83% at 168 h and 48 h, respectively. According to Table 1 and Eq. (3), the diffusion coefficients are Dl = 1.593 × 10−10 cm2/s under 0.1 MPa and Dh = 5.007 × 10−10 cm2/s under 3.5 MPa. The high HP not only accelerates the rate of the water diffusion, but also increase the saturated water absorption value of the coating. Thus, it is reasonable to assume that high HP opens some hard diffusion zones inside the epoxy coating where the water cannot reach under atmospheric pressure. According to the thickness of the EG coating, the size of the coating was set to 200 μm by 200 μm by 200 μm. Some researchers have found that the coating has separate diffusion regions in which water molecules exhibit different mobilities [9]. Therefore, the EG coating was assumed to be separated into two parallel parts: a) the easy diffusion zone where the water can reach under any HP; and b) the hard diffusion zone where the water can only reach under high HP (on the left and right sides of the simulated coating). Since the high HP increases the saturated value of the water absorption by opening the hard diffusion zone, the volume ratio of the easy diffusion zone versus the hard one can be set to 1.79: 0.04 referring to the values of the saturated water uptake in Table 1. In addition, high HP acts as an external pushing force to help water diffuse into the hard diffusion-n zone. Once HP is high enough, the water will be pushed into the hard diffusion zone, then the pressure in the hard diffusion zone and easy diffusion zone will be similar. Given the similar stress conditions of the two diffusion zones under HP, we assumed that the diffusion coefficients of these two diffusion zones were equal when they were under high HP. For the initial conditions, the water concentration was 0% (mass %) inside the coating, and 1.83% both on the upper and lower surface. The periodic boundary was set on the side surface, considering the infinite large of the coating in contrast to the simulation area. Then water diffusion in the coating obeys following rules: 1) The water diffusion in the EG coating follows Fick’s law during the whole immersion time, and the equation is:
∂c = D∇2 c ∂t
(4)
2) When the HP is 3.5 MPa, Dh = 5.007 × 10−1° cm2/s, both in the easy and hard diffusion zone. 3) When the HP is 0.1 MPa, Dl = 1.593 × 10−1° cm2/s in the easy diffusion zone, whereas Dl = 0 cm2/s in the hard one.
1) Only when the pressure changes, the diffusion coefficient will change. 2) Each zone follows Fick’s law independently according to the Ref. [26]. 3) In the easy diffusion zone
3.2. Water absorption modelling under AHP The experimental results of the water absorption under the AHP of 3.5 MPa are shown in Table 2. The saturated water absorption under the AHP of 3.5 MPa increases to 2.53%, higher than the value under the
Table 2 The computational results of the water absorption compared with gravimetric tests under AHP (%). t (h)
0
12
24
36
48
60
72
84
96
108
120
132
156
168
180
192
Experiment data Simulation data
0 0.10
1.16 1.26
1.27 1.64
1.44 1.87
1.74 2.01
1.82 2.10
1.99 2.16
1.99 2.20
2.04 2.22
2.07 2.24
2.21 2.33
2.26 2.41
2.43 2.44
2.53 2.49
2.53 2.52
2.53 2.52
170
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Fig. 2. Schematic diagram of a water diffusion model with a multi-cracking zone under AHP: a) no cracks happening, b) surface cracking and c) inner cracking. The number symbol represents different areas: 1) the easy diffusion zone, 2) the hard diffusion zone, 3) the surface cracking zone and 4) the inner cracking zone.
When the HP is 3.5 MPa, Dh = 5.007 × 10−10 cm2/s. Whereas, Dl = 1.593 × 10−10 cm2/s with the pressure of 0.1 MPa. 4) In the hard diffusion zone When the HP is 3.5 MPa, Dh = 5.007 × 10−10 cm2/s. Whereas, Dl = 0 cm2/s with the pressure 0.1 of MPa. 5) In the multi-cracking zone We assumed that the crack first occurred on the surface of the coating, and then took place inside the coating. When t < t1, the diffusion coefficients in the surface and inner cracking zone are 0 cm2/s (Fig. 2a). When t1 < t < t2, the cracking first occurred in the surface cracking zone (Fig. 2b). The diffusion coefficient in the surface cracking zone is 1 × 10−9 cm2/s and 0 cm2/s in the inner. When t > t2, the whole cracking zone is open. The diffusion coefficient of the surface cracking zone is 1 × 10−9 cm2/s, and the diffusion coefficient of the inner cracking zone is 5.007 × 10-10 cm2/s under 3.5 MPa, and 1.593 × 10-10 cm2/s under 0.1 MPa. (Fig. 2c). Where t1 is the immersion time when the first cracking occurs in the surface cracking zone and t2 is the time cracking happens in the inner zone.
Fig. 3. The computational results and the gravimetric test results of the water absorption under the HP of 0.1 MPa and 3.5 MPa.
4. Results and discussion
the water into the hard diffusion zone of the EG coating, although the increase is small. Due to the water absorption of the coating, the EG coating inevitably swells during the water diffusion, and the swelling has also been found in other polymer matrices [13,41]. The swelling accelerates the water diffusion and increases the water absorption in case 2, and makes the experimental results larger than the calculated results. Moreover, the effect of swelling under 3.5 MPa on the water absorption of the EG coating is smaller than that under 0.1 MPa. The high HP also increase the diffusion rate. It can be speculated that the high HP reduces the effect of swelling on the water diffusion by accelerating the rate of the water diffusion.
4.1. Water absorption under HP The computational results of the water uptake under 0.1 MPa and 3.5 MPa compared with the gravimetric tests are shown in Fig. 3. The water absorption versus the immersion time is separated into three cases, and the simulation results of the case 1 and the case 3 are consistent with gravimetric tests (Table 1). For the case 1, the water absorption curve under the HP of 3.5 MPa ends at 30 h earlier than the time of 60 h under 0.1 MPa (Fig. 3a), and the water absorption ratio (Mt/M∞) has a linear relationship with the square root of immersion time (Fig. 3b). For the case 3, the water absorption curve under 3.5 MPa starts from 96 h earlier than 240 h under 0.1 MPa. The calculated saturated water absorption is 1.79% and 1.83% under 0.1 MPa and 3.5 MPa, respectively (Fig. 3a), and the value of Mt/ M∞ reaches 1, which means that the coating does not crack under the HP of 3.5 MPa (Fig. 3b). Whereas, the computational water absorption values of the case 2 under these two HPs are lower than the experimental values, and the water diffusion does not follow Fick’s law. The results show that the HP of 3.5 MPa accelerates the rate of water diffusion by increasing the diffusion coefficient of the EG coating. The 3.5 MPa HP also increases the saturated water absorption by pushing
4.2. Water absorption under AHP The process of the calculated water uptake of the EG coating under AHP is shown in Fig. 4, with the presets that the surface and inner areas are cracking at 0 h and 108 h, respectively. At 2 h, the water first diffuses into the cracking zone and easy diffusion zone (Fig. 4a). The maximum concentration is 1.8% on the surface, and the surface cracking area is full of water with the minimum concentration of 0.5% at the top of the surface crack. With the increase of the water concentration in the surface cracking zone, the water also spreads from the 171
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Fig. 4. The calculated water absorption of the EG coating under AHP at different times with the preset cracking time of 0 h and 108 h: a) 2 h, b) 20 h, c) 108 h and d) 110 h.
1.8%. Once the inner zone cracks, the water will immediately diffuse into it so that the water concentration near the crack decreases from 1.8% to 1.1% shown in Fig. 4d. It shows that the cracking zone is the most popular area for water diffusion followed by easy diffusion and hard diffusion zone. A statistic method, R-square, was used to measure the fitness of the model in explaining the variation of experimental data. It can take on any value between 0 and 1, with a value closer to 1 indicating that a greater proportion of variance is accounted by the model. The summed square of residuals (SSE) is defined as: n
SSE =
∑ wi (yi − yˆi )2 i=1
(5)
the sum of squares about the mean (SST) is defined as: n
SST =
∑ wi (yi − yi )2 i=1
(6)
where wi is weight coefficient, yi is the observed value, yˆi is the predicted value, and yi is the mean of the observed data. R-square is expressed as:
Fig. 5. The computational results of the water absorption for the EG coating and the EM coating compared with the corresponding experimental results under AHP.
R-square = 1 −
SSE SST
(7)
The calculated and experimental results of the water absorption of the EG coating under the AHP of 3.5 MPa are shown in Fig. 5. The calculated water absorption first increases with increasing time and then stabilizes at 2.53%, which fits to the gravimetric results with the Rsquare value of 0.8983. From 12 h to 72 h, the calculated water absorption is higher than the experimental ones, which is caused by a little higher volume fraction of the surface cracking zone. However, from 0 h to 12 h and after 108 h, the calculated water absorption is consistent with the experiment. It is reasonable to speculate that the
crack to the easy diffusion zone. In the easy diffusion zone, the water reaches 40 μm below the surface with the minimum concentration of 0.2%. While, no water diffuses into the hard diffusion zone during the first half AHP cycle. At 20 h (Fig. 4b), all the areas are filled with water with the minimum concentration of 0.7% in the middle of the coating and the maximum concentration of 1.8% on the surface, except the inner cracking zone. Until 108 h (Fig. 4c), the water spreads uniformly in the coating and the surface cracking zone with the concentration of
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Fig. 6. The computational results of the lifetimes as a function of the interval between the first two cracks happening for the EG coating under AHP. The first cracking occurred respectively at a) 0 h, b) 25 h, c) 50 h, d) 75 h, e) 100 h and f) 125 h.
coating under HP and AHP. Therefore, the sizes of the EM coating with a crack and the crack inside were set to 220 μm by 220 μm by 200 μm and 40 μm by 210 μm by 200 μm, respectively. The diffusion coefficient of the EM coating under high HP and atmospheric pressure is 6.431 × 10−10 cm2/s and 3.992 × 10−10 cm2/s [25]. For the initial conditions, the water concentration was 0% (mass %) inside the coating, and 1.62% both on the upper and lower surface. The calculated and experimental water absorption of the EM coating under AHP is shown in Fig. 5. The calculated water absorption stabilizes at 1.96%, and the EM coating cracks between 72 h and 84 h and between 108 h and 120 h, respectively, with the R-square value of 0.9766. Therefore, the water diffusion behaviour for pigmented epoxy coatings under AHP is a non-Fickian diffusion, which is superposed by multiple independent Fickian diffusion processes accompanied by cracking. As soon as a crack happens, the diffusion will deviate from the Fickian type significantly at once.
cracks occur between 0 h and 12 h, and between 108 h and 120 h, respectively. The increased water absorption under AHP is attributed to the occurrence of the cracks in the coating. To verify the accuracy of the water absorption model, the water absorption of the epoxy mica (EM) coating has also been calculated. The protective organic coatings used in the deep ocean with high HP is relatively distinct — epoxy with various inorganic fillers, such as glass flakes and mica. Under different applied environments, the applied coating will change the ratio of various fillers, but not change the main composition in the coating. Therefore, for the coatings used in the deep ocean, we assumed that the water absorptions of different coatings under AHP are related. The degree of the cracking influences the water absorption of the coating under AHP, and it can be described by the ratio of the saturated water absorption of the coating under AHP and its saturated water absorption under HP. The higher the saturated water absorption under HP, the poorer the protection of the coating, the more severe the coating cracks under AHP, which results in a much higher water absorption of the coating under AHP. Therefore, we assumed that the saturated water absorption of the coating under HP is proportional to the ratio of the saturated water absorption of the coating under AHP and the saturated water absorption under HP, if the coating cracks under AHP (the maximum pressure of the AHP equals to the HP):
QHP = k
QAHP QHP
4.3. A lifetime determination method for shortening the experiment time For the pigmented epoxy coatings under AHP, the cracking time is largely depended on the coating itself and AHP. The coating cracking under AHP is caused by the combined effect of water and pressure. The repeated movement of water in the coating under AHP separates the fillers from the matrix, leading to coating cracking [24], which makes it difficult to determine the cracking time by mechanical analysis. However, the cracks in the coating contribute largely to the water absorption of the coating under AHP. Thus, the lifetime of the coating under AHP can be predicted if the cracking time is obtained from the experiment, which can significantly shorten the time required for immersion experiments. According to the Ref. [24,25], after the wet adhesion of the coating drops dramatically, the water absorption of the coating under AHP increases by 27% compared to the water absorption under the HP (the HP equals to the maximum pressure of the AHP). Thus, the water
(8)
where QHP is the saturated water absorption under HP, QAHP is the saturated water absorption under AHP, and k represents the environment factor. Due to the AHP conditions of the EM coating and the EG coating were similar [24,25], we assumed that the environment factors of the EG coating and the EM coating are the same. If the crack does not occur under HP, the expected value of the saturated water absorption of the EM coating under HP is 1.62% (Fig. 5). Then, the ratio of the saturated water absorption of the EM coating under AHP and HP is calculated as 1.21 by Eq. (8), according to the saturated water absorptions of the EG 173
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Table 3 The relative differences of the lifetimes between the ratio of 0.5 and 0.25. t1 (h)
0 25 50 75 100 125
The interval between the first two cracks happening, Δt, (h) 1
10
25
50
75
100
125
150
175
18% 12% 7% 6% 6% 3%
15% 10% 1% 8% 7% 4%
12% 11% 10% 8% 5% 5%
13% 11% 8% 8% 6% 5%
11% 7% 7% 5% 5% —
9% 7% 5% 5% — —
8% 6% 6% — — —
7% 5% — — — —
5% — — — — —
cracks of the coating under AHP are obtained from the experiment. 5. Conclusion The water absorption of the pigmented epoxy coating under HP and AHP was studied by the FEM. The calculated water absorption of the coating shows the following results: 1 HP accelerates the rate of the water diffusion by increasing the diffusion coefficient of the coating, and increases the saturated water absorption by opening some hard diffusion area inside the coating. 2 The non-Fickian diffusion caused by cracks in the coating under AHP is superposed by the occurrences of many Fickian diffusion processes. 3 The lifetime of the coating under AHP can be quickly determined when the time of the cracking is measured from the experiment, which can shorten the time required for immersion experiments. Acknowledgements Fig. 7. The lifetimes of the EG coating as a function of the occurrence interval between the first two cracking with the volume ratio of the surface cracks of 0.5.
The investigation was supported by the National Natural Science Foundation of China under the Contract No. 51622106, the National Key R&D Program of China No. 2017YFB0702303, and the National Key Basic Research and Development Plan of China under the Contract No. 2014CB643303.
absorption value of 127% under AHP, more than that under the corresponding HP, can be assumed to be a criterion to define the failure of the coatings under AHP. For the EG coating, the criterion is 2.32% and the lifetime can be considered as the time when the water absorption reaches the value. Since the volume ratio of the surface and inner cracking zone affects the water absorption of the coating, finding a suitable volume ratio of the cracking zones to represent most of the ratios (making the method universal) is necessary. Therefore, the volume ratio of the surface crack to the whole cracking zone was set to 0.25 and 0.50. The lifetimes of the EG coating for the different volume ratios as a function of the interval between the occurrences of the first two cracks (Δt) under the AHP of 3.5 MPa are shown in Fig. 6. The first cracking time and the interval reflect the protective performance of the coating under AHP. The lifetime at the ratio of 0.5 is shorter than the ratio of 0.25, and increase with increasing interval. When the second cracking time (t 1 + Δt) is after 125 h, the lifetimes with different first cracking times are similar. It shows that the lifetime is controlled by the occurrence of the inner crack, when the interior of the coating cracks late enough. Comparing the lifetimes with the volume ratio of 0.5 to 0.25, the relative differences of them are shown in Table 3. The relative difference decreases as the cracking time increases. The proportion of the relative differences that does not exceed 10% is 79.5%. Thus, the cracking zone divided in half for the surface and inner crack can be used to predict the lifetime of the pigmented epoxy coating under AHP. The lifetimes of the EG coating under AHP are predicted for the different assumed cracking times (t1 and t2), as shown in Fig. 7. Obviously, the lifetime can be quickly determined if the occurrences of the first two
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Finite element analysis of the water diffusion behaviour in pigmented epoxy coatings under alternating hydrostatic pressure
T
⁎
Rui Liua,b, Li Liua,c, , Fandi Mengc, Wenliang Tiana, Ying Liua, Ying Lia, Fuhui Wangc a Corrosion and Protection Division, Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Wencui Road 62, Shenyang 110016, China b School of Materials Science and Engineering, University of Science and Technology of China, Hefei 230000, China c Key Laboratory for Anisotropy and Texture of Materials (MoE), School of Materials Science and Engineering, Northeastern University, NO. 3-11, Wenhua Road, Heping District, Shenyang 110819, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Epoxy coatings Water diffusion Alternating hydrostatic pressure Finite element analysis
The water diffusion behaviour in pigmented epoxy coatings under alternating hydrostatic pressure (AHP) has been calculated by the finite element method. The water diffusion in the coating under AHP follows the nonFickian diffusion, which is caused by cracks and superposed by the occurrences of many independent Fickian diffusion processes. The cracks largely contribute to the water absorption of the coating. If the occurrence time of the cracks is measured by the experiment, the lifetime of the coating under AHP can be quickly determined.
1. Introduction Epoxy resin based organic coatings have been widely and effectively used in marine environments to protect structure materials from corrosion [1–3]. Unfortunately, organic coatings are confronted with the tough challenges and dramatic deterioration served in deep-ocean environments because of significant water uptake. The diffusion of water marks the start of coating failure and it has been demonstrated to induce swelling [4], cracking [5] and so on. In order to solve these problems, a lot of researchers have studied the behaviour of water diffusion in organic coatings. For Fickian diffusion, many researchers discussed the free volume in a polymer which contributed to water diffusion in coatings [6–8]. Wong and Broutman [9] draw attention to the water diffusion mechanism in epoxy resins and proposed a model where epoxy network was consisted of two regions in which water molecules possessed different mobilities by the differences of free volume. However, the non-Fickian diffusion is more common in the water diffusion process of coatings, because of the swelling and cracking of the coatings [10–12]. Van Westing et al. [13] proposed the swelling coefficient to adjust the ideal Fickian relation for water-epoxy system. After that Nguyen et al. [10] and Zhu et al. [14] studied the transition from Fickian to non-Fickian behaviour during water uptake. They suggested that the diffusion was rate-determined first in the ideal Fickian sorption then the amorphisation crystallisation of the polymer
started and water transport processes got disordered inducing nonFickian sorption. Based on the studies of Berens [15] and Van Westing et al. [13], Reichinger et al. [16] pointed out the substrate-dependent water transport with a focus on the polymer-pigment coating by quantifying the transport rates and found that the diffusion rate of the pigmented coating on steel was five times slower than the clear coating. In deep-ocean environments, one of the principal determinants of the coating failure is the high hydrostatic pressure (HP) [17–20]. The coupled effects of water diffusion and stress lead to the premature failure of the coatings [12]. In recent years, a series of studies on water diffusion of epoxy coatings under HP and alternating hydrostatic pressure (AHP) have been performed [21–23]. Liu et al. [21] have studied the water diffusion behaviour of the epoxy coating under HP and indicated that its ability of water absorption is enhanced by high HP. While under AHP, the water diffusion becomes more complex and is totally different, because the cracks in the coating contribute largely to the water absorption [24]. Meanwhile, AHP also makes the water diffusion coefficient of the coating change frequently [25]. Therefore, it is difficult to describe the water diffusion process under AHP by a simple equation originating from the theories proposed by Berens and Van Westing [13,15]. A two-Fickian-stage absorption curve decomposed into two simultaneous or successive Fickian sub-curves is able to provide useful insights into the mechanism of water diffusion within organic coatings [26] and it was used in this study. Due to the complex structure inside epoxy coatings and variable
⁎ Corresponding author at: Corrosion and Protection Division, Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Wencui Road 62, Shenyang, 110016, China. Fax: +86-24-2392-5323; Tel: +86-24-8108-3918. E-mail address: [email protected] (L. Liu).
https://doi.org/10.1016/j.porgcoat.2018.07.011 Received 28 December 2017; Received in revised form 22 June 2018; Accepted 5 July 2018 Available online 20 July 2018 0300-9440/ © 2018 Published by Elsevier B.V.
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HPs, it is helpful to solve the water diffusion process in the organic coating under AHP by means of the finite element method (FEM). Numerical investigations have been done with the FEM to study Fickian and non-Fickian process in composites [27–29]. Gagani and Echtermeyer [30] studied the effect of cracks on water diffusion of the glass fiber/epoxy composite by FEM and found that the cracks on the external layer of the sample increase the rate of water uptake, while cracks in the internal layer do not influence diffusivity. However, the cracks in their simulations were caused by tensile stress, which is different from HP in deep-sea environments. The nature of water diffusion under AHP is still ambiguous, because the high HP limits the in-situ observation of the changes of coating morphology and the in-situ measurement of the water uptake. In this study, a detailed description on the non-Fickian behaviour of water diffusion in epoxy coatings under AHP was made by FEM according to Wong’s model and Coniglio’s idea. Based on the model, the lifetime of a coating under AHP can be quickly determined. 2. Experiment 2.1. Sample preparation Fig. 1. Schematic diagram of the automatic deep ocean simulation device: 1) nitrogen cylinder, 2) valve, 3) solid reference electrode, 4) thermocouple, 5) working electrode, 6) counter electrode, 7) pressure controller, 8) automatic elevator, 9) circulation line, 10) temperature controller, 11) temperature measuring and 12) autoclave.
The epoxy glass flake (EG) coating consisted of E-44 epoxy resin (biphenol A, Wuxi Resin Factory, China) as the binder, polyamide (TY650, Tianjin Yanhai Chemical Co., Ltd., China) as the hardening agent, dimethylbenzene as the solvent, and glass flake (produced by Wen’an Huaxing Glass Flake Factory, China) as a single component pigmented (thickness: 2∼5 μm, diameter: 30∼40 μm). They were mixed in a weight proportion of 1:0.8:0.3:0.3 for stoichiometric reaction, and then stirred using a commercial magnetic stirrer machine (Thermostat magnetic stirrer hotplate 85-2 of Guohua Electric Appliance Co., Ltd.) for 2 h to mix sufficiently. Finally, the coating was left for 0.5 h to make them partially cure before painting. The free film sample was made by brushing the coating on a silica gel plate. After being cured in an oven at 40 °C for 4 h and 60 °C for 2 h, the film was peeled off from the plate and cut into the dimension of 20 mm by 20 mm by 0.2 mm for gravimetric experiment. Subsequently, the film was cured at 60 °C for 20 h and room temperature (25 °C, 30% RH) for 7 days. The coating thickness was measured by a portable electronic gauge (PosiTector 6000 of Defelsko) and the average thickness of 200 ± 10 μm was obtained according to ISO 2808-2007 [31]. Prior to gravimetric tests, samples were stored in a desiccator to keep dry and avoid any change in properties due to adsorption of moisture from the atmosphere.
for each test and the average of the water absorption was calculated according to the following equation [32]:
Qt =
mt − m 0 × 100% m0
(1)
where mt is the mass of free film at time t, m0 is the initial mass before immersion and Qt is the water absorption (mass%) at time t. Different models have been developed to describe the water diffusion process inside materials [33–36]. If it follows an ideal Fickian process, water absorption data for a plane sheet geometry is represented by the equation [2,37,38]:
Mt 8 =1− 2 M∞ π
∞
∑ n=0
1 −(2n + 1)2Dπ 2 ⎤ exp ⎡ t ⎥ ⎢ (2n + 1)2 l2 ⎦ ⎣
(2)
where Mt is the mass of absorbed water, which equals to mt -m0, at time t, M∞ is the mass of water absorbed at saturation, l is the thickness of the free film, D is the diffusion coefficient which is considered to be constant during the immersion time. For a free film, when Mt/ M∞ < 0.6, the water uptake is proportional to the square root of immersion, and Eq. (2) can be approximated as the following equation [10,39,40].
2.2. Experimental setup and gravimetric tests The immersion experiments were separated in two parts according to the different experimental environments (HP and AHP). For the HP environments, the experiments under 0.1 MPa were conducted in atmospheric pressure. While for the HP of 3.5 MPa and AHP, the experiments were carried out in a specially designed deep ocean simulation system, shown in Fig. 1. The high HP was obtained by pumping high purity nitrogen into the autoclave, which was controlled by a pressure valve. AHP was obtained by the instant alternation between HP at 3.5 MPa (equal to 350 m depth HP in the ocean) and atmospheric pressure at 0.1 MPa. Each alternating pressure cycle (12 h) consisted of high static pressure for 6 h and atmosphere pressure for 6 h, and started from 0.1 MPa. The testing solution was de-ionized water and the temperature kept 25 ± 1 °C. Free films were immersed under the HP of 0.1 MPa and 3.5 MPa, and the AHP of 3.5 MPa, respectively. After removal from the test solution, each sample was dried quickly with a filter paper. Then, the mass gain was measured using a Sartorius MC5 microbalance (1 μg resolution) at different immersion time. Three replications were needed
Mt 4 D = M∞ l π
t
(3)
3. Water absorption modelling In this work, COMSOL multiphysics was applied to establish a model to satisfy the experiment results of the water diffusion in the epoxy coating under AHP. 3.1. Water absorption modelling under HP Due to the variation of the diffusion coefficient and the occurrence of cracks, the model of the water absorption under AHP is difficult to set up directly. Fortunately, the fact that water diffusion under HP follows Fick’s law has been confirmed [22]. Thus, the water absorption of the EG coating under different HPs was firstly calculated by Eq. (3), 169
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Table 1 The gravimetric test results of the average water absorption under different HPs (%). t (h)
0
2
4
8
10
12
24
36
48
72
96
120
168
240
360
0.1 (MPa) 3.5 (MPa)
0 0
0.21 0.19
0.37 0.51
0.49 0.76
0.55 0.89
0.61 0.99
0.84 1.34
0.97 1.54
1.13 1.78
1.36 1.80
1.52 1.82
1.67 1.82
1.80 1.83
1.78 1.72
1.81 1.84
HP of 3.5 MPa. It can be speculated that there must be other factors affected by AHP that increase the water absorption in addition to HP. Some researchers have found that some cracks occurred inside the epoxy coatings under AHP [24], and the occurrence of the cracks can improve the water absorption. Therefore, a model with easy and hard diffusion zone plus a cracking zone to represent the EG coating under AHP was established. We assumed that the volume of the coating after addition of cracks under AHP was the same as the volume of the coating without cracks under HP. The size of the coating with a crack and the size of the crack were set to 235.5 μm by 235.5 μm by 200 μm and 70.3 μm by 213.7 μm by 200 μm respectively, according to the values of the saturated water absorption under HP and AHP (Tables 1 and 2). Under AHP, the cracks were located at the coating/pigment interface and on the coating surface. However, because of a large number of the coating/pigment interfaces in the coating [24], the inner crack fraction is more than the surface crack fraction. Therefore, the cracking zone was assumed to be separated in half into two parts for the most severe scenario: a) the surface cracking zone, where the cracks occur near the surface of the coating, and b) the inner cracking zone, where the cracks occur at the interface between the coating and pigments. For the surface cracking zone, the water enters the cracks from the outside of the coating. The water diffusion coefficient here is obviously larger than that in the coating. We assumed that the diffusion coefficient in the surface cracking zone was 1 × 10−9 cm2/s. Whereas, for the inner cracking zone, the water enters the cracks from the coating, and the diffusion rate of the inner cracking zone is controlled by the water diffusion in the coating [30]. We assumed that the water diffusion coefficient in the inner cracking zone was equal to the diffusion coefficient in the easy diffusion zone of the coating. Distinguished from the model under HP, the pressure changes periodically with AHP. In accordance with the immersion experiment under AHP, each simulated AHP cycle (12 h) consisted of the static pressure of 3.5 MPa and the atmosphere pressure of 0.1 MPa. The AHP cycle started from 0.1 MPa for 6 h, then instantly increased to 3.5 MPa for remaining 6 h. Due to the variation of the pressure, the diffusion coefficient of the coating is not constant any more, which has been confirmed by the fact that the diffusion coefficient immediately changes when HP increases [25]. Moreover, the volume ratio of the easy diffusion zone versus the hard diffusion zone, the initial conditions and the boundary conditions under the AHP of 3.5 MPa were the same as under the HP of 3.5 MPa. The following assumptions were made for the simulation under AHP, and the schematic diagram of water absorption under AHP is shown in Fig. 2.
according to the experimental results (Table 1). In Table 1, the water absorption values of the EG coating first increase obviously and then approach a steady state. The water absorption under 0.1 MPa and 3.5 MPa reaches the saturated value of 1.79% and 1.83% at 168 h and 48 h, respectively. According to Table 1 and Eq. (3), the diffusion coefficients are Dl = 1.593 × 10−10 cm2/s under 0.1 MPa and Dh = 5.007 × 10−10 cm2/s under 3.5 MPa. The high HP not only accelerates the rate of the water diffusion, but also increase the saturated water absorption value of the coating. Thus, it is reasonable to assume that high HP opens some hard diffusion zones inside the epoxy coating where the water cannot reach under atmospheric pressure. According to the thickness of the EG coating, the size of the coating was set to 200 μm by 200 μm by 200 μm. Some researchers have found that the coating has separate diffusion regions in which water molecules exhibit different mobilities [9]. Therefore, the EG coating was assumed to be separated into two parallel parts: a) the easy diffusion zone where the water can reach under any HP; and b) the hard diffusion zone where the water can only reach under high HP (on the left and right sides of the simulated coating). Since the high HP increases the saturated value of the water absorption by opening the hard diffusion zone, the volume ratio of the easy diffusion zone versus the hard one can be set to 1.79: 0.04 referring to the values of the saturated water uptake in Table 1. In addition, high HP acts as an external pushing force to help water diffuse into the hard diffusion-n zone. Once HP is high enough, the water will be pushed into the hard diffusion zone, then the pressure in the hard diffusion zone and easy diffusion zone will be similar. Given the similar stress conditions of the two diffusion zones under HP, we assumed that the diffusion coefficients of these two diffusion zones were equal when they were under high HP. For the initial conditions, the water concentration was 0% (mass %) inside the coating, and 1.83% both on the upper and lower surface. The periodic boundary was set on the side surface, considering the infinite large of the coating in contrast to the simulation area. Then water diffusion in the coating obeys following rules: 1) The water diffusion in the EG coating follows Fick’s law during the whole immersion time, and the equation is:
∂c = D∇2 c ∂t
(4)
2) When the HP is 3.5 MPa, Dh = 5.007 × 10−1° cm2/s, both in the easy and hard diffusion zone. 3) When the HP is 0.1 MPa, Dl = 1.593 × 10−1° cm2/s in the easy diffusion zone, whereas Dl = 0 cm2/s in the hard one.
1) Only when the pressure changes, the diffusion coefficient will change. 2) Each zone follows Fick’s law independently according to the Ref. [26]. 3) In the easy diffusion zone
3.2. Water absorption modelling under AHP The experimental results of the water absorption under the AHP of 3.5 MPa are shown in Table 2. The saturated water absorption under the AHP of 3.5 MPa increases to 2.53%, higher than the value under the
Table 2 The computational results of the water absorption compared with gravimetric tests under AHP (%). t (h)
0
12
24
36
48
60
72
84
96
108
120
132
156
168
180
192
Experiment data Simulation data
0 0.10
1.16 1.26
1.27 1.64
1.44 1.87
1.74 2.01
1.82 2.10
1.99 2.16
1.99 2.20
2.04 2.22
2.07 2.24
2.21 2.33
2.26 2.41
2.43 2.44
2.53 2.49
2.53 2.52
2.53 2.52
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Fig. 2. Schematic diagram of a water diffusion model with a multi-cracking zone under AHP: a) no cracks happening, b) surface cracking and c) inner cracking. The number symbol represents different areas: 1) the easy diffusion zone, 2) the hard diffusion zone, 3) the surface cracking zone and 4) the inner cracking zone.
When the HP is 3.5 MPa, Dh = 5.007 × 10−10 cm2/s. Whereas, Dl = 1.593 × 10−10 cm2/s with the pressure of 0.1 MPa. 4) In the hard diffusion zone When the HP is 3.5 MPa, Dh = 5.007 × 10−10 cm2/s. Whereas, Dl = 0 cm2/s with the pressure 0.1 of MPa. 5) In the multi-cracking zone We assumed that the crack first occurred on the surface of the coating, and then took place inside the coating. When t < t1, the diffusion coefficients in the surface and inner cracking zone are 0 cm2/s (Fig. 2a). When t1 < t < t2, the cracking first occurred in the surface cracking zone (Fig. 2b). The diffusion coefficient in the surface cracking zone is 1 × 10−9 cm2/s and 0 cm2/s in the inner. When t > t2, the whole cracking zone is open. The diffusion coefficient of the surface cracking zone is 1 × 10−9 cm2/s, and the diffusion coefficient of the inner cracking zone is 5.007 × 10-10 cm2/s under 3.5 MPa, and 1.593 × 10-10 cm2/s under 0.1 MPa. (Fig. 2c). Where t1 is the immersion time when the first cracking occurs in the surface cracking zone and t2 is the time cracking happens in the inner zone.
Fig. 3. The computational results and the gravimetric test results of the water absorption under the HP of 0.1 MPa and 3.5 MPa.
4. Results and discussion
the water into the hard diffusion zone of the EG coating, although the increase is small. Due to the water absorption of the coating, the EG coating inevitably swells during the water diffusion, and the swelling has also been found in other polymer matrices [13,41]. The swelling accelerates the water diffusion and increases the water absorption in case 2, and makes the experimental results larger than the calculated results. Moreover, the effect of swelling under 3.5 MPa on the water absorption of the EG coating is smaller than that under 0.1 MPa. The high HP also increase the diffusion rate. It can be speculated that the high HP reduces the effect of swelling on the water diffusion by accelerating the rate of the water diffusion.
4.1. Water absorption under HP The computational results of the water uptake under 0.1 MPa and 3.5 MPa compared with the gravimetric tests are shown in Fig. 3. The water absorption versus the immersion time is separated into three cases, and the simulation results of the case 1 and the case 3 are consistent with gravimetric tests (Table 1). For the case 1, the water absorption curve under the HP of 3.5 MPa ends at 30 h earlier than the time of 60 h under 0.1 MPa (Fig. 3a), and the water absorption ratio (Mt/M∞) has a linear relationship with the square root of immersion time (Fig. 3b). For the case 3, the water absorption curve under 3.5 MPa starts from 96 h earlier than 240 h under 0.1 MPa. The calculated saturated water absorption is 1.79% and 1.83% under 0.1 MPa and 3.5 MPa, respectively (Fig. 3a), and the value of Mt/ M∞ reaches 1, which means that the coating does not crack under the HP of 3.5 MPa (Fig. 3b). Whereas, the computational water absorption values of the case 2 under these two HPs are lower than the experimental values, and the water diffusion does not follow Fick’s law. The results show that the HP of 3.5 MPa accelerates the rate of water diffusion by increasing the diffusion coefficient of the EG coating. The 3.5 MPa HP also increases the saturated water absorption by pushing
4.2. Water absorption under AHP The process of the calculated water uptake of the EG coating under AHP is shown in Fig. 4, with the presets that the surface and inner areas are cracking at 0 h and 108 h, respectively. At 2 h, the water first diffuses into the cracking zone and easy diffusion zone (Fig. 4a). The maximum concentration is 1.8% on the surface, and the surface cracking area is full of water with the minimum concentration of 0.5% at the top of the surface crack. With the increase of the water concentration in the surface cracking zone, the water also spreads from the 171
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Fig. 4. The calculated water absorption of the EG coating under AHP at different times with the preset cracking time of 0 h and 108 h: a) 2 h, b) 20 h, c) 108 h and d) 110 h.
1.8%. Once the inner zone cracks, the water will immediately diffuse into it so that the water concentration near the crack decreases from 1.8% to 1.1% shown in Fig. 4d. It shows that the cracking zone is the most popular area for water diffusion followed by easy diffusion and hard diffusion zone. A statistic method, R-square, was used to measure the fitness of the model in explaining the variation of experimental data. It can take on any value between 0 and 1, with a value closer to 1 indicating that a greater proportion of variance is accounted by the model. The summed square of residuals (SSE) is defined as: n
SSE =
∑ wi (yi − yˆi )2 i=1
(5)
the sum of squares about the mean (SST) is defined as: n
SST =
∑ wi (yi − yi )2 i=1
(6)
where wi is weight coefficient, yi is the observed value, yˆi is the predicted value, and yi is the mean of the observed data. R-square is expressed as:
Fig. 5. The computational results of the water absorption for the EG coating and the EM coating compared with the corresponding experimental results under AHP.
R-square = 1 −
SSE SST
(7)
The calculated and experimental results of the water absorption of the EG coating under the AHP of 3.5 MPa are shown in Fig. 5. The calculated water absorption first increases with increasing time and then stabilizes at 2.53%, which fits to the gravimetric results with the Rsquare value of 0.8983. From 12 h to 72 h, the calculated water absorption is higher than the experimental ones, which is caused by a little higher volume fraction of the surface cracking zone. However, from 0 h to 12 h and after 108 h, the calculated water absorption is consistent with the experiment. It is reasonable to speculate that the
crack to the easy diffusion zone. In the easy diffusion zone, the water reaches 40 μm below the surface with the minimum concentration of 0.2%. While, no water diffuses into the hard diffusion zone during the first half AHP cycle. At 20 h (Fig. 4b), all the areas are filled with water with the minimum concentration of 0.7% in the middle of the coating and the maximum concentration of 1.8% on the surface, except the inner cracking zone. Until 108 h (Fig. 4c), the water spreads uniformly in the coating and the surface cracking zone with the concentration of
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Fig. 6. The computational results of the lifetimes as a function of the interval between the first two cracks happening for the EG coating under AHP. The first cracking occurred respectively at a) 0 h, b) 25 h, c) 50 h, d) 75 h, e) 100 h and f) 125 h.
coating under HP and AHP. Therefore, the sizes of the EM coating with a crack and the crack inside were set to 220 μm by 220 μm by 200 μm and 40 μm by 210 μm by 200 μm, respectively. The diffusion coefficient of the EM coating under high HP and atmospheric pressure is 6.431 × 10−10 cm2/s and 3.992 × 10−10 cm2/s [25]. For the initial conditions, the water concentration was 0% (mass %) inside the coating, and 1.62% both on the upper and lower surface. The calculated and experimental water absorption of the EM coating under AHP is shown in Fig. 5. The calculated water absorption stabilizes at 1.96%, and the EM coating cracks between 72 h and 84 h and between 108 h and 120 h, respectively, with the R-square value of 0.9766. Therefore, the water diffusion behaviour for pigmented epoxy coatings under AHP is a non-Fickian diffusion, which is superposed by multiple independent Fickian diffusion processes accompanied by cracking. As soon as a crack happens, the diffusion will deviate from the Fickian type significantly at once.
cracks occur between 0 h and 12 h, and between 108 h and 120 h, respectively. The increased water absorption under AHP is attributed to the occurrence of the cracks in the coating. To verify the accuracy of the water absorption model, the water absorption of the epoxy mica (EM) coating has also been calculated. The protective organic coatings used in the deep ocean with high HP is relatively distinct — epoxy with various inorganic fillers, such as glass flakes and mica. Under different applied environments, the applied coating will change the ratio of various fillers, but not change the main composition in the coating. Therefore, for the coatings used in the deep ocean, we assumed that the water absorptions of different coatings under AHP are related. The degree of the cracking influences the water absorption of the coating under AHP, and it can be described by the ratio of the saturated water absorption of the coating under AHP and its saturated water absorption under HP. The higher the saturated water absorption under HP, the poorer the protection of the coating, the more severe the coating cracks under AHP, which results in a much higher water absorption of the coating under AHP. Therefore, we assumed that the saturated water absorption of the coating under HP is proportional to the ratio of the saturated water absorption of the coating under AHP and the saturated water absorption under HP, if the coating cracks under AHP (the maximum pressure of the AHP equals to the HP):
QHP = k
QAHP QHP
4.3. A lifetime determination method for shortening the experiment time For the pigmented epoxy coatings under AHP, the cracking time is largely depended on the coating itself and AHP. The coating cracking under AHP is caused by the combined effect of water and pressure. The repeated movement of water in the coating under AHP separates the fillers from the matrix, leading to coating cracking [24], which makes it difficult to determine the cracking time by mechanical analysis. However, the cracks in the coating contribute largely to the water absorption of the coating under AHP. Thus, the lifetime of the coating under AHP can be predicted if the cracking time is obtained from the experiment, which can significantly shorten the time required for immersion experiments. According to the Ref. [24,25], after the wet adhesion of the coating drops dramatically, the water absorption of the coating under AHP increases by 27% compared to the water absorption under the HP (the HP equals to the maximum pressure of the AHP). Thus, the water
(8)
where QHP is the saturated water absorption under HP, QAHP is the saturated water absorption under AHP, and k represents the environment factor. Due to the AHP conditions of the EM coating and the EG coating were similar [24,25], we assumed that the environment factors of the EG coating and the EM coating are the same. If the crack does not occur under HP, the expected value of the saturated water absorption of the EM coating under HP is 1.62% (Fig. 5). Then, the ratio of the saturated water absorption of the EM coating under AHP and HP is calculated as 1.21 by Eq. (8), according to the saturated water absorptions of the EG 173
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Table 3 The relative differences of the lifetimes between the ratio of 0.5 and 0.25. t1 (h)
0 25 50 75 100 125
The interval between the first two cracks happening, Δt, (h) 1
10
25
50
75
100
125
150
175
18% 12% 7% 6% 6% 3%
15% 10% 1% 8% 7% 4%
12% 11% 10% 8% 5% 5%
13% 11% 8% 8% 6% 5%
11% 7% 7% 5% 5% —
9% 7% 5% 5% — —
8% 6% 6% — — —
7% 5% — — — —
5% — — — — —
cracks of the coating under AHP are obtained from the experiment. 5. Conclusion The water absorption of the pigmented epoxy coating under HP and AHP was studied by the FEM. The calculated water absorption of the coating shows the following results: 1 HP accelerates the rate of the water diffusion by increasing the diffusion coefficient of the coating, and increases the saturated water absorption by opening some hard diffusion area inside the coating. 2 The non-Fickian diffusion caused by cracks in the coating under AHP is superposed by the occurrences of many Fickian diffusion processes. 3 The lifetime of the coating under AHP can be quickly determined when the time of the cracking is measured from the experiment, which can shorten the time required for immersion experiments. Acknowledgements Fig. 7. The lifetimes of the EG coating as a function of the occurrence interval between the first two cracking with the volume ratio of the surface cracks of 0.5.
The investigation was supported by the National Natural Science Foundation of China under the Contract No. 51622106, the National Key R&D Program of China No. 2017YFB0702303, and the National Key Basic Research and Development Plan of China under the Contract No. 2014CB643303.
absorption value of 127% under AHP, more than that under the corresponding HP, can be assumed to be a criterion to define the failure of the coatings under AHP. For the EG coating, the criterion is 2.32% and the lifetime can be considered as the time when the water absorption reaches the value. Since the volume ratio of the surface and inner cracking zone affects the water absorption of the coating, finding a suitable volume ratio of the cracking zones to represent most of the ratios (making the method universal) is necessary. Therefore, the volume ratio of the surface crack to the whole cracking zone was set to 0.25 and 0.50. The lifetimes of the EG coating for the different volume ratios as a function of the interval between the occurrences of the first two cracks (Δt) under the AHP of 3.5 MPa are shown in Fig. 6. The first cracking time and the interval reflect the protective performance of the coating under AHP. The lifetime at the ratio of 0.5 is shorter than the ratio of 0.25, and increase with increasing interval. When the second cracking time (t 1 + Δt) is after 125 h, the lifetimes with different first cracking times are similar. It shows that the lifetime is controlled by the occurrence of the inner crack, when the interior of the coating cracks late enough. Comparing the lifetimes with the volume ratio of 0.5 to 0.25, the relative differences of them are shown in Table 3. The relative difference decreases as the cracking time increases. The proportion of the relative differences that does not exceed 10% is 79.5%. Thus, the cracking zone divided in half for the surface and inner crack can be used to predict the lifetime of the pigmented epoxy coating under AHP. The lifetimes of the EG coating under AHP are predicted for the different assumed cracking times (t1 and t2), as shown in Fig. 7. Obviously, the lifetime can be quickly determined if the occurrences of the first two
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