Research on The Chloride Diffusion Modiﬁed Model for Marine Concretes with Nanoparticles under The Action of Multiple Environmental Factors

: Marine concrete structures are subject to the action of multiple environments during their service time. This leads to increased deterioration in the durability of marine concretes under the combined action of bending load and dry–wet cycles, salt freeze–thaw cycles, and salt spray erosion. The main reason for the damage of concrete under the action of the above three environments is Cl - attack. The free Cl - content ( Cl -f ) and the free Cl - diffusion coefﬁcient ( D f ) of concrete can explain the diffusion of Cl - in concrete. This paper considers the actual environment of marine concrete structures and develops the Cl - diffusion modiﬁed model for nano-marine concretes under the action of dry–wet cycles, salt freeze–thaw cycles, and bending load and salt spray erosion. The nano-SiO 2 , nano-Fe 2 O 3 , and nano-Fe 3 O 4 were ﬁrstly incorporated into ordinary marine concrete, then the Cl - content of each group of marine concrete was measured at different depths, and the Cl - diffusion coefﬁcients were calculated; ﬁnally, the Cl - diffusion modiﬁed model was established under different environmental factors. The test results show that the total and free Cl - diffusion coefﬁcients of nano-marine concretes were lower than those of ordinary marine concrete, and the nano-SiO 2 , nano-Fe 2 O 3 , and nano-Fe 3 O 4 of the optimum dosage were 2%, 1%, and 2%, respectively. The ﬁtting results of Cl - content have a good correlation, and the correlation coefﬁcient ( R ) is basically above 0.98.


Introduction
Cldiffusion can cause reinforcement corrosion for marine concrete structures in long-term marine environments such as cross-sea bridges and harbors, which is the main reason for the shortened service life of marine concrete structures [1][2][3]. Marine concretes in different positions are exposed in the atmosphere zone (salt spray zone), tidal zone, splash zone, and submerge zone, respectively. In addition to the Clerosion, the structures are also influenced by the damage of salt freeze-thaw cycles in cold regions [4][5][6][7]. The studies reported that the structural damage in the submerge zone is more serious under the combined effects of environments and Clerosion [8].
When Cldiffuses into concrete, part of the Clproduces a series of chemical reactions with cement hydration products called binding Cl -. The unreacted Clwill exist with a freedom that is called free Cl -. Several studies have reported [9,10] that free Clis one of the main reasons for reinforcement erosion. Therefore, investigating the distribution of free Clin concrete can effectively evaluate the durability performance of marine concrete structures. The Cldiffusion coefficient is an important parameter for predicting the service life of concrete structures [11].
The Cldiffusion coefficient will change by time, temperature, water-cement ratio, and other factors. The basic Fick's second law is too idealistic to consider Cldiffusion.
(2) Fine aggregate: medium sand with a fineness modulus of 2.42.
(3) Coarse aggregate: selected continuous gravel gradation with particle size 5-31.5 mm. (4) Defoamer agent: Tributyl phosphate was selected as the defoamer agent. (5) Water reducing agent: FDN-type naphthalene high-efficiency water reducing agent was used, its dosage according to the method specified in the Concrete Admixture (GB8076-2008). (6) Nanoparticles: Nano-SiO 2 has a strong pozzolanic effect, which can react with cement secondary hydration, and can effectively improve the microstructure of concrete. Although nano-Fe 2 O 3 and nano-Fe 3 O 4 do not have a pozzolanic effect, their surface has high activity and strong adsorption, respectively, and can also react with cement hydration products. So, nano-SiO 2 and nano-Fe 3 O 4 were used for the dry-wet cycles and salt freeze-thaw cycles tests under no load condition, and nano-SiO 2 and nano-Fe 2 O 3 were used for the bending load and salt spray erosion test. This selection method can effectively reflect the difference between nano-Fe 2 O 3 and nano-Fe 3 O 4 in the improvement of concrete durability. According to their different properties, the chloride content of concrete in three different environments is studied. And the results of the modified model are also different. The nano-SiO 2 , nano-Fe 2 O 3, and nano-Fe 3 O 4 was produced by Anhui Kerun Nanotechnology Co. The properties of nanoparticles are shown in Table 1.

Marine Concrete Mix Proportioning
According to the Code for Durability Design of Concrete Structures (GB/T50476-2019) [47] and the Code for Mix Proportions Design Procedure of Ordinary Concrete (JGJ55-2011) [48], the environmental action grade is III-C under the marine chloride environment. With the slump requirements taken into consideration, the concrete design strength level in this test is C45, the water-binder ratio is 0.44, the sand rate is 33%, the water reducing agent level is 0.25% of the cement amount, and the dosage of defoamer is 4% of the water reducing agent amount. Nano-marine concretes are based on the concrete ratio for ordinary marine concrete structures, the water-cement ratio and unit water consumption were maintained, and the cement was substituted by the equal quality of nanoparticles. The dosage of nanoparticles was used 0.5%, 1.0%, 2.0%, and 3.0% in this paper. The marine concretes ratios are shown in Table 2. The dry-wet cycle is a period every 24 h. The specific operation is as follows: After curing for 28 days, the specimen is immersed in NaCl solution (concentration 5%) for 11 h. After taking it out for natural air drying for 1 h, it is put into the oven for 11 h (oven temperature is set at 80 • C), and then taken out for cooling to room temperature (1 h). The NaCl solution was changed every 30 days to keep the concentration constant. The specimens were immersed in NaCl solution and replaced every 50 days under the action of full immersion. A total of 108 specimens were used for the determination of Clcontent under the two test conditions, and the size of the specimen was 100 × 100 × 100 mm.

The Combined Action of Freeze-Thaw Cycles and Cl -Erosion Test
The salt freeze-thaw cycles test was conducted for the fast-freezing method according to the Concrete long-term Performance and Durability Test Method Standard (GB/T 50082-2019) [49]; The circulating medium of salt freeze-thaw cycles was 5% NaCl solution. The specimens were immersed in NaCl solution, and each freeze-thaw cycle was completed within 2~4 h, and the melting time should not be less than 1/4 of the freeze-thaw cycles. The minimum and maximum temperature at the center of the specimen were set at −18 • C and 5 • C, respectively, and they were automatically controlled by the freeze-thaw testing machine. A total of 72 specimens were used for the determination of the Clcontent, and the size of the specimen was 100 × 100 × 100 mm.

The Combined Action of Bending Load and Salt Spray Erosion Test
(1) Test conditions: Referring to the neutral salt spray test (NSS test), a concrete durability test chamber [50] under the action of salt spray erosion was independently developed (application number ZL 202120853835.7). The test blocks of preloaded concrete, together with the loading device, were put into the concrete durability test chamber (see Figure 1) for the salt spray erosion test. In Figure 1, a [51], the test in this paper self-assembled a set of bending loading devices [50] (see Figure 2). Its advantages include small size and being easy to operate during the process of loading and unloading, intuitive control of stress ratios, and can be directly placed in the salt spray test chamber. Figure 1. The concrete durability test chamber for the salt spray erosion test [50]. Reprinted from Ref. [50]. 2022, Maohua Zhang

Cl -Sampling and Determination
Cl -Sampling (1) The combined action of dry-wet cycles and Clerosion: After 25 dry-wet cycles and sampling, Cldetermination was carried out. After the test blocks were dried, two parts of powder (10 g each) were taken as a group at different depths, and the drilling depths were 2, 5, 10, 15, 20, 25, and 30 mm, respectively. (2) The combined action of salt freeze-thaw cycles and Clerosion: After 25 salt freezethaw cycles, the test blocks were taken. They were used to drill the powder on the four parts. The salt freeze-thaw cycles caused the surface layer of concrete to peel off, so the drilling depths were 5, 10, 15, 20, 25, and 30 mm, respectively. (3) The combined action of bending load and salt spray erosion: After the test blocks arrived at the corresponding test age, each group of them was used to drill the powder from the tensile zone and the compressive zone (100 mm), respectively (see Figure 2b). After screening, they were put into the oven for 2 h (temperature 105 ± 5 °C) and then cooled to nature temperature for the Clcontent test. The drilling depths were 3, 5, 10, 15, 20, 25, and 30 mm, respectively. The process of Clsampling, sorting, and drying is shown in Figure 3. The concrete durability test chamber for the salt spray erosion test [50]. Reprinted from Ref. [50]. 2022, Maohua Zhang.

Cl -Sampling and Determination
Cl -Sampling (1) The combined action of dry-wet cycles and Clerosion: After 25 dry-wet cycles and sampling, Cldetermination was carried out. After the test blocks were dried, two parts of powder (10 g each) were taken as a group at different depths, and the drilling depths were 2, 5, 10, 15, 20, 25, and 30 mm, respectively. (2) The combined action of salt freeze-thaw cycles and Clerosion: After 25 salt freezethaw cycles, the test blocks were taken. They were used to drill the powder on the four parts. The salt freeze-thaw cycles caused the surface layer of concrete to peel off, so the drilling depths were 5, 10, 15, 20, 25, and 30 mm, respectively. (3) The combined action of bending load and salt spray erosion: After the test blocks arrived at the corresponding test age, each group of them was used to drill the powder from the tensile zone and the compressive zone (100 mm), respectively (see Figure 2b). After screening, they were put into the oven for 2 h (temperature 105 ± 5 • C) and then cooled to nature temperature for the Clcontent test. The drilling depths were 3, 5, 10, 15, 20, 25, and 30 mm, respectively.
The process of Clsampling, sorting, and drying is shown in Figure 3.    The Determination of Cl - The free Clcontent of each group of samples was determined according to the Clcontent test method in the Test Procedure for Hydraulic Concrete (SL 352-2020) [52]. Binding Clcontent ( The determination process of total Clcontent is shown in Figure 4. The free Clcontent determination process is shown in Figure 5. After the test is finished, the total Clcontent and free Clcontent formula are shown in Equations (1) and (2).    The Determination of Cl - The free Clcontent of each group of samples was determined according to the Clcontent test method in the Test Procedure for Hydraulic Concrete (SL 352-2020) [52]. Binding Clcontent (Clb ) = total Clcontent (Clt ) -free Clcontent (Clf ). The determination process of total Clcontent is shown in Figure 4. The free Clcontent determination process is shown in Figure 5. After the test is finished, the total Clcontent and free Clcontent formula are shown in Equations (1) and (2).  The Determination of Cl - The free Clcontent of each group of samples was determined according to the Clcontent test method in the Test Procedure for Hydraulic Concrete (SL 352-2020) [52]. Binding Clcontent (Clb) = total Clcontent (Clt) -free Clcontent (Clf).
The determination process of total Clcontent is shown in Figure 4. The free Clcontent determination process is shown in Figure 5. After the test is finished, the total Clcontent and free Clcontent formula are shown in Equations (1) and (2).     The Determination of Cl - The free Clcontent of each group of samples was determined according to the Clcontent test method in the Test Procedure for Hydraulic Concrete (SL 352-2020) [52]. Binding Clcontent (Clb) = total Clcontent (Clt) -free Clcontent (Clf).
The determination process of total Clcontent is shown in Figure 4. The free Clcontent determination process is shown in Figure 5. After the test is finished, the total Clcontent and free Clcontent formula are shown in Equations (1) and (2).    The flow chart of the methodology in this paper is shown in Figure 6. The flow chart of the methodology in this paper is shown in Figure 6.

Test Results and Discussion
3.1. The Difference of Cl -Diffusion Process under the Action of Dry-Wet Cycles and Full Immersion Figure 7 shows the variation curves of the total and free Clcontent of PC under the action of dry-wet cycles and full immersion with depth (x) and age (G and Q), where G and Q represent the number of dry-wet cycles and the time of full immersion, respectively (one period of dry-wet cycle is equal to full immersion for 1 day). The variation trend of total and free Clcontent in concrete with age and depth is basically the same under the action of dry-wet cycles and full immersion, but the total Clcontent is higher than the free Clcontent under the same condition. The content of Clat each age decreases gradually with the increase of depth until it becomes stable under the action of full immersion. When depth and age are constant, compared with the two test conditions, the Clcontent is higher under the action of dry-wet cycles, and the difference of Clcontent becomes sharper with the increase of G and Q.

Test Results and Discussion
3.1. The Difference of Cl -Diffusion Process under the Action of Dry-Wet Cycles and Full Immersion Figure 7 shows the variation curves of the total and free Clcontent of PC under the action of dry-wet cycles and full immersion with depth (x) and age (G and Q), where G and Q represent the number of dry-wet cycles and the time of full immersion, respectively (one period of dry-wet cycle is equal to full immersion for 1 day). The variation trend of total and free Clcontent in concrete with age and depth is basically the same under the action of dry-wet cycles and full immersion, but the total Clcontent is higher than the free Clcontent under the same condition. The content of Clat each age decreases gradually with the increase of depth until it becomes stable under the action of full immersion. When depth and age are constant, compared with the two test conditions, the Clcontent is higher under the action of dry-wet cycles, and the difference of Clcontent becomes sharper with the increase of G and Q.

The Cl -Diffusion Performance of Marine Concretes under the Action of Dry-wet Cycles
Figures 8-10 show the variation of Clcontent of marine concretes under the action of dry-wet cycles and full immersion, respectively (Take x = 2 mm as an example, the other depth has the same pattern). It can be seen that the content of total and free Clin NS, NF (II), and PC increased gradually with the increase of Q. When Q ≤ 100, the increase of Clcontent was higher than that of when Q > 100. The total and free Clcontent of NS and NF(II) with different dosages are is lower than that of PC. This indicated that nano-SiO2 and nano-Fe3O4 particles can improve the Clerosion resistance of concrete. When the dosage and ag-e are constant, the binding Clcontent and binding Clcapacity of NS are higher than that of NF(II) under the action of wet-dry cycles and full immersion, which proves that nano-SiO2 has a better effect on improving Clerosion performance. When the number of dry-wet cycles and the age of full immersion are the same, the total and free Clcontent of NS is lower than that of NF(II). This indicated that nano-SiO2 has a better effect on improving the Clerosion resistance of concrete than that of nano-Fe3O4. The total and free Clcontents of both NS and NF(II) initially decreased, then increased with additional admixture, and the optimal dosage of nano-SiO2 and nano-Fe3O4 is about 2%. When the dosage and age are constant, the binding Clcontent and binding Clcapacity of NS are higher than those of NF(II) under the action of dry-wet cycles and full immersion. The binding Clcontent bottomed out at the optimal dosage.

The Cl -Diffusion Performance of Marine Concretes under the Action of Dry-wet Cycles
Figures 8-10 show the variation of Clcontent of marine concretes under the action of dry-wet cycles and full immersion, respectively (Take x = 2 mm as an example, the other depth has the same pattern). It can be seen that the content of total and free Clin NS, NF (II), and PC increased gradually with the increase of Q. When Q ≤ 100, the increase of Clcontent was higher than that of when Q > 100. The total and free Clcontent of NS and NF(II) with different dosages are is lower than that of PC. This indicated that nano-SiO 2 and nano-Fe 3 O 4 particles can improve the Clerosion resistance of concrete. When the dosage and ag-e are constant, the binding Clcontent and binding Clcapacity of NS are higher than that of NF(II) under the action of wet-dry cycles and full immersion, which proves that nano-SiO 2 has a better effect on improving Clerosion performance. When the number of dry-wet cycles and the age of full immersion are the same, the total and free Clcontent of NS is lower than that of NF(II). This indicated that nano-SiO 2 has a better effect on improving the Clerosion resistance of concrete than that of nano-Fe 3 O 4 . The total and free Clcontents of both NS and NF(II) initially decreased, then increased with additional admixture, and the optimal dosage of nano-SiO 2 and nano-Fe 3 O 4 is about 2%. When the dosage and age are constant, the binding Clcontent and binding Clcapacity of NS are higher than those of NF(II) under the action of dry-wet cycles and full immersion. The binding Clcontent bottomed out at the optimal dosage.

The Cl -Diffusion Performance of Marine Concretes under the Action of Dry-wet Cycles
Figures 8-10 show the variation of Clcontent of marine concretes under the action of dry-wet cycles and full immersion, respectively (Take x = 2 mm as an example, the other depth has the same pattern). It can be seen that the content of total and free Clin NS, NF (II), and PC increased gradually with the increase of Q. When Q ≤ 100, the increase of Clcontent was higher than that of when Q > 100. The total and free Clcontent of NS and NF(II) with different dosages are is lower than that of PC. This indicated that nano-SiO2 and nano-Fe3O4 particles can improve the Clerosion resistance of concrete. When the dosage and ag-e are constant, the binding Clcontent and binding Clcapacity of NS are higher than that of NF(II) under the action of wet-dry cycles and full immersion, which proves that nano-SiO2 has a better effect on improving Clerosion performance. When the number of dry-wet cycles and the age of full immersion are the same, the total and free Clcontent of NS is lower than that of NF(II). This indicated that nano-SiO2 has a better effect on improving the Clerosion resistance of concrete than that of nano-Fe3O4. The total and free Clcontents of both NS and NF(II) initially decreased, then increased with additional admixture, and the optimal dosage of nano-SiO2 and nano-Fe3O4 is about 2%. When the dosage and age are constant, the binding Clcontent and binding Clcapacity of NS are higher than those of NF(II) under the action of dry-wet cycles and full immersion. The binding Clcontent bottomed out at the optimal dosage.  (Taking 5 mm depth as an example, the other depth has the same pattern). The variation regulation of free Clcontent is similar to that of total Clcontent. It can be seen that the total Clcontent of NS and NF(II) is lower than that of PC, while the binding Clcontent is higher than that of PC. This indicated that nanoparticles can solidify the Clof pores in concrete, reduce free Clcontent, and improve Clerosion resistance of concrete under the action of freeze-thaw cycles. The total Clcontent of NS and NF(II) initially decreased, then increased with the additional admixture, and the binding Clcontent initially increased and then decreased with the additional admixture. When A = 2%, the total Clcontent of NS and NF(II) was the lowest, and the binding Clcontent was the highest. This indicated that the effect of improving the salt freeze-thaw resistance of concrete was the best at A = 2%. When the dosage and age are constant, the total Clcontent of NS is lower than NF(II), and the binding Clcontent of NS is higher than NF(II). This indicated that (Taking 5 mm depth as an example, the other depth has the same pattern). The variation regulation of free Clcontent is similar to that of total Clcontent. It can be seen that the total Clcontent of NS and NF(II) is lower than that of PC, while the binding Clcontent is higher than that of PC. This indicated that nanoparticles can solidify the Clof pores in concrete, reduce free Clcontent, and improve Clerosion resistance of concrete under the action of freeze-thaw cycles. The total Clcontent of NS and NF(II) initially decreased, then increased with the additional admixture, and the binding Clcontent initially increased and then decreased with the additional admixture. When A = 2%, the total Clcontent of NS and NF(II) was the lowest, and the binding Clcontent was the highest. This indicated that the effect of improving the salt freeze-thaw resistance of concrete was the best at A = 2%. When the dosage and age are constant, the total Clcontent of NS is lower than NF(II), and the binding Clcontent of NS is higher than NF(II). This indicated that  (Taking 5 mm depth as an example, the other depth has the same pattern). The variation regulation of free Clcontent is similar to that of total Clcontent. It can be seen that the total Clcontent of NS and NF(II) is lower than that of PC, while the binding Clcontent is higher than that of PC. This indicated that nanoparticles can solidify the Clof pores in concrete, reduce free Clcontent, and improve Clerosion resistance of concrete under the action of freeze-thaw cycles. The total Clcontent of NS and NF(II) initially decreased, then increased with the additional admixture, and the binding Clcontent initially increased and then decreased with the additional admixture. When A = 2%, the total Clcontent of NS and NF(II) was the lowest, and the binding Clcontent was the highest. This indicated that the effect of improving the salt freeze-thaw resistance of concrete was the best at A = 2%. When the dosage and age are constant, the total Clcontent of NS is lower than NF(II), and the binding Clcontent of NS is higher than NF(II). This indicated that the effect of improving the salt freeze-thaw resistance of nano-SiO 2 is better than that of nano-Fe 3 O 4 . the effect of improving the salt freeze-thaw resistance of nano-SiO2 is better than that of nano-Fe3O4.   Figure 13 shows the relationship between free Cl -, the content of NS and NF(I), and the nanoparticle admixture (x = 5 mm, 7 d and 30 d was used as an example, and the rest of the depth range has the same trend with age). T and C in the Figure represent the tensile and compressive zones, respectively; 0, 0.2, 0.3, 0.5 and 0.6 in the Figure represent the stress ratio, respectively. It can be seen that the free Clcontent in the tensile and compressive zones of nano-SiO2 and nano-Fe2O3 concrete with different dosages under different stress conditions are lower than those of PC. This indicated that the amount of nano-SiO2 and nano-Fe2O3 improved the durability performance of marine concretes. The free Clcontent of both NS and NF(I) initially decreased, then increased with the increase of dosages, and the free Clcontents of nano-SiO2 and nano-Fe2O3 are the lowest at A = 2% and A = 1% under different stress conditions. With increasing stress, the free Clcontent in the the effect of improving the salt freeze-thaw resistance of nano-SiO2 is better than that of nano-Fe3O4. Figure 11. Total Clcontent (x = 5 mm).  Figure 13 shows the relationship between free Cl -, the content of NS and NF(I), and the nanoparticle admixture (x = 5 mm, 7 d and 30 d was used as an example, and the rest of the depth range has the same trend with age). T and C in the Figure represent the tensile and compressive zones, respectively; 0, 0.2, 0.3, 0.5 and 0.6 in the Figure represent the stress ratio, respectively. It can be seen that the free Clcontent in the tensile and compressive zones of nano-SiO2 and nano-Fe2O3 concrete with different dosages under different stress conditions are lower than those of PC. This indicated that the amount of nano-SiO2 and nano-Fe2O3 improved the durability performance of marine concretes. The free Clcontent of both NS and NF(I) initially decreased, then increased with the increase of dosages, and the free Clcontents of nano-SiO2 and nano-Fe2O3 are the lowest at A = 2% and A = 1% under different stress conditions. With increasing stress, the free Clcontent in the  Figure 13 shows the relationship between free Cl -, the content of NS and NF(I), and the nanoparticle admixture (x = 5 mm, 7 d and 30 d was used as an example, and the rest of the depth range has the same trend with age). T and C in the Figure represent the tensile and compressive zones, respectively; 0, 0.2, 0.3, 0.5 and 0.6 in the Figure represent the stress ratio, respectively. It can be seen that the free Clcontent in the tensile and compressive zones of nano-SiO 2 and nano-Fe 2 O 3 concrete with different dosages under different stress conditions are lower than those of PC. This indicated that the amount of nano-SiO 2 and nano-Fe 2 O 3 improved the durability performance of marine concretes. The free Clcontent of both NS and NF(I) initially decreased, then increased with the increase of dosages, and the free Clcontents of nano-SiO 2 and nano-Fe 2 O 3 are the lowest at A = 2% and A = 1% under different stress conditions. With increasing stress, the free Clcontent in the tensile zone of concretes (PC, NS, NF(I)) gradually increased at the same dosage, and the free Clin the compressive zone gradually decreased. Compared with the unloaded condition, when the test age was at 30 d, the free Cllevels in the marine concretes increased less when the stress ratio was 0.2. When σ ≥ 0.3, free Clcontents significantly increased, while free Clin the compressive zone clearly decreased when the stress ratio was small. tensile zone of concretes (PC, NS, NF(I)) gradually increased at the same dosage, and the free Clin the compressive zone gradually decreased. Compared with the unloaded condition, when the test age was at 30 d, the free Cllevels in the marine concretes increased less when the stress ratio was 0.2. When σ ≥ 0.3, free Clcontents significantly increased, while free Clin the compressive zone clearly decreased when the stress ratio was small.

Basic Model
The studies reported that [53] Fick's second law can explain the diffusion path of Clin concrete, and its expression is as follows: where C is the chloride content (%); T is the time of concrete exposed to Clenvironment (s); x is depth (mm); D is Cldiffusion coefficient of concrete(m 2 /s).

Basic Model
The studies reported that [53] Fick's second law can explain the diffusion path of Clin concrete, and its expression is as follows: where C is the chloride content (%); T is the time of concrete exposed to Clenvironment (s); x is depth (mm); D is Cldiffusion coefficient of concrete(m 2 /s). When the boundary conditions are: C(0,T) = Cs; When C(x,0) = 0, Equation (4) can be written: where C 0 is the initial chloride content (%) in concrete; Cs is the apparent chloride content (%) in concrete; T is the test age of concrete(s); erf is the error function, er f u = 2 √ π u 0 e −t 2 dt. However, the actual diffusion path of Clis complicated. Concrete is a kind of porous and heterogeneous material, and the chloride diffusion coefficient is not constant. Moreover, the diffusion path of Clin concrete is influenced by multiple environmental factors, such as temperature, loads, and other factors on the action of diffusion and binding of Cl -. Therefore, Fick's second law needs to be modified based on the above problems in this paper.

Cl -Diffusion Coefficient
Cldiffusion coefficient is an important parameter that describes the diffusion speed of Clin concrete. The free Clcontent of each group of marine concretes at different depths was substituted into the Equation (4), the total and free Cldiffusion coefficient of them can be deduced inversely. Figures 14-16 show the variation of the free Cldiffusion coefficient with A for NS, NF(I), and NF(II) under three environmental conditions, respectively. The total Cldiffusion coefficient has the same trend.
(%) in concrete; T is the test age of concrete(s); erf is the error function, However, the actual diffusion path of Clis complicated. Concrete is a kind of porous and heterogeneous material, and the chloride diffusion coefficient is not constant. Moreover, the diffusion path of Clin concrete is influenced by multiple environmental factors, such as temperature, loads, and other factors on the action of diffusion and binding of Cl -. Therefore, Fick's second law needs to be modified based on the above problems in this paper.

Cl -Diffusion Coefficient
Cldiffusion coefficient is an important parameter that describes the diffusion speed of Clin concrete. The free Clcontent of each group of marine concretes at different depths was substituted into the Equation (4), the total and free Cldiffusion coefficient of them can be deduced inversely. Figures 14-16 show the variation of the free Cldiffusion coefficient with A for NS, NF(I), and NF(II) under three environmental conditions, respectively. The total Cldiffusion coefficient has the same trend.

Environmental Coefficient
The diffusion path of Clunder different environmental conditions (submerge zone, tidal zone, splash zone, and salt spray zone) is different. Referring to the research results [54], the environmental coefficient (fh) selected in this paper is shown in Table 3. Table 3. Values of environmental coefficient [54]. Reprinted with permission from Ref. [54]. 2004, Hongfa Yu".

Clbinding Coefficient
In the actual environment, the damage mechanism of salt spray erosion to marine concrete structures is similar to that of dry-wet cycles. There are not only considerations of salt spray erosion, but also the influence of bending load in this paper, and the diffusion of free Clin concrete under the coupling effect of bending load and salt spray erosion is

Environmental Coefficient
The diffusion path of Clunder different environmental conditions (submerge zone, tidal zone, splash zone, and salt spray zone) is different. Referring to the research results [54], the environmental coefficient (f h ) selected in this paper is shown in Table 3.

Cl -Binding Coefficient
In the actual environment, the damage mechanism of salt spray erosion to marine concrete structures is similar to that of dry-wet cycles. There are not only considerations of salt spray erosion, but also the influence of bending load in this paper, and the diffusion of free Clin concrete under the coupling effect of bending load and salt spray erosion is discussed. Therefore, in this paper, the Clbinding coefficient (K) is only introduced when marine concretes are subjected to dry-wet cycles and salt freeze-thaw cycles.
The Fitting of Cl -Binding Coefficient K can directly reflect the Clbinding capacity of concrete, and the expression is as follows: where K is the binding coefficient; Clb is the binding Clcontent (%); Clf is the free Clcontent (%).
The Clbinding mechanism mainly includes three types: a linear binding mechanism, a Fangmuir binding mechanism, and a Langmuir binding mechanism. The linear binding mechanism is not only more concise and direct but also has a high correlation with the test results. Hence, this paper employed the linear binding mechanism to fit K. The free Clcontent and binding Clcontent at different ages and depths were substituted into Equation (5) and obtained the K value of NS and NF(II) under the action of dry-wet cycles and full immersion, as shown in Figures 17 and 18. (Taking A = 2% as an example, the fitting results of each dosage are similar). It can be seen that the fitting results can all reach above 0.99. mechanism is not only more concise and direct but also has a high correlation with the test results. Hence, this paper employed the linear binding mechanism to fit K. The free Clcontent and binding Clcontent at different ages and depths were substituted into Equation (5) and obtained the K value of NS and NF(II) under the action of dry-wet cycles and full immersion, as shown in Figures 17 and 18. (Taking A = 2% as an example, the fitting results of each dosage are similar). It can be seen that the fitting results can all reach above 0.99.    20 show that the fitting results of the relationship between the K value and A value for NS and NF(II) under the action of the dry wet cycle (full immersion) and salt freeze-thaw cycles, respectively, The fitting equation of K value and A value can be seen in Equation (6).
where e, f and g are the fitting coefficients, and the fitting results are shown in Tables 4 and 5. and A value for NS and NF(II) under the action of the dry wet cycle (full immersion) and salt freeze-thaw cycles, respectively, The fitting equation of K value and A value can be seen in Equation (6).
where e, f and g are the fitting coefficients, and the fitting results are shown in Tables 4  and 5.

Depth of Convection
According to the test results in Figure 7, the diffusion of Clin concrete under the action of dry-wet cycles and full immersion exists in the convection zone. Therefore, the convective depth (x c ) should be considered in the Cldiffusion correction model under the action of dry-wet cycles and full immersion.

Load Coefficient
According to the free Clcontent in three groups of marine concretes in Figure 12, the diffusion speed of Claccelerated in the tensile zone, and the diffusion coefficient will also increase. The Cldiffusion increases are more evident when the tensile stress increases. However, the diffusion speed of Clin the compressive zone will be lower with increasing stress. In order to reflect the relationship between bending load and Cldiffusion, the load coefficient f (σ) is introduced. According to the test results, each group of marine concretes under different tensile and compressive stresses (set D/D 0 = f (σ)) were fitted that can obtain the relation curve in the tensile zone and compressive zone between the bending load and Cldiffusion coefficient, as shown in Equations (7) and (8).
where h, i, g and k are all fitting coefficients, and the fitting results are shown in Tables 6 and 7.

The Age Attenuation Coefficient
Mangat et al. [55] reported that the Cldiffusion coefficient was dependent on time to a certain extent. They proposed to use m to represent the age attenuation coefficient, and established a modified model of the relationship between the Cldiffusion coefficient and time: Costa et al. [56][57][58] provided m values in different types of concrete under different marine environments, as shown in Table 8. Due to the nano-particles used in this paper, the m value in Table 8 is not consistent with materials in this test. So, it is necessary to re-fit the m value of three groups of marine concrete under the action of three environmental factors. The m value is fitted under three conditions according to Equation (9). The Combined Action of Dry-Wet Cycles and Cl - Figure 21 shows that the fitting results of relationship between m and A of NS and NF(II) under the action of dry-wet cycles and full immersion. The fitting formula of m and A is shown in Equation (10).
where a, b, c and d are fitting coefficients, and the fitting results are shown in Table 9. The Combined Action of Dry-Wet Cycles and Cl - Figure 21 shows that the fitting results of relationship between m and A of NS and NF(II) under the action of dry-wet cycles and full immersion. The fitting formula of m and A is shown in Equation (10).
where a, b, c and d are fitting coefficients, and the fitting results are shown in Table 9.    The Combined Action of Salt Freeze-Thaw Cycles and Cl - Figure 22 shows the fitting results of the relationship between m and A of NS and NF(II) under the action of salt freeze-thaw cycles. The fitting formula of m and A is shown in Equation (10). The fitting results are shown in Table 10. The Combined Action of Salt Freeze-Thaw Cycles and Cl - Figure 22 shows the fitting results of the relationship between m and A of NS and NF(II) under the action of salt freeze-thaw cycles. The fitting formula of m and A is shown in Equation (10). The fitting results are shown in Table 10.  The Combined Action of Bending Load and Salt Spray Erosion Figure 23 shows the fitting results of the relationship between m and A of NS and NF(I) under combined the action of bending load and salt spray erosion. The fitting formula of m and A is shown in Equation (11). The fitting results are shown in Table 11.  The Combined Action of Bending Load and Salt Spray Erosion Figure 23 shows the fitting results of the relationship between m and A of NS and NF(I) under combined the action of bending load and salt spray erosion. The fitting formula of m and A is shown in Equation (11). The fitting results are shown in Table 11. m= l + rA + jA 2 +zA 3 (11) The Combined Action of Bending Load and Salt Spray Erosion Figure 23 shows the fitting results of the relationship between m and A of NS and NF(I) under combined the action of bending load and salt spray erosion. The fitting formula of m and A is shown in Equation (11). The fitting results are shown in Table 11.

The Action of Dry-Wet Cycles and Full Immersion
Considering f h , m and x c to modify Fick's second law, Equations (6), (8) and (9) are substituted into Equation (4), respectively, to obtain the Cldiffusion correction model of nano-marine concretes under the action of dry-wet cycles and full immersion: where C is Clcontent (%); C 0 is the initial Clcontent (%); Cs is the apparent Clcontent (%); erf is error function; D 0 is the Cldiffusion coefficient (m 2 /s) at the initial time; t 0 is the initial time (s); f h is the environmental coefficient; X c is the depth of convective zone (mm); X is the depth from the concrete surface (mm); T is the age (s); A is nanoparticles dosage (%); a, b, c, d, e, f, and g are the fitting coefficients. According to the actual service environment of marine concretes, the f h value is chosen as 1.32 and 0.92 in the modified model under the action of full immersion and dry-wet cycles, respectively. Taking different ages and depths as an example, the test results of NS at Q = 25 d, x = 10 m, G = 75, x = 10 mm, and NF(II) at Q = 100 d, x = 2 mm, G = 150, x = 20 mm were fitted to the modified model. Figures 24 and 25 show the fitting results of the relationship between the Cldiffusion correction formula and the content (A) (the rest of the ages and depths have the same law). The fitting curve of the modified model under the action of dry-wet cycles has a high correlation with the test results, and R can reach above 0.99. wet cycles, respectively. Taking different ages and depths as an example, the test results of NS at Q = 25 d, x = 10 m, G = 75, x = 10 mm, and NF(II) at Q = 100 d, x = 2 mm, G = 150, x = 20 mm were fitted to the modified model. Figures 24 and 25 show the fitting results of the relationship between the Cldiffusion correction formula and the content (A) (the rest of the ages and depths have the same law). The fitting curve of the modified model under the action of dry-wet cycles has a high correlation with the test results, and R can reach above 0.99.

The Action of Salt Freeze-Thaw Cycles Modified Model
The influence of salt freeze-thaw cycles on the durability of marine concretes are shown as follows: firstly, cracks on the surface of increase until they become cracking, and then the surface of concrete begins to peel off. Therefore, not only considering the effects of fh, K, and m, but also the effects of salt freeze-thaw cycle damage and the thickness of the exfoliation layer should be considered in the Cldiffusion correction model under the action of salt freeze-thaw cycles.

Influence of Salt Freeze-Thaw Cycles
(1) Salt freeze-thaw cycles damage degree of concrete: According to the concept of damage mechanics [57], the damage degree of concrete structures subjected to salt freezethaw cycles is usually shown as salt freeze-thaw damage degree (F). The deterioration of concrete structure durability is more serious with the gradual increase of the F value. The expression of the F value is as follows: where F(N) denotes the damage degree of concrete after a number of N (%); ED0 denotes the dynamic modulus of elasticity of concrete before salt freeze-thaw cycles (MPa); EDN denotes the dynamic modulus of elasticity of concrete after the number of N (%) (MPa). The salt freeze-thaw cycles damage degree of marine concretes can be calculated as shown in Table 12.

The Action of Salt Freeze-Thaw Cycles Modified Model
The influence of salt freeze-thaw cycles on the durability of marine concretes are shown as follows: firstly, cracks on the surface of increase until they become cracking, and then the surface of concrete begins to peel off. Therefore, not only considering the effects of fh, K, and m, but also the effects of salt freeze-thaw cycle damage and the thickness of the exfoliation layer should be considered in the Cldiffusion correction model under the action of salt freeze-thaw cycles.

Influence of Salt Freeze-Thaw Cycles
(1) Salt freeze-thaw cycles damage degree of concrete: According to the concept of damage mechanics [57], the damage degree of concrete structures subjected to salt freezethaw cycles is usually shown as salt freeze-thaw damage degree (F). The deterioration of concrete structure durability is more serious with the gradual increase of the F value. The expression of the F value is as follows: where F(N) denotes the damage degree of concrete after a number of N (%); E D0 denotes the dynamic modulus of elasticity of concrete before salt freeze-thaw cycles (MPa); E DN denotes the dynamic modulus of elasticity of concrete after the number of N (%) (MPa).
The salt freeze-thaw cycles damage degree of marine concretes can be calculated as shown in Table 12. Fitting the relationship between the F value and N value: the fitting formula for the relationship between the F value and N value for NS and NF(II) under the action of salt freeze-thaw cycles is as follows: where u, v, w are the fitting coefficients, and the fitting results are shown in Table 13. The relationship between F and Cldiffusion coefficient: Salt freeze-thaw cycles will cause concrete structures to deteriorate and generate cracks, thus accelerating the migration rate of Clin the concrete. Studies have reported [55] that the F and Cldiffusion coefficient approximately obey the exponential function relationship, the equation is as follows: where D 0 is the Cldiffusion coefficient (m 2 /s) of sound concrete; D F is the Cl-diffusion coefficient (m 2 /s) after the action of salt freeze-thaw cycles; F is the damage degree of salt freeze-thaw cycles (%); ω is the effect coefficient of salt freeze-thaw cycles. The fitting results are shown in Table 14.

Influence of Peeling Layer Thickness
Studies have shown [59] that the peeling layer thickness of concrete is proportional to the mass loss rate, the expression is as follows: where ∆B is the peeling layer thickness of concrete after the action of salt freeze-thaw cycles (mm); λ is the coefficient of test conditions (e.g., specimen shape, peeling uniformity, material properties, and the effect of salt freeze-thaw cycles, etc.; Rm is the mass loss rate of concrete after the action of salt freeze-thaw cycles (%). In the literature [60], the relationship between Rm and N of concrete under the action of salt freeze-thaw cycles was investigated and the following relationship was obtained.
where R m is the mass loss rate of specimens under the action of salt freeze-thaw cycles; N is the number of salt freeze-thaw cycles; p and q are material characteristic parameters. After summarizing and analyzing the above factors, Fick's second law is modified. Considering the influence of f h, m, K, F, and ∆B, the Cldiffusion correction model of nano-marine concretes is obtained as follows: where C is the Clcontent (%) in concrete; C 0 is the initial Clcontent (%) in concrete; Cs is the surface Clcontent of concrete (%); erf is error function; T is the time of concrete structure exposed to Clenvironment (s); x is the depth from the concrete surface (mm); D is Cldiffusion coefficient (m 2 /s); f h is the environmental coefficient; K is the Clbinding coefficient; ∆B is the thickness of the exfoliated layer (mm); ω is the influence coefficient of salt freeze-thaw cycles damage; F is the damage degree of salt freeze-thaw cycles (%). Considering the influence of the actual environment and referring to the existing research results, fh is 1.32; λ, p and q are 0.03, 2.5 and 0.2, respectively; C 0 are 0.03% and 0.5%; D 0 is 0.5 × 10 −12 m 2 /s; T is 3.456 × 10 5 s (4 d). Taking different ages and depths as examples, the test results of NS at N = 25 and x = 5 mm, NF (II) at N = 125 and x = 15 mm were fitted, respectively. Figure 26 shows the fitting results of relationship between the Cldiffusion correction formula and A of nano-marine concretes under the action of salt freeze-thaw cycles (other ages and depths have the same trend). The fitting curve of the modified model under the action of salt freeze-thaw cycles has a high correlation with the test results, and R can reach above 0.99. where Rm is the mass loss rate of specimens under the action of salt freeze-thaw cycles; N is the number of salt freeze-thaw cycles; p and q are material characteristic parameters. After summarizing and analyzing the above factors, Fick's second law is modified. Considering the influence of fh, m, K, F, and ΔB, the Cldiffusion correction model of nanomarine concretes is obtained as follows: where C is the Clcontent (%) in concrete; C0 is the initial Clcontent (%) in concrete; Cs is the surface Clcontent of concrete (%); erf is error function; T is the time of concrete structure exposed to Clenvironment (s); x is the depth from the concrete surface (mm); D is Cldiffusion coefficient (m 2 /s); fh is the environmental coefficient; K is the Clbinding coefficient; ΔB is the thickness of the exfoliated layer (mm); ω is the influence coefficient of salt freeze-thaw cycles damage; F is the damage degree of salt freeze-thaw cycles (%). Considering the influence of the actual environment and referring to the existing research results, fh is 1.32; λ, p and q are 0.03, 2.5 and 0.2, respectively; C0 are 0.03% and 0.5%; D0 is 0.5 × 10 −12 m 2 /s; T is 3.456 × 10 5 s (4 d). Taking different ages and depths as examples, the test results of NS at N = 25 and x = 5 mm, NF (II) at N = 125 and x = 15 mm were fitted, respectively. Figure 26 shows the fitting results of relationship between the Cldiffusion correction formula and A of nano-marine concretes under the action of salt freeze-thaw cycles (other ages and depths have the same trend). The fitting curve of the modified model under the action of salt freeze-thaw cycles has a high correlation with the test results, and R can reach above 0.99.

Combined Action of Bending Load and Salt Spray Erosion
The load coefficient f(σ), environmental influence coefficient fh, and age attenuation coefficient m were taken into account to modify Fick's second law. Equations (7), (8), and (10) were substituted into Equation (4), and the Cldiffusion correction model of nanomarine concretes was obtained as follows:

Combined Action of Bending Load and Salt Spray Erosion
The load coefficient f (σ), environmental influence coefficient fh, and age attenuation coefficient m were taken into account to modify Fick's second law. Equations (7), (8), and (10) were substituted into Equation (4), and the Cldiffusion correction model of nano-marine concretes was obtained as follows:

Conclusions
Four types of marine concretes were prepared in this paper, including OPC, NS, NF(I), and NF(II). According to their service characteristics under the action of the above three environmental factors, the Clcontent was tested. And based on Fick's second law, the Cldiffusion modified model for nano-marine concretes was also proposed. The main conclusions are as follows: (1) Under the action of dry-wet cycles, the convection zone appears due to the action of capillary adsorption, which is different from the erosion mechanism of the other two environmental actions. So, the erosion speed of Clwas significantly accelerated under the action of dry-wet cycles. (2) With the increase in the number of salt freeze-thaw cycles, the icing expansion pressure, osmotic pressure and salt crystallization pressure in concrete were increased, which lead to the deterioration of concrete durability. (3) Under the action of three different environments, the nano-SiO 2 , nano-Fe 2 O 3 , and nano-Fe 3 O 4 of the optimum dosage were 2%, 1%, and 2%, respectively. Compared with the other two nanoparticles, nano-SiO 2 had the best effect on improving concrete. (4) Considering the characteristics of Cldiffusion under the action of three environmental factors and its relative parameters, Fick's second law is modified, and the Cldiffusion modified model of nano-marine concretes is obtained. It can be used to calculate the Clcontent in different dosages and depths of nanoparticles. The correlation coefficient R value of the fitting curve and the test results is basically above 0.99, which can provide a reference for the life prediction of actual marine concretes. (5) Under the coupling effect of bending load and salt spray erosion, the fitting results correlation coefficient R of the load influence coefficient f (σ) for marine concretes is basically above 0.95, which has a high correlation. Compared with the unloaded condition, the load influence coefficient values in the tensile zone of marine concretes are increased to different degrees, while their values in the compressive zone are decreased. It is proved that the Clcontent is closely related to bending load.

Patents
The patent number used in this article (ZL202120853835.7), and the patent name is "A concrete durability test chamber under salt spray erosion".