Modeling of Chloride Distribution in Cement-Based Materials with Neumann Boundary Condition

*e nonstable transport of chloride in cement-based materials, including diffusion, convection, and chloride binding, is described as a general form of Fick’s law. Inspired by the heat transport of concrete, the second boundary condition called the Neumann boundary condition is considered in the chloride transport of concrete. *e theoretical deduction of one-dimensional chloride distribution with the Neumann boundary condition is performed, while a virtual boundary is introduced to carry out the approximate treatment. Finally, the comparison between the general Dirichlet boundary condition and the present Neumann boundary condition and the influence of the thickness of the virtual boundary as well as the influence of the water-cement ratio (w/c) are discussed.


Introduction
It is well known that concrete structures exposed to the chloride environment, e.g., seawater and deicing salts, might be seriously damaged due to the chloride-induced corrosion of internal steel.With the continuous penetration of chloride, the pH value around steel is decreased, while the chloride value is increased.When chloride concentration reaches some given level called the threshold value, steel would be corroded [1].erefore, the determination of the transport of chloride [2][3][4] and threshold value [5][6][7][8] as well as improvement of chloride resistance [9][10][11] is the hot topic for decades.
It is reasonable to consider that the determination of the transport of chloride is the basic and most important research to promote theories from the experimental results [3,12] and develop life prediction models of concrete structures [13][14][15][16].
e diffusion caused by the chemical potential gradient [2] and the convection caused by the collective movement of groups or aggregates of molecules within fluids [17] are two main mechanisms of chloride transport in general conditions.eoretically, they follow Fick's law [4] and Darcy's law [18,19], respectively.In fact, the chloride transport, especially nonstable transport, is a complicated process while one cannot distinguish clearly their two mechanisms from the chloride distribution [20][21][22].Furthermore, chemical/physical chloride binding by the hydration products reduces the free chloride concentration and retards the penetration of chloride to the level of the steel [23][24][25].
e discussion between free chloride and total chloride that which one can be responsible for the corrosion initiation is still existing [26][27][28][29].
us, free chloride is more used to evaluate the corrosion risk of reinforce concrete, while total chloride is collected and analyzed as a reference.Experimental studies have been performed by different researchers, and the main conclusions are as follows [30][31][32]: (i) it is hard to determine exactly the chloride concentration at depths smaller than about 5 mm, (ii) the distributions of total chloride and free chloride have the similar tendency, and (iii) with the increase of depth, the chloride concentration decreases rapidly while its distribution follows the partial differential equation as zC/zt � α∇ 2 C. Figure 1 shows the general chloride concentration profiles.
It is well known that the nonstable diffusion follows Fick's second law with zC/zt � D∇ 2 C, where D is the diffusion coefficient.Note that the actual transport of chloride needs to take into account several different mechanisms and the coupling between them; it is unsuitable to describe its transport character as the diffusion coefficient.e actual transport of chloride is normally defined as zC/zt � D app ∇ 2 C, where D app is the apparent diffusion coefficient generally obtained by the curve fitting of free chloride concentration with Fick's second law [12,[33][34][35].Also, a series of models based on this basic law have been developed to take into account the multi-ionic effect [36], temperature [37,38], stress [39][40][41], electrical field [42], curing age [14,43,44], etc.And the relevant development is based on the mathematical calibration of various forms of partial differential equation with the first boundary condition called the Dirichlet condition.With the Dirichlet boundary condition, the surface chloride concentration of concrete is prescribed as a given value, e.g., chloride concentration of the external chloride environment.However, the Dirichlet condition might not be accurate especially in the zone of severe changes.In this type of zone, the distribution of chloride concentration varies drastically that one cannot determine the exact value.us, the surface chloride concentration of concrete is not equal to chloride concentration of the external chloride environment [31].In fact, the evolution of chloride concentration is controlled by the flux of chloride ions across the surface, while it is proportional to the chloride concentration difference between the surface and the surrounding medium.is boundary condition is called as the Neumann boundary condition, and it is widely applied to the thermal stress and temperature analysis of mass concrete [45,46].Considering the similar conditions that temperature/chloride concentration is violently changed at the surface zone of concrete, it is reasonable to assume that the boundary condition of chloride transport might be considered as the Neumann boundary condition as better.
e present research is organized as follows: first, the differential equation of nonstable transport of chloride with two different boundary conditions is summarized.en, we focus on the one-dimensional case with the Neumann boundary condition, and its exact solution with the constant apparent diffusion coefficient is deduced by the mathematical method.Considering the disadvantage of resolving its exact solution, a virtual boundary is introduced to simplify the approximate deduction.According to relevant experimental results, the comparison between the general Dirichlet boundary condition and the present Neumann boundary condition and the influence of the thickness of the virtual boundary as well as the influence of the water-cement ratio (w/c) are discussed.

Description of Chloride Transport with Two Boundary Conditions
2.1.General Expression.As mentioned above, the nonstable chloride transport can be described as Fick's second law and one-dimensional expression is as follows: where D app is the apparent diffusion coefficient (m 2 /s), t is the exposure time (s), and C f (x, t) is the free chloride concentration at depth x (m) after time t (%/m 3 ).D app can be determined as a constant or a variable considering the multi-ionic effect, temperature, stress, electrical field, curing age, etc.And also, the initial chloride concentration of the structure is given as follows: is paper focuses on the influence of different boundary conditions so that one simplified example with D app as a constant is performed later.

First Boundary Condition.
e first kind of boundary condition is called the Dirichlet boundary condition.It is the prescribed chloride concentration given as follows: where C s is the surface chloride concentration.
For the ideal concrete of one semi-infinite medium, the exact solution of Equation (1) with the Dirichlet boundary condition can be derived as an error function solution [47]: where erf is the error function.D app and C s can be achieved using Equation (4) as a constant.is model is widely used to determine chloride distribution in concrete as well.

Second Boundary Condition.
e second boundary condition is called the Neumann boundary condition.In the application of heat transport of concrete, it is prescribed that heat flux across the surface is proportional to the 2 Advances in Materials Science and Engineering concentration difference between the surface and the surrounding medium: where T a is the environment air temperature, T is the surface temperature of concrete, λ is the conductivity of concrete, β is the surface conductance, and n is the outward normal of the surface.In this paper, the Neumann boundary condition is extended to the application of chloride transport as follows: where C s is the peak value near the boundary instead of the environment chloride concentration, C is the chloride concentration of the boundary, λ is the chloride conductivity of concrete, and β is the surface chloride conductance.In the application of chloride transport, it is hard to determine λ and β compared with the case of heat transport.

Exact Solution of Chloride Transport with the Second
Boundary Condition.In this study, an ideal infinite concrete plat (Figure 2) is considered while D app is a constant and thickness is 2L.e chloride can penetrate into this concrete from both the left and right sides.
It can be defined as follows: where describe the chloride diffusion along the negative direction of x-axis and positive direction of x-axis, respectively, and t is the exposure time.
In order to get the formed solution of Equation ( 7), firstly we study the following problem defined on [0, L]: We assume that u(x, 0) is of the form u(x, t) � X(x)T(t) with nontrivial functions X(x) and T(t).Inserting it into the homogeneous equation and the homogeneous boundary condition, we get the eigenproblem: and the ordinary differential equation for T(t): e parameter λ arising in Equation ( 9) is called the eigenvalue of the problem; the corresponding nontrivial solution X(x) is the so-called eigenfunction.It is well known that the eigenvalues and the eigenfunctions are as follows: where μ n are the positive solution of the equation x tan x � L/d.en, using the ordinary differential Equation (10), we have en, the function , satisfies the homogeneous equation and the homogeneous boundary equation.So does the linear combination of μ n , denoted by u(0, t): By the orthogonality of the eigenfunctions, we have

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Hence, we get the desired form u(x, t) of the problem (8) as follows: Next, we construct the solution to the one-dimensional general chloride transport Equation (7) as follows: Finally, the exact solution of Equation ( 7) can be expressed as follows: where μ n is the infinitely many solutions of In the case of heat transport problems, the number of roots of the characteristic equation is less than 10 to obtain the converging solution.In the present chloride transport problems, the number of roots of the characteristic equation must be selected as 10 5 and even more to ensure the convergence of (17).Furthermore, λ and β cannot be easily determined as mentioned above.It is considered that direct application of the second boundary condition for chloride transport is unavailable.

2.5.
e Approximate Treatment of the Second Boundary Condition.When the surface chloride concentration C is changed from C 1 to C 2 , (6) will be As shown in Figure 3(a), the tangents to the chloride curves at the surface will always pass through point A, and the distance between point A and the surface of concrete is e distribution of chloride with the Neumann boundary condition is presented in Figure 3(b).For the second boundary condition, if a virtual thickness d � λ/β is introduced to concrete, a virtual boundary is obtained.e chloride concentration on this virtual boundary is equal to C s . is means that the introduced virtual boundary satisfies the Dirichlet boundary condition with the prescribed chloride concentration C s shown in Figure 3(c).
Considering that d is added to the surface, the chloride distribution with the second boundary condition can be rewritten as the general expression of the Dirichlet boundary condition: Note that the chloride transport with the Neumann boundary condition is simplified to chloride transport with the Dirichlet boundary condition while an additional parameter d is introduced.Determination of D app , C s , and d can be easily solved.

Experimental Validation
Diffusion tests and non-steady-state migration tests are carried out to verify the above theoretic results.e apparent diffusion coefficient (D app ), surface chloride concentration (C s ), and additional parameter d are obtained from diffusion tests by data fitting.Non-steady-state migration tests are carried out for verifying the variation of the non-steady-state diffusion coefficient (D nssm ) in hardened concrete.e variation of the non-steady-state diffusion coefficient (D nssm ) can reflect the changes of the apparent diffusion coefficient (D app ) indirectly.

Specimen Preparation.
In this study, the samples of mortar were used.Mortar mixtures consisted of Type I Portland cement (according to ASTM 2005b), river sand with a maximum size of 5 mm and a fineness modulus of 2.90, and tap water.e chemical composition of the cement is given in Table 1, and mixing proportions are given in Table 2. e mixture was put into 100 × 100 × 100 mm and Φ 100 mm × 150 mm molds for diffusion tests and migration tests, respectively.All the samples were stored at 20 °C and 90% relative humidity for 24 h and then were demolded and immersed in 20 °C water for 7 days; afterwards, the samples were cured in a room at 20 °C and 90% relative humidity for 90 days to reduce the time-dependent influence based on the hydration process.

Diffusion Tests.
Cubes with the size of 100 × 100 × 100 mm were selected for diffusion tests.Before testing, one side surface of 100 × 100 mm was selected as the aggressive surface exposed to the NaCl solution, while the other five surfaces were coated with epoxy resin to obtain the one-dimensional transport of the chloride in the concrete.
en, all samples were kept immersed in the 2.8 mol/L NaCl solution for 28, 56, and 90 days (Figure 4).e concentration of NaCl solution was kept constant by sealing the immersion container.
After removing from the NaCl solution, the first layer of 2 mm was cut off to avoid the buildup of salt crystals at the surface, and the rest of the samples was grounded into powder every 3 mm by an electric grinding machine 4 Advances in Materials Science and Engineering (Figure 5) to obtain the chloride concentration profile.All the powder was dried in a stove for 24 h, and chloride was then extracted by distilled water.e free chloride concentration was measured by AgNO 3 titration with an automatic potentiometric titrator (Figure 6) related to the mass of the mortar.

Non-Steady-State Migration Tests.
Cylinders with the size of Φ 100 mm × 150 mm were cut into Φ 100 mm × 50 mm after demolding.Non-steady-state migration tests were conducted at the age of 7, 14, 28, 56, and 90 days based on NT BUILD 492.10% NaCl solution and 0.3 mol/L NaOH solution were used at the cathode and anode, respectively.An external electrical potential of 30 V was applied to force chloride ions to migrate into the specimens.
After the tests, the specimens were axially split and 0.1 mol/L AgNO 3 solution was sprayed on the newly split sections to obtain the chloride penetration depth.e non-steady-state chloride migration coefficient D nssm can be calculated with the chloride penetration depth, as described in NT BUILD 492, with where

Results and Discussion
Chloride content (%, binder) Chloride content (%, binder)  Figure 7 shows the chloride profiles obtained from the diffusion tests.It can be seen that most experimental data are in accordance with the regulation except the data at 3.5 mm.As shown in Figures 7(b) and 7(c), at the depth of 3.5 mm, chloride content with longer days of exposure is higher than that with shorter days of exposure.is is mainly caused by the rapid variation near the surface.us, considering that the accuracy and regularity of experimental data at 3.5 mm are problematic, data at 3.5 mm are not shown in Figure 8.  3; values of d fall between 1.5 mm and 2 mm and change slightly with the water-cement ratio. is is probably due to the reduction of time-dependent influence, and in this paper, the value of d is only influenced slightly by the water-cement ratio.According to the simulation results calculated by the Intel Visual Fortran, the change of relative error (difference value between simulation data and experimental data) with the width of the virtual boundary d in case of w/b � 0.45 is shown in Figure 9.As shown in Figure 9, it is obvious that the value of relative error reaches the minimum degree while d is between 1.5 mm and 2 mm, and the same tendency with w/b � 0.4 and w/b � 0.5 is also obtained.e results imply that the calculated range of d provides an important reference for further study.

Variation of D nssm with Curing
Age. Variation of D nssm with curing age is shown in Figure 10; it can be seen that the value of D nssm changes violently before the curing age of 28 days and then remains almost stable between the curing age of 56 and 90 days.It can be explained that the hydration reaction reduces with the curing age; thus, the value of D nssm tends to be stable, which indirectly proves that D app is almost stable at the age of 90 days.In our study, samples were cured for 90 days, and then we perform the diffusion tests.
erefore, the example with D app as a constant can be considered available.

Influence of Water-Cement Ratio (w/c).
In this study, experimental study is based on three water-cement ratios.As shown in Table 3, with the increase of water-cement ratio, values of D app and C s increase.is is caused by the higher porosity with the higher water-cement ratio.However, the value of d changed slightly with the water-cement ratio, which implies that d is influenced insignificantly by the water-cement ratio.

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D app is considered constant.As shown in Figure 11, it is obvious that, at the same depth, the chloride content obtained from the proposed model is higher, to achieve the same content like 0.1%.At 5 years, the penetration depths of the proposed model and tradition model are 114 mm and 101 mm, respectively, and at 10 years, the penetration depths are 162 mm and 143 mm, respectively.e prediction of the proposed model is 13% higher at both 5 years and 10 years.Considering that the chloride content is underestimated and the service life is overestimated by the tradition model, the present model might be better applied in the case of longterm prediction.

Conclusions
By investigating the theoretical model and experimental data with two boundary conditions, the following conclusions are made: (1) Due to a complex mechanism, the chloride concentration decreases rapidly with the increasing depth.

Figure 3 :
Figure 3: Explanation of the Dirichlet boundary condition and Neumann boundary condition: (a) approximate treatment of the Neumann boundary condition; (b) chloride profile of the Neumann boundary condition; (c) chloride profile of the Dirichlet boundary condition.

Figure 8
shows the experimental chloride profile with the Dirichlet boundary condition and simulation chloride profile with the Neumann boundary condition, and experimental data are also shown in this figure for comparison.It is obvious that the simulated results are in accordance with the regulation and agree well with the experimental data, which demonstrates that the proposed theoretical model with the Neumann boundary condition is reasonable for predicting the chloride diffusion and service life of cementbased materials.4.2.Influence of d on Neumann Condition.Values of d with the Neumann condition are also shown in Table

Figure 8 :
Figure 8: Chloride profile over time obtained from experimental data and model in the present study: (a) w/b � 0.4; (b) w/c � 0.45; (c) w/b � 0.5.

( 3 )Figure 9 :Figure 10 :Figure 11 :
Figure 9: Variation of relative error with the width of the virtual boundary.

Table 1 :
Chemical composition of cement.
Advances in Materials Science and Engineering values of D app and C s change slightly at different exposure times; consequently, the simulation values of D app and C s are considered stable.As shown in Table 3, it is obvious that both D app and C s with the Neumann boundary condition are higher than those with the Dirichlet boundary condition.is means when using the Dirichlet boundary condition, values of D app and C s are underestimated and lead to overestimation of service life.

Table 2 :
Mixing proportions of mortar.

Table 3 :
Values of D app , C s , and d with two boundary conditions.