COMSOL Multiphysics Modeling

COMSOL Multiphysics User's Conference 2009, January 12-13, Dhahran, Saudi Arabia

COMSOL Multiphysics Modeling of Chloride Binding in Diffusive Transport of Chlorides in Concrete M.A. Shazali1, W.A. Al-Kutti2, M.K. Rahman3, A.H. Al-Gadhib2, M.H. Baluch2 1

2

INCO Precast Engineering, Industrial Contractors Co. Ltd., Al-Khobar, Saudi Arabia Department of Civil Engineering, King Fahd Univ. of Petroleum & Minerals, Saudi Arabia 3 Research Institute, King Fahd Univ. of Petroleum & Minerals, Saudi Arabia

Abstract chlorides ingress in concrete under field service conditions is often ignored for formulation and analytical difficulty reasons. Underlying effort of this paper is therefore to demonstrate the use of COMSOL Multiphysics Software [2], an interactive 3D-partial differential equation development and simulation environment previously known as FEMLAB, in effectively solving the diffusive transport of chloride in concrete. In this regard, treatment of the time-to-corrosion initiation problem is decisively done in a manner averse to unrealistic assumptions of constant diffusivity associated with null binding often enforced for solution expediency.

One of the most revealing causes of deterioration leading to durability loss in concrete structures is premature time-tocorrosion ignition due to chloride induced corrosion associated with diffusive transport of chloride in concrete. To permit a more realistic modeling of the diffusive transport of chloride in concrete, chloride binding capacity model based on experimental isotherm describing the relation between total, bound, and free chlorides have to be considered. The aim of this work is to demonstrate the capability of using COMSOL Multiphysics modeling to provide insight regarding a more realistic consideration of chloride penetration behavior in concrete in view of chloride binding. Simulations of the problem in COMSOL allow main process of the chloride binding to be closely captured in the light of its nonlinear concentration dependence and influence on evolution of chloride diffusivity governing the transport problem. Good agreement of the model results with experimental data was established to provide a basis for reasonable service life prediction in view of the time-to-corrosion initiation paradigm. Keywords: COMSOL, multiphysics, modeling, concrete, durability, chloride, diffusion, binding, corrosion. Mohammed A. Shazali, PhD Senior Structural Engineer, INCO Precast Division P.O. Box 437 Al-Khobar 31952, Kingdom of Saudi Arabia E-mail: [email protected]

Figure 1: Conceptual corrosion sequence in concrete [1].

Introduction

Chloride Diffusion Equations

The impact of environmental exposures in shortening the service life or durability of concrete infrastructures in the field has continued to be substantial worldwide. For chloride exposures, the forecast to the service life loss problem is closely related to ascertaining the corrosion initiation time of the steel reinforcement in concrete. In accordance with conceptual steel corrosion sequence model advocated by Tuutti [1], the durability capacity of the concrete structure is significantly limited to the corrosion initiation time that is very much dependent on diffusive transport of chloride in concrete.

The methodology proposed in this paper accounts for external chloride penetration into concrete considering influence of physical diffusion and chemical binding of chloride transported in to concrete. Considering onedimensional space problem, the transport problem of chlorides can be modeled according to the following form of Fick’s second law equation.

∂ (η .S .C f ) ⎞ ∂C a ∂ ⎛ ⎟⎟ = ⎜⎜ Dce ∂t ∂x ⎝ ∂x ⎠

(1)

where Ca is the total chloride concentration in kg/m3 concrete, Cf if the free chloride concentration in kg/m3 pore solution, Dce is the effective diffusivity in m2/s, η is the volumetric void fraction or the capillary porosity in m3

Although most corrosion initiation time studies in the literature are observed to focus on pure physical diffusion process of the problem, the role of binding mechanisms for 1


pore-voids /m3 concrete, and S is the degree of moisture saturation in the foregoing voids in the concrete. From here onwards the quantity S is dropped for taking a unit value in the case of chloride diffusing through saturated voids considered in this study.

aluminoferrite (C4AF) present as part of the cement or binder component of the concrete substrate. According to the anion exchange hypothesis, the formation of the Friedel salt (Fr) in which unreacted aluminates react with the intruding free chloride ions (Cl-) in the presence of portlandite (calcium hydroxide, CH) can be represented as follows.

Assuming constant diffusivity and null binding capacity, (Ca = Cf) the solution to Equation (1) for semi infinite concrete with initial and boundary condition set to Cf(x,0)=0 and Cf(0,t)=Cs respectively is given by

⎛ ⎡ x ⎤⎞ C f ( x, t ) = C s ⎜1 − Erf ⎢ ⎥⎟ ⎜ 4 D t ⎥ ⎟⎠ ce ⎦ ⎣⎢ ⎝

2Cl

(

)

CH +10 H 2O + C 3 A ⎯⎯ ⎯ ⎯ ⎯→ C 3 A.CaCl 2 .10 H 2 O + 2OH −

(4)

Classical empirical isotherms of the Linear, Langmuir, Freundlich, and Langmuir-Freundlich types are commonly employed to characterize binding behavior of this nature, independent of the chloride transport. It is further assumed in this paper that both the C3A and C4AF aluminates content of the cement paste portion of the concrete substrate contribute equally in reacting with the free chloride ions (Cl-), and thus both equally serve as reactants for the reaction product formation.

(2)

Although Equation (2) is widely used, the solution to the chloride transport problem is more complex because whereas the chloride diffusivity is not constant, the role of chloride binding is significant owing to adsorption affinity of cementitious material to chlorides. Based on model proposed by Bazant [3], Saetta et al. [4, 5], and adopted elsewhere [6], a chloride diffusivity parameter Dc can be defined using a multivariate law characterized by use of empirical functions predicating the influence of parameters embraced in the following form: Dc = Dco .F Tc , H c , Bc , d c = Dco .Fct .Fch .Fcb .Fcd

−

Following the Langmuir isotherm, the chloride binding in which the chloride concentration in the pore phases (Cf) is in steady state equilibrium with the chloride concentration on the concrete adsorbent (Cb) can be expressed with the following algebraic expression:

(3)

Cb =

in which Fct, Fch, Fcb, and Fcd are chloride diffusivity influence functions arising from effects of temperature (T), humidity (h), binding (b) and chemical damage (d) respectively on diffusivity of free chloride in concrete. For isothermal and saturated moisture conditions, only the effect of binding on chloride diffusivity (Fcb) needs to be considered together with the Fcd influence which however remains to be the focus of ongoing further studies. The diffusivity parameter Dco, represents the effective diffusivity quantity obtained at reference or unit influence function conditions. The Fcb influence function calls for an additional formulation to account for the rate of formation of bound chlorides in a form of algebraic equation governing conservation of the binding equilibrium process. This requires experimental identification of chloride binding isotherm relevant to the cementitious constituent of the concrete.

αC f 1 + βC f

(5)

in which Cb is the concentration of the bound chlorides in kg/ m3 of concrete and the parameters α and β are empirical binding constants that vary according to the concrete binder composition. The relationship between the total (Ca), bound (Cb), and free (Cf) chloride contents in concrete can be expressed as:

C a = ηC f + Cb

(6)

By virtue of Equations (3) and (6), Equation (1) needs to be modified to express Ca in Cf terms to become:

∂C f ∂t

=

∂ ⎛ ∂ (η .S .C f ) ⎞ ⎟⎟ ⎜ Dc ∂x ⎜⎝ ∂x ⎠

(7)

For the Langmuir isotherm, the binding capacity influence function Fcb drawn on Equations (3), (6) and (7), can then be expressed as:

Chloride Binding Equations Many investigators have reported about the chloride binding affinity of cementitious materials and its dependence on various concrete mix-proportioning parameters [7-9]. The studies have adequately highlighted the importance of proper binding representation and hence its direct impact on the reliability of conclusions drawn from results of simulation involving chloride transport problems in concrete. In terms of reaction mechanism, the main form of chloride binding is generally reported as a reaction with the aluminate and iron phases to produce calcium chloro-aluminate of the Friedel Salt type. The aluminate and iron phases are essentially those of tricalcium aluminate (C3A) and tetra-calcium

Fcb =

η

η + (∂C b / ∂C f )

(8)

where ∂Cb/ Cf is the slope of the chosen isotherm function, which from Equation (5) is given by

∂C b α = ∂C f (1 + β C f

)

2

(9)

and thus the binding capacity influence function Fcb becomes: 2


Fcb =

η

η + α / (1 + β C f )2

(10)

It follows that Ca can be obtained from Equation (6), once Equation (7), augmented by Equations (3) and (10), is solved for Cf. The problem is non-linear for which analytical solution is complicated to evaluate because of the high nonlinearity involved. Analytical treatment of the augmented formulation requires solution that depends upon integrals of Bessel functions or inverse Laplace transforms. Appropriate COMSOL Multiphysics procedures were accordingly drawn and implemented to seamlessly achieve the required solution for Cf and hence that of Cb as well as Ca accordingly.

2.

Declare consistent units by selecting "None" at [Model Navigator] > [Settings] tab > [Unit system:] section.

3.

At the [Model Navigator] > [New] tab, select 1D from [Space dimension] drop down list.

4.

In the list under [Application Modes], select [COMSOL Mutiphysics] > [Convection and Diffusion] > [Diffusion] > [Transient analysis]. And click [OK] to display the COMSOL Multiphysics top window.

5.

Under [Draw] > [Specify Object] > [Line], enter "0 (space) 50" in the [Coordinates] > [x:] edit field. Click [OK].

6.

Under the [Mesh] > [Free Mesh Parameters] > [Global] tab, specify 0.5 in the [Maximum element size] edit field. Click [Remesh] > [OK] to have the mesh have 100 elements.

7.

Click on [Zoom Extents] button on the [Main Toolbar] to adjust the coordinate system to the size of the line.

8.

Under the [Options] > [Constants], in the [Constant] dialog box enter the following Name-Expression pairs:

COMSOL Multiphysics Procedures Study Problem To assess the predictive capability of deploying COMSOL Multiphysics to model diffusive transport of chlorides in view of the chloride binding, chloride concentration profile with measured chloride binding data reported by Sergi et al. [10] was simulated as a case study problem. Available information and known parameter values for the measured chloride profiles include the use of OPC cement based paste samples made at water to cement ratio (w/c) equal to 0.5 by weight. The paste samples were submitted to external chlorides concentration of 1M NaCl for and an exposure period of a total of 100 days. The samples were tested for their chlorides concentration profiles, over a penetration or cover depth of 50mm, and chloride binding analysis to accord with Langmuir relation at α=1.67 mL/g and β = 4.08L/mol. Base on best fit of the COMSOL model with the experimental data, a chloride diffusivity (Dco) associated with the given Langmuir binding constant is found here to equal 1.85 mm2/day. The saturated porosity value was not reported but is commensurately taken here to equal 0.31 considering the w/c information given for the test sample.

Name eta alpha beta Cs Ci Dco 9.

Expression 0.31 1.67 4.08 1 0 1.85

Under the [Options] > [Expressions] > [Subdomain Expressions], in the [Subdomain Expressions] dialog box, with [Subdomains] tab > [Subdomain selection] at 1, enter the following Name-Expression pairs: Name dbdf Fcb

Expression alpha/(1+beta*c)^2 1/(1 + dbdf/eta)

10. Under the [Physics] > [Subdomain Settings], the following equation is displayed:

Problem in COMSOL Based on the governing differential-algebraic equations, Equations (7) and (9), modeling the chloride transport and chloride binding problem involves three unknown variables, namely Cf, Cb, and Ca, but with only Cf seen to be independent prime mover to the problem formulation. When Equation (3) defining the dependency of the diffusivity coefficient on the binding parameters (α, β) and the free chlorides (Cf) are included in Equation (7), the resulting governing equation involved becomes strongly nonlinear and difficult to solve. In this regard, recourse to numerical solution technique offered by COMSOL Multiphysics has been sought and adopted to completely cope with complexity and nonlinearity of the onedimensional problem in a pragmatic manner.

δ ts

∂c + ∇.(− D∇c) = R ∂t

(11)

where δts is the time scale coefficient, c is the concentration, D is the diffusion coefficient, and R is a reaction rate. Make this equation to match up with Equation (7). 11. Under the [c] tab from the [Subdomain Settings], with [Subdomains] tab > [Subdomain selection] at 1, enter the corresponding parameters: Parameter δts D R

The 1D model of the physical problem is developed in COMSOL according to the following key modeling steps: 1. Start COMSOL Multiphysics to display the [Model Navigator] startup window.

Value/Expression 1 Dco*Fcb 0

12. Under the [init] tab from the [Subdomain Settings], with [Subdomains] tab > [Subdomain selection] at 1, 3


enter the corresponding parameters for the initial condition c(x, t=to=0) = Ci as follows: Parameter C(to)

The complete set of associated chloride profiles comprising free, bound and total components is shown in Figure (4), normalized to same unit of measurement for comparison purposes.

Value Ci

13. Under the [Physics] > [Boundary Settings], in the [Boundary Settings] dialog box , with [Boundaries] tab > [Boundary selection] at 1, enter the corresponding parameters for the boundary condition c(x=0, t) = Cs as follows: Parameter C0

Value/Expression Cs

14. Condition at right boundary c(x=50, t) defaults to the insulated flux condition; Assert this by selecting "Insulation/Symmetry" with [Boundaries] tab > [Boundary selection] at 2. [Apply] > [OK]. 15. Under the [Solve] > [Solver Parameters] > [General] tab, enter "0:0.1:100" in the [Times:] edit field. [OK]. 16. Under the [Postprocessing] > [Plot Parameters] > [general] tab, select "100" in the [solution at time:] drop down list.

Figure 2. COMSOL predicted Cf chloride profiles.

17. In the [Line] tab, select "Concentration c" and click [OK] to obtain the plot of the concentration displayed in the drawing area of the COMSOL screen.

1.0 Time = 100 days : Sergi et al. [10] 0.8

content, [mol/L]

18. Under the [Postprocessing] > [Domain Plot Parameters] > select "stored output times" from the [select via:] drop down list. 19. In the [Solution to use] section, choose "100" and click [Apply] to obtain plot of the concentration displayed in a separate window for the chosen time, t=100 days.

COMSOL model Analytical Eq.(2)

0.6

0.4

0.2

Results and Discussions

0.0 0

5

10

15

20

25

30

35

40

45

depth [mm]

The last stage of the COMSOL modeling steps enumerated above provides the plot of the concentration, c, displayed in a separate [Figure] window. Thereafter the plot was formatted further to obtain the style of the one shown as Figure (2). This figure thus represents COMSOL solution of the free chlorides concentration, Cf, to the problem. Comparison of the COMSOL solution with the experimental data to the modeled problem as reported by Sergi et al. [10] is shown in Figure (3). The good agreement observed between the measured and predicted results in the figure suggests the ability of the COMSOL thus far to predict experimentally observed behavior.

Figure 3. Comparison of Cf profiles with measured data.

Also shown in Figure (3) is the analytical solution provided by use of Equation (2) under constant diffusivity assumption. In this case the solution obtained with stationary Dc = Dco value could not reflect the nonlinearity evidently accruable from the associated binding behavior and concentration dependence. The implication of the mismatch between the experimental and the analytical results in this regard is seen to be quite significant. It inadvertently exposes the inherent inaccuracy associated with the constant diffusivity assumption underlying validity of such analytical equations.

Figure 4. COMSOL predicted Cf, Cb, and Ca profiles.

4

50


[2] COMSOL User guide (http://www.comsol.com)

Once the free chloride content (Cf) is known from the COMSOL solution, the corresponding bound chloride (Cb) can readily be post-processed by algebraic relation given in Equation (5). Likewise, post processing on Equation (6) provides for the total chlorides (Ca). Concurrent increase in the level of the total chlorides (Ca) is observed throughout the penetrated zone. This simultaneous change in Ca is evidently attributable to the chloride binding capacity front active within the zone.

[3] Z.P. Bazant, "Physical model for steel corrosion in concrete sea structures-theory," ASCE Journal of Structural Division, 105(6), 1979: 1137-1153. [4] A.V. Saetta, R.V. Scotta, and R.V. Viataliani, "Analysis of chloride diffusion into partially saturated concrete," ACI Material Journal, 90(5), 1993: 441-451. [5] A.V. Saetta, R.V. Scotta, and R.V. Viataliani, "Reliability of reinforced concrete structures under chemical-physical attack," The Arabian Journal of Science and Engineering, 23(2C), 1998: 41-56.

Conclusions A model of chloride transport in concrete considering chloride diffusion and chloride binding has been presented. Concordance of experimental data with results obtained from the COMSOL Multiphysics solutions illustrates the suitability and robustness of the COMSOL simulations approach to completely cope with complexity and nonlinearity of the problem. Modeling to account for chloride binding in diffusive transport of chlorides in such a pragmatic manner would have otherwise proved cumbersome and far much difficult to accomplish. The contrasting behavior observed between the free and bound chlorides levels underscores the importance of including chloride binding as a pre-requisite to realistic modeling of chloride transport in concrete. Further efforts are under way to promote exploration and evaluation of the software in terms of higher space dimension, established material properties, and synergetic coupling of other physics crucial to holistic rather than isolated study of the problem within the realm of concrete durability mechanics.

[6] M.A. Shazali, "Computational Chemodamage Transport Modeling of Durability Synergies in Concrete", Ph.D. Dissertation, Civil Engineering Dept., King Fahd Univ. of Petroleum & Minerals, Saudi Arabia, 2004: 524pp. [7] Rasheeduzzafar, S.E. Hussain, and S.S. Al-Saadoun, "Effect of tricalcium aluminate content of cement on chloride binding and corrosion of reinforcing steel in concrete," ACI Material Journal, 1992:3-12. [8] Rasheeduzzafar, S.S Al-Saadoun, A.S. Al-Gahtani, and F.H. Dakhil, "Effect of tricalcium aluminate content of cement on corrosion of reinforcing steel in concrete," Cement and Concrete Research, 20(5), 1990: 723-738. [9] Rasheeduzzafar, S.E. Hussain, and S.S. Al-Saadoun, "Effect of cement composition on chloride binding and corrosion of reinforcing steel in concrete," Cement and Concrete Research, 21(1), 1991: 777-794.

References

[10] G. Sergi, S. Yu, and C. Page, "Diffusion of chloride and hydroxyl ions in cementitious materials exposed to a saline environment," Magazine of Concrete Research, 44(158), 1992: 63–69.

[1] K. Tuutti, "Corrosion of Steel in Concrete", Swedish Cement and Concrete Research Institute, Stochkolm, 1982.

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COMSOL Multiphysics Modeling

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