A mathematical model for modeling the formation of calcareous deposits on cathodically protected steel in seawater

Electrochimica Acta 78 (2012) 597–608

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A mathematical model for modeling the formation of calcareous deposits on cathodically protected steel in seawater Wen Sun, Guichang Liu ∗ , Lida Wang, Yu Li Department of Materials Science and Chemical Engineering, School of Chemical Engineering, Dalian University of Technology, 2 Linggong Road, Dalian 116024, China

a r t i c l e

i n f o

Article history: Received 5 April 2012 Received in revised form 20 June 2012 Accepted 20 June 2012 Available online 29 June 2012 Keywords: Calcareous deposits Cathodic protection Modeling studies ALE method

a b s t r a c t A 1D mathematical model, which aims at modeling the formation of calcareous deposits on the surface of cathodically polarized steel in seawater, was developed in this paper. The current model is related to mass transport phenomenon, electrochemical reactions, precipitation reactions and homogenous reactions. The model is also capable of tracking the growth interface of the calcareous deposits via the arbitrary Lagrangian–Eulerian method. The current model predicted time-dependent changes of the physical properties of calcareous deposits, including thickness, deposit porosity, coverage rate and electric resistance, and the numerical results are in good agreement with existing experiments. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Cathodic protection (CP) is an effective and commonly used means for preventing offshore metallic facilities from corrosion which is caused by seawater. Under CP, the electrochemical reactions [1–4] that occur on the surface of metallic structures are the reduction of oxygen for applied protection potentials varying from −0.8 to −1.2 V vs. SCE O2 + 4e− + 2H2 O → 4OH−

(1)

and the reduction of water leading to hydrogen evolution for more negative cathodic potentials. 2H2 O + 2e → 2OH− + H2 ↑

(2)

Both reactions can generate OH− ions near the surface of metallic structures resulting in the formation of calcareous deposits on the metal surface. Moreover, the existing experiments show that the calcareous deposits are mixtures of CaCO3 and Mg(OH)2 [5]. The mechanism for the formation of calcareous deposits [6] is believed to be Mg2+ + 2OH− → Mg(OH)2 ↓

(3)

HCO3 − + OH− ↔ H2 O + CO3 2−

(4)

Ca2+ + CO3 2− → CaCO3 ↓

(5)

∗ Corresponding author. Tel.: +86 411 84986047; fax: +86 411 84986047. E-mail address: [email protected] (G. Liu). 0013-4686/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.electacta.2012.06.056

The formed compact calcareous deposits act as a physical barrier impede solute diffusion (e.g. oxygen), and thus the current density which is needed to keep CP decreases with the increase of time. Hence, understanding the formation kinetics and the physical–chemical properties of such layers is essential to improve cathodic protection monitoring. So far, there are numerous papers [1–3,7–15] devoted to studying how the factors such as CP, physics and chemistry of seawater, and surface preparation influence the formation of calcareous deposits through experiments in seawater. However, there are few papers studying this phenomenon via numerical method. The first available model on simulating the formation of calcareous deposits was presented by Sadasivan [16]. In his master’s thesis, a 1D model was developed to simulate this phenomenon by considering diffusion as the mass transport mechanism. Moreover, the Tafel equation and the limiting current density were used to present hydrogen evolution and oxygen reduction in his model, respectively. Subsequently, Dexter and Lin [17] presented a steady state model to calculate the pH at metal surface under a cathodic polarization in quiescent saline waters in the presence of both calcareous deposits and biofilms. In 1993, Yan et al. [5,18] published two papers on mathematical modeling of the formation of calcareous deposits on cathodically protected steel rotating disk in artificial seawater. The model was used to study the influences of parameters such as applied potential, rotation speed, temperature, salinity and depth on the formation of calcareous deposits and their ability to change the cathodic current density in CP systems. Deslouis et al. [12] only considered Mg(OH)2 deposition and presented a model which is capable of evaluating of the thickness of deposits at short and long time scales.

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Nomenclature a c ci,0

anode cathode concentration of species i on the electrode surface (mol/m3 ) ci,bulk concentration of species i in bulk solution (mol/m3 ) ci,interface concentration of species i at interface (mol/m3 ) reference concentration of species i (mol/m3 ) ci,ref Di diffusion coefficient of species i (m2 /s) Di,e effective diffusion coefficient of species i (m2 /s) E electric filed strength (V/m) F Faraday’s constant (96,487 C/mol) i0j exchange current density of electrochemical reaction j (A/m2 ) J current density vector (A/m2 ) jj current density of electrochemical reaction j (A/m2 ) proportionality constant k k forward reaction constant for the homogenous reaction (m3 /(mol s) k៭ backward reaction constant for the homogenous reaction (1/s) reaction rate constant for the precipitation reaction kCaCO3 of CaCO3 (mol/(m2 s)) kCaCO3 , interface surface reaction rate constant for the precipitation reaction of CaCO3 (mol/(m3 s)) equilibrium constant for the homogenous reaction Keq (m3 /mol) kMg(OH)2 reaction rate constant for the precipitation reaction of Mg(OH)2 (m7 /(mol2 s)) kMg(OH)2 , interface surface reaction rate constant for the precipitation reaction of Mg(OH)2 (m6 /(mol2 s)) ksp,Mg(OH)2 apparent solubility product constant of Mg(OH)2 (mol2 /m6 ) ksp,CaCO3 apparent solubility product constant of CaCO3 (mol3 /m9 ) l thickness of calcareous deposits (m) thickness of diffusion layer (m) L m cementation exponent reaction order of precipitation reaction of CaCO3 m Mi molecular weight of species i (kg/mol) n saturation coefficient Ni molar flux of species i (mol m2 /s) nj number of electrons in reaction j anodic reaction order of species i in reaction j pij qij cathodic reaction order of species i in reaction j R universal gas constant (8.3145 J/(K mol)) reaction rate of species i (mol/(m3 s)) Ri Ri,e effective reaction rate of species i (mol/(m3 s)) Ri, interface surface reaction rate of species i (mol/(m2 s)) sL fluid saturation temperature (K) T umi mobility (mol m2 /(J s)) effective mobility (mol m2 /(J s)) umi,e v growth velocity of calcareous deposits (m/s) volume of porous deposit layer (m3 ) V Vc solution volume in the porous deposit (m3 ) Vp volume of porous deposit (m3 ) volume of deposit (m3 ) Vs Vu volume of diffusion layer (m3 ) xCaCO3 /Mg(OH)2 molar ratio of CaCO3 to Mg(OH)2 zf surface resistance ( m2 ) Zf porosity resistance ( m2 ) charge number of species i zi

Greek letters transfer coefficient for reaction j ˛j j overpotential for reaction j (V)  current density (A/m2 )  potential (V) charge density (C/m3 )  i density of species i (kg/m3 ) ϑ relative permittivity coverage rate ε deposit porosity 0 initial deposit porosity

electric conductivity (S/m) effective electric conductivity (S/m)

e

e∗ effective conductivity in calcareous deposits (S/m) ij stoichiometric coefficient for the species i in reaction j tortuosity of porous layer

Based on these existing models and current development in simulation, we believe that a more advanced model can be developed. An overall view of recently published papers, which aim to simulate the electrochemical phenomena, shows that there are two main ways to introduce the electrochemical reactions into mathematical models: (1) application of Butler–Volmer equation [19–22] or Tafel equation [23,24] jj = i0j

  i

jj = i0j

ci,0 ci,ref

pij

 exp



˛j F ci,0 exp  ci,ref RT j

˛aj F RT



j

  qij ci,0 −

ci,ref

 exp

˛cj F RT

 j



(6)

i

(7)

(2) employment of polarization data obtained from potentiodynamic scan experiments [25]. For both above-mentioned methods, if the following expression:  = f ()

(8)

is used to stand for the polarization behavior, then the schematic of a fictitious computational domain along with the governing equations and the boundary conditions for most electrochemical reactions involved in simulation can be shown in Fig. 1. Published papers [26–30] also show that these two methods can afford satisfactory numerical results for simple or simplified systems. However, the physical, chemical and biological phenomena, taking place on the cathodic surface in seawater is time-dependent and quite complex [31,32]. Hence, both methods would run out when come to simulate such a phenomenon existing in longrunning systems. Nevertheless, measurement errors and defects for potentiodynamic scan experiments are unavoidable. Fig. 2 gives a schematic of the obtained polarization curve via potentiodynamic scan experiment. Clearly, the obtained polarization data only include information of the points located on the “dash dot line”. And that means lots of useful information about the electrochemical behavior of tested material would be missed (e.g. information of point A, B, C, etc.). Moreover, a number of factors will introduce other errors into the measurement results [33], such as the interfacial capacitance at the electrode/solution interface, irreversible changes to the interfacial structure caused by long-time polarization, the scan rate and mode used in potentiodynamic scan experiments, immersion time, and so on. Obviously, without a unified test standard for potentiodynamic scan experiments, the obtained polarization data vary from person to person even for the same system. Hence, it is open to objection that whether or

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599

Fig. 1. The schematic of a fictitious computational domain along with the governing equation and the boundary conditions.

not the experimental data obtained from potentiodynamic scan experiments are available for numerical modeling. In order to take the place of those two above-mentioned methods, developing a new method to introduce electrochemical phenomena into a mathematical model seems urgently needed. Fortunately, there are several researches aiming at modeling the behavior of cathodic surfaces by time-dependent polarization curves through boundary element method (BEM) [34,35]. These papers show a captivating potentiality in modeling time-dependent electrochemical phenomena by time-dependent polarization curves obtained by either potentiostatic data or synthesized current–potential curves. The main purpose of this work is to establish a mathematical model for modeling the time-dependent formation of calcareous deposits on the surface of cathodically polarized steel in seawater by 3D polarization surface which is determined by potentiostatic data obtained from in situ measurements. The model will be capable of predicting the time-dependent changes of current density and the physical properties of calcareous deposits, including thickness, coverage rate, porosity and composition. As the growth of calcareous deposits is so considerable that it significantly affects the mass transport in the diffusion layer, therefore, the current model is expected to be capable of tacking the growth interface of the calcareous deposits by moving mesh technique. We believe that the model will be helpful in acknowledging the

formation kinetics and physical–chemical properties of calcareous deposits. 2. Numerical model development In this paper, we assume that there are intermediate states in the precipitation reactions of Mg(OH)2 and CaCO3 , which are respectively denoted by Mg(OH)2,ads and CaCO3,ads . Moreover, we further suggest that the lifetime of intermediate state should be extremely short. Then related precipitation reactions can be shown as follows: Mg2+ + 2OH− ↔ Mg(OH)2,ads → Mg(OH)2 ↓ Ca2+ + CO3 2− ↔ CaCO3,ads → CaCO3 ↓

(9) (10)

Then the proposed process for the formation of calcareous deposits on cathodically polarized steel in seawater can be presented as shown in Fig. 3. To simplify the simulation, the geometry of the process is abstracted to a one-dimensional model. Herein,

1 denotes the electrode surface, 2 is the surface of the calcareous deposits and 3 is the limit of bulk solution. represents the calcareous deposits domain, while is the domain of the solution between deposits interface and bulk solution interface. Nevertheless, the following assumptions are also necessary to be made before presenting the mathematical equations:

Fig. 2. Schematic of the measurement of polarization curve.

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W. Sun et al. / Electrochimica Acta 78 (2012) 597–608 Table 1 Fix physical parameters for electricity. Parameter

Value

sL m n

ϑ

1 1 2 4.67 S/m 80

2.2. Governing equations and boundary conditions for mass transport

Fig. 3. Schematic of the formation of calcareous deposits in the diffusion layer and 1D geometry.

(1) Dilute solution theory is applicable; (2) Corrosion of the steel has only little influence on the calcareous deposition and is neglected; (3) The precipitation intermediates form in both calcareous deposits layer and solution; (4) The intermediates attach to the surface of metal electrode forming calcareous deposits; (5) The homogeneous reactions occur inside the diffusion layer; (6) Electrochemical reactions occur only on the uncovered surface of electrode; (7) The tortuosity and relative permittivity of the porous layer is assumed to be a constant. 2.1. Governing equations and boundary conditions for electricity The fundamental equations involved are Ohm’s law J = E

(11)

the equation of continuity ∂ +∇ ·J=0 ∂t

(15)



 

= fc (, t)

(16)

1

While, for the interface of bulk solution, it is set to be

3

=0

Physical parameters are presented in Table 1.

Ni = −Di ∇ ci − zi umi Fci ∇ 

(20)

In Eq. (19), the effective diffusion coefficient in the porous layer is calculated by the following equation Di,e = Di (1 − ) +

Di NM,PE

(21)

in which the MacMullin number is defined as NM,PE =

(22)

And the mobility in Eqs. (23) and (24) is respectively expressed as Di,e RT

(23)

Di RT

(24)

For domain , following discussion is presented. The homogeneous reaction (4) is rapid equilibrium among OH− , CO3 2− and HCO3 − as Keq =

k HCO

−

k៭ HCO

−

3 3

cCO

=

3

2−

cOH− cHCO

3

(25)

−

And the consumption rate for HCO3 − is suggested to be governed by

Hence, for the surface of cathodically polarized electrode, the boundary condition satisfies ∂ 

e ∂n 

(19)

(14)

Fig. 4 shows three dimension polarization surface determined from potentiostatic data obtained from in situ measurements in Ref. [37]. Here, the following equation is used to denote the polarization surface c = fc (, t)

Ni = −Di,e ∇ ci − zi umi,e Fci ∇ 

umi = (13)

(18)

For domain and domain , the flux for each of the ions in the electrolyte is respectively given by the Nernst–Planck equation as follows:

(12)

, however, the electric Eqs. (11)–(13) are available for domain conductivity should be revised to make sure that these equations are also available for domain . Based on the Archie’s low [36], the effective electric conductivity for domain is presented as follow

e = ε m sLn + (1 − ε)

∂ci + ∇ · Ni = Ri ∂t

umi,e =

and Gauss’ law

∇ · (ϑE) = 

Concentrations of seven components are taken into account in this model. They are OH− , CO3 2− , HCO3 − , Mg(OH)2,ads , CaCO3,ads , Mg2+ and Ca2+ , respectively. The material balance equation for the species i is given by

(17)

RHCO

3

2−

= k HCO

3

−

cHCO

3

−

cOH− − k៭ HCO

3

−

cCO

3

2−

(26)

For precipitation reactions (3) and (5), the consumption rates for Mg2+ and Ca2+ can be expressed as RMg2+ = kMg(OH)2 (cMg2+ cOH− 2 − ksp,Mg(OH)2 ) · ı()

RCa2+ = kCaCO3

cCa2+ cCO

3

ksp,CaCO3

with cMg2+ cOH− −1 = ksp,Mg(OH)2 =

cCa2+ cCO

3

2−

ksp,CaCO3

2−

−1

(27)

m −1

· ı()

(28)

(29)

(30)

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Fig. 4. The 3D polarization surface obtained from potentiostatic data presented in Ref. [37].

Table 2 Fix physical parameters for mass transport.

respectively. Here, ı() is a step function and is presented as



ı() =

0 for  ≤ 0

(31)

1 for  > 0

Based on Eqs. (26)–(28), the consumption rate for OH− and CO3 2− is ROH− = 2RMg2+ + RHCO

−

(32)

= RCa2+ − RHCO

−

(33)

3

RCO

3

2−

3

respectively. While for domain , the homogenous reactions and precipitation reactions only occur in the solution filling the pores of the calcareous layer. Hence, relative reaction rates should be revised as Ri,e = [(1 − ε) + ε]Ri



1

=

ij fc (, t) nj F[ ε + (1 − ε)]

(35)

For other species that are not involved in electrochemical reactions the boundary condition at 1 is



N i · n

1

=0

(36)

The concentration for species i at the bulk interface 3 is placed equal to its bulk concentration:



ci 

3

= ci,bulk

Keq k HCO3 − − k៭ HCO3

kMg(OH)2 ksp,Mg(OH)2 kCaCO3 ksp,CaCO3 m ij nj R T F

Value

Source

3

84 m /mol 0.1 m3 /(s mol)

Ref. [5] Chosen arbitrarily

1.19 × 10−3 1/s 7.4 × 10−4 m6 /(s mol2 ) 0.45 mol3 /m9 2.26 × 10−5 mol/(m3 s) 0.696 mol2 /m6 1.7 4 4 1 8.3145 J/(mol K) 298.15 K 96,487 (s A/mol)

Eq. (25) Calculated from Ref. [5] Ref. [5] Calculated from Ref. [5] Ref. [5] Ref. [5] Eq. (1) Eq. (1) Ref. [5] – – –

(34)

The production rate of species i due to electrochemical reaction j at boundary 1 is given as Ni · n

Parameter

(37)

Physical parameters applied in mass transport are given in following tables. 2.3. Arbitrary Lagrangian Eulerian (ALE) method for the growth of the calcareous deposits ALE method is a moving mesh technique which is capable of combining the best interesting aspects of two classical mesh methods, the Lagrangian method and the Eulerian method, while minimizing as far as possible their drawbacks. In the ALE method, the nodes of the computational mesh may be moved with the continuum in Lagrangian manner, or be fixed in Eulerian manner, or

be moved in some arbitrarily specified ways to give a continuous rezoning capability. Because of this freedom in moving the computational mesh offered by the ALE method, greater deformations can be captured than that allowed by the Lagrangian method, with higher resolution than that afforded by the Eulerian method Tables 2 and 3. In this paper, a material frame with X reference coordinates for a 1-D formulation and a spatial frame with x spatial coordinates are introduced. For the domains with free displacement in the transient case, the mesh displacement is obtained by solving the following equation ∂2 ∂x =0 ∂X 2 ∂t

(38)

Table 3 Diffusion coefficients and bulk concentrations of the components in seawater. Species

Diffusion coefficients Di /m2 /s

ci,bulk /mol L−1

Reference

OH− CO3 2− HCO3 − Mg(OH)2,ads CaCO3,ads Mg2+ Ca2+

5.27 × 10−9 9.55 × 10−10 1.19 × 10−9 1 × 10−9 1 × 10−9 7.05 × 10−10 9.55 × 10−10

1.6 × 10−6 2.07 × 10−4 1.54 × 10−3 0 0 5.45 × 10−2 1.05 × 10−2

[5,12] [5,12] [5,12] Chosen arbitrarily Chosen arbitrarily [5,12] [5,12]

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W. Sun et al. / Electrochimica Acta 78 (2012) 597–608

Fig. 5. Schematic of the electrode surface and electrical equivalent circuit [2].

Based on considering the constraints that are placed on the boundaries, Eq. (38) smoothly deforms the mesh through Laplace smoothing which introduces deformed mesh positions x and y as degrees of freedom in the model. The normal component n of velocity vector v of the calcareous deposits surface 2 is calculated using Eq. (34) and represented as n·v =

∂l 1 = ∂t (1 − 0 )ε +



3

3

MMg(OH)2 RMg2+ ,interface

2− , interface

2−

Mg(OH)2

= kCaCO3 , interface

c

c

Ca2+ , interface CO3 2− , interface

ksp,CaCO3

 =  =

c − interface OH , interface ksp,Mg(OH)2

cCa2+ , interface cCO

3

2−

, interface

ksp,CaCO3

Relative parameters are presented in Table 4.

−1

(39)

As presented in Appendices A and B, the coverage rate and porosity of the calcareous deposits satisfy the following expressions respectively



(40)

1 ∂ε 1 = 1 − ε ∂t (1 − 0 )l

· ı( )

(42)

−1

(43)

Whereas the velocity vector of the cathodically polarized steel is considered to be zero as the corrosion of cathode surface 1 is neglected n·v=0

L 0

RCO

3

2−

d

CaCO3

+

MMg(OH)2

L 0

RMg2+ d

−

Mg(OH)2

MCaCO3 RCO 2− MMg(OH)2 RMg2+ 1 ∂ 3 + = Mg(OH)2 CaCO3 ∂t

0.24

0.18

l l k =k m n =

e∗

sL ε

m sLn ε

 0

t

∂l dt ∂t

(48)

And the surface resistance of uncovered electrode is presented as l k Z=k = (1 − ε)

(1 − ε)

0.15

(47)

Moreover, as the equivalent circuit shown in Fig. 5, the porous resistance of the calcareous deposits is denoted as Zf and can be calculated by Eq. (48) and shown as follow

(44)

Simulated results Experimental data [37]



(46)

Zf = k

Current density J / A m^-2

MCaCO3

(41)

−1

0.21

(45)



m

RMg2+ , interface = kMg(OH)2 , interface (cMg2+ , interface cOH− , interface 2 − ksp,Mg(OH)2 ) · ı(  )

cMg2+ ,

dx = 0

2.4. Governing equations for the physical properties of calcareous deposits

,interface

CaCO3

with RCO

MCaCO3 RCO

Moreover, for 3 , the prescribed displacement satisfies

 0

t

∂l dt ∂t

(49)

0.12 Table 4 Parameters fixed in ALE method.

0.09 0.06 0.03

0

200000

400000

600000

800000

1000000

Time t / s Fig. 6. Numerical and experimental results of current density on cathodically polarized steel with time.

Parameter

Value

Reference

MCaCO3 MMg(OH)2 CaCO3 Mg(OH)2 0 kCaCO3 ,interface kMg(OH)2 ,interface

0.1 kg/mol 0.058 kg/mol 2720 kg/m3 2360 kg/m3 0.25 1.13 × 10−8 mol/(m2 s) 3.7 × 10−7 m7 /(mol2 s)

– – – – [5] [5] [5]

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603

Fig. 7. Three dimension concentration profile of OH− ions inside the diffusion layer.

The composition of calcareous deposits is characterized by the molar ratio of the CaCO3 to Mg(OH)2 in the deposits at time t. The molar ratio is calculated by Eq. (49) shown as

Lt xCaCO3 /Mg(OH)2 =

 L0  t0 0

0

RCaCO3 dt d

(50)

RMg(OH)2 dt d

3. Results and discussion Fig. 6 shows the comparison of time-depended changes of the simulated current density and the experimental results on the surface of cathodically protected steel. It can be seen that the simulated results agree well with experimental data. It implies that

such an idea that simulating time-dependent electrochemical phenomenon via 3D polarization surface, which is determined from potentiostatic data, is feasible. Figs. 7–11 show the concentration profiles of some components in seawater inside the diffusion layer. In the preliminary stage of CP, the concentration of OH− ions on the electrode surface is pretty high due to the high cathodic current in this period and gradually decreases to the bulk concentration as distance increases as shown in Fig. 7. However, it also can be seen from Fig. 7 that the concentration of OH− ions inside the diffusion layer decreases due to the descending of cathodic current, which is caused by the formation of calcareous deposits, as polarization time prolonging. Since OH− and HCO3 − are reactants of the homogenous reactions, a high concentration of OH− ions would result in a low concentration of HCO3 −

Fig. 8. Three dimension concentration profile of HCO3 − ions inside the diffusion layer.

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Fig. 9. Three dimension concentration profile of CO3 2− ions inside the diffusion layer.

inside the diffusion layer just as shown in Fig. 8. However, CO3 2− ions concentrate on the surface of electrode due to the homogenous reactions, especially in the preliminary stage of CP as shown in Fig. 9. Fig. 10 shows the concentration profile of Ca2+ ions inside the diffusion layer. Numerical results indicate that the concentration of Ca2+ doesn’t change much. Therefore, the supersaturation of CaCO3 is governed by CO3 2− ions. Fig. 11 shows that the situation of Mg2+ ions is similar with that of Ca2+ ions. And the formation of Mg(OH)2 is controlled by the concentration of OH− ions inside the diffusion layer. Furthermore, there is little change in the concentration of Mg2+ ions for a long time indicating that Mg(OH)2 does not generate within this period and that is believed to be indirectly

influenced by the descending of cathodic current which is caused by the formation of calcareous deposits. Fig. 12 shows the concentration profile of OH− ions inside the calcareous deposits. The distance coordination of the concentration surface increases with the increase of time and that implies that the calcareous deposits are continuously growing all the time. Fig. 13 presents the concentration profile of OH− ions along the diffusion layer after cathodically polarizing for 70,000 s. It can be seen from Fig. 13 that the calcareous deposits increase the resistant of mass transport in the calcareous deposits layer. Species concentrations play a very important role in the formation of calcareous deposits and the thickness of calcareous deposits is considerable inside the diffusion layer as observed in previously experimental research

Fig. 10. Three dimension concentration profile of Ca2+ ions inside the diffusion layer.

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605

Fig. 11. Three dimension concentration profile of Mg2+ ions inside the diffusion layer.

[5], hence modeling the growth kinetics of the calcareous deposits by ALE method is of significance to predict the time-dependent changes of the physical properties of calcareous deposits. Previously numerical and experimental results argued that the thickness and coverage rate of calcareous deposits increase quickly at the beginning of CP and tend to be constant with the increase of polarization time [5,18,38]. Fig. 14 shows the thickness and growth rate of calcareous deposits. It can be seen that the calcareous deposits grow rapidly in the preliminary stage of CP. However, the growth rate of the calcareous deposits drastically decreases after cathodically polarizing for about 30,000 s. Fig. 14 also indicates that the run time of the CP system is 4 × 105 –1.08 × 106 s, the calcareous deposits is thickening at a constant velocity. However, the growth rate curve in Fig. 14 shows that the velocity keeps descending as time goes. Though the slope of the curve seems rather constant and far from zero at 1.08 × 106 s, based on change tendency of the

growth rate, the related experiments and mathematical analysis of Eq. (46), it can be inferred that the thickness of calcareous deposits has no significant change in when the CP systems have been run for a long time which might be much larger than 1.08 × 106 s in current model. Fig. 15 presents the coverage rate and deposit porosity of calcareous deposits. The numerical results show that the surface of electrode is almost covered by calcareous deposits after cathodically polarizing for 300 h. Similarly with the thickness, the coverage rate approximates a constant value 1.0 as CP time prolonging. Moreover, the numerical results also show that the calcareous deposits are becoming denser with the time of CP prolonging. However, the deposit porosity decreases very slowly with time. To sum up the above arguments, we suggest that there should be no noticeable changes of the physical properties of deposits including the thickness, the coverage rate and the porosity when the CP system has run for a certain long time. Furthermore, this approximately stable

Fig. 12. Three dimension concentration profile of OH− ions inside the calcareous deposits.

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1E-3

0.06

0.8

Thickness l / cm

1E-4

Mass transport in solution

0.04

0.6 1E-5 0.4 1E-6

Thickness, -0.9 V vs. SCE Thickness, -0.75 V vs. SCE

0.02

Deposit porosity, -0.9 V vs. SCE Deposit porosity, -0.75 V vs. SCE

1E-7

0.00 0.00

0.01

0.02

0.03

0.04

0.05

1E-8

Distance L / cm

200000

400000

1.20E-008 1.00E-008 8.00E-009 0.002

6.00E-009 4.00E-009 2.00E-009

0.001

Growth rate v / m s^-1

0.00E+000 -2.00E-009

0.000 0

200000

400000

600000

800000

1000000

Time t / s Fig. 14. Changes of the thickness and growth rate of calcareous deposits with time.

calcareous layer will result in a basically unchanged constant cathodic current density in long-running systems as shown in Fig. 6. Polarized potential plays a very important role in the formation of calcareous deposits [37,39]. Sun et al. [37] pointed out that the thickness of calcareous deposits formed on the electrode surface is very thin and the coverage rate is small under the polarized potential of −0.75 V vs. SCE in quiescent seawater, while the thickness and coverage rate increase when more negative polarized

1000000

potentials are imposed. Fig. 16 shows the thickness and deposit porosity of calcareous deposits under different polarized potential. Both the thickness and deposit porosity of protective calcareous deposits for the −0.75 V vs. SCE situation is far less than those for the −0.9 V vs. SCE situation. Therefore, when the cathodic potentials loaded on the steel are in a certain range, the more negative the cathodic potentials are, the more protective the calcareous deposits are. As shown in Fig. 5, the surface of electrode is partially covered by porous calcareous deposits. And, it is clear that porosity resistance Zf and surface resistance Z are associated with the deposits porosity, thickness or coverage rate. For the calcareous deposits with a certain thickness, a larger porosity and coverage rate will result in a smaller porosity resistance according to Eq. (47), while a larger coverage rate is in favor of a larger surface resistance according to Eq. (48). Fig. 17 shows changes of the surface resistance of calcareous deposits with time. At the beginning of CP, both the coverage rate and thickness of calcareous deposits increase rapidly. However, the impact of the coverage rate on Zf is larger than that of the thickness. Hence, Zf decreases in this period as shown in Fig. 17. While both the thickness and coverage rate are not in favor of Z, Z keeps ascending with polarization time increasing. In addition, in agreement with the previous studies [40,41], the total resistance, which excludes solution contact resistance, steadily increases with time due to the

800

1.0

600

0.2500

0.2485 0.4 0.2480 0.2

Deposit porosity θ

0.2490

0.6

0.2475 Coverage rate Deposit porosity

0.0

Z Zf Total resistance

400

400

200000

300

200 200 100

0.2470

0 0

500

600

Porosity resistance Zf / ohm cm^2

0.2495

0.8

Coverage rate ε

800000

Time t / s

Surface resistance Z / ohm cm^2

Thickness of calcareous deposits l / cm

1.40E-008

0.003

600000

Fig. 16. Changes of the thickness and deposit porosity of calcareous deposits with time under different polarization potential.

1.60E-008

Thickness Growth rate

0.2

0.0 0

Fig. 13. Concentration profile of OH− ions along the diffusion layer, t = 70,000 s.

0.004

Deposit porosity θ

Concentration of cOH- / mol L^-1

1.0

0.01

cOH-

Mass transport in calcareous deposits

0.08

400000

600000

800000

1000000

0

200000

400000

600000

800000

1000000

0

Time t / s

Time t / s

Fig. 15. Changes of the coverage rate and deposit porosity of calcareous deposits with time.

Fig. 17. Changes of the porosity and surface resistance of calcareous deposits with time, k = 1000.

W. Sun et al. / Electrochimica Acta 78 (2012) 597–608

607

40 8.0x10 -7

30 6.0x10 -7

20 4.0x10 -7

10

0

2.0x10 -7

Reaction rate Ri / mol m^-2 s^-1

Molar ratio xCaCO3/Mg(OH)2

nCaCO3/nMg(OH)2 RMg(OH)2 RCaCO3

0.0 0

200000

400000

600000

800000

1000000

Time t / s Fig. 18. Changes of the reaction rates and mole ration of CaCO3 and Mg(OH)2 of calcareous deposits with time.

deposits of calcium carbonate forming on the surface as shown in Fig. 17. When CP is imposed on metal structures in seawater, calcareous deposits are formed with slower kinetics than Mg(OH)2 at the beginning of CP and the content of Mg(OH)2 included in calcareous deposits is relatively high as shown in Fig. 18. However, just as discussed previously, the supersaturation of Mg(OH)2 largely depends on the concentration of OH− ions or solution pH. Once the concentration of OH− ions or solution pH is not high enough, Mg(OH)2 would not form on the surface of electrode just as shown in Fig. 18. Consequently, CaCO3 forms much faster than Mg(OH)2 and the primary ingredient of deposits is calcium carbonate as shown in Fig. 18. Such a phenomenon is also discussed in existing Refs. [12,42].

Fig. 19. The changes happened to calcareous deposits inside the diffusion layer during cathodic polarization.

Assuming that the newly formed calcareous deposits on the inner surface of porous deposits is solid, then the increased solid volume of the porous deposits in a quite short time can be expressed as



Appendix A. Governing equation for the deposits porosity of calcareous deposits During cathodic polarization, the physical properties of the calcareous deposits change with time as shown in Fig. 19. In this paper, the solution is divided into four parts: (1) bulk solution; (2) interface solution; (3) covered solution; (4) solution inside the pores of the deposit. We believe that precipitation reactions occur in the four parts of the solution. Furthermore, precipitation reactions occurring in solutions 1 and 3 contribute to the ascending of coverage rate and those in solution 2 are in favor of ascending the thickness of calcareous deposits, while the precipitation in solution 4 is associated with the decrease of porosity.

3

2−

Vc

CaCO3

+

MMg(OH)2 RMg2+ Vc Mg(OH)2

· ıt

(A.1)

Here, the volume of the solution 4 in the porous deposit is Vc = ε sl

(A.2)

The definition of deposit porosity is =

Vp − Vs Vp

(A.3)

with the volume of porous deposit Vp = εsl

4. Conclusions 1. Simulating time-dependent formation of the calcareous deposits via 3D polarization surface, which is determined from potentiostatic data, is feasible and can offer satisfactory numerical results. 2. ALE method can primly track the time-dependent influences of the calcareous deposits on mass transport and the formation kinetics of calcareous deposits. 3. The thickness, coverage rate and electric resistance of calcareous deposits increase with time, while the porosity decreases with time. 4. A more negative CP potential will result in the deposition of thicker and more compact calcareous deposits. 5. The calcareous deposits contain more Mg(OH)2 than CaCO3 in the preliminary stage of CP, while the content of CaCO3 increases with polarization time prolonging.

MCaCO3 RCO

ıVs =

(A.4)

Then, the following expression can be obtained + ı =

Vp − (Vs + ıVs ) Vp

(A.5)

Combining Eqs. (A.3) and (A.5), then the following formula is obtained ı = −

ıVs Vp

(A.6)

Furthermore, combining Eqs. (A.1), (A.2), (A.4) and (A.6), we have



ı = −

MCaCO3 RCO CaCO3

3

2−

+

MMg(OH)2 RMg2+ Mg(OH)2

· ıt

(A.7)

Finally, Eq. (A.7) is converted into the following expression −

MCaCO3 RCO 2− MMg(OH)2 RMg2+ 1 ∂ 3 = + CaCO3 Mg(OH)2 ∂t

(A.8)

Appendix B. Governing equation for the coverage rate of calcareous deposits We suggest that the calcareous deposits that formed on the surface of electrode and the outer surface of previously formed calcareous deposits are porous with a constant porosity of 0 . Then the increased volume of calcareous deposits in ıt is presented as ıVp =

1 1 − 0



MCaCO3 RCO

3

CaCO3

2−

Vu

+

MMg(OH)2 RMg2+ Vu Mg(OH)2

· ıt

(B.1)

Here, Ri Vu represents the total reaction rates in solutions 1 and 3 of species i which is much greater than that in solution 4. Then

608

W. Sun et al. / Electrochimica Acta 78 (2012) 597–608

Ri Vu is close to the total reaction rate inside the diffusion layer and can be proximately calculated by following formula



L

Ri Vu =

Ri s d

(B.2)

0

The definition of coverage rate is ε=

Vp V

(B.3)

Here V = sl

(B.4)

Then, in a short time, the following formula can be obtained ε + ıε =

Vp + ıVp V

(B.5)

Combining Eqs. (B.3) and (B.5), then we have ıε =

ıVp V

(B.6)

Combining Eqs. (B.1), (B.2), (B.4) and (B.6), then



1 ıε = (1 − ε) (1 − 0 )l

+

MMg(OH)2

L 0

MCaCO3

L 0

RCO

3

2−

d

CaCO3

RMg 2+ d

 · ıt

Mg(OH)2

(B.7)

Furthermore, the differential form Eq. (B.7) can be presented as following



1 ∂ε 1 = 1 − ε ∂t (1 − 0 )l

+

MCaCO3

L 0

RCO

3

2−

d

CaCO3

MMg(OH)2

L 0

RMg2+ d

Mg(OH)2

 (B.8)

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