modeling the effect of residual and diffusion-induced ... · modeling the effect of residual and...

Modeling the Effect of Residual andDiffusion-Induced Stresses on Corrosionat the Interface of Coating and Substrate

M.H. Nazir,‡,* Z.A. Khan,* A. Saeed,* and K. Stokes**

ABSTRACT

The effect of residual and diffusion-induced stresses oncorrosion at the interface of coating and substrate has beenanalyzed within a multidisciplinary approach, i.e., materialscience, solid mechanics, and electrochemistry. A self-consistentequation for corrosion current density, involving the combinedeffect of residual stress and diffusion-induced stress is devel-oped. The influences of temperature, moduli ratio, thicknessratio, thermal mismatch ratio, and residual stress gradient ofcoating and substrate on the corrosion current density arethen discussed. Results indicate that when the thermal ex-pansion of coating is greater than substrate, the decrease intemperature from fabrication temperature accounts for the samedirection of both the residual and the diffusion stresses. Thisbehavior increases the deflection of the coating-substratesystem and results in the evolution of tensile residual stressin the coating. The tensile stress opens the pre-existing coatingmicrocrack, allowing the diffusion of corrosive agents andtherefore, accelerating the corrosion damage to the coating/substrate interface. The model is based on experimentalobservations conducted to understand the behavior of corrosionat the coating/substrate interface in the presence of tensile orcompressive residual stresses. At the end, the model wasvalidated against the experimental results showing a goodquantitative agreement between the predicted theoretical andexperimental trends.

KEY WORDS: blister, coating, coating failure, corrosion,delamination, diffusion-induced stress, iterative algorithm,mathematical modeling, partial differential equations,residual stress

INTRODUCTION

Recent studies have shown that the corrosion of steel-based infrastructures is estimated to cost about 3% ofglobal gross domestic product (GDP) each year.1 Tocombat and control the corrosion of metals, variousmethods are deployed to mitigate the high direct andindirect costs of corrosion.2-4 Themost commonmethodof prevention against corrosion uses organic coat-ings.5-6 The organic coatings are used to isolate themetal from its surrounding environment by using aphysical barrier.

Based on experimentation performed in thispaper, it was concluded that the failure of coating-substrate system depends on three interdependentcauses. (i) Corrosion current density “i”: a corrosionsensor adhered to the substrate, beneath the coating,measures the corrosion current density. (ii) Tensile/compressive residual stress: the corrosion currentdensity directly depends upon the development of ten-sile or compressive residual stresses on the coating.These stresses develop, as a result of mismatches incoefficients of thermal expansion (CTE), when sub-jected to the change in temperature, ΔT, from fabricationtemperature. (iii) Electrolyte diffusion: temperaturechange directly controls the electrolyte diffusion towardthe interface, subject to CTE mismatch. All of thesethree causes are interdependent but fall under “three”

Submitted for publication: June 7, 2015. Revised and accepted:December 9, 2015. Preprint available online: December 9, 2015,http://dx.doi.org/10.5006/1804.

‡ Corresponding author. E-mail: [email protected].* Bournemouth University, NanoCorrr, Energy and Modeling (NCEM)Research Group, Faculty of Science and Technology, BournemouthUniversity, United Kingdom.

** Defence Science and Technology Laboratory (DSTL), Ministry ofDefence (MoD), Salisbury, United Kingdom.

500ISSN 0010-9312 (print), 1938-159X (online)

16/000073/$5.00+$0.50/0 © 2016, NACE International CORROSION—APRIL 2016

CORROSION SCIENCE SECTION

different disciplines. Corrosion current density useselectrochemistry concepts, tensile/compressive resid-ual stress uses solid mechanics concepts, and elec-trolyte diffusion uses material science concepts.

Although corrosion is an electrochemical reac-tion, the process is strongly influenced by themechanical and microstructural properties of mate-rials.7 In addition, the corrosion resistance of a coatedsteel sample is also affected by microcracks in thecoating.8 These microcracks may act as pathways forcorrosive agents to diffuse through the coating bar-rier, which may result in the initiation of localizedcorrosion in the less noble metal substrate.9 Stresscorrosion cracking (SCC), resulting from growth of crackformation in a corrosive environment, results whenthe metals are subjected to tensile residual stress in acorrosive environment.10-12 Contrary to this, com-pressive residual stress results in the inhibition of thecorrosion of metals.13-14 In addition to residualstress, the chemical stress (or diffusion-induced stress)also influences the SCC and, therefore, affects thestability and reliability of materials.15 The couplingeffect of residual stress and diffusion-induced stresson the propagation of corrosion at the coating/substrate interface has not been investigated in de-tail in the existing literature. It is therefore essential toanalyze the propagation of corrosion at the interfaceof coating and substrate in consideration of the cou-pling effects of residual and diffusion-inducedstresses.

For the residual stress (strain) analysis, the Stoneyformula has been extensively used in the literature to infercoating stress changes from experimental measure-ment of system curvature changes.16 Based on Stoney’sformula, some of the modified models have been de-veloped by Huang and Zhang,17 Soliman and Waheed,18

Clyne and Gill,19 Widjaja, et al.,20 Zhang, et al.,21

Wisnom, et al.,22 and Bansal, et al.,23 to address advanceand better closed-form solutions of the uniformed re-sidual strain in the coatings. Later on it was identified,with the help of finite element analysis, that thepropagation of corrosion at the interface of coating andsubstrate is inhibited by compressive residual stress,which causes the contraction of open corrosion paths inthe coatings.24 However, the results from the model24

showed only the limited effect of compressive residualstress on the cross-sectional area of corrosion pits. Themodel did not address the effects of the compressiveresidual stress over pre-existing structuralmicrocracksin the coatings. These microcracks in the presence ofcompressive residual stress contract, and can inhibitthe diffusion of corrosive agents through the coating andresult in lower corrosion rate at the interface of coatingand substrate.

Meanwhile, the diffusion-induced stress hasdrawn a considerable interest in the previous dec-ades.25-26 For the diffusion-induced stress, variousnumerical models have been available since the

pioneering work of Podstrigach and Shevchuk.27

Later on, some modified numerical models were devel-oped by researchers28-32 after a series of studies ondiffusion-induced stress in the coating-substratesystem. In recent research work, various numericalmethods and advanced techniques for the coating-substrate delamination involving diffusion-inducedstresses have been developed by Nguyen, et al.,33

Prawoto, et al.,34 Zhang, et al.,21 Yang and Li,35 andRusanov.36 However, the occurrence of the residualstress in the coating-substrate system resulting frommismatch in thermal expansion cannot be neglected;therefore, both the diffusivity and concentration ofcorrosive agents will be enhanced by the hydrostaticstress. The existing stress may speed up or hinder thediffusivity depending upon the direction of stressgradient.

Previous research hasmodeled37-49 to address thecoating-substrate failure in the individual disciplines.However, there is a missing link in modeling betweenthe three distinct disciplines, which can combine thethree disciplines to address majority of coating failurerelated issues. In order to address the missing linkbetween these three disciplines, amathematicalmodelhas been developed in the next sections. The newlydeveloped mathematical model follows a multidisci-plinary approach which fosters a close collaborationbetween three major disciplines, i.e., material science,solid mechanics, and electrochemistry. The purpose ofthis work is to analyze corrosion at the interface ofcoating and substrate, induced by diffusion in thepresence of residual stress. The model involves thecoupling effects of electrochemical reactions resultingfrom corrosion at the interface, with the residual stressand diffusion-induced stress in a coating-substratesystem. Then, the solution of corrosion current den-sity at the interface is derived involving the feedback ofresidual stress to the diffusion of corrosive agentthrough coating. The evolution of corrosion currentdensity corresponding to different moduli ratio,thickness ratio, thermal expansion mismatch ratio, andresidual stress gradients of coating and substrate havebeen discussed in terms of Newton-Rhapson method.Finally, validation of the developedmodel is performedand a comparison is developed between simulated andexperimental results.

EXPERIMENTAL PROCEDURES

Sample PreparationA thin carbon steel panel with a thickness of

s = 1/20 in (1 mm) was used to prepare three testsamples with dimensions 6 in × 4 in each. The coef-ficient of thermal expansion (CTE) and Young’smodulusfor the steel are αs = 11.7 × 10−6 K−1 and Es = 200 GPa,respectively.50 Prior to coating deposition, surfaceconditioning is deployed by using a polishing wheelwith emery paper of 200 grit size. After polishing, the

CORROSION—Vol. 72, No. 4 501


conditioned samples were cleaned with a 35 min im-mersion into a constantly stirred solution of 50 g/LTurco 4215 NC-LT†. After completing this alkalinecleaning, the samples were rinsed with deionized waterand air dried. Weight was recorded to the nearest fifthsignificant digit and the steel samples were stored in adesiccator. To ensure the repeatability, the experi-mental data was collected from three samples eachadhered with corrosion sensor with dimensions of40 mm × 20 mm × 0.1 mm. These sensors51 consisted ofmultiple plates made from the material of interestwhich formed the two electrodes. The electrodes wereused in conjunction with a potentiostat for conductinglinear polarization resistance (LPR) measurements. Theuse of a relatively large counter electrode minimizedpolarization effects at the counter electrode to ensurethat a stable reference potential was maintainedthroughout the experiments. Potential step-sweeps wereperformed by applying a series of 30 steps over a rangeof ±10 mV spanning a period of 2.6 s.52

Corrosion sensors measure the polarization re-sistance Rp (Ω) between the corrosive agents (electrolyticsolution) and the steel samples. The polarization re-sistance is then used to calculate the corrosion currentdensity “i” at the interface by using Stern-Gearyequation as:53

i=B=Rp (1)

where B is the Stern-Geary constant. The corrosioncurrent density is measured from the polarizationresistance values by using a Stern-Geary constant of30 mV for carbon steel.52

The corrosion sensors were adhered to the con-ditioned face of the steel samples with industrialstrength epoxy, as shown in Figure 1. The bondingagent (industrial strength epoxy) was placed on theopposite edges of the sensors so as to adhere thesensors to the conditioned surface of the steel samplesin a manner such that the ambient environment isallowed to rapidly diffuse between the sensors and thesteel. The sensor array included at least two inter-laced inert electrodes that were manufactured from anoble metal. The noble metal options were Au, Pt, andPd because of the low contact resistances. The noblemetals were principally inert, such that the sensorarray did not readily corrode in typical ambientenvironments.

After installation of corrosion sensors on steelsamples, the samples were applied with the thincoating (primer red oxide) by using a conventional spraygun at the temperature of 318 K (45°C); this is shownin Figure 1. The thickness of the coating wash = 0.020 mm. The coefficient of thermal expansionCTE (αc = 21.6 × 10−6 K−1) and Young’s modulus(Ec = 6.14 GPa) for the thin coating were measuredby using thermomechanical analysis test according

to the procedures of ASTM E831-1454 and ASTME2769-13,55 respectively. The coating was allowed tofully seal over a 24 h period at 318 K before testing.The temperature of 318 K is referred to as the fabrica-tion temperature of the coating-substrate system.

Accelerated Corrosion TestingAccelerated corrosion testing was performed in a

corrosive atmosphere chamber. The electrolyte solutioncomposing the fog was 5 parts of sodium chloridein 95 parts of deionized water. The samples were posi-tioned at a 60° angle inside the chamber with thecoated face downward, as to avoid the direct pathway forcondensate into the coating. Electrical connectionsfor the corrosion sensors were made to a data acquisi-tion unit (DAU) positioned outside the chamber bypassing extension cables through a bulkhead, as shownin Figure 2. Temperature and corrosion data wereacquired at 1 min intervals.

Accelerated corrosion testing was conducted for atotal time of 50 h and consisted of five steps each(10 h per step), as shown in Figure 3. The purpose of thesesteps was to analyze the corrosion current density atcoating/substrate interface corresponding to variousstress levels in coating at different temperatures. Thefirst step involved exposing the samples to a salt fogfor a period of 10 h at fabrication temperature,

Back side of the sensor

Corrosion sensor

Substrate Thin carbon steel panel

Thin primer coatingCoating

SubstrateThin carbon steel panel

40 mm

Thickness = 0.1 mm

LPR/structure pads

Gold electrodes

(a)

(b)

(c)

20 mm

FIGURE 1. (a) The corrosion sensor.52 (b) Installation of corrosionsensor on steel sample, (c) applied with the thin coating (referred toas primer) by using a conventional spraying gun at temperature of318 K (45°C).

† Trade name.

502 CORROSION—APRIL 2016


i.e., 318 K (45°C). For this period of time, ΔT is equal to0±1 K, as the final temperature (318±1 K) and fabrica-tion temperature of coating-substrate system(318 K) were almost same. ΔT is calculated by usingEquation (1).

ΔT =Final Temperature − Fabrication Temperature (2)

The second step involved reducing the temperaturefrom 318 K to 303±2 K (29.8±2°C) and maintaining thistemperature for a period of 10 h, while keeping thesame level of fog inside the chamber. The third stepinvolved increasing the temperature from 303±2 K to

318 K and maintaining this temperature for a period of10 h. The fourth step involved again reducing thetemperature from 318 K to 311±3 K and maintaining for10 h, followed by the last step in which the temperaturewas again increased andmaintained at 318K for last 10 hof the experiment.

Experimental ObservationsThe corrosion sensors adhered to the substrate

beneath the coating determine the corrosion currentdensity based upon electrolyte conductivity, ionicstrength of electrolyte, and temperature. Corrosion at

Diffusion of ionic agents

Substrate

Coating

Microcracks (defects)

x

ylc = ls

s

h

Substrate

Coating

Coating

Substrate

Sensor signal out

Substrate

Coating Diffusion of ionic agents

αc > αs

ΔT < 0

Cracks opening

Data acquisition unit (DAU)

Microcrack

Tensile direction

Cracks opening

Diffusion layer

(a)

(b)

Diffusion layer

25 μm

25 μm

FIGURE 2. (a) Accelerated corrosion testing of primer coated steel sample in a corrosive atmosphere chamber. (b) Becausethe CTE of thin primer is greater than the steel substrate, i.e., αc> αs, negative temperature change (ΔT< 0) accounts for thetensile residual stress in the coating. The tensile residual stress allows the opening of pre-existing coating defects resulting inthe evolvement of corrosion at the interface. The corrosion rate at the coating/substrate interface is monitored by a corrosionsensor connected to computer through DAU.



the interface of the coating and substrate is primarilycontrolled by the concentration of an electrolyte solutionand the temperature. The concentration is controlledby the residual stress in the coating, which is caused bytemperature difference, ΔT. The residual stress caneither block the pathway of corrosive agents (as a resultof compressive behavior) by constricting the pre-existing coating microcracks or open the pathway ofcorrosive agents (as a result of tensile behavior) byexposing the pre-existing coating microcracks.

The elastic response of coating suffered from theresidual stress and diffusion-induced stress is analyzedin terms of the principle of linear superposition.56

Because the CTE of thin primer is greater than the steelsubstrate, i.e., αc > αs, the positive temperaturechange (ΔT > 0) leads to the opposite direction of residualstress compared to the direction of diffusion stress.This behavior reduces the deflection of coating-substrate system resulting in the compressive re-sidual stress in the coating, allowing the contraction ofpre-existing coating microcracks. This shows thatcompressive residual stress blocks the diffusion ofcorrosive agents toward the interface and, therefore,avoids corrosion at the interface, measured by corrosionsensors. On the other hand, negative temperaturechange (ΔT < 0) accounts for the same direction of boththe residual and diffusion stresses. This behaviorincreases the deflection of coating-substrate system andresults in tensile residual stress in the coating at agiven time. With increasing tensile loading, the coatingmicrocracks open, the crack tips penetrate into thediffusion layer, and the Kirkendall voids then intersectwith the crack tip.57 After substantial deformation,normally 10%, the macrocracks are visibly wide andtheir tips, after penetrating through the diffusionlayer, arrive at the diffusion layer/substrate interfaceas shown in Figures 2(a) and (b). This type of crackpropagation, normal to the interface, is identified asMode I path, leading to Type I opening cracks.Widening of these types of microcracks give rise to

unfavorable exposure of the carbon steel substrate.This indicates that tensile residual stress allows theopening of microcracks resulting in the diffusion ofcorrosive agents, which accelerates the corrosion at thecoating/substrate interface as shown in Figure 2.

In the above experimentation, when the coating-substrate system is cooled (ΔT < 0) to the final temper-ature from its fabrication temperature (318 K [45°C]),tensile residual stress resulting from the mismatchin the coefficients of thermal expansion (Δα = αs − αc)between thin steel substrate and thin primer is evolved.The possible maximum residual stress that candevelop in the coating can be determined fromEquation (3) as:58

σrc =Ec

�1 −

4Ec hEs s

�ðαs − αcÞΔT (3)

The tensile residual stress that is developed in thecoating by reducing the temperature (in the secondand fourth steps of experimentation) inside the chamberis shown in Figure 3.

Figures 4(a) through (c) show the corrosion currentdensity for all three samples adhered with corrosionsensors. It can be seen that for all three samples duringthe first step (10 h) when ΔT is equal to 0±1 K, the residualstress was almost zero with no effect on microcracks,resulting in the minimum diffusion of corrosive agentstoward the interface. Therefore, corresponding corro-sion current density during the first step is negligible forall of the samples, as shown in Figures 4(a) through (c).For the second step (10 h), at point A in Figures 4(a)through (c), ΔT started decreasing such that ΔT reached−15±2 K at about 11 h. This decrease in ΔT resulted in thedevelopment of tensile residual stress in the coatingcausing the opening of microcracks. There was a 3 h lagobserved in all of the samples. It is believed that this lagis a result of the diffusion process through the exposedcracks before the corrosion current density starts in-creasing, indicated as point B in Figures 4(a) through (c).By the time the temperature reached its lowest pointduring the second step, the corrosion current almostreached its highest point, indicated as point C inFigures 4(a) through (c). During the third step (10 h),when ΔT increased to 0±1 K and maintained for 10 h,a negligible corrosion current density was observed forall of the samples, similar to step 1. For the fourthstep, ΔT again started decreasing and reached −7±3 K atabout 31 h. It is worth noting that this time there wasonly a lag of almost 45 min (instead of the previous 3 h)before the corrosion current density started increas-ing, indicated as point D. The possible reason for thisreduced lag at point D during the fourth cycle couldbe that the coating degradation resulting from acceler-ated testing with time created newmicrocracks. Thesenewly developed microcracks, along with the othermicrocracks, now result in a large diffusion rate ofcorrosive agents because of the large number ofmicrocracks now opened up when the coating is

0.00081st step for 10 h 2nd step for 10 h 3rd step for 10 h 4th step for 10 h 5th step for 10 h

ΔT1 = 318±1 K – 318 K = 0±1 K

ΔT2 = 308±3 K – 315 K = –7±3 K

ΔT3 = 308±2 K – 323 K = –15±2 K

Tensile residual stressTemperature [K]

Tensile stress

Temperaturereduced

ΔT < 0 K

ΔT = 0 K

Tensile stress

5

0

–5

–10

–15

–20

–25

0.0007

0.0006

0.0005

0.0004

0.0003

0.0002

0.0001

Ten

sile

Res

idu

al S

tres

s (G

Pa)

Total 50 h of Experiment

Time (h)

00 5 10 15 20 25 30 35 40 45 50

–0.0001

–0.0002

FIGURE 3. Graph showing the tensile residual stress, which isdeveloped in the coating by reducing the temperature (in the secondand fourth step of experimentation) inside the chamber.



subjected to tensile residual stress at low ΔT. The large

diffusion rate resulted in the early corrosion initiali-

zation, which explains the possible reason for reduced

lag time. Likewise, because of the large number of

microcracks and high diffusion rate, high peaks ofcorrosion current density far greater than the secondstep were observed, indicated as point E; even at ΔT thistime was reduced to −7±3 K, which is lower than thesecond step. For the fifth step, ΔT increased to 0±1 Kand maintained for 10 h with negligible corrosioncurrent density for all of the samples, the same as insteps 1 and 3.

Based on the observation from the 5 step experi-mentation, amathematical model has been developedin the next section.

MATHEMATICAL MODEL

A multidisciplinary approach has been adoptedduring this research to develop a mathematical model.This multidisciplinary technique is unique in terms ofcombining three distinct fields of material science, solidmechanics, and electrochemistry. A holistic approachto the modeling technique is shown in Figure 5.

Diffusion Model of Coating—Utilization of MaterialScience Concepts

This part of modeling is specifically developed byconsidering the diffusion of corrosive species k when theprimer coated steel sample is exposed to a corrosiveenvironment. The type and concentration of the speciesdecide the rate of coating delamination. The mini-mum threshold concentration of the corrosive species isrequired at the coating defect in order to start theprocess of delamination.59 This part of modeling follows

(a)

(b)

ΔT = 0 K

ΔT < 0 K

ΔT = 0 K

ΔT < 0 K

Temperaturereduced

Temperaturereduced

A

C

B D

E

A

C

BD

E

ΔT = 308±3 K – 315 K = –7±3 K

ΔT = 308±2 K – 323 K = –15±2 K

Corrosion sensor 1

Temperature (K)

Corrosion sensor 2

Temperature (K)

ΔT = 318±1 K – 318 K = 0±1 K

ΔT = 318±1 K – 318 K = 0±1 K

ΔT = 308±3 K – 315 K = –7±3 K

ΔT = 308±2 K – 323 K = –15±2 K

0.007 5

0

–5

–10

–20

–25

5

0

–5

–10

–20

–25

1st step for 10 h 2nd step for 10 h 3rd step for 10 h 4th step for 10 h 5th step for 10 h

1st step for 10 h



Co

rro

sio

n C

urr

ent

Den

sity

, i (

mA

)C

orr

osi

on

Cu

rren

t D

ensi

ty,

i (m

A)

Time (h)

Time (h)

2nd step for 10 h 3rd step for 10 h 4th step for 10 h 5th step for 10 h

0.0045

0.004

0.0035

0.003

0.0025

0.002

0.0015

0.001

0.0005

0

0.006

0.005

0.004

0.003

0.002

0.001

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

–15

–15

0

Temperaturereduced

ΔT < 0 K

ΔT = 0 K

A

C

B D

E

Co

rro

sio

n C

urr

ent

Den

sity

, i (

mA

)


Time (h)

1st step for 10 h 2nd step for 10 h 3rd step for 10 h 4th step for 10 h 5th step for 10 h

ΔT = 318±1 K – 318 K = 0±1 K

ΔT = 311±3 K – 318 K = –7±3 K

Corrosion sensor 3

Temperature (K)

ΔT = 303±2 K – 318 K = –15±2 K

0.007 5

0

–5

–10

–15

–20

–25

0.006

0.005

0.004

0.003

0.002

0.001

00 5 10 15 20 25 30 35 40 45 50

(c)

FIGURE 4. Graph showing that, for the case of the second and fourthstep (for 10 h) when ΔT is reduced to −7±3 and −15±2 K, this results inthe same direction of residual stress gradient as of the diffusiondirection. This behavior increases the deflection of coating-substratesystem at a given time and results in the evolvement of tensile residualstress in the coating. The tensile residual stress opens the pre-existingstructural defects in the coating, which allows the corrosive agents todiffuse toward the interface and resulting in the evolvement of corrosioncurrent density.

Material Science

Solid Mechanics

Electrochemistry

Corrosion current density “i” along

the coating/substrate interface

1. Residual stress 2. Diffusion inducedstress 3. Bending curvature of coating-substrate system

1. Temperature 2. Electrolyte salinity 3. Ionic diffusivity

1. Oxygen reduction 2. Substrate oxidation

Diffusion model of coatings

Stress model of coatings

Electrochemical model of coatings

FIGURE 5. Modeling methodology for the developed mathematicalmodel.



the assumptions of: (i) bulk concentration of corrosivespecies ckB

exists at the defect in coating, and (ii) theelectrolyte solution potential Φ follows the electro-neutrality condition at all of the boundaries as:

Φ=Xyk=1

zk ck =0 (4)

where k = 1, 2 : : : , y represents the number of cor-rosive species involved, and zk represents the chargenumber of the species with concentration ck. Thediffusing corrosive species k maintains a concentrationck over an entire exposed surface of the coating.

The rate of delamination is decided by the con-centration of the corrosive species (cation) along thecoating/substrate interface, which is found by usinga well-known solution of the differential equationfor Fick’s second law:60

ck = ckB

�1 − erf

�x

2ffiffiffiffiffiffiffiffiffiffiffiffiDktD

p��

; ck = ckBwhen

�x=0tD > 0

(5)

where x defines the distance to the coating defect, tD is thediffusion time,59 and Dk is the average of the standarddiffusion coefficient of all of the corrosive species:

Dk = ½Dk,STAionF1ðTÞF2ðRHÞF3ðtexpÞ�−2 (6)

The diffusion coefficient DS,STAionof each corrosive

species along the coating/substrate interface dependsupon the temperature (T), pore relative humidity (RH),and time of exposure (texp). The diffusion coefficientDS,STAion

of the corrosive species along the interface istwo orders smaller than in aqueous solution.61 The termsF1(T), F2(RH), and F3(texp) in Equation (6) are thetemperature variation function, RH variation function,and aging function, respectively:62

F1ðTÞ= exp��

Ga

R

��1

TSTA−1T

��(6a)

F2ðRHÞ= 1

1þ�

1−RH1−RHSTA

�m (6b)

F3ðtexpÞ=�tSTAta

�nag

(6c)

In Equation (6a), TSTA is the standard temperaturevalue, Ga is the activation energy of corrosivespecies during the diffusion process, and R is theuniversal gas constant. In Equation (6b), RH is theactual pore relative humidity, RHSTA is the standardrelative humidity at which Dk drops betweenmaximum and minimum values, and m is theparameter that characterizes the spread of drop inRHSTA.

60 In Equation (6c), tSTA is the time of exposure

at which Dk is measured (normally 1month), ta is theactual time of exposure, and nag is the age reductionfactor.

The threshold concentration of corrosive speciesckTH

along the interface to start the process of delami-nation can be derived from Equation (5) as:

CkTH= cko

�1 − erf

�η

ffiffiffiffiffitp

p2ffiffiffiffiffiffiffiffiffiffiffiffiDktD

p��

=kckeqckB

�1 − erf

�η

ffiffiffiffiffitp

p2ffiffiffiffiffiffiffiffiffiffiffiffiDktD

p��

; tp = t − tn (7)

where η represents the mobility constant of cations,t represents time passed after the coating defectconfronts the electrolyte solution, tn represents the timerequired to activate the defect, and cko

represents theequilibrium concentration of species at the interface incontact with the bulk electrolyte solution ckB

. Theequilibrium constant kckeq

is used to relate ckoand ckB

,

such that kckeq=�ckockB

.

For the coating-substrate system exposed to anisotropic electrolyte solution, the chemical potentialcorresponding to the stressed state of the system canbe represented as:42

μQk =μo

k þ RT lnðckÞ − VPk

ðσm

0dσm; σm =

13

ði=3

i=1σi (8)

where μok is the chemical potential in the given

standard state, ck is the concentration of corrosivespecies derived fromEquation (5), and σm is the stresstensor and is equal to the mean of the sum of threeprinciple stresses σi (i = 1, 2, and 3), where σi can bewritten as a sum of diffusion-induced stresses σdi

andresidual stresses σri , i.e., σi =σdi

þ σri . VPkis a scaler

term (without the inclusion of stress tensor effect)representing the partial molar volume of diffusingcorrosive species k and can be found by using Euler’sfirst theorem for homogeneous functions:64

VPk=�∂Vmk

∂nk

�T,P,nk

=

Vmk2

− Vmk1

nk2−nk1

!T,P,nk≠i

=

0@mk2

ρk2−

mk1ρk2

nk2− nk1

1A

T,P,nk≠i

(9)

where VPkdepends on the change in molar volume of

solution Vmk. The change in molar volume Vmk

depends on temperature T, pressure P, and molarconcentration of diffusing corrosive species nk. Con-sider the case with constant T and P, Vmk2

− Vmk1is

the change in molar volume corresponding to thechange in molar concentration of diffusing speciesk from nk1

to nk2.

For the case of an inhomogeneous distribution ofsolute particles in a non-ideal electrolyte solution, Fick’s



second law of diffusion ½∇ × Jk!

= − ∇ × ðDkckRT μQ

k Þ�,65 inconjunction with the law of conservation of mass, can beincorporated with Equations (8) and (9) as:

∇ × Jk!

= − Dk∇2ck þDkVPk

3RT∇ck∇

�ði=3

i=1ðσdi

þ σriÞ�

þDkVPk

3RTck∇2

�ði=3

i=1ðσdi

þ σriÞ�

(10)

where ~Jk is the diffusion flux of corrosive species k andrepresents the change in concentration of a corrosivespecies k with respect to time within the stressedcoating-substrate system as:

∇ × Jk!

= −∂ck∂t

(11)

Substituting Equation (11) into Equation (10) gives:

∂ck∂t

=Dk∇2ck −

DkVPk

3RT∇ck∇

�ði=3

i=1ðσdi

þ σriÞ�

−DkVPk

3RTck∇2

�ði=3

i=1ðσdi

þ σriÞ�

(12)

Equation (12) follows the law of conservation of mass.It is worth noting that first part of Equation (12) on theright hand side represents the chemical contributionwithout the inclusion of stress. The second part of theequation includes the contribution of stress alongwith chemical effects. The third part of equation is theresult of stress-induced diffusion.

Stress Model of Coating—Utilization of SolidMechanics Concepts

This part of modeling is based on solid me-chanics concepts integrating the principles of bilayercantilever beam. In the current research, one layer ofthe bilayer cantilever beam is considered as coating,while the other layer is considered as substrate. Itshould be noted that both of the layers of bilayercantilever exhibit bending as a result of the diffusion-induced stress σd and residual stress σr. Coating/substrate interface is located at location x = 0, thefree surface of the coating is located at x = h, and thefree surface of substrate is located at x = −s, as shownin Figure 2.

The analysis in this part of modeling is based onthe following assumptions: (i) thickness of coating h isvery small compared to thickness of substrates,(ii) the material properties of coating and substratesuch as Young’s modulus, diffusivity, and chemicalpotential are homogeneous, isotropic, and invariable,(iii) the strain tensor corresponding to principlestrains is very small, and (iv) the diffusion-induced

stress σd corresponding to time t = 0 is zero. Thediffusion-induced stress evolves after the diffusionof corrosive species k in both layers of cantileverbeam at time t > 0, which results in the bendingof beam.

The strain distribution E in a system is the sum ofuniform component Euj

and a bending componentEb as:58

E = Eujþ Eb = Euj

þ x − tbζj

ðfor − s ≤ x ≤ hÞ (13)

where x = tb dictates the neutral point of the bendingaxis where Eb is zero and ζj represents the radius ofcurvature of bilayer cantilever. The term j = r, d where ris the residual component and d is diffusion com-ponent. Equation (13) will be utilized to derive therelations for residual stresses σr and diffusion-induced stresses σd in a bilayer cantilever beam. Thisequation is analogous to the strain continuity equa-tion in the Timoshenko shear model.66

The relation for the residual stresses in a bilayercantilever beam is given as:42

σrc =Ec

�Eur

þ x−tbζr

− αcΔT

σrs =Es

�Eur

þ x−tbζr

− αsΔT9>=>;Residual stresses

(14a)

(14b)

where Ec, Es, αc, and αs are the elastic moduli andcoefficient of thermal expansion (CTE) of coating andsubstrate, respectively, and ΔT is the change intemperature from fabrication temperature or duringapplication. The terms Eur

, tb, and 1/ζr can be ex-panded as:67

Eur=

ðEsαssþ EcαchÞΔTðEssþ EchÞ

(15a)

tb =− Ess2 þ Ech2

2ðEssþ EchÞ(15b)

1ζr

=3½Ess2ðEu − αsΔTÞ − Ech2ðEu − αcΔTÞ�

Ess2ð2sþ 3tbÞ þ Ech2ð2h − 3tbÞ(15c)

The average diffusion-induced stresses distributionin a bilayer cantilever beam are related to strains byusing a modified form of stress-strain relation as:

σdc=Ec

�Eud

þ x−tbζd

− 13 ckc

VPkc

σds=Es

�Eud

þ x−tbζd

− 13 cks

VPks

9>>=>>;Diffusion-

(16a)

induced stresses(16b)

The bi-axial strains are identical along y and z planes forthe case of planar geometry; therefore, Em can bereplaced by Em/(1−vm), where vm is the Poisson’s ratio



and m= c, s. The equations for the diffusion-inducedstresses (Equations [16a] and [b]) are similar to theequations for the residual stresses (Equations [14a]and [b]). However, for diffusion processes, the coatingproperties may be assumed to change linearly68 withthe concentration of diffusing corrosive species k, whichresults in diffusion-induced stresses. Therefore, it ispossible to calculate diffusion-induced stresses byanalogy to thermal stresses, which has been previ-ously calculated by Prussin69 and Zhang.70 Prussin andZhang treated concentration gradients ckc

analogousto those generated by temperature gradients ΔT andpartial molar volume VPk

analogous to thermal ex-pansion α.

The strain and stress distribution in bilayercantilever beam (Equations [13], [16a], and [16b])resulting from diffusion are dependent on the solu-tion of two parameters, i.e., Eud

and 1/ζd. It is possible tofind Eud

and 1/ζd by using the following two boundaryconditions.

At first, the resultant force corresponding touniform strain component is zero, such that:

ðZc

Ec

�Eud

−13VPkc

ckc

�dZc

þðZs

Es

�Eud

−13VPks

cks

�dZs =0 (17)

The terms Zc = bh, Zs = bs, and b = bs = bh. The solu-tion of Equation (17) gives:

Eud=

13

�EchVPkc

ckcþEssVPks

cks

ðEchþ EssÞ�

(18)

where ckc= 1

h ∫h0ckc

dx and cks= 1

s ∫0−scks

dx.Second, the summation of bending moments

of cantilever layers with respect to neutral point,i.e., x = tb is zero, such that:

ðZc

σdcðx − tbÞdZc þ

ðZs

σdsðx − tbÞdZs =0 (19)

The bending curvature of cantilever layers 1/ζdresulting from diffusion-induced stress can bedetermined from the solution of Equations (13), (16a),(16b), (18), and (19):

1ζd

=2½EcEshsð2sÞðVPkc

ckc− VPks

cksÞ�

E2ch4 þ E2

ss4 þ 2EcEshsð2h2 þ 3hsþ 2s2Þ (20)

Substituting the residual stresses Equations (14a)and (b) and diffusion-induced stresses Equations (16a)and (b) into Equation (12) forms a coupling relationshipbetween stress model of cantilever layers and diffusion

model as:

∂ckc

∂t=Dkc

�1þ EcVPkc

2

9RTckc

�∂2ckc

∂x2 þDkc

EcVPkc

2

9RT

�∂ckc

∂x

�2

−Dkc

EcVPkc

3RT∂ckc

∂x

�1ζd

þ 1ζr

�(21a)

∂cks

∂t=Dks

�1þ EsVPks

2

9RTcks

�∂2cks

∂x2 þDks

EsVPks

2

9RT

�∂cks

∂x

�2

−Dks

EsVPks

3RT∂cks

∂x

�1ζd

þ 1ζr

�(21b)

In Equations (21a) and (b), in the third part on right

hand side, the terms can be written as: Ecζd

= ∂σdc∂t ,

Ecζr

= ∂σrc∂x ,

Esζd

= ∂σds∂t , and

Esζr

= ∂σrs∂x , where ∂σdc

∂t and ∂σrc∂x , and

∂σds∂t and

∂σrs∂x represent the rate of change of diffusion-inducedstresses and residual stresses in coating and sub-strate, respectively.

The total bending curvature of cantilever layers�1ζdþ 1

ζr

in Equations (21a) and (b) as a result of

the diffusion-induced stress and residual stresscan be written as the summation of Equations (15c)and (20):

1ζd

þ 1ζr

=2½EcEshsð2sÞðVPkc

ckc− VPks

cksÞ�

E2ch4 þ E2

ss4 þ 2EcEshsð2h2 þ 3hsþ 2s2Þ

þ 3½Ess2ðEu − αsΔTÞ − Ech2ðEu − αcΔTÞ�Ess2ð2sþ 3tbÞ þ Ech2ð2h − 3tbÞ

(22)

Equation (22) shows that the total bending curvature ofcantilever layers is the sum of bending curvature ofcantilever layers resulting from diffusion-induced stress1/ζd and bending curvature of cantilever layersresulting from residual stress 1/ζr. The bending cur-vatures as a result of diffusion-induced stress andresidual stress are dependent upon the concentration ofcorrosive species ck and CTE mismatch resultingfrom temperature change ΔT, respectively.

Electrochemical Model of Coating—Utilizationof Electrochemistry Concepts

This part of modeling gives the design for corro-sion current density resulting from the electrochemicalreactions along coating/substrate interface. Thecorrosion current density depends on the electro-chemical reactions as a result of substrate dissolutionand oxygen reduction. The conventional equation ofcorrosion current density from literature71-72 cannow be incorporated with the derived equation of totalbending curvature (Equation [22]). This indicates thatthe electrochemical reactions along the coating/sub-strate interface also depend on the total bendingcurvature resulting from diffusion-induced stress (1/ζd)



and residual stress (1/ζr). No homogeneous reactionsare taken into account; only electrochemical reactionsare considered. The expressions of the polarizationkinetics for oxygen reduction and substrate dissolutionare developed, which are presented next.

The polarization kinetics for the reaction involv-ing oxygen reduction is expressed as: O2 + 2H2O +4e−→ 4OH−.

By incorporating the total bending curvature1ζdþ 1

ζrderived in Equation (22) with the conventional

equation of corrosion current density,71-72 a newequation has been developed for the current densityresulting from the oxygen reduction at any positionalong the coating/substrate interface:

icoatO2=−

0BBBBB@

1

−nFDO2cO2

0B@

�1

ζddþ 1

ζrd

�1

ζdndþ 1

ζrnd

�

1ζdd

þ 1ζrd

hndþ

�1

ζdndþ 1

ζrnd

hd

1CA

−10

V−EoO2

βO2

1CCCCCA

−1

;

V=E−Φ

(23)

where EoO2, E, Φ, and βO2

are the equilibrium potentialof oxygen, substrate potential, solution potential, andoxygen Tafel slope, respectively. The terms hd and hnd

are the coating thickness for delaminated andnondelaminated coating, respectively. The terms 1

ζdd

and 1ζdnd

, and 1ζrd

and 1ζrnd

are the bending curvaturesresulting from diffusion-induced stress and residualstress for delaminated and nondelaminatedcoating-substrate system, respectively. Theterm cO2

represents the dissolved concentrationof oxygen at coating surface, DO2

is the diffusioncoefficients for oxygen, and n is the number ofelectrons transferred.73

The polarization kinetics for the forward reactioninvolving substrate dissolution is expressedas: S → Snþ þ ne−

The current density is at the coating/substrateinterface as a result of substrate dissolution is given as:63

is = io,s10V−Eosβs ; V=E − Φ (24)

where Eos, βs, and io,s are the equilibrium potential of

substrate, substrate-Tafel slope, and exchange currentdensity for the substrate dissolution reaction,respectively.

The equations ([23] and [24]) for both the oxygenreduction and substrate dissolution can be combined togenerate the net corrosion current density “i” alongthe coating/substrate interface as:

i= icoatO2þ is (25)

It is clear that the above equation agrees withAllahar,72 when the total bending curvature

�1ζdþ 1

ζr

resulting from diffusion-induced stress and residual

stress is replaced by interface porosity (gap at the in-terface between coating and substrate) p1.5 in acoating-substrate system.

IMPLEMENTATION OF MATHEMATICAL MODEL

The initial and boundary conditions for thestressed bilayer cantilever are expressed as follows:

ckcðxÞ= cks

ðxÞ=0 at time t=0 (26)

∂cks

∂x

x= s

=0; ckcðhÞ= cko

at time t > 0 (27)

For numerical simulation work, all of the parametersand variables, which are used in the equations, shouldbe converted in to dimensionless forms as:

D 0k =Dkc

=Dks; D 0

O2=DO2

=DO2O(28a)

x 0 =xb=

xhþ s

(28b)

c 0kc= ckc

=cko; c 0

ks= cks

=cko; c 0

O2= cO2

=cO2O(28c)

VPkc=VPks

=1 (28d)

t 0 =Dkct=h2 (28e)

1ζ 0r=

hζr

2EcVPk

3RT=

4Λ−1

�1η

��1þ 1

η

�2ðαc − αsÞΔTEcVPk�

Λ−1

�1η

�2− 1�

2þ 4Λ−1 1

η

�1η þ 1

�2RT

(28f)

1ζ0d

=b

ζdVPkcko

=4ð1þ1=ηÞ3½1þðΛ−1=ηÞ�

�Λ−1Ð0− 1=η1=ηþ1

c0ksx 0dx 0þÐ 1

1=ηþ1

0 c0kcx 0dx 0

�hΛ−1�1η

2−1i2þ4Λ−1

�1η

�1ηþ1

2

−

2h1−Λ−1

�1η

2i�1þ1

η

2�Λ−1

Ð0− 1=η1ηþ1

c0ksx 0dx 0þÐ 1

1=ηþ1

0 c0kcx 0dx 0

�hΛ−1�1η

2−1i2þ4Λ−1

�1η

�1ηþ1

2

(28g)

i 0 = icoat0

O2þ is

0

= −

8>>>>><>>>>>:

1

− D 0O2c 0O2

26664

�1

ζ 0dd

þ 1ζ 0rd

�1

ζ 0dnd

þ 1ζ 0rnd

�

1ζ 0dd

þ 1ζ 0rd

hnds þ�

1ζ 0dnd

þ 1ζ 0rnd

hds

37775− 10

V−EoO2

βO2

9>>>>>=>>>>>;

−1

þ 10V–Eosβs

io,sio,sO

(28h)



where Λ−1 = Ec/Es and 1/η = h/s, and it is assumedthat VPk

=VPkc=VPks

.

The dimensionless partial differential equation(PDE) Equation (28h) is then solved by developing ageneralized Equation (23) representing the time it-erative algorithm shown in Figure 6.

∂i 0

∂tðnÞ= i 0t

0 ðnÞ (29)

Temperature ΔT influences the “time discretization”of “i” during the application of coating-substratesystem. The iterations of parameter ΔT is repre-sented by “n” and the time states are representedby superscript “t 0” which is generalized in Equa-tion 29. Newton-Rhapson method74 is utilized tosolve the time-varying PDE ∂i 0

∂t . The governing PDE ∂i 0∂t

for normalized corrosion current density i 0 in thedomain is discretized in time and space, and castinto the matrix form by using Newton-Rhapsonmethod.

The algorithm follows the following steps.(i) The algorithm initiates the iterative process

for a parameter and computes Δn as:

Δn=n − ni where ni =0 (30)

(ii) The initially calculated Δn in Equation (24) isused every time to upgrade n to n + 1 by using:

nþ 1=nþ Δn (31)

(iii) The value of i 0t0 ðnÞ at time t 0i =0 is assumed as

i′0(n). This is in accordance with the boundarycondition in stressed cantilever, as mentionedin Equation (26).

(iv) The algorithm then performs the time discreti-zation of i 0 which is then cast into the matrixform as:

Ki 0 · i 0t 0þ1ðniÞ=Li 0 (32)

where Ki 0 represents the coefficient matrixwhich is a function of i 0t

0þ1 at time state t′ + 1.The vector Li 0 is the load vector which is afunction of i 0t

0 ðnÞ at time state t′. Both of thematrices Ki 0 and Li 0 are used in the computationof the value of i 0t

0þ1.(v) The time convergence criterion ξt 0 is calculated

as:

ξt 0 =100�f t

0þ1 − f t0

f t0

�(33)

where f is a generic term representing time t′.The value of f for each time state in the domain,excluding that on the boundary conditions, wascomputed and the maximum value of ξmax

was determined. On the comparison of ξmax withξt 0 , if ξmax > ξt 0 , the value of t 0 is updated,and control returned to step (iv). For the con-dition ξmax < ξt 0 , the algorithm moves tonext step.

(vi) The convergence criterion ϑn for variable pa-rameter n is calculated as:

ϑn =100�mnþ1 − mn

mn

�(34)

where m is a generic term representing variableparameter n. The value of n is updated for thecondition if ϑmax > ϑn by returning the control tostep (ii). For the condition ϑmax < ϑn, the algo-rithm ends.

RESULTS AND DISCUSSION

Based on the numerical simulation by usingNewton-Rhapson method, the effect of residual stressgradient 1=ζ 0

r (resulting from the change in temper-ature ΔT) on corrosion current density is discussed inthis section.

The corrosion current density is significantlyaffected by the residual stress and diffusion-inducedstress. Figure 7 illustrates the evolution of nor-malized corrosion current density incorporating theeffects of diffusion-induced gradient 1=ζ 0

d and re-sidual stress gradient 1=ζ 0

r of the coating. It is clearthat the normalized corrosion current densitydepends upon the diffusion of corrosive agentsthrough the coating microcracks, which accounts

Start

Initial assumptionn1 = 0 ; t′i = 0 ; i′t ′i (ni)

Assemble Ki ′ and Li ′

Solve for (n)using Equation (32)

Calculate ξmax using Equation (33)

Update

No

Yes

End

max > n ?

No

Yes ϑ ϑ

n = n + 1

Update n to n + 1 usingEquation (31)

Iterative mehod

i′t +1′

ξmax > ξt′ ?

(n + 1) =i′t′

(n + 1)i′t +1′

FIGURE 6. Newly developed algorithm utilizing Newton-Rhapson method in order to solve and implement themathematical model.



for the diffusion-induced gradient 1=ζ 0d. However, the

diffusion-induced gradient 1=ζ 0d is changed by the

residual stress gradient depending upon the mis-match in CTE of coating and substrate. Because theCTE of the coating is twice that of the substrate, i.e.,αc = 2αs, therefore, the positive temperature changeΔT = 7 K (> 0) leads to the opposite direction of residualstress (1=ζ 0

r = −4.2) compared to the diffusion stressdirection. Thus, the total gradient can be written as:1=ζ 0

d − 1=ζ 0r. This type of stress gradient results in the

evolution of the compressive residual stress at a giventime, as shown in Figure 7. The compressive be-havior of the coating constricts the pre-existingcoating microcracks, which in turn creates obsta-cles for the diffusion of corrosive agents and reducesthe corrosion current density at the interface. Onthe other hand, the negative temperature change ΔT =−7 K (< 0) leads to the same direction of both thestresses, i.e., residual stress (1=ζ 0

r =4.2) and diffusionstress. Thus, the total gradient can be written as:1=ζ 0

d þ 1=ζ 0r. This type of stress gradient results in

the evolution of the tensile residual stress. Thetensile behavior opens the pre-existing coatingmicrocracks and allows the diffusion of corrosiveagents before reaching the saturation state of corro-sive agents in the coating. This results in the in-crease of corrosion current density. The results indi-cate that a 14° change in temperature can result in a100% shift in the corrosion current density at a givendiffusing time. For the case when ΔT = 0 K, the effectof residual stress gradient 1=ζ 0

r is negligible: only thediffusion-induced gradient 1=ζ 0

d influences thecorrosion current density. This indicates that the re-sidual stress plays a vital role in the corrosioncurrent density; however, the effects of residual stress

on the corrosion current density graduallydisappear when the saturation condition of corrosiveagents is reached. From Figure 7 it can be seenthat for the coating under tensile behavior ΔT = −7 K,the saturation condition is reached well beforeDkc

t=h2 =3.5; however, for coating undercompressive behavior ΔT = 7 K, the saturationcondition is reached at some point in time af-ter Dkc

t=h2 =3.5.Effects of various temperature values ΔT on the

corrosion current density are shown in Figure 8,where VPkc

0 =VPkc=VPks

=0.5, Λ−1 = Ec/Es = 0.03,s/h = 10, αc/αs = 2, and Dkc

=Dks=102. From Figure 8,

it is clear that as the temperature increases fromΔT = −17 K to ΔT = −7 K (tensile regime), the corrosioncurrent density decreases. However, this decrease in thecorrosion current density is observed only for a timeperiod of Dkc

t=h2 < 3.5. After this time period, the sat-uration condition of corrosive agents is reached andthe corrosion current density becomes independent ofthe change in temperature ΔT. For the increase intemperature from ΔT = 7 K to ΔT = 17 K (compressiveregime), the corrosion current density still decreasesbut the saturation condition of corrosive agents isreached at some point after time Dkc

t=h2 =3.5. Thissaturation behavior of corrosive agents in compressiveregime shows that the saturation time is higher whenthe coating is subjected to compressive residual stress.In conclusion, the residual stress plays a vital roleonly at the start of exposure; however, the effect ofresidual stress disappears when the saturation isreached.

Figure 9 shows the evolution of corrosion currentdensity corresponding to various values of moduliratio Ec/Es, and the residual stress gradient 1=ζ 0

r at two

1/ζr′ = –4.2 (ΔT = 7 K) → Compressive behavior of coating

1/ζr′ = 0.05 (ΔT = 0 K) → Negligible residual stress

1/ζr′ = 4.2 (ΔT = –7 K) → Tensile behavior of coating

Dkc t/h2 = 3.5

Λ–1 = Ec/Es = 0.03s/h = 10αc/αs = 2Dkc

/Dks= 102

1/ζd′ + 1/ζr′

1/ζd′ – 1/ζr′1/ζd′

0.11

0.10

0.09

0.08

0.07

0.060.0 0.5 1.0 1.5

Normalized Time Dkct/h2

No

rmal

ized

Co

rro

sio

n C

urr

ent

Den

sity

, i

2.0 2.5 3.0 3.5

FIGURE 7. Figure illustrating the evolvement of normalized corrosioncurrent density “i” incorporating the effects of diffusion-inducedgradient 1=ζ 0

d and residual stress gradient 1=ζ 0r of the coating. The

residual stress 1=ζ 0r in the coating could be compressive (when ΔT =

7 K [> 0]) and could be tensile (when ΔT = –7 K [< 0]).

Λ–1 = Ec/Es = 0.03s/h = 10αc/αs = 2Dkc

/Dks= 102

VPkc′ =

VPkc

/VPks= 0.5

ΔT = –7 K

7 K

–7 K

___ ___ ___

–12 K

–17 K Tensile behavior of coating

12 K 17 K

ΔT increasing

Compressive behavior of coating

ΔT = –17 K

ΔT = –12 K

ΔT = 7 K

ΔT = 12 KΔT = 17 K

No

rmal

ized

Co

rro

sio

n C

urr

ent

Den

sity

, i


0.0

0.100

0.098

0.096

0.094

0.092

0.090

0.088

0.086

0.084

0.082

0.0800.5 1.0 1.5 2.0 2.5 3.0 3.5

FIGURE 8. Figure showing the effects of various temperature valuesΔT on the corrosion current density “i.” It is clear that as thetemperature increases from ΔT =−17 K (tensile behavior of coating)to ΔT = 17 K (compressive behavior of coating), the corrosion currentdensity decreases.



different temperatures, i.e., ΔT = −7 K and ΔT = 7 K.The parameters are set as VPkc

0 =VPkc=VPks

=0.5,s/h = 10, αc/αs = 2, and Dkc

=Dks=102. It is clear in

Figure 9 that the corrosion current density graduallyincreases and then stabilizes with the increasing timefor various ratios of moduli. It is worth noting, that thecorrosion current density increases with the increasein the ratios of modulus, Ec/Es. This indicates that, fora given value of thickness h/s and concentrationof corrosive species, increasing the flexibility of sub-strate can increase the corrosion current density. Itis also clear that the corrosion current densityincreases or decreases with the magnitude of resid-ual stress gradient 1=ζ 0

r. The negative temperaturechange (ΔT = −7 K) accounts for the increase in totalgradient (1=ζ 0

d þ 1=ζ 0r) because of the same direction of

residual stress and the diffusion stress. This be-havior in turn increases in the corrosion currentdensity. However, the positive temperature change(ΔT = 7 K) accounts for the decrease in total gradient(1=ζ 0

d − 1=ζ 0r), which in turn decreases in the corro-

sion current density.For the case considering the total thickness to be

constant, the effects of thickness ratio s/h on cor-rosion current density are shown in Figure 10, whereVPkc

0 =VPkc=VPks

=0.5, Δ−1 = Ec/Es = 0.03, αc/αs = 2,and Dkc

=Dks=102. From Figure 10, it is observed that

when ΔT < 0 K under tensile condition, the corrosioncurrent density decreases for the thickness ratios/h < 4.5, and then becomes stable for s/h > 4.5. Thisclearly indicates that the tensile residual stress incoating assists the opening of coating microcracksonly if the thickness ratio s/h is less than 4.5. However,

when the thickness ratio s/h exceeds 4.5 (i.e., for thincoatings), microcracks opening is no longer influencedby tensile residual stress in the coating. On the otherhand, for the case of compressive residual stress whenΔT > 0 K, the thickness ratio s/h does not significantlyaffect the microcracks and therefore results in constantcorrosion current density for all thickness ratios s/h.It can be concluded that the thickness ratio plays asignificant role in controlling corrosion current den-sity only if the coating is under tensile residual effect,while if the coating is under compressive residualeffect, the thickness ratio s/h does not affectthe corrosion current density. It is also clear fromFigure 11 that for the condition ΔT = 7 K, when thecoating is under compression, the corrosion currentdensity is independent of time Dkc

t=h2 for any ratio ofthickness s/h.

Figure 12 illustrates the evolution of corrosioncurrent density involving the effects of CTE mis-match between coating and substrate. For a qualita-tive discussion two modulus ratios have been con-sidered (i) when Λ−1 = Ec/Es = 0.2 and (ii) when Λ−1 =Ec/Es = 0.03. Now considering the first case withΛ−1 =Ec/Es = 0.2, it is clear from Figure 12(a) that for thecondition, when ΔT = −7 K, the corrosion currentdensity increases with the decrease in the CTEmismatch between coating and substrate. For in-stance, considering ΔT = −7 K, for the case whenαc/αs = 2, the corrosion current density is higher thanthe corrosion current density for αc/αs = 10 andαc/αs = 50. This behavior is a result of the residualstress gradient 1=ζ 0

r, which decreases with theincrease in αc/αs, and thus, results in the decrease of

VPkc′ =

VPkc

/VPks= 0.5

___ ___ ___

0.102Ec/Es = 0.15 Ec/Es = 0.09 Ec/Es = 0.03

0.100

0.098

0.096

0.094

0.092

0.090

No

rmal

ized

Co

rro

sio

n C

urr

ent

Den

sity

, i


ΔT = –7 K1/ζ′r = 4.4

ΔT = –7 K1/ζ′r = 4.4ΔT = –7 K1/ζ′r = 4.4

ΔT = 7 K1/ζ′r = –3.8ΔT = 7 K1/ζ′r = –3.8ΔT = 7 K1/ζ′r = –3.8

s/h = 10αc/αs = 2Dkc

/Dks= 102

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

FIGURE 9. Figure showing the evolvement of corrosion currentdensity “i” corresponding to various values of moduli ratioEc/Es and the residual stress gradient 1=ζ 0

r at two different tempera-tures, i.e., ΔT =−7 K and ΔT = 7 K. The corrosion current densityincreases with the increase in the ratios of modulus, Ec/Es. Thisindicates that increasing the flexibility of substrate can increase thecorrosion current density.

VPkc′ =

VPkc

/VPks= 0.5

___ ___ ___

Λ–1 = Ec/Es = 0.03αc/αs = 2Dkc

/Dks= 102

No

rmal

ized

Co

rro

sio

n C

urr

ent

Den

sity

, i

0.18

Saturation pointΔT = 17 KΔT = 12 KΔT = 7 K

ΔT = 0 K

ΔT = –17 KΔT = –12 KΔT = –7 K

ΔT increasing

0.16

0.14

0.12

0.10

0.08

0.06

0.044.0 4.5 5.0 5.5

Relative Coating Thickness, s/h

6.0 6.5 7.0 7.5 8.0

FIGURE 10. Figure showing the effects of relative thickness s/h oncorrosion current “i” density. It is observed that when ΔT< 0 K undertensile condition, the corrosion current density decreases for thethickness ratio s/h< 4.5, and then becomes stable for s/h> 4.5. Forthe case of compressive residual stress when ΔT> 0 K, the thicknessratio s/h does not have a significant effect and results in constantcorrosion current density for all thickness ratios s/h.



total stress gradient 1=ζ 0d þ 1=ζ 0

r. This decrease in thetotal stress gradient results in the decrease of corro-sion current density. However, for ΔT = 7 K, the corrosioncurrent density is low for smaller CTE mismatch,i.e., the corrosion current density for αc/αs = 2 is lowerthan the corrosion current density for αc/αs = 10 andαc/αs = 50. This behavior is a result of the residual stressgradient 1=ζ 0

r, which increases with the increasein αc/αs, and thus, results in the increase of total stressgradient 1=ζ 0

d − 1=ζ 0r. This increase in the total stress

gradient results in the increase of corrosion currentdensity.

Now considering the second case with modulusratio Λ−1 = Ec/Es = 0.03, it is clear that when ΔT =−7 K, behavior similar to the previous case (whenΛ−1=Ec/Es = 0.2) was again observed in which thecorrosion current density was high for small CTEmismatch and was low for large CTE mismatch, asshown in Figure 12(b). For ΔT = 7 K, the behavior ofcorrosion current density was again similar to theprevious case.

However, the notable difference between both ofthese cases was that the values of corrosion currentdensity corresponding to all of the CTE mismatches(αc/αs equal to 50, 10, and 2) were found to be higher forlargermodulus ratio (e.g., Ec/Es = 0.2) in the first casecompared to the smaller modulus ratio (e.g., Ec/Es =0.03) in the second case. The reason behind thisbehavior is that for a given ΔT, thickness, and con-centration of diffusion species, higher flexibility ofsubstrate (smaller Es), as in case 1, can increase thedeflection of coating-substrate system. This behaviorcan result in the evolution of higher tensile residualstress in the coating, allowing the microcracks toopen up wide, resulting in higher corrosion currentdensity. On the other hand, lower flexibility of

substrate (higher Es), as in case 2, can decrease thedeflection of coating-substrate system. This behaviorcan result in the evolution of smaller tensile residualstress in the coating openingmicrocracks that are notwide enough, resulting in lower corrosion currentdensity.

Therefore, it can be concluded that for coating-substrate system, higher modulus ratio accounts forlarger corrosion current density compared to lowermodulus ratio. However, for a given modulus ratio,larger corrosion current density can be avoided ifhigher αc/αs is used for the applications with negativetemperature change (ΔT < 0) and lower αc/αs is usedfor the applications with positive temperature change(ΔT > 0).

(a)

(b)


αc/αs = 2 (ΔT = –7 K)

αc/αs = 2 (ΔT = –7 K)

αc/αs = 10 (ΔT = –7 K)

αc/αs = 10 (ΔT = –7 K)

αc/αs = 50 (ΔT = –7 K)

αc/αs = 50 (ΔT = –7 K)

αc/αs = 50 (ΔT = 7 K)

αc/αs = 50 (ΔT = 7 K)

αc/αs = 10 (ΔT = 7 K)

αc/αs = 10 (ΔT = 7 K)

Λ–1 = Ec/Es = 0.03s/h = 10Dkc

/Dks= 102

Λ–1 = Ec/Es = 0.2s/h = 10Dkc

/Dks= 102

VPkc′ =

VPkc

/VPks= 0.5

___ ___ ___

VPkc′ =

VPkc

/VPks= 0.5

___ ___ ___αc/αs = 2 (ΔT = 7 K)

αc/αs = 2 (ΔT = 7 K)

ΔT = –7 K1/ζ′r = 4.5

ΔT = –7 K1/ζ′r = 4.4

ΔT = –7 K1/ζ′r = 4.2

ΔT = –7 K1/ζ′r = 4.15

ΔT = 7 K1/ζ′r = –3.8ΔT = 7 K1/ζ′r = –4.15

ΔT = 7 K1/ζ′r = –4.2

ΔT = –7 K1/ζ′r = 1.6

ΔT = –7 K1/ζ′r = 0.16

ΔT = 7 K1/ζ′r = –3.8

ΔT = 7 K1/ζ′r = –1.6

ΔT = 7 K1/ζ′r = –0.16


No

rmal

ized

Co

rro

sio

n C

urr

ent

Den

sity

, iN

orm

aliz

ed C

orr

osi

on

Cu

rren

t D

ensi

ty, i

0.105

0.100

0.095

0.090

0.085

0.080

0.075

0.100

0.095

0.090

0.085

0.080

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.05 0.10 0.15 0.20 0.25 0.30 0.35

FIGURE 12. Figure showing the evolution of corrosion currentdensity “i” involving the effects of CTE mismatch between coatingand substrate for two cases (a) when Δ−1 = Ec/Es = 0.2, and (b) whenΔ−1 = Ec/Es = 0.03. For the case when Ec/Es = 0.2, higher modulusratio accounts for larger corrosion current density compared to lowermodulus ratio Ec/Es = 0.03. However, for a given modulus ratio, largercorrosion current density can be avoided if higher Δc/Δs is used forthe applications with negative temperature change (ΔT< 0) andlower Δc/Δs is used for the applications with positive temperaturechange (ΔT> 0).

No

rmal

ized

Co

rro

sio

n C

urr

ent

Den

sity

, i

Relative Thickness, s/h

0.11Dkct/h

2 increasing

Λ–1 = Ec/Es = 0.03αc/αs = 2Dkc

/Dks= 102

ΔT = 7 K

VPkc′ =

VPkc

/VPks= 0.5

___ ___ ___Dkct/h

2 = 0.5

Dkct/h2 = 1.0

Dkct/h2 = 2.0

Dkct/h2 = 3.0

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.034.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

FIGURE 11. Figure showing the corrosion current density “i” forvarious values of time Dkc t=h

2 by considering temperature ΔT = 7 K. Itis clear that when the coating is under compression, the corrosioncurrent density is independent of time Dkc t=h

2 for any ratio ofthickness s/h.



VALIDATION OF MODEL

Another goal of this work is to make a quanti-tative comparison between the experiments and thetheoretical model of the previous sections. It wasemphasized in the Experimental Procedures andResults and Discussion sections that the residualstress change in the coating under the temperaturevariation has a profound influence on the corrosionrate at the coating/substrate interface. Corrosion ratein mm/y is calculated by using:75

Corrosion rate=3.27 × 10−3

�i x EW

ρ

�(35)

where i is the corrosion current density calculatedfrom Equation (25), ρ is the density of steel in g/cm3,and EW is the equivalent weight in grams of steel.Figure 13 shows the experimental result plottedusing the mean of data values from three corrosionsensors for the initial two 20 h steps. Figure 13 alsoshows a typical simulation result at specific condi-tions for which an experiment has been performed.A comparative analysis between experimental andsimulation results showed that for the first stepwhen ΔT is equal to 0±1 K, the corrosion rate isnegligible for both experimental and simulationresults. For the second step during 10 h to 14 h,indicated as point A when ΔT starts to decrease, thesimulation result showed a slight increase in thecorrosion rate. However, during the experiment forthis short interval of time (10 h to 14 h), no change inthe corrosion rate was observed. This perhaps canbe attributed to the low sensitivity of corrosion sensorsto the incipient electrochemical activity. For thenext 14 h to 17 h, indicated as point B when thetemperature reaches its lowest point for the wholeexperiment, the simulation result showed a suddenrise-fall trend maintaining the maximum corrosionrate during this time. A similar behavior with high

corrosion rate was also observed for the experi-mental result, but the rise-fall behavior was not assharp as for the simulation result. From 17 h to19 h, represented as point C, the trends for both thesimulation and experimental results showed similarbehavior in a way that the corrosion rate was highfor low-temperature peaks and low for high-temperature peaks. And finally after 19 h at point D,the corrosion rate started to decrease with in-creasing temperature. The comparison between ex-perimental data and simulation results in Figure 13shows a satisfactory agreement. In the simulation, themodel overpredicts the corrosion rate for short-termexperiments (20 h), which makes some of the datapoints in the simulation graph deviate from theexperimental data points.

CONCLUSIONS

v The performance of coating-substrate system hasbeen investigated in the presence of residual stressunder the frame work of material science, solidmechanics, and electrochemistry. Based on novel self-consistent equations for corrosion current density,the effects of temperature, moduli ratio, thickness ra-tio, thermal mismatch ratio, and residual stressgradient of coating and substrate are addressed withthe help of the Newton Rhapson method. The resultsshow that for the case when coefficient of thermalexpansion (CTE) of the coating is greater than sub-strate, the negative change in temperature (ΔT < 0 K)accounts for the same direction of residual stress asof the diffusion stress. This behavior increases thedeflection of coating-substrate system and results intensile residual stress in the coating. The tensile be-havior opens the pre-existing microcracks in thecoating and, therefore, allows the diffusion of corrosiveagents before reaching the saturation state of cor-rosive agents in coating. The increase in the diffusionrate also increases the corrosion current density atthe interface of coating and substrate. After reachingthe saturation state at a certain time, the corrosioncurrent density becomes constant; therefore, residualstress plays a critical role in the corrosion currentdensity only before the time period when the saturationcondition is reached. However, the effect of residualstress on the corrosion current density gradually dis-appears after the saturation condition of corrosiveagents is reached.v The results also show that a larger modulus ratioEc/Es accounts for higher corrosion current density;therefore, to achieve the better performance ofcoating-substrate system, it is always better to use asmaller modulus ratio. For the case when ΔT < 0 K,the corrosion current density decreases first, andthen becomes stable with increasing ratio of thick-ness s/h. However, for the case when ΔT > 0 K,

B

D

CA

Temperature (K)

Simulation

Experimental

0.002

1st Step for 10 h

ΔT = 318±1 K – 318 K = 0±1 K

ΔT = 308±2 K – 323 K = –15±2 K

2nd Step for 10 h

0.0016

0.0012

0.0008

0.0004

00 2 4 6 8 10

Time (h)

Co

rro

sio

n R

ate

(mm

/y)

12 14 16 18 20–20

–18

–16

–14

–12

–10

–8

–6

–4

–2

0

2

FIGURE 13. Comparative analysis between experimental and simu-lation results to validate the model.



the corrosion current density is constant for allthickness ratios s/h. Therefore, increasing thethickness of coating is not necessarily better forachieving higher performance of coating-substratesystem in terms of corrosion. For the conditionwhere ΔT < 0 K, the corrosion current density increaseswith the decrease in the CTE mismatch betweencoating and substrate. However, for the conditionwhere ΔT > 0 K, the corrosion current densitydecreases with the decrease in the CTE mismatch.Therefore, for higher performance in terms of cor-rosion, it is better to use higher αc/αs for the appli-cations with negative temperature change (ΔT < 0)and lower αc/αs for the applications with positivetemperature change (ΔT > 0).v A model experiment was performed in conjunctionwith the analysis. The experiment showed good,quantitative agreement with the trends predicted bythe theory. Furthermore, the experiment emphasizedthe important role that the residual stress resultingfrom temperature change plays in the evolution ofcorrosion current density at the coating/substrateinterface.

ACKNOWLEDGMENTS

This research is joint funded by Defence Scienceand Technology Laboratory (DSTL), Ministry ofDefence (MoD), and Bournemouth University U.K.The authors acknowledge their support andcontributions.

REFERENCES

1. A. Stierle, Science 321, 5887 (2008): p. 349-350.2. D.A. Jones, Principles and Prevention of Corrosion (New York, NY:

Pearson Education, 2014).3. B. Wessling, J. Posdorfer, Electrochim. Acta 44, 12 (1999):

p. 2139-2147.4. F.L. LaQue, Marine Corrosion: Causes and Prevention (New York,

NY: Wiley-Blackwell, 1975).5. C.G. Munger, L.D. Vincent, Corrosion Prevention by Protective

Coatings (Houston, TX: NACE International, 1999).6. W.-K. Lu, R.L. Elsenbaumer, B. Wessling, Synthetic Metals 71,

1 (1995): p. 2163-2166.7. X. Zhao, P. Munroe, D. Habibi, Z. Xie, J. Asian Ceramic Societies 1,

1 (2013): p. 86-94.8. D.P. Riemer, M.E. Orazem, “Modeling Coating Flaws with Non-

Linear Polarization Curves for Long Pipelines,” in Corrosion andCathodic Protection Modeling and Simulation, ed. R.A. Adey,Advances in Boundary Elements, vol. 12 (Southampton, UnitedKingdom: WIT Press, 2006): p. 225-259.

9. K. Holmberg, A. Mathews, Thin Solid Films 253, 1 (1994):p. 173-178.

10. M. Kaufman, J. Fink, Acta Metall. 36, 8 (1988): p. 2213-2228.11. T. Magnin, R. Chieragatti, R. Oltra, Acta Metall. Mater. 38, 7 (1990):

p. 1313-1319.12. K. Sieradzki, R. Newman, J. Phys. Chem. Solids 48, 11 (1987):

p. 1101-1113.13. W.-Y. Chu, J. Yao, C.-M. Hsiao, Corrosion 40, 6 (1984):

p. 302-306.14. X. Liu, G. Frankel, Corros. Sci. 48, 10 (2006): p. 3309-3329.15. D.S. Campbell, “Mechanical Properties of Thin Films,” inHandbook

of Thin Film Technology, eds. L.I. Maissel, R. Glang, chpt. 12(New York, NY: McGraw-Hill, 1970).

16. X. Feng, Y. Huang, A. Rosakis, J. Appl. Mech. 74, 6 (2007):p. 1276-1281.

17. S. Huang, X. Zhang, Sensors and Actuators A 134, 1 (2007):p. 177-185.

18. H. Soliman, A. Waheed, J. Mater. Sci. Technol. 15, 5 (1999):p. 457-462.

19. T. Clyne, S. Gill, J. Thermal Spray Technol. 5, 4 (1996):p. 401-418.

20. S. Widjaja, A.M. Limarga, T.H. Yip, Thin Solid Films 434, 1 (2003):p. 216-227.

21. X. Zhang, B. Xu, H. Wang, Y. Wu, Thin Solid Films 488, 1 (2005):p. 274-282.

22. M. Wisnom, M. Gigliotti, N. Ersoy, M. Campbell, K. Potter,Composites Part A: Appl. Sci. Manufacturing 37, 4 (2006):p. 522-529.

23. P. Bansal, P. Shipway, S. Leen, Acta Mater. 55, 15 (2007):p. 5089-5101.

24. M.S. Ahmed, P. Munroe, Z.-T. Jiang, X. Zhao, W. Rickard,Z.-F. Zhou, L.K.Y. Li, Z. Xie, Corros. Sci. 53, 11 (2011):p. 3678-3687.

25. E. Busso, J. Lin, S. Sakurai, M. Nakayama, Acta Mater. 49, 9(2001): p. 1515-1528.

26. I. Spitsberg, D. Mumm, A. Evans, Mater. Sci. Eng. A 394, 1 (2005):p. 176-191.

27. Y.S. Podstrigach, P. Shevchuk, Soviet Materials Science: atranslation of Fiziko-Khimicheskaya Mekhanika Materialov/Academy of Sciences of the Ukrainian SSR 3, 5 (1968):p. 420-426.

28. Y. Ida, J. Geophysical Research 74, 12 (1969): p. 3208-3218.29. F. Larché, J. Cahn, Acta Metall. 21, 8 (1973): p. 1051-1063.30. W.B. Kamb, J. Geophysical Research 66, 1 (1961): p. 259-271.31. F. Larcht’e, J. Cahn, Acta Metall. 30, 10 (1982): p. 1835-1845.32. G.B. Stephenson, Acta Metall. 36, 10 (1988): p. 2663-2683.33. T. Nguyen, J. Hubbard, J. Pommersheim, J. Coatings Technol. 68,

855 (1996): p. 45-56.34. Y. Prawoto, N. Kamsah, M.M. Yajid, Z. Ahmad, Theoretical Appl.

Fracture Mech. 56, 2 (2011): p. 89-94.35. F. Yang, J. Li, J. Appl. Phys. 93, 11 (2003): p. 9304-9309.36. A.I. Rusanov, Surf. Sci. Reports 58, 5 (2005): p. 111-239.37. A. Saeed, Z. Khan, M. Clark, M. Nel, R. Smith, Insight-Non-De-

structive Testing and Condition Monitoring 53, 7 (2011): p. 382-386.38. A. Saeed, Z.A. Khan, M. Hadfield, S. Davies, Tribology Trans. 56, 4

(2013): p. 637-644.39. A. Saeed, Z.A. Khan, E. Montgomery, Mater. Perform. Characteri-

zation 2, 1 (2013): p. 1-16.40. Z.A. Khan, P. Pashaei, R.S. Bajwa, M.H. Nazir, M. Camak, Int. J.

Computational Methods & Experimental Measurements 3, 2 (2015):p. 165-174.

41. M.H. Nazir, Z. Khan, K. Stokes, J. Adhesion Sci. Technol. 29, 5(2014): p. 392-423.

42. M.H. Nazir, Z.A. Khan, K. Stokes, J. Adhesion Sci. Technol. 29, 14(2015): p. 1415-1445.



45. M.H. Nazir, Z.A. Khan, Int. J. Computational Methods & Experi-mental Measurements 3, 4 (2015): p. 316-328.

46. Z.A. Khan, M. Grover, M.H. Nazir, “The Implications of Wet and DryTurning on the Surface Quality of EN8 Steel,” in Transactionson Engineering Technologies, eds. G.-C. Yang, S.-L. Ao, L. Gelman(New York, NY: Springer, 2015), p. 413-423.

47. M.H. Nazir, Z.A. Khan, A. Saeed, K. Stokes, Eng. Failure Anal. 63(2016): p. 43-60.

48. A. Saeed, Z.A. Khan, M.H. Nazir, Sustainability 7, 12 (2015):p. 16451-16464.

49. M.H. Nazir, Z.A. Khan, A. Saeed, K. Stokes, “A Model for CathodicBlister Growth in Coating Degradation Using MesomechanicsApproach,” Werkst. Korros. in press (2016): doi: http://dx.doi.org/10.1002/maco.201508562.

50. D.R. Lide, CRC Handbook of Chemistry and Physics (Boca Raton,FL: CRC Press, 2004).

51. D.W. Brown, R.J. Connolly, D.R. Darr, B. Laskowski, “LinearPolarization Resistance Sensor Using the Structure as a WorkingElectrode,” European Conference of the Prognostics and HealthManagement Society (Rochester, NY: PHM Society, 2014):p. 1-7.

52. D. Brown, D. Darr, J. Morse, B. Laskowski, “Theoretical andExperimental Evaluation of a Real-Time Corrosion Monitoring



http://dx.doi.org/10.1002/maco.201508562






System forMeasuring Pitting in Aircraft Structures,” First EuropeanConference of the Prognostics and Health Management Society(Rochester, NY: PHM Society, 2012): p. 1-9.

53. V. Feliu, J. González, C. Adrade, S. Feliu, Corros. Sci. 40, 6 (1998):p. 995-1006.

54. ASTM E831-1451, “Standard Test Method for Linear ThermalExpansion of Solid Materials by Thermomechanical Analysis”(West Conshohocken, PA: ASTM, 2014).

55. ASTM E2769-13, “Standard Test Method for Elastic Modulus byThermomechanical Analysis Using Three-Point Bending andControlled Rate of Loading” (West Conshohocken, PA: ASTM,2013).

56. W. Wilson, I.-W. Yu, Int. J. Fracture 15, 4 (1979): p. 377-387.57. Z. Tang, F. Wang, W. Wu, Oxidation of Met. 48, 5-6 (1997):

p. 511-525.58. C.-H. Hsueh, J. Appl. Phys. 91, 12 (2002): p. 9652-9656.59. M. Stratmann, A. Leng, W. Fürbeth, H. Streckel, H. Gehmecker,

K.-H. Groβe-Brinkhaus, Progress in Organic Coatings 27, 1 (1996):p. 261-267.

60. A. Leng, H. Streckel, M. Stratmann, Corros. Sci. 41, 3 (1998):p. 579-597.

61. M. Stratmann, R. Feser, A. Leng, Electrochim. Acta 39, 8 (1994):p. 1207-1214.

62. K.J. Laidler, J. Chemical Education 61, 6 (1984): p. 494.63. M.-W. Huang, C. Allely, K. Ogle, M.E. Orazem, J. Electrochem. Soc.

155, 5 (2008): p. C279-C292.64. O.D. Kellogg, Foundations of Potential Theory (New York, NY:

Springer, 1929).65. L. Mejlbro, “The Complete Solution of Fick’s Second Law of Dif-

fusion with Time-Dependent Diffusion Coefficient and SurfaceConcentration,” Proc. of the Durability of Concrete in Saline Envi-ronment, ed. P. Sandberg (Lund, Sweden: Cementa AB, 1996),p. 127-158.

66. G. Cowper, J. Appl. Mech. 33, 2 (1966): p. 335-340.67. F.-Z. Xuan, L.-Q. Cao, Z. Wang, S.-T. Tu, Computational Mater. Sci.

49, 1 (2010): p. 104-111.68. W. David, M. Thackeray, L. De Picciotto, J. Goodenough, J. Solid

State Chem. 67, 2 (1987): p. 316-323.69. S. Prussin, J. Appl. Phys. 32, 10 (1961): p. 1876-1881.70. X. Zhang, W. Shyy, A.M. Sastry, J. Electrochem. Soc. 154, 10

(2007): p. A910-A916.71. K.N. Allahar, M.E. Orazem, K. Ogle, Corros. Sci. 49, 9 (2007):

p. 3638-3658.72. K.N. Allahar, “Mathematical Modeling of Disbonded Coating and

Cathodic Delamination Systems” (Ph.D. diss., University of Florida,2003).

73. J. Newman, Electrochemical Engineering (Englewood Cliffs, NJ:Prentice-Hall, 1991).

74. D. Sheen, Introduction to Numerical Analysis (New York, NY:Springer, 1980).

75. ASTM G59-97, “Standard Test Method for Conducting Potentio-dynamic Polarization Resistance Measurements” (West Consho-hocken, PA: ASTM International, 2009).

NOMENCLATURE

Unless otherwise specified, the following nomenclatureis used in this paper.

Notation DescriptionΔT Temperature difference/changeαc and αs Coefficient of thermal expansion (CTE)

of coating and substrate, respectivelyEc and Es Young’s modulus of coating and

substrate, respectivelyh Thickness of coatings Thickness of substrateσrc Residual stressRp Polarization resistance

B Stern-Geary constanti = B/Rp Corrosion current densityckB

Bulk ionic concentrationΦ Solution potentialzk Charge number of corrosive speciesx Distance to the defecttD Diffusion timeDS,STAion

Diffusion coefficient of the corrosivespecies along the interface

RH Pore relative humiditytexp Time of exposureDk Average of standard diffusion coefficient

of all of the corrosive speciesRHSTA Standard relative humidityTSTA Standard temperaturetSTA Time of exposure at which Dk is

measured (normally 1 month)Ga Activation energy of corrosive speciesR Universal gas constantta Actual time of exposurenag Age reduction factorckTH

Threshold concentration of speciesη The mobility constant of cationst Time passed after the defect confronts

the solutiontn The time required to activate the defectcko

The equilibrium concentration of ionsat the interface in contact with the bulksolution ckB

kckeq= cko

=ckBEquilibrium constant

μok Chemical potential in the given

standard stateσm Stress tensorσi =σdi

þ σri Principle stresses (sum of residual stressand diffusion induces stress)

VPkPartial molar volume of diffusingspecies k

VmkMolar volume of solution

P Pressurenk Molar concentration of diffusing speciesJk Diffusion flux of species kE Strain distributionEur

Uniform component of residual strainEb Bending component of strainζr Radius of curvature resulting from

residual stressEud

Uniform component of diffusion-inducedstrain

ζd Radius of curvature resulting fromdiffusion-induced stress

EoO2

Equilibrium potential of oxygenE Substrate potentialβO2

Oxygen Tafel slope1ζdd

and 1ζdnd

Bending curvatures resultingfrom diffusion-induced stress fordelaminated and nondelaminatedcoating, respectively



1ζrd

and 1ζrnd

Bending curvatures resulting fromresidual stress for delaminated andnondelaminated coating, respectively

cO2Dissolved concentration of oxygen atcoating surface

DO2Diffusion coefficients for oxygen

n The number of electrons transferred

icoatO2Current density resulting from theoxygen reduction at any positionalong the coating/substrate interface

is Current density resulting from substratedissolution

i= icoatO2þ is Corrosion current density along the

interface of coating and substrate



modeling the effect of residual and diffusion-induced ... · modeling the effect of residual and...

Documents