Reference

NATL INST. OF STAND & TECH

NBS

Publications

A111QS 17Mmi

NBS TECHNICAL NOTE 1150

/ W \

a

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

-UC

1UU

1150

i^6Z

Mathematical

Models for the

Corrosion Protective

Performance of

Organic Coatings
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

Absolute Physical Quantities2 — Radiation Research — Chemical Physics —

Analytical Chemistry — Materials Science

THE NATIONAL ENGINEERING LABORATORY provides technology and technical ser- vices to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement capabilities;

provides engineering measurement traceability services; develops test methods and proposes

engineering standards and code changes; develops and proposes new engineering practices;

and develops and improves mechanisms to transfer results of its research to the ultimate user. The Laboratory consists of the following centers:

Applied Mathematics — Electronics and Electrical Engineering2 — Manufacturing

Engineering — Building Technology — Fire Research — Chemical Engineering2

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

guidelines, and managing Federal participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

provides the technical foundation for computer-related policies of the Federal Government.

The Institute consists of the following centers:

Programming Science and Technology — Computer Systems Engineering.

' Headquarters and Laboratories at Gaithersburg, M D, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
Mathematical Models for the Corrosion

Protective Performance of Organic

Coatings

James M. Pommersheim

Department of Chemical Engineering

Bucknell University

Lewisburg, PA 17837

and

Paul G. Campbell and Mary E. McKnight

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

Waucnai Surest* et Standard* SEP 3 1982

7£~0

40* °F %

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued September 1982
•

! i

National Bureau of Standards Technical Note 1150

Natl. Bur. Stand. (U.S.), Tech. Note 1150, 99 pages (Sept. 1982)

CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1982

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $5.50

(Add 25 percent for other than U.S. mailing)
TABLE OF CONTENTS

Page

LIST OF TABLES v

LIST OF FIGURES vi

ABSTRACT vii

1. INTRODUCTION 1

2. CONCEPTUAL MODELS 3

2.1 Water and Oxygen Permeability *

2.2 Blister Formation and Growth ^

2.3 Corrosion Phenomena -*

3. MATHEMATICAL MODELS 9

3.1 Water and Oxygen Permeability

3.1.1 Volume Balance 10

3.1.2 Rate Laws for Sorption and Diffusion l x 3.1.2.1 Rate Laws for Sorption (Solubilities) H

3.1.2.2 Rate Laws for Diffusion **

3.1.3 Conservation Balances

^

3.1.4 Model Solutions 16

3.2 Blister Formation and Growth. 22

3.2.1 Blister Growth - Concentrated Spots of Impurity 28

3.2.2 Blister Growth - Uniform Distribution of Impurity. ... ^1

3.3 Corrosion Phenomena 33

3.3.1 Potential Difference Model 35

3.3.2 Equivalent Circuit Model 37

4. DISCUSSION OF RESULTS 46

4.1 Water and Oxygen Permeability ^°

4.2 Blister Formation and Growth 51

4.3 Corrosion Phenomena 56

5. SUMMARY AND CONCLUSIONS 60

in
Page

6. NOMENCLATURE 65

7. REFERENCES 69

APPENDICES

A. SOLUTION TO THE MATHEMATICAL MODEL FOR DIFFUSION OF OXYGEN

AND WATER THROUGH ORGANIC COATINGS 70

A. 1 Long Time Model Solutions 72

A. 2 Short Time Model Solutions 77

B. SOLUTION TO THE MATHEMATICAL MODEL FOR CORROSION PHENOMENA

BENEATH AN ORGANIC COATING 81

B.l Potential Difference Model 81

B.2 Equivalent Circuit Models 84

B.2.1 Constant Product Resistance 85

B.2. 2 Langmuir Adsorption of Corrosion Products 86

B.2. 3 Accumulation of Solid Products 89

xv
LIST OF TABLES

Table No. Page

1 Concentration of Water and Oxygen in Organic Coatings

(Summary of Models) 19

2 Assumptions of the Mathematical Models for Blister Growth ... 26

v
LIST OF FIGURES

Figure No. Page

1 Schematic Representation (Cross-Section) of

Organic Polymer Coating 2

2 Schematic Representation of a Blister Within an Organic Coating 24

2A Concentrated Spot of Impurity 24

2B Uniform Distribution of Impurity 24

3 Equivalent Electrical Circuit for Polarization 38

VI
ABSTRACT

Mathematical models were developed for conceptual models describing

the principal phenomena that occur in the corrosion performance of polymeric coatings. These include models for water and oxygen permeability

through organic coatings, models for the growth of blisters beneath

coatings, and preliminary models for the polarization occurring at the

electrode surfaces. Results predicted by the models are discussed in

terms of the improvement of the protective function of the membrane.

Keywords: absorption; adhesion; adsorption; conceptual models; corrosion;

mathematical models; organic coating; osmosis; osmotic pressure; oxygen;

permeability; pigment; protective performance; substrate; vehicle; water.

VI

1 . INTRODUCTION

In order to assess the corrosion protective performance of organic coatings the phenomena that lead to corrosion must be understood. Corrosion

itself is the destruction of a metal surface by electrochemical action.

Aqueous corrosion occurs when three things are present: oxygen, water and an

electrolyte. There will be no corrosion unless all three are present at the

corrosion site.

Figure 1 shows a cross-section of a typical organic coating. A coating

having a uniform thickness L (typically 0.1 mm) covers a metal substrate. The

organic coating consists of a vehicle (or binder) phase and a particle (or

pigment) phase. Water and oxygen can penetrate through the coating to the

surface of the substrate. The rate of passage of the penetrants (water and

oxygen) will depend on many factors: the availability at the coating surface,

their solubility in the coating, their diffusivity through the coating, their

affinity for the pigment particles their rate of consumption at the corrosion

site and the osmotic pressure of the electrolyte-membrane combination. In

addition to depending on the chemical and physical nature of the organic

coating used, all of these factors are temperature sensitive.

The electrolyte can be a foreign ion such as chloride or sulfate or it

can be the corrosion products themselves. Since corrosion does not begin

without an electrolyte, a foreign ion must be initially present as an impurity

on the metal surface, or as a leachable ion present in the coating, or as an

ion present in an external aqueous environment which can diffuse through the

coating to the metal surface. In most applications, the surface of the metal

is not perfectly clean and an accumulation of impurity ions may initiate
4 Ii

i-i s • ** H \ \* 3 A

0>

i— /38 U

-^ oo

•H

+J

&i cniw1§

•H

-Piw£U

•H

4Juw

0) H

fa
localized corrosion there. The presence of these ions leads to osmotic

pressure differences between the inside and outside of the coating. This

augments the diffusion of water through the coating and can lead to the formation of blisters at the metal surface.

In this paper, the key conceptual phenomena affecting the corrosion protective performance of organic coatings are discussed, and mathematical models

describing these phenomena are formulated and solved.

2. CONCEPTUAL MODELS

The model for corrosion protective performance of organic coatings was

conceived in terms of three different conceptual sub-models:

(i) submodel for water and oxygen permeability

(ii) submodel for blister formation and growth

(iii) submodel for corrosion phenomena

2.1 Water and Oxygen Permeability

Both water and an oxidizing chemical species must be present at the metal

surface in order for corrosion to occur. Compounds like halogens, sulfur

dioxide and oxygen can all serve as oxidizers but molecular oxygen is by far

the most prevalent. The organic coating (refer to Figure 1) serves as a

barrier to the diffusion of water and oxygen. If the coating were not present

they would have ready access to the metal surface. The presence of the coating

helps prevent the initiation of corrosion, and, in addition, slows down the

rate of corrosion once it has begun. Most coatings, are relatively permeable

to the penetration of water and oxygen, especially water [1] . To arrive at

the metal surface both water and oxygen must first dissolve (or absorb) into

the outside of the coating and then diffuse through the coating towards the
metallic corrosion site. Some of the water and oxygen does not arrive at the

metal surface since they are absorbed within the coating. In addition some

will not arrive because it will be adsorbed by the pigment particles within

the coating.

For a given type of organic coating, metal substrate and corrosive

environment, the overall permeation of the organic coating to oxygen and water

depends on the concentrations of oxygen and water outside the coating, the

thickness of the coating, the solubility of the compounds into the film, the

diffusivity through the film, the pigment to vehicle ratio, the pigment size

and shape, and the temperature. With more pigment particles, it is more

difficulty for species to diffuse. At the same time, more retention of penetrants occurs because of the greater particle density. Thus, raising the

pigment to vehicle ratio will lower the permeation by lowering the diffusivity

and increasing the amount of adsorption within the coating. Imperfect wetting

or adhesion of the binder to the pigment particles during formulation of the

coating can also give rise to pockets surrounding the particles into which

adsorbed water and oxygen can go. Raising the temperature will raise the

diffusivity, lower the solubility, and raise the adsorption. For most coatings

the overall permeability of the film increases with increasing temperature

[2] . This indicates that the diffusion coefficient or diffusivity of the

oxygen and water through the film are temperature sensitive parameters.

2.2 Blister Formation and Growth

When water has first penetrated through to the metal surface it mixes

with impurity ions present at the surface and small liquid cells of locally

high concentration are formed. The large anions there, typically CI and
SO. , cannot diffuse out through the coating, which now acts as a semipermeable membrane. Alkali pigments can supply additional species to the anion

pool within the blister and may change the nature of the corrosion process.

Osmotic pressure builds up and more water diffuses into the cells, gradually

enlarging them. In regions where the adhesion between the coating and the

metal is poor, or where the concentration of impurity is especially high, the

coating can disbond from the metal substrate and a blister can form. The

blister continues to enlarge but at a slower and slower rate as the impurity

ion is diluted and the osmotic pressure drops. If the enlarging blister

exposes additional amounts of surface impurity ions, the rate of blister

growth does not decrease as fast. A similar effect will occur when ionic

corrosion products (such as hydroxyl ions) are formed adjacent to the blister,

In general, blister growth is slow enough that it is not limited by the

availability of water. The blister formation and initial water incursion

problems are uncoupled since they do not occur on the same time scale. Of

course, water permeability (refer to previous section) is essential to the

initiation of the cells which lead to the formation of blisters.

After sufficient time, the blisters may enlarge to such a point that

adjacent blisters link together. This can lead to complete detachment of the

coating from the substrate causing failure of the organic coating.

2.3 Corrosion Phenomena

Corrosion reactions initiate when oxygen, water and ions are present

locally at the metal surface either in a detached film or within a blister.
For iron-based metals, the most common half-cell reactions are:

I

I _ Anodic Reaction : Fe -* Fe + 2e (1)

(oxidation)

Cathodic Reaction : HO + tO + 2e~ -> 2 (0H)~ (2)

(reduction)

The anodic and cathodic reactions occur at different sites on the metal

surface. The electrons released in the anodic reactions travel through the

metal to the location of the cathode. This completes the circuit. Electron

flow is retarded (or equivalently, circuit resistance is increased) by several

mechanisms. These include:

1) Build-up of oxide layers or layers of other solid corrosion products

such as Fe(OH)

3 or Fe

2) Formation of an electrical double layer adjacent to the metal

surface .

3) Formation of gases such as hydrogen at the cathode surface or oxygen

at the anode surface.

4) Adsorption of hydrogen or oxygen ions which result in an overvoltage

because of polarization of the anode or cathode.

The anodic and cathodic sites may shift over the surface with time as the

coating becomes detached. This can permit substantially uniform corrosion to

occur. The liquid films of corrosion products formed beneath detached coatings

may be basic [3]

With a blistered surface, the anodic reaction is more likely to occur in

the center of a blister where a pit may subsequently develop. The cathodic

reaction is more likely to occur at the periphery of the blister where the

oxygen needed for this reaction is more readily accessible to depolarize the

cathode. The locally high basicity generated by the formation of hydroxyl

ions can weaken the coating near the cathode surface. Accumulation of solid
corrosion products at the cathode surface also occurs. Both factors can cause

"cathodic delamination" with more rapid coating detachment than would otherwise be observed.

For the half-cell reactions considered, the principal solid corrosion

product is ferric hydroxide, Fe(OH) . The original corrosion product is

ferrous hydroxide, Fe(OH)„, which forms when ferrous, Fe , and hydroxyl,

(OH) , ions diffuse towards one another through solution. Ferrous hydroxide

undergoes further oxidation to ferric hydroxide. In strongly oxidative

environments, films of ferric oxide (Fe ) also form. The corrosion products

can build-up uniformly as may occur under a detached film, or they may precipitate closer to the cathode than to the anode as can occur with a blistered

coating.

Corrosion is viewed as being basically an electrochemical phenomenon.

The rate of corrosion is considered to be a function of the electromotive

force and the resistance of the circuit. The greater the potential between

the anode and cathode the greater is the driving force for corrosion. The

driving force is initially proportional to the potential as determined by the

galvanic series for the metal in solution. As corrosion proceeds, the potential

difference between the anode and cathode decreases and the accumulating

corrosion products cut down on the electron flow. The potential of the cathode

slowly drifts towards that of the anode (cathodic polarization) , while the

potential of the anode drifts slowly towards that of the cathode (anodic

polarization). In general, cathodic polarization exerts more of a controlling

influence on the overall corrosion process than does anodic polarization. For

example, when hydrogen adsorbs at the cathode it can polarize the surface

reducing the electron flow and thus decreasing corrosion at the anode.
Corrosion can be accelerated again if oxygen is introduced into the system.

In this case, adsorbed hydrogen ions are removed from the cathode surface by

the reaction:

4 H+ +

2 + 4e +2 H

2 (3)

Corrosion increases because the cathode potential drops, increasing the

driving force between the cathode and anode and raising the electron flow.

Oxygen in this case acts as a cathodic depolarizer.

The area of the cathode surface relative to that of the anode is also an

important parameter in corrosion models. Returning to the previous example,

the hydrogen ions reaching a large cathode will be spread out over a larger

area than with a small cathode and are easier to remove by reaction (3)

.

Thus, in oxidative environments, such as those present beneath a corrosion

protective coating, an enlarging cathode contributes to accelerated corrosion

rates.

The corrosion phenomena can also be viewed as being part of an electrical

circuit. The difference in potential between the anode and cathode create an

electron flow which passes through several series resistances. These include

the resistance of the electrolyte, the resistance of the corrosion products

and the resistance of the metal itself. For isothermal systems, the metal

resistance remains constant and should be low. Corrosion product resistance

increases with time as the products accumulate (either as an electrical double

layer, adsorbed gases, or solid products).

The electrical double layer that forms at the cathode consists of an

inner layer of oriented water dipoles next to the electrode surface and an
outer layer of solvated ions in solution (outer Helmholtz plane) . These

layers provide additional resistance by slowing down the electron flow. This

cathodic polarization decreases the circuit potential. Eventually the resistance provided by the double layer approaches some limit or ultimate value and

the potential differences between the anode and the cathode also become constant

Solid films of corrosion products can cause similar effects.

The resistance provided by the conducting solution can also slowly change

as its composition changes. As ion concentrations increase, the higher ionic

mobilities which result lower the electrolyte resistance. However, it can

still be an appreciable part of the total circuit resistance, especially in

solutions which remain relatively dilute.

3. MATHEMATICAL MODELS

The overall mathematical model for the corrosion protective performance

of organic coatings was formulated based upon the previously developed conceptual model describing the principal phenomena that occur. Mathematical

sub-models were developed for each of the conceptual models discussed in the

previous section. These included mathematical sub-models for water and oxygen

permeability, for blister formation and growth, and for the corrosion phenomenon itself.

3.1 Water and Oxygen Permeability

Since water and oxygen must be present at the metal surface in order for

corrosion to occur, it is important to know the rate at which they pass

through the coating. Mathematical models for this process are based on a
knowledge of the rate laws for absorption and adsorption and diffusion and the

mass conservation balances for water and oxygen. Input to the conservation

balances consists of the rate laws for absorption into the surface of the

coating, for diffusion through the coating and for adsorption on the surface

of the pigment particles. The rate laws and conservation equations must both

account for the effect of the pigment particles on the permeation process.

3.1.1 Volume Balance

The relations between pigment to vehicle ratio, pigment particle size,

and volume fraction vehicle (porosity) can be determined by making an overall

volume balance on the coating.

Consider the system depicted in figure 1. Pigment particles of uniform

size are evenly dispersed throughout the coating. The coating thickness is L.

Z is the coating depth measured normal to the free surface. n, the number

density of particles, or pigment particles per unit volume of coating, is calculable from the relation:

6 1 " £ ST\ n = n -j- (1)

where (1 - e) is the volume fraction of the pigment particles and d is their

diameter (equivalent sphere) . If the porosity e (volume fraction vehicle) is

unknown it can be found by making a volume balance on the coating:

£

1 + (p/v) (Pv

/P

p

) (2)

where (p/v) is the pigment to vehicle weight ratio and p /p is the density

ratio.

10
3.1.2 Rate Laws for Sorption and Diffusion

3.1.2.1 Rate Laws for Sorption (Solubilities)

Absorption into the coating surface was assumed to follow Henry's law

which states that the solubility into the coating is proportional to the

concentration (or partial pressure) of penetrant in the external phase outside

the coating. This presumes that there is an equilibrium between phases at the

interface. For coatings exposed to vapors, the constant of proportionality is

the Henry's law constant H.. Thus:

Pio H

i

Clo (3)

where

p . is the partial pressure of component i in the external phase

C. is the concentration of penetrant i just inside the coating

surface (Z = 0)

For coatings exposed to liquids, a similar equation applies. Aqueous

systems favor the penetration of water through the coating while vapor systems

favor the penetration of oxygen. At higher humidities in vapor systems, the

penetration of water increases because of the higher partial pressure of

water. Oxygen transfer to organic coatings in aqueous systems increases as

the amount of oxygen dissolved in the liquid increases. In any case, as a

result of exposure to the external environment, it is presumed that a constant

concentration of penetrant C. is present just inside the coating. It is

11
important to re-emphasize that if either oxygen or water are absent from the

external phase, only a negligible amount of corrosion occurs since C. =0.

10

Within the coating, pigment particles also have an affinity for water and

oxygen. For a single pigment particle, the water and oxygen present in the

vicinity of its surface adsorb on the surface at a rate proportional to the

product of the particle's surface area (S ) and the concentration of penetrant

(C.) present in the vehicle near the pigment particle. Thus:

r. = k . S C. (4) is ax p 1

where

and

r. is the rate of adsorption of penetrant i per particle

is

k . is the adsorption rate constant for penetrant i. ai

3.1.2.2 Rate Laws for Diffusion

The diffusion of oxygen and water through the coating was based on Fick's

Law. This states that the flux of chemical species i, (N.) is proportional to

the concentration gradient. The constant of proportionality in Fick's Law is

the diffusion coefficient or diffusivity (D.). Thus:

9C

N. = - D. VC. = - D. ^r1 (5)

l l l l dZ

In equation (5) the gradient VC . is expressed as 9C./SZ. This assumes that

diffusion only occurs in the Z direction. This is a reasonable assumption

considering that most coatings are thin and have large flat surfaces which are

12
exposed to uniform environments. Thus there is little flux across the film in

the directions parallel to the surfaces.

The diffusion coefficient D. will in general be much less than the

molecular or free stream diffusivity of molecules moving unimpeded through

solution. D. is thus regarded as an effective diffusivity rather than a true

diffusion coefficient. The diffusion of molecules is slowed because:

1) The effective area of diffusion is lowered due to the presence of

the pigment phase.

2) The diffusion path between and within the vehicle are tortuous so that it takes longer for a given molecule to pass from one point to

another.

3) Any given diffusion path has constricted regions (bottlenecks) where

there is a greater resistance to the diffusive flow of molecules.

The effects of these constraining factors are mathematically expressed in

the relationship [4]

D. = D* e 2 (6)

1 1 T

where

D. is the molecular (or Knudsen) diffusivity

l

G is the constriction factor

T is the tortuosity factor

The porosity e and constriction factor a are both less than unity while

the tortuosity factor T is greater than unity. All three factors act to lower

the actual diffusion coefficient D.. In general, equation (6) cannot be used

to predict the value of D. since the factors on the right hand side of this

equation are unknowns which can only be determined from experiments.

13
Bruggetnan [5] has shown theoretically, using an electrical analog model,

that g/t varies approximately as /e. According to equation (6), this predicts

3/2 that the diffusion coefficient should vary as £ . However the useful range

of Bruggeman's approximation is only from 0.4 < £ < 1. At lower porosities,

3/2

the dependence of D. on £ is even more pronounced than £ , although actual

dependence has not been determined theoretically. Greater dependence of the

effective diffusivity on the porosity can occur with clumping of pigment

particles caused by poor dispersion or nesting of smaller particles within

spaces between larger particles.

3.1.3 Conservation Balances

The overall conservation equations governing the rate at which water and

oxygen penetrate an organic coating and arrive at corrosion sites will depend

on the rate laws describing the processes which are occurring within the film.

The rate laws comprise a set of constitutive relations which serve as input to

the conservation equations.

An overall conservation balance for each penetrant takes the form:

Rate of Accumulation = Rate of Diffusive Input - Rate of

(7)

Diffusive Output - Rate of Disappearance by Sorption

These molar balances are written on the vehicle, so that the pigment is not

part of the system. However, the number density of pigment particles is an

important parameter since this influences the rate of removal of penetrant by

sorption.

14
Application of equation (7) leads to:

3C. 3N.

dt dZ IS

Equation (8) is the conservation equation for species i. All terms have units

of moles of i/time, volume of vehicle. The term on the left represents the

accumulation of penetrant within the vehicle (refer to equation (7)). The

first term on the right represents the net influx of penetrant in any local

region of the vehicle. The second term on the right gives an expression for

the rate at which the penetrant is removed from the vehicle by adsorption.

Substitution into equation (7) of the rate laws for diffusion (equation (5))

and adsorption (equation (4)) and equation (1) for n results in:

3C. 3 2 C. 6k

oL

In equation (9) it is assumed that the effective diffusivity D. is constant.

This is a reasonable assumption for most coating systems, since, in general,

for any given coating system D., will only be a function of temperature.

Equation (9) is a linear second order non-homogeneous partial differential equation. Its solution gives C.(Z,t), the concentration of penetrant

as a function of film depth (Z) and time (t). In order to effect a solution,

it is necessary to specify three conditions; two boundary conditions (on Z)

and one initial condition (on t) . These are given by equations (10) , (11) and

(12), respectively:

15
C.(0,t) = C. (10) i ' 10

3C.

g^" (L,t) = (11)

C.(Z,0) = (12)

One boundary condition (equation (10)) is written for the outside coating

surface (next to the external environment) and one for the coating-metal

substrate interface. At the coating surface (Z = 0) , it is presumed that the

concentration C. is equal to the solubility C. of water (or oxygen) in the

vehicle of the organic coating. At the coating-substrate interface (Z = L)

,

the flux of water (and oxygen) is equal to zero. This is expressed mathematically by equation (11) and corresponds physically to the fact that the

metal surface is an impermeable boundary to diffusion. Equation (12), the

initial condition, expresses mathematically the fact that the coating was

initially free of water or oxygen.

Equations (9) , (10) , (11) and (12) comprise the complete mathematical

model for permeation. There is one partial differential equation for oxygen

and one for water, both having the same form but different parameters. The

parameters of the model are D., (p/v), k , C. and d. D., k and C.

i' VK " ai io l ai io

differ for water and oxygen.

3.1.4 Model Solutions

It is important to be able to predict concentration profiles and fluxes

within the coating over the entire time range from the point when the pene16
trants first start to reach the metal substrate to the time when corrosion has

initiated and a steady-state concentration profile has developed.

Solutions to equation (9) for C.(Z,t) can be effected by several different

methods. These include:

a) separation of variables (boundary value methods)

b) Laplace Transform methods

c) computer methods

The analytical methods, (a) and (b) , are able to show the precise relations between the variables and parameters of the model. In this sense, they

are superior to computer methods. Both analytical methods produce solutions

expressed as an infinite series of terms, each of which is a function of both

time and position. The separation of variables method gives solutions which

converge most rapidly at long dimensionless times. This implies that fewer

terms in the series are needed for convergence at long times. Laplace

Transform methods yield solutions which converge quickest at shorter

dimensionless time.

The dimensionless time referred to here can be obtained by expressing

equation (9) in terms of dimensionless groups. Such an analysis shows that:

tD./L

2 f=-r~ (13)

t

1 represents the dimensionless time. Equation (13) lumps together the

quantities t, D., L and e into a single dimensionless group. t

1 is large at

long absolute times t, for small coating thicknesses L, for large values of

the effective diffusion coefficient D., and for films having low porosities.

17
Penetrant concentrations have been found as a function of depth into the

coating and time for both large and small dimensionless times. For both cases

concentrations at the coating-metal interface were found. In addition steady

state mathematical solutions for penetrant concentrations and solutions in

which adsorption is not significant were derived.

The solutions to these different cases are presented below. A summary by

equation number is provided in Table 1. Details of the derivations are presented in Appendix A. Further discussion of the solutions obtained is provided in Section 4 of this report. All of the solutions are given in terms of

the dimensionless concentration C = C./C. , expressed as a function of the

1 xo

r

dimensionless coating depth x = Z/L, the dimensionless time t' (equation 13)

and the dimensionless adsorption-diffusion constant a, as defined by (equation 15)

18
c CD

CU T3

too

O

>>|5 Xo <4-l O

TdC >,

rjM

CO

m P

0)

g

•u 3

to 01 3 v^

14-1 CO60C C •H O 4-1 H CO 4J

cO

U

(-1 4J CJ C •H

cu

C

CO c M

)-< O• rH

0) iH

rO

cO H

CO

}-> 4-1 CO3

en

Co

•H . 4-1 O,o

COoa o co

•H

4Jo

CO 43

4-1 •H

U

(3O

•H

4-1 exMO

COo

b$

C

er

•H

4J

COOuc

•H43

4-1 •H* g

•H

4-1 a.mo

CO

a o

•H

toO| 8asH

00O+

CO

cuS

•H

4->

toO CO

CN

CN

CNO

CN

CTiUO

43

CO t

CN

CO

CN

CO +J

CO /-> 8

t^

T) II CO

ai 4_i 4-> w

CO

4-1 X

CU

4-1

OJ 4=

4-1 O

4J

O

T3 •H

01

O

>,

CU CO ^ •H

cu CO

5-1 CU

co CO

CO

CO

a)CO CU

OJ CO 4= CU 4J 4=

ti 4J

a)M

a

CO •H

Cu

a

a

•H •H

4-> c CO

CU S-i > 4-1 •rl c

toO CUa

CO fl M

ai 43e cu 3 4-1 G CO

4-1 CI CO OH >,

4-1 XI

CO cO 3 CU cr 4-1 0) CO

19
Long Time Model Solutions

Concentration-Time Profiles

C = cosh a (1 - x)

cosh a

- TT n=l

2

(-l)

n+1

(2n-l )exp

- [a2 + (2n- 1} I

]t ' C0S

^^

1

(1 'X)]

2 a + (2n-l) T4

(14)

where

a = L

6k

d D ^ £

y

1/2

= L

6k

-fj* (P/V)(pv

/P

p

)

1/2

(15)

-Adsorption Insignificant (a = 0)

4 r

C = 1 -- I n=l

r _,n+l 2 (2n_D

exP [-(2n-l)

Z J

t'] cos [(2n-l) f

(1-x) ] (16)

Concentration-History at Metal Substrate Interface (x = 1)

C(l) = sech a - it I n=l

n+1 -[a

2 + (2n-l)2 - ] t' (-1)

1 (2n-l)exp ^

2 2 ir a + (2n-l) |

2 (17)

-Adsorption Insignificant (a = 0)

00 n ? ? rm - 1 ^ Y (-D r -(2n-i)

z -z C(D - 1 -

^ I ^TT exp [- t']

n=l

(18)

20
Short Time Model Solutions

Concentration-Time Profiles

. r (2+2n-x)

2

, 2 . 1 oo t

1 exp -[-—

T

+ a u]

C = A? £ (-l) n{(2 + 2n - x) /

^

du n=o o u

2 ., r (2n+x) 2 , t exp - L

t + a u]

+ (2n + x) /

^

du } (19)

o u

-Adsorption Insignificant (a = 0)

1 oo t' 2

r,— v /-,\ii r,~ „ vf r (2+2n-x) , du

C = Arf I (-1) {(2 + 2n - x) / exp -[-—7^—

H

—/2 n=o o u

+ (2n + x) J exp - [^f^] % (20)

o u

Concentration-History at Metal Substrate Interface (x = 1)

C =

I (-l) n

[ e (2n+l)a erfc ( 2n+l

+ &

^ ) n=o 2/P"

(21)

, -(2n+l)a j. , 2n+l >r-r ., + e ' erfc ( - a /t' )]

2/T"

-Adsorption Insignificant (a = 0)

.211+1,

C = 2 I (-l) n erfc F5^) (22) n=o 2/t1

"

21
Steady State Solutions (t = «)

Concentration Profiles

_ cosh a(l - x) , cosh a

Concentration at Substrate

C = sech a (24)

3.2 Blister Formation and Growth

As discussed in Section 2.2, blister formation and growth occurs as a

result of the diffusion of water into the osmotic cell created at the coatingmetal interface by impurity ions. Blistering is considered to initiate at the

metal substrate interface at weak spots randomly distributed over the substrate surface. These can be rough spots, spots with high impurity ion

concentrations or places with incipient adhesion loss. The weakest spots are

ones where blisters first initiate. Blister growth occurs when there is

sufficient water present in the vicinity of the metal substrate. Osmotically

driven diffusion of water through the coating controls the rate of growth. As

the blister becomes diluted with water the rate of blister growth slows.

To account for these phenomena several mathematical models are presented

here. In the first (developed in subsection 3.2.1), the impurity ions (such

as chloride or sulfate) are considered to be concentrated at specific spots on

the metal surface where blister initiation occurs. The rest of the surface is

presumed to be impurity free. In the second scenario modeled (developed in

22
subsection 3.2.2), the impurity ions are considered to be spread uniformly

over the whole surface. Blister initiation still takes place over weak spots

on the surface, but as the blister ages, new impurity ions are introduced at

the periphery. In both models it is presumed that the adhesion loss and the

stiffness of the organic coating do not control the rate of blister growth

[6, 7]. This means that the effect of the over pressure created by tension

in the disbonding film can be neglected. Thus, adhesion loss is considered to

be an effect rather than a cause. At later times the osmotic pressure may

fall to the point that the tension in the film cannot be overcome. At this

point blister growth will stop.

The mathematical models for growth of a blister on a metal substrate are

based upon a modified Fick's Law for the diffusion of water which takes the

flux of water (N) to be proportional to the product of the effective diffusivity (D) through the coating, the concentration of water in the coating (C)

and the gradient of the logarithm of the water activity coefficient (d Ina/dZ)

across the coating between the external environment and the interior of the

blister [6, 7], This is expressed mathematically by the relation

N = - DC ^f (25)

Figure 2 is a schematic representation of a blister within an organic coating.

It shows the profile of a single hemispherical blister which has grown to a

radius r within the coating. The water activity coefficient within the blister

is a , while that in the external environment above the free surface is denoted

by a„ . The distance Z is measured normal to the coating from the metal surface.

The blister is assumed to remain small relative to the coating thickness

23
2A: Concentrated Spot of Impurity

a 2 Free surface

Coating

Blister

/ / / / /

Metal substrate

/ / / /

Impurity

2B: Uniform Distribution of Impurity

Free surface

T

Coating

Blister

~

/ /»/ VP 7 / / 7 /

\

Impurity

Metal substrate

Figure 2: Schematic Representation of a Blister within an Organic Coating

24
(r << L) . This remains true for the initial and middle stages of blister

growth. In the later stages of growth, the blister may enlarge to the point

where its size approaches or exceeds the initial film thickness L. For this

stage, the models presented in this report will not apply exactly.

The assumptions made in the course of the development of these models are

summarized in Table 2. Because of assumption 4 in Table 2 the blistering

process is uncoupled with the process of molecular diffusion. Diffusion of

water during blistering is driven by the osmotic pressure and not by concentration gradients (Fick's law) as described in the previous section. In

effect we are assuming, in this scenario that molecular diffusion is much

faster and occurs much sooner than the formation of blisters. By the time the

blisters begin to form there is plenty of water in their vicinty. In addition

the demand of the few small blisters which form for more water is minimal.

This implies that C, the total concentration of the concentration of water in

the coating in the vicinity of the blister, can be treated as a constant.

It can be shown [8, 9] that the gradient of the log of the activity coefficient is proportional to the gradient of the molarity (m) of the impurity

ion, i.e., (dm/dZ) . m is expressed as moles solute/kg solvent. In this case,

in the light of the assumptions given in Table 2, equation (25) can be integrated to give:

N = lTIooo Am (26)

25
TABLE 2: Assumptions of the Mathematical

Models for Blister Growth

1. . Blister formation does not occur until water is present at the

metal - coating interface.

2. Blister does not change shape during growth (presumed hemispherical).

3. Dissolution of impurity ions into the blister from the metal surface

does not control growth.

4. Mass transfer of water to the blister surface is driven by osmotic

pressure, not by molecular diffusion.

5. Diffusion is unidirectional.

6. Blister size is much less than coating thickness.

7. Desorption of water into the blister from the bottom of the coating

is rapid enough that it does not influence growth.

8. Adhesion loss and stiffness of the organic coating do not control

blister growth.

9. The coating functions as a semipermeable membrane, allowing water to

diffuse osmotically inward but preventing impurity ions from diffusing

outward.

10. The molar osmotic coefficient <p is constant.

11. Only one impurity (solute) is present.

12. Pigment particles have no effect on the osmosis process.

13. The diffusion coefficient D (of water in the coating) is constant

during osmosis.

26
where

£ is the diffusion path length

V is the number of ions formed from one molecule of the impurity ion

M is the solvent molecular weight (18 for water)

<J> is the molar osmotic coefficient (assumed constant)

The multiplier of Am on the right hand side of equation (26) is a constant

which will be specific to the kind of impurity ion and coating involved. Am

represents the change in molarity of the impurity ion between the blister and

the outside of the coating. When the coating is exposed to water containing

on impurity ion Am is equal to m the molarity of the impurity ion within the

blister. The diffusion process can be viewed as being driven by the osmotic

pressure difference, Ait, which exists between the inside of the blister and

the external environment. Thus:

. RT . a2 RT vMd> , . /0 -, N ATT= ^

ln

a7 "(-"lOOO* Am (27)

where

R is the gas constant

T is the absolute temperature

V is the partial molal volume of the solvent (water)

When no impurity ions are present in the external solution, then a„ = 1 and

Am = m. As the blister becomes diluted with water, a„ rises and the osmotic

pressure within the blister drops. As a result the diffusion process slows.

27
3.2.1. Blister Growth - Concentrated Spots of Impurity

When a concentration of impurity ions such as chloride or sulfate is

present at a metal surface together with a sufficient amount of water, then

blister initiation and growth occurs. Consider the metal to have a small spot

where a high concentration of impurity ions are located. This is shown

schematically in Figure 2A.

A water balance equates the increase in mass of the water within the

blister to the amount which fluxes through the mantle (curved) surface:

P § - ft - 5 S > (28)

where

p is the density of water within the blister

V is the volume of the blister

M is the mass of water within the blister

S is the surface area of the mantle

With the presumption that the blister retains its shape: — = S — (29) dt

b

dt Uy;

where

r is a characteristic dimension of the blister

Substituting equation 29 and equation 26 into equation 28 and rearranging:

4^ = [M DC vM<j> - . dt

P I Tooo ] Am (30)

28
Equation 30 gives an expression for the rate dr/dt at which the blister

dimension r increases with time. The volumetric blister growth rate is

S dr./dt, while the mass blister growth rate is p S dr./dt.

The molarity m is a function of the amount of salt present at the blister

initiation site, M , and the amount of water which has diffused through the

blister surface. An electrolyte or salt balance made on the blister yields:

dM

"at

= Ar

d

(31)

where

A is the (flat) area of the blister on the metal surface

2 r is the rate of dissolution based on surface (grams/mm , s)

In general, the dissolution rate r, is quite high when compared to the time

d

scale for water diffusion. The impurity ion are concentrated right at the

bottom of the blister and they have ready access to the blister interior. The

dissolution is effectively complete before much water has had a chance to

penetrate the blister. In this case the molality within the blister can be

expressed directly as:

1000 M

m = r

-5- (32)

M M

s

Substituting equation 32 into equation 30, with Am = m (no impurity ion in the

external phase), and integrating, expressions for the blister radius r, mantle

surface area S and blister volume V are obtained as a function of time:

1/4

r = k

r

t (33)

29
where

1/2

S = k t

' (34)

s

3/4

V = k t J/H

(35) v

M DC 9 n ,.

k = 2tt k

2 (37)

s r

k = 4 tt k

3

(38) v 3 r

k , k and k are constants for the radius, surface area, and volume, defined

r s v

respectively by equations (36), (37) and (38). In the development of these

models, the blister was assumed to retain the shape of a hemisphere throughout

its growth. The characteristic dimension r is the radius of the hemisphere.

The modal can be simply extended to other blister shapes which are surfaces of

revolution, such as the segment of a sphere or an ellipsoidal shape.

Equation (35) predicts that the blister mass (or volume) increases as the

0.75 power of time. If the time is doubled the blister volume increases by 68

per cent. The blister radius increases as the 0.25 power of time (equation 33)

The blister radius doubles after the time has increased by a factor of 16.

These results also predict that the rate of blister growth slows with time,

being greater at the beginning when the blister is small and contains a concentrated salt solution, and less later when the solution in the blister has

become diluted with water.

30
Combining equations 27, 32 and 35 an expression can be derived for the

osmotic pressure as a function of time:

1/4 -3/4

Att = k Mc /Ht

' (39)

p S

where

3/4 1/4

k

P

= 2^

" [-T~] ^

(40)

p M V 6 MDC

Equation 39 predicts that the osmotic pressure varies the 0.25 of the

amount of salt in the blister and reciprocally with the 0.75 power of the

time. At time zero the model predicts very large osmotic pressures within the

blister. At small times the model predicts very large osmotic pressures

within the blister. The molarity, however, cannot exceed the saturation

value, m*, of the impurity ion in water. This places an upper limit on the

osmotic pressure as predicted by equation 27 with Am = m*.

The dimension £ contained in the equations represents a characteristic

diffusion length. In cases where water must be drawn in from the external

environment, Z = L, the coating thickness. When water is already present as a

result of the diffusion process then £ = 6, where 6 represents a characteristic

film thickness (6 < L) for the diffusion process. With no diffusional resistance, water in the vicinity of the blister would pass freely across the

mantle surface and blister growth would be quite rapid.

3.2.2 Blister Growth - Uniform Distribution of Impurity

The second type of blister growth model considers the impurity ion to be

uniformly spread across the surface. This is shown schematically in Figure 2B.

31
For the growth of a single blister, this differs from the previous model

development in one important aspect - as the blister grows, fresh impurity

enters through its base. In this case the amount of impurity ion, M , added.

to the blister is proportional to the flat or base area, A, of the blister

2 (equal to Trr for a hemispherical blister) . The greater the area of the

peeled blister, the greater is M . Thus

M = k'A = k'TTr

2 (41)

The proportionality constant k' in equation 41 is equal to the surface con2 centration of impurity ions, g/mm .

As in the previous model, this model presumes that the rate of dissolution of impurity ions into the blister is quite rapid compared to the rate of

blister growth. Thus, as the blister peels from the surface, the fresh

impurity ions exposed on the surface can be considered to immediately transfer

into the solution. By substituting equation 41 and equation 32 into equation 30,

with Am = m, and integrating, expressions are obtained for the blister radius,

mantle surface area and blister volume as functions of time:

r - k' t 1/2

(42)

where

S = k» t (43)

s V = k' t 3/2

(44)

K- [fK'-^h V*] 1/2

(45)

r M C I

s

32
k» = 2tt k' (46)

. = 1 „ ,,,3 k' = -=- TT k'^ (47) v 3 r

k

1 , k1 and k', are the constants in equations (42), (43) and (44), defined,

respectively by equations (45) , (46) and (47)

.

An expression for the osmotic pressure within a blister growing over a

surface containing a uniform amount of impurity is obtained by combining

equations 27, 32, 41, 42, 45 and 47:

Att = k' t

1/2

(48)

P

where

| k

f v<J> I

1/2

k» = 1000 [- —

]

(49)

P MM DC

s

The specific values of the osmotic pressure depend on the value of the osmotic

rate constants, k' given by equation 49.

3.3 Corrosion Phenomena

Corrosion is possible and, after sufficient time probable, whenever

water, oxygen and an electrolyte are in proximity to a metal surface. Because

the metal is at a higher free energy level than the corrosion products, there

is a driving force for corrosion. The potential for corrosion can be evaluated

by considering the difference in potential between the half-cell reactions

which occur at the anodic and cathodic sites on the metal surface. For any

33
given coating-metal system, the tendency to corrode varies with the temperature, and concentrations of water, oxygen and electrolyte in the vicinity of

the metal. Higher values of any of these variables can act to accelerate the

corrosion process. When concentrations of one of the chemical species are

low, e.g. oxygen, corrosion rates depend on the rate at which it is transported to the cathodic site. Since oxygen permeation controls the observed

corrosion rate, it can then be considered to be the rate-limiting step. In a

similar fashion, water permeation would be rate-controlling if the concentration of water at the corrosion site were low. This could result from: a) low

relative humidity outside the coating, b) low affinity of the coating for

water (low C. ), c) high affinity of the pigment for water (high k )

, d) slow

10 a diffusion of water through a relatively thick coating.

The electrolyte itself would control observed rates whenever its concentration in the water solution were low. This would result from low initial

concentrations of impurity ions on the metal surface (clean surface) and

dilution of the electrolyte by water within a blister because of osmotic flow.

Since the corrosion reactions themselves can generate electrolytes (e.g.,

hydroxyl ions), the process can be self-accelerating or autocatalytic, with

more and more ions being released into solution. This, in turn, may raise the

ionic strength within the blister leading to the incursion of more water. The

enlarging cathode area associated with the growing blister can accelerate the

corrosion even more, both because of the larger area and the differential

aeration which occurs where oxygen has easier access at the periphery of the

blister.

34
3.3.1 Potential Difference Model

The greater the potential difference between the anode and cathode, the

greater will be the driving force for corrosion. As the corrosion reactions

proceed, the driving force diminishes because of the build-up of reaction

products in the vicinity of the electrode surfaces. Thus the driving force

for corrosion is the potential difference between the anode and the cathode.

The initial value of the driving force is determined from the thermodynamic

potentials of the half cell reactions (refer to Section 2.3). The existence

of this potential causes electron flow and corrosion proceeds. As the potential of the cathode, E , drifts towards that of the anode, E , and vice versa,

polarization occurs. Polarization was modeled in terms of the potential

difference 6 between the anode and the cathode, where 6 = E - E . Thus the

cathode potential rises (and the anode potential falls) at a rate which is

proportional to this difference in potential.

Mathematically, this is expressed as

dE

c dT " k

c

6 (50)

" dE

A

= kA 6 (51) dt A

k and k are unknown polarization rate constants for the cathode and anode,

respectively. Their units are reciprocal time, s . They vary with temperature and anode and cathode surface area. Thus:

k

A

= V S A

eXP [ _LA

/RT ] (52)

k

c = k

coo s c exp c "L

c /RT ] (53)

35
where

S., S are the anode (A) and cathode (C) surface areas

L , L are the activation energies for the anode and cathode

R is the gas constant

and

k A , k^ are specific rate constants (based on area) for the anode

and cathode.

Equations 52 and 53 are based on the presumption that the rate constants

follow an Arrhenius - law type temperature dependence, and, in fact, k and

k are analogous to Arrhenius frequency factors. When the temperature and

surface areas of the anode and cathode are constant, and the electrode has a

uniform potential across it, both k and k are fixed and equations 50 and 51

A Li can be solved analytically to give :

k + k

^A

F A r F k F

f£ = r_ 1I0C \_ _ foA -<k + k ) t (54)

E

L k+lc J k+k U E

; h

0C

R

A + kC

k

A + kC

b

0C

k + k

^A

E

A r A C E

0C . \ , E

0A .. -(k. + k_) t

r — . = [ ] + (- 1) e A C (55)

h

0C

K

A + kC

k

A

R

C \)C

6 - (E0A

- E oc

) e"

(kA

+ V C (56)

+

The complete derivation of these equations is given in Appendix B.l,

36
E^ A and E^„ are the initial potentials of the anode and the cathode, respec- OA OC

tively. (EnA

- Enr) is equal to initial potential difference, which can be

found from the thermodynamic half cell potentials.

Equations 54 and 55 predict that the anode potential decreases and the

cathode potential rises at ever decreasing rates towards a common final

potential E

f given by:

k

A

E

OC

+ k

C

E0A ,__. Ef \^~ (57)

Equation 56 gives the driving force 5 for the polarization process. It

decreases exponentially with time. After a suitably long time polarization is

complete, 6 -* 0, and both anode and cathode have the potential given by

equation 57.

3.3.2 Equivalent Circuit Model

An alternative model can be developed in which the polarization process

can be modeled in terms of an equivalent electrical circuit. Figure 3 shows a

schematic of the circuit. The anode and the cathode are taken to be at the

opposite end of a charged capacitor C having an initial thermodynamic potencL tlal 6 = E0A " E0C

As with the potential difference model developed in the previous section,

this model is based on a voltage difference 6 between half cell reactions.

This falls with time and eventually reaches some steady state value. The

existence of a potential causes electron flow through the solution and polarization proceeds. The driving force is represented in the model by a charged

capacitor. Initially the capacitor is fully charged. As shown in figure 3,

37
CM

a

AA/WWVW

dfco

<o

r

T

coft

co

I

CM CO

oh

H

idN

a

J!

9

•H

pa

*

CO

•H

fa

38
at time zero the switch S-. is opened, the switch S„ is closed, and the circuit

is completed. The capacitor then slowly discharges through the resistances.

The electrolyte serves as a charge carrier for the electron flow. But

the presence of the electrolytic solution contributes to the overall resistance of the electrical circuit. The resistance R will remain the same as

long as the composition and conductivity of the solution remains the same. At

the beginning of the polarization process, before the products of corrosion

have accumulated, this is likely to be true. Also at later times when the

corrosion products have come to a pseudo-equilibrium with the electrolyte

solution, R would also tend to be constant. In between, R would decrease

from its old level towards a new level. The decrease in resistance is caused

by an increase in ionic conductivity as product ions, particularly OH and

Fe, are released into the solution. When the concentration of these ions

reach the solubility limit, solid reaction products precipitate from solution

and the conductivity becomes constant again.

The initial potential decreases because of the electron transfer or flow

of current. The initial current decreases both because of the decreasing

potential, and the increasing resistance R (refer to figure 3), attributable

to the corrosion products. The products can form an adsorbed film of ions or

gases on the cathode surface, or a solid layer of corrosion products such as

Fe(OH) or Fe 0, can deposit on the surface. In either case, the resistance

increases with time, although film formation would be more rapid than product

layer formation.

With film formation either of two mechanisms can operate:

a) a monolayer of ions or gases can adsorb on the electrode surface

(Langmuir adsorption mechanism)

.

b) an electrical double layer can form in the vicinity of the surface.

39
For mechanism (a) , ions (or gases) adsorb on active sites located on the metal

surface at a rate which is proportional to the product of the number of

unoccupied sites with the concentration of ions in the bulk solution. They

desorb at a rate proportional to the number of occupied sites. These assumptions are analogous to those of the classical Langmuir theory for the physical

adsorption of gases. The electrical resistance of the adsorbed ions (R

9

) will

be proportional to the fraction of the total surface which is occupied. Based

on the Langmuir theory, it can be shown that

R

2 = ( Vu U "

6

at]

(58)

where

(R ) is the ultimate or final resistance

I u and

a is the rate constant for film formation by Langmuir adsorption

The development of equation 58 presumes that the concentration of adsorbing

ions in the conducting fluid medium does not change with time. This is

reasonable considering the relatively small number of ions which can generally

supply monolayer coverage to a surface. Equation 58 predicts that R~ increases

linearly with time at the beginning of the corrosion process when the surface

has only a small coverage of adsorbed ions. At later times, when the surface

The complete derivation of equation 58, including the determinations of the

values of (R~) and a in terms of Langmuir theory, is presented in Appendix B.2.2,

40
is nearly covered, the resistance approaches a constant value. If the rate of

increase is high, corresponding to a large value of the constant a, then

monolayer coverage is rapid and R„ is constant during most of the corrosion

process.

An electrical double layer can also form next to the electrode surface as

a result of the electric field in the interface region between the metal

surface and the solution [10]. Water, which has a large dipole moment, can

form an ionic cloud near the metal surface. The water dipoles next to the

metal surface orient themselves depending on the charge on the metal. This

inner layer of water molecules is known as the "hydration sheath." The second

layer out from the surface contains saturated ions and is known as the "outer

Helmholtz plane." Together the hydration sheath and Helmholtz plane comprise

an electrical double layer. The formation of a double layer increases the

resistance to the transfer of electrical charge, and specifically, R_ . At

present, the precise way in which R„ varies with the charge and composition of

the electrical double layer has not been determined.

Solid corrosion products such as Fe_0„ or Fe(0H)_ can deposit on the surface,

These, too, will increase the resistance R and cut down on the charge transfer.

If the products accumulate uniformly over the surface of the metal, the resistance increases in direct proportion to the thickness of the corrosion layer.

Thus:

t R_ = 6 % = y / idt = y (Q - Q) = YC (5-6) (59)

41
where

£ is the thickness of the corrosion products

$,y are empirical constants of proportionality

i is the current

Q is the charge

Q is the initial charge

The thickness I can only be measured accurately if a tarnish layer

accumulates, but it can be implied from the theory developed here.

Equation 59 expresses the fact that the total amount of corrosion product

laid down, by Faraday's Law, are proportional to the total corrosion current

t which has flowed until time t, i.e. / idt. This in turn will be equal to the

o change in charge, (Q - Q) or C (6 - 6) . Thus, the resistance at any time is

o a o proportional to the degree of polarization.

Referring again to Figure 3, it is possible to derive expressions for the

charge Q, the current i, and the voltage drop 6, provided the resistance of

the electrolytic solution, R.. , and the resistance of the corrosion products,

R„, are known as functions of system variables.

Applying Kirchoff's Second Law (the algebraic sum of the potential

differences around any closed loop is zero) to the electrical circuit shown in

figure 3:

i (R

1 + R2

) + - / idt = (60) a o since

i - §

42
Equation 60 can be rewritten as

(R

i

+ V f+

c •

° < 62> a

with the initial condition that

Q = C 6 (63)

o a o

v '

With assumptions made about R- and R , these circuit equations were

solved to give the charge, Q, current, i, and potential difference, 6, as

functions of time. In all of the developments, both C , the capacitance of

3. the charged electrodes, and R1

, the electrolyte resistance, were taken as

constants.

Three different sub-models were used for R„, the resistance of the corrosion products. These included:

(a) a constant value of R_

(b) R„ based on film formation by Langmuir adsorption (equation 58)

(c) R based on the uniform accumulation of a layer of corrosion products

(equation 59)

Solutions to each of these sub-models for Q, i and 6 are summarized here.

They are based on the circuit-model equations, 60, 61, 62 and 63. Derivations

are supplied in Appendix B.2.

When R~ is constant, sub-model (a) predicts that:

§

" I =1 = £XP [ (R, + V C

] " »» ( " t/T) (M> o o o 12a

43
where

*o T^ <«>

and

T = (R.12a + RjC (66)

Equation 64 predicts that the relative charge, current and potential difference

all fall exponentially at the same rate, t = (R.. + R„)C represents a time

X L. 3. constant for this process. In general, T will be an unknown model constant

since C is unknown. The initial current i is calculated using equation 65, a o

b

from the initial potential, 6 , and total circuit resistance.

o When R- is based on film formation by Langmuir adsorption (sub-model (b))

the corresponding formulas for Q, 6, and i are:

1

at

jc R e R ax

o o 1 1

at

R e R ax

[(i + ^) -^i

i

R

i

R

i

i rn R

o [1 + u ,- -at N n j£ (1 - e )]

(68)

where

T = C (R. + R ) (69) u a 1 u

T is a system time constant based on the final resistance (R. + R ) which

u 1 u corresponds to equilibrium monolayer coverage of the electrode surface.

Equation 67 predicts that the relative charge and potential difference

should both fall in the same manner. Equation 68 gives an analogous expres44
sion for the current. The initial current i is equal to 6 /R- in this case,

o o 1 since R_ is initially zero. A comparison of equations 67 and 68 shows that:

D | =f U + /> (70)

o o 1

Since R increases with time equation 66 predicts that the relative current,

i/i , falls faster than the potential difference ratio, -* . At large values

o of the constant a:

R

" t/T

ft-|-a + ^) (7D

o o 1

while

f

- a + jp) (72)

o 1

Physically large values of the constant a corresponds to the rapid

attainment of equilibrium of the adsorbing species with the metal surface.

When R„ is based on the accumulation of a layer of solid corrosion products, the corresponding formulas for Q, 6, and i are:

C YQo d-Q/Q

o

) -t

1\ Ca[Ri + yQq

. yfi

j O

(74)

45
Equation 73 is transcendental in Q/Q , and must be solved numerically for

specific values of model parameters. Once Q/Q is known, the current ratio

i/i can be found from equation 74.

o

4. DISCUSSION OF RESULTS

4.1 Water and Oxygen Permeability

Equations 14 through 24 present expressions for the water and oxygen

concentrations within the coating and at the metal substrate as a function of

time. The equations for the concentration are expressed in terms of dimensionless variables and parameters: dimensionless concentration, C = C./C. , l 10

2 dimensionless depth, x = z/L, dimensionless time, t' = tD./eL , and dimensionless parameter, a, defined by equation 15. It is advantageous to express the

model in terms of dimensionless groups since in effect, all possible cases can

be condensed into a single mathematical form.

Even though the models have the same mathematical structure, they predict

that each species penetrates the film at different rates. This is because

specific model parameters will differ for each penetrant. These parameters

are C. , the concentration of penetrant dissolved at the top of the coating,

10 r °

D., the effective diffusion coefficient of penetrant through the coating and

k . , the adsorption coefficient of the penetrant on the surface of the pigment

3.1 particles.

For a given coating (fixed L and e) , the actual values of C. , D. and

k . will determine the concentration of the penetrant at the metal substrate ai

as a function of time. Each penetrant's concentration rises towards its final

or steady-state value. The rate of rise depends on the value of the model

parameters

46
The concentrations at the coating-substrate interface depend on t' and

the degree to which pigment particles are effective in removing water and/or

oxygen. Permeation to the surface becomes important at times t when the

dimensionless time (t') is of the order of unity (t 1 ~ 1). It is important to

emphasize that it is t' which affects the rate at which steady-state is

attained rather than the absolute time t.

For t' < 1 the short time model solutions (equations 19, 20, 21 and 22)

should be used. For t' > 1 the long time model solutions (equations 14, 15,

16, 17 and 18) should be used. For dimensionless times t' ~ 1, either set of

solutions produce equivalent answers, with a similar number of terms needed

for convergence in the infinite series. At such times, the penetrant concentrations in the coating will have risen to within a significant fraction of

their final or steady-state values. Thus the protective performance of the

film will be enhanced if D. is small, L is large and £ is small. The most

sensitive of these parameters is the film thickness, L. If L is doubled it

takes four times as long for the penetrant to reach a given fraction of its

final concentration. This is one of the reasons why the use of multiple coats

of paint is preferred. From this single viewpoint, using five coats of paint

is over thirty times as effective as a single coat.

In addition to dimensionless time t', concentrations at the coatingsubstrate interface also depend on the affinity of the penetrants for the

pigment particles. A quantitative measure of this effect is provided by the

dimensionless parameter a, as defined by equation 15.

6kai 1p 1/2 6k . p 1/2 a - Lld-DT<¥» =L hTi7(p/v)^] < 15 > 1 ± pp

47
The higher the value of the parameter a, the greater is the retention of

the penetrant within the organic coating. Water and oxygen have different

values of a, mainly because of the different chemical attraction they have for

the pigment particles when dissolved in the vehicle. The parameter a is a

function of pigment loading, (p/v)(p /p ), pigment particle diameter, d,

diffusivity, D., affinity of the pigment particle for the penetrant, as measured by the adsorption coefficient k . , and coating thickness L. a can take

on any positive value, including zero. Realistically, a can be zero if no

pigment is used (as in varnishes) or if the pigment has no affinity for the

penetrant, i.e. k . = 0. For example, a = if oxygen absorbed in the coating

3.1 does not adsorb on the surface of titanium dioxide (TiO ) pigment particles.

In order to keep the concentrations of the penetrants small at the metal

surface and, thus minimize the potential for corrosion, it is best to design a

coating system which has a high value of the parameter a. According to

equation 15 this will occur with a suitable combination of the following

factors:

1. lots of small sized pigment particles (p/v high, d low)

2. retarded diffusion through the coating (D low)

3. rapid adsorption onto pigment particles (k . high)

3.1

4. thick coating (L large)

The parameter a is most sensitive to this last factor. a varies in direct

proportion to coating thickness, doubling the thickness doubles the value of

a. This is in contrast with the other parameters which vary directly or

inversely with the square root, which is a less sensitive function. All of

the factors listed act to retain the penetrant within the coating, not allow48
ing it to reach the metal surface. Two additional factors will also act to

keep penetrant concentrations low at the metal substrate:

5. low solubility of the penetrant into the vehicle (low C. )

6. short contact times (low t')

Thus, even in the worst possible case, where a = 0, it is still possible to

have low concentrations at the metal surface if the solubility of the penetrant into the vehicle is low, for the concentration at the surface may never

exceed this value. Also, the concentrations at the metal surface do not start

to reach an appreciable fraction of their final or steady-state values until

the dimensionless contact time t' is approximately unity. Thus, the surface

concentration relative to its final value remains low until time 0, where the

characteristic time Q is given by:

9 = £L

2 /D. (75)

l

For a typical film with L = 0.1 mm, D = 0.5 x 10 mm /s (an estimate) and

£ = 0.5, this leads to = 1000s. Typical service lives of most organic

coatings are many orders of magnitude greater than this. This implies that

steady state will be attained in a matter of hours (at least for this value of

the dif fusivity) . Thus short contact times may not be an important contri-

-10 2 buting factor. However, for lower values of the dif fusivity (say D. ~ 10 mm /s,

which would be typical of solid-state dif fusivities) the contact time would

become an important factor [11, 12].

Examining the diffusion process in terms of the overall protective function

of the coating, both water and oxygen must be present in concentrations which

49
are sufficient to initiate and accelerate corrosion. The steady state concentrations of oxygen and water at the metal surface are given by equation 24 as

C. = C. sech a. (24) 1 io 1

'

Whether or not corrosion will actually occur will depend on the temperature,

pH, nature of the metal and electrolyte and the electrochemical potentials of

the half-cell reactions. In some instances corrosion will not occur regardless of the oxygen concentration because it is thermodynamically unfavorable

or favors the formation of a passive oxide film. Pourbaix diagrams of potential

vs pH are convenient ways to summarize such information [13]. In some regions

it is possible to go from a condition where corrosion is not possible to one

where it is, by simply increasing the oxygen concentration. At the point

where this occurs the specific concentration is denoted as C . whether or 2,c

not corrosion will actually occur cannot be determined a priori but this

concentration represents a threshold above which corrosion is possible. In

addition as the oxygen concentration is raised the probability that corrosion

will occur becomes more likely. Since the concentrations at the substrate

rise with time at suitably long times the limit C may be exceeded for a 2,c

given coating and environment. Thus the possibility for corrosion will exist

when C is less than the steady state value. The absence of corrosion is

2,c

indicated by:

C. sech a. < C. (76) io i ic

where

50
To insure that corrosion will not occur, equation 76 need only be valid for

oxygen or water. For such a system, corrosion is prevented if a can be

raised, or C lowered, to the point where the inequality is obeyed for oxygen.

For an existing coating system, a can be raised most conveniently by applying

extra layers of coating, while C can be lowered most conveniently by lowering

the concentration of penetrant in the external phase. In laboratory situations

the partial pressure of oxygen can be lowered, e.g., by dilution with nitrogen,

or in aqueous systems by bubbling nitrogen through the solution. Coatings

exposed to the atmosphere will have the partial pressure fixed at 0.21 atm.

In terms of the design of a coating system, the model predicts that better

protection will be afforded if:

1. a large fraction of pigment is used in the formulation

2. smaller particles are used

3. a denser, thicker coating is applied

4. the coating is made oxygen and water repellant

Using the model, the exact quantitative effort of each of these factors can be

assessed.

4.2 Blister Formation and Growth

In section 3.2 two mathematical models were developed for the growth of

water-filled blisters at the metal substrate under the coating. Both models

were based on diffusion and osmotic theory and had a single adjustable constant, the osmotic rate constant. Equations were given for the radius, mantle

surface area, and volume of a hemispherical blister as a function of time.

In the first model (refer to section 3.2.1), blister growth was considered to occur over concentrated spots of impurity located at the point of

51
blister initiation. No new impurity ions come into the blister as it grew.

This model was mathematically expressed by equations 33 through 40, with the

osmotic pressure relationships given by equations 39 and 40. In the second

model (refer to section 3.2.2), blister growth was considered to occur over an

unclean surface which contained a uniform distribution of impurity. As the

blister grew, fresh impurity entered through its base. This model is mathematically expressed by equations 42 through 49, with the osmotic pressure

relationships given by equations 48 and 49.

Both models have the same mathematical form - the radius, surface area

and blister volume all vary directly as the time raised to various fractional

powers. Also, for both models, the blister volume varies as the cube of the

blister radius, while the mantle and flat surface areas both vary with the

square of the radius. Comparing the model solutions for concentrated impurity

distribution (equations 33, 34 and 35), with those found for uniform salt

distribution (equations 42, 43 and 44), shows that the blister size increases

more rapidly with the uniform salt distribution. Blister size will double

when the time has increased by a factor of 1.6, compared to a factor of 2.5

for concentrated salt distribution. As time proceeds, the rate of blister

growth actually increases (as /t~~) . This is unlike the concentrated salt

distribution case where a decrease in volumetric growth rate with time is

predicted

.

Of course a true comparison of the absolute (rather than relative) sizes

of the blisters for the two models needs to be made based on the release of

equivalent amounts of salt into the blister. In many cases the salt spot on

the surface is present as a small patch. Surface diffusion gives a more or

less uniform salt concentration over the patch. It is likely that the center

52
of such a spot has a slightly higher salt concentration so that blister

initiation is favorable there. The model developed for uniform salt distribution is applicable until the blister has grown to the size of the patch,

after which the model for concentrated salt distribution applies. Data of

blister volume vs time would show a sharp change in slope at this time.

Equations 39 and 48 present expressions for the osmotic pressure within

blisters for the concentrated impurity and uniform impurity models, respectively. For both models, the osmotic pressure is predicted to fall with time

as the blister solution becomes diluted with water. The models incorrectly

predict an infinite osmotic pressure at time zero. In actuality, the highest

osmotic pressure would be that corresponding to the saturation value of the

impurity salt in water at the temperature of the system. Comparing equation 39

with 48 shows that the osmotic pressure falls off most rapidly when the salt

is initially concentrated within the blister rather than being uniformly

spread across the metal. This will be true for both changes in the absolute

value of the osmotic pressure, d(AiT), and the relative value, d(A7r)/A7T.

Thus from equations 39 and 48 it can be shown that:

•d(A-rr) . _1_ (11 . Air 2t K ' and

Aim _ 3

(7g)

Air 4t

for the uniform and concentrated salt models, respectively. Comparing

equation 78 to equation 77 shows that the relative change in the osmotic

pressure -d (Ait/Ait) is greater when the salt is concentrated, since any given

time t, equation 78 predicts the relative rate of fall of osmotic pressure

will be larger.

53
Of course when all the salt is concentrated within the blister, the

initial osmotic pressure is much greater than when the salt is uniformly

distributed. In this case the osmotic pressure has further to fall. For

uniform salt distribution, osmotic pressures do not fall as fast because the

concentration of fresh salt is being added at the periphery of the blister.

As with blister growth rates, it is best to compare osmotic pressures based on

the availability to the blister of the same total amount of salt.

At later times the osmotic pressure can fall to the point where the

disbonding tension in the film cannot be overcome and blister growth will

cease. Blister initiation does not occur when the surface is relatively

clean, when the adhesion of the coating to the substrate is good or where

water has not penetrated to the surface. The osmotic pressure developed is

insufficient to destroy the adhesion, and the blister does not form.

Blister initiation can also be prevented by reversing the concentration

gradient. Thus if a concentrated salt solution is kept in the external

solution outside the coating, water may be drawn from the coating. This is

the principle which makes it possible to store acids and concentrated salts in

painted vessels made of metals which would normally corrode. This helps

prevent the initiation of blisters and can collapse existing blisters, although

wrinkling of the coating may be observed. The models developed in this report

for blister growth can easily be modified to treat blister collapse.

It is important to understand the extent to which the blister initiation

and growth processes are coupled with the absorption and diffusion processes

discussed in the previous section. The interaction of these processes depends

on both time and distance scales. Thus, if blister initiation and growth is

54
slow compared to the time taken for water to diffuse to the metal-coating

interface, then blister formation is more likely and initial blister growth

more rapid because of the proximity of the water to the surface. The diffusion

path for water will then be much less than the thickness of the coating and

diffusion times considerably shortened. If, on the other hand, blister growth

is inherently faster than the diffusion rate of water, then water will need to

be drawn into the blister all the way from the surface of the coating. The

observed rate of blister growth will then be controlled by the absorption and

diffusion rates of water within the coating.

For a surface containing only a few small blisters, the amount of water

drawn to the coating-metal interface will not depend on the number of blisters.

This is because the total demand of these blisters for water is small compared

to the amount of water which can be absorbed by the vehicle or adsorbed by the

pigment. Thus, even though water diffusion has a pronounced effect on blister

growth, the reverse is not true. However, as the number of blisters per unit

area of coating (number density of blisters) increases, a point will be

reached where a significant fraction of the water which reaches the metalcoating interface will be brought there as a result of osmotic effects.

For systems with multiple blisters, the number density of blisters is an

important parameter. The models developed here for single blisters must be

extended in order to treat blister clusters. Additional factors to consider

include:

1. the possibility of blister impingement and coelescence

with other blisters

2. diffusion of water along the surface in addition to normal

to the surface

3. different blisters initiating at different times and growing

at different rates

55
This last factor is possible if the surface has a distribution of weak spots

having different adhesive strengths or if there is a non-uniform concentration

of impurity ions on the metal.

4.3 Corrosion Phenomena

In section 4.2 two mathematical models were developed for the corrosion

phenomena which occur in organic coating systems. Both models treated the

polarization which occurs when an electrical potential is dissipated in an

electrochemical cell. These models are the potential difference model (PDM)

and the equivalent circuit model (ECM) . The (ECM) itself has three separate

submodels, one based on a constant value of the resistance of the corrosion

products, one based on Langmuir adsorption (or film formation) of the corrosion products on the cathode surface, and one based on a uniform accumulation

of layers of corrosion products. Both the (DCM) and (ECM) models consider

that an electrochemical cell having an anode and cathode is formed beneath the

coating. The models are not specific to any particular corroding system.

They presume surface uniformity and do not account for changes in size or

location of the anode or cathode surfaces.

In the potential difference model (PDM) (presented in section 3.3.1) the

driving force for polarization is the potential difference between the anode

and the cathode. The potential difference between cathode and anode was found

to decrease exponentially with time (equation 56) , while the cathode and anode

potentials themselves (equations 54 and 55) , exponentially approached a final

common potential (given by equation 57) . The models contain two unknown

polarization rate constants, which vary directly with electrode surface area

and follow an Arrhenius temperature dependency.

56
The equivalent circuit models (ECM) , as presented in section 3.3-2, are

based upon the dissipation of the initial potential across two series resistances, one corresponding to the resistance of the electrolytic solution (R.. ) ,

and the second corresponding to the resistance of the corrosion products (R

9

)

.

Solutions are presented for the charge, current and potential difference of

the circuit. When R? is constant the potential difference predicted across

the circuit, (equation 60), has the same mathematical form as that for the

(PDM) , equation 56. In fact a comparison of the two equations shows that:

T " < R

1

+ VCa

- VTk^ (79)

Thus the circuit time constant (R.. + R„)C is analogous to the reciprocal of

the sum of the polarization rate constants.

For the other (ECM) submodels for R~ , the potential difference also

decreases with time, but the mathematical analogy to the (PDM) is no longer

valid.

For all (ECM) models the circuit current and charge also decreased with

time. When R„ is a constant, the charge, current and potential difference all

decrease exponentially from their initial values in the same way (refer to

equation 60) . For the film formation and uniform product accumulation submodels, the potential and charge decrease together, but the current falls

faster because of the increase of R„ with time.

The models developed in this report can be used to fit polarization data

for two dissimilar metals immersed in a fluid media containing ionized acid,

base or salt. Because polarization is fairly rapid (scale of minutes) compared

to the time scale of other corrosion phenomena, such as pitting or precipitation

57
of corrosion products, it can be considered to be an important phenomenon

during the earlier stages of the corrosion process. The corrosion reactions

themselves produce charge by generating electrons at the anode as the metal is

consumed. This is not accounted for in the models. In fact, the polarization

models predict a decrease in charge with time.

In addition, the increase in solute or electrolyte concentration with

time due to the accumulation of ions of corrosion product has not been modeled.

In the models this would result in a decrease in R , the resistance of the

electrolytic solution or conducting medium. When ions in the electrolyte

"soup" reach critical concentrations determined by the ionic field and the

solubility products, precipitation occurs. Precipitation acts as a sink for

ions so that the resistance of the electrolyte may approach a constant value.

Refinements in the model should take these factors into account.

For any given coating - metal system the tendency to corrode varies with

the temperature and concentrations of water, oxygen and electrolyte in the

vicinity of the metal. Higher values of any of these variables act to accelerate the corrosion process. When concentrations of one of the chemical

species are low, e.g. oxygen, corrosion rates depend on the rate at which it

is transported to the cathodic site. Oxygen permeation through the film can

then be considered to be the rate-limiting step since it controls the observed

corrosion rate. In a similar fashion, water permeation would be rate-controlling

if the concentration of water in the vicinity of the corrosion site were low.

This could result from: a) low relative humidity outside the coating, b) low

affinity of the coating for water (low C. ), c) high affinity of the pigment

for water (high k. ) d) slow diffusion of water through a relatively thick

xa coating. The solute or electrolyte itself would control rates whenever its

58
concentration in the water solution were low. This can result from low initial

concentrations of impurity ions on the metal surface (clean surface) , or

dilution of the electrolyte by water within a blister because of osmotic flow.

Since the corrosion reactions themselves can generate electrolytes (e.g.,

hydroxyl ions) the process can be autocatalytic, with more and more ions being

released into solution. This, in turn, can raise the ionic strength within

the blister leading to the incursion of more water. The enlarging cathode

area associated with the growing blister can accelerate the corrosion even

more, both because of its larger area and the differential aeration which

occurs at the periphery of the blister where oxygen has easier access.

As discussed in the previous section, blister formation and growth will

depend on the location and amount of water which has penetrated the coating.

In the same way, the accumulation of corrosion products can alter the osmotic

pressure within the blister. This, in turn, can affect the rate at which

water is drawn to the blister. Decreases in ion concentrations raise the

circuit resistance and cut down on the rate of corrosion.

In summary, the overall corrosion process is complex. The separate

mathematical models developed in this paper for permeation of water and

oxygen, for the growth of blisters and for the polarization of electrode

surfaces were all modeled independently of one another, as if these processes

were uncoupled. Yet both diffusion and osmotic pressure can indirectly

influence the amount of corrosion which can occur beneath a blistered coating.

Future modeling and experimental efforts should be directed at a more complete

understanding of the coupling which occurs between these phenomena.

59
5. SUMMARY AND CONCLUSIONS

Mathematical models were developed for conceptual models describing the

main phenomena which occur when corrosion occurs beneath an organic coating.

The potential for corrosion exists when oxygen, water and ionic impurities are

present together in the vicinity of the metal substrate. This situation is

likely to occur in blisters, which are water filled cavities located at the

metal-coating surface. Blisters grow because of osmosis of water from the

coating. Their rate of growth depends on their size and the concentration of

impurity ions within them and outside the coating, as well as the location and

distribution of water within the coating. The penetrants water and oxygen

both enter the coating by absorption and then pass through the coating by

diffusion. Not all of the penetrant may reach the metal surface because of

adsorption on the surface of the pigment particles. All of these processes

are a function of the nature of the coating system and temperature. The rates

of absorption, diffusion and adsorption differ for water and oxygen.

Within blisters, corrosion reactions can initiate if sufficient dissolved

oxygen is present. For iron-based metals oxidation at the anode produces

ferrous ions, while reduction at the cathode consumes water and oxygen to form

hydroxyl ions. Electrons generated at the anode flow through the metal to the

cathode where they are accepted by the cathodic reaction. Pitting of the

anode can occur, while corrosion products generally form in the vicinity of

the cathode. This can cause polarization of the cathode surface, as can the

formation of adsorbed films of gases and ions or the formation of electrical

double layers

.

60
Mathematical models were developed and solved for:

a) the permeation of water and oxygen through the coating (absorption,

diffusion and adsorption)

b) the growth of blisters on the metal substrate

c) polarization of the anode and cathode surfaces

Permeation models were based on the flux laws and conservation laws for

the diffusing species. At the outside coating surface the water and oxygen

were assumed to absorb by Henry's Law. Adsorption of penetrants onto the

pigment surface was assumed to occur at a rate proportional to the local

concentration and the concentration of pigment particles. A volume balance

related the porosity of the coating (volume fraction vehicle) to the pigment

to vehicle weight ratio. Diffusion was modeled based on Fick's Law. For both

oxygen and water the separate rate laws established for adsorption, diffusion

and absorption were incorporated into the overall conservation equations.

This formulation resulted in two linear partial differential equations; one

for oxygen and one for water. Both equations had the same mathematical form.

The dependent variable was the concentration and the independent variables

were the depth into the coating and time. Parameters in the model included

the pigment to vehicle ratio, the pigment particle size, the diffusivity of

the penetrant through the coating (D.) its solubility in the coating (C. ) and

the adsorption coefficient of the penetrant onto the pigment particles (k.).

Since C. , D. and k. have different values for water and oxygen, the rates of 10 11

absorption, diffusion and adsorption differ.

Model solutions for concentration as a function of coating depth and time

were expressed in terms of dimensionless variables and parameters. Both short

and long time solutions to the mathematical model were presented, as were

61
steady state profiles and concentrations at the metal-coating surface. The

model predicted that the permeation of water and oxygen would be inhibited

and, thus, the potential for corrosion lessened, with a suitable combination

of:

1. lots of small sized pigment particles

2. retarded diffusion through the coating

3. rapid adsorption onto pigment particles

4. thick (or multiple) coating(s)

5. low solubility of penetrant into vehicle (water and oxygen repellancy)

6. short dimensionless contact times

Of all of these factors the model was found to be most sensitive to the coating

thickness L. Increases in L gave much lower concentrations of water and

oxygen at the metal surface as well as lower dimensionless contact times. The

protective function of the coating is greatly enhanced if membrane thickness

2 is raised. It was shown that for short contact times, given by eL /D.,

penetration to the surface is unlikely to initiate blister growth and subsequent corrosion.

Corrosion will not occur unless oxygen and a conducting solution are

present at a metal surface. Even then, corrosion is not possible unless it is

thermodynamically favored, as indicated, for example, by conditions of oxygen

concentration, pH, temperature and electromotive potential which place it in

the active region of the appropriate Pourbaix diagram. As a result of exposure

of a coating to an outside oxygen environment, concentrations at a potential

corrosion site increase. Corrosion can then become thermodynamically favored

as passage from a passive to an active region occurs. At this point a critical

62
value of oxygen concentraiton, C . , is reached. It was established, that even

if a steady state were attained, oxygen might not be present at the metal

surface in this concentration. Mathematically this was expressed by C. sech

a. < C. , where a is a dimensionless parameter which is a measure of the

1 1C

retention of oxygen by the coating. Under these conditions corrosion cannot

occur although the converse does not hold.

Blister growth was considered to occur either over concentrated spots of

impurity located at the point of blister initiation or over a surface containing a uniform concentration of impurity. The solutions to both models had

similar mathematical forms: the blister radius, surface area, volume and

osmotic pressure were all found to vary directly as fractional powers of the

time. Blister volume varied as t

* for concentrated spots of impurity and

as t

* for a uniform distribution of impurity. The relative value of the

osmotic pressure was predicted to fall more rapidly for the concentrated

distribution of impurity, due to the rapid dilution of the blister with water.

For a uniform distribution of impurity the osmotic pressure does not fall as

fast because fresh salt is continually being added at the periphery of the

blister. The coupling between blister formation and the permeation of water

was discussed in terms of the time and distance scales characteristic of each

process.

Two mathematical models were developed for the corrosion phenomena

occurring in organic coatings. These were the potential difference model and

the equivalent circuit model. The models treated the polarization occurring

when an electrical potential is dissipated in an electrochemical cell. In the

polarization model, the driving force was the potential difference between the

anode and the cathode. Model solutions predicted that the potential difference

63
decreases exponentially with time so that both anode and cathode potentials

slowly approach a common final potential.

In the second model, corrosion was modeled in terms of an equivalent

electrical circuit. Polarization rates were taken to be functions of the

electromotive force and the circuit resistance. The potential between the

anodic and cathodic sites within the corrosion cell was considered to be

dissipated across several series resistances. These included the resistance

of the electrolyte and the resistance of the corrosion products. Corrosion

product resistance was attributable to the formation of product layers next to

the cathode surface. The resistance of the electrolyte was taken as a constant while the resistance of the corrosion products (R~) was modeled using

several different submodels. This included a constant value of R~ , one based

on Langmuir adsorption (or film formation) of corrosion products and one based

on uniform accumulation of layers of corrosion products.

All of the models predicted that the potential between electrodes, the

charge and the current decrease with time. The equivalent circuit submodel

for constant R„ was shown to analogous to the potential difference submodel.

For the film formation model the potential and charge decreased together but

the current fell faster because of the increase of R„ with time. Refinements

in the model were suggested which would take into account the nature of the

corrosion reactions, the charge they generate and the increase in electrolyte

concentration with time.

The coupling between the corrosion process, the blistering process, and

the permeation process was discussed. The coupling is complex and only the

separate processes were modeled in this paper. Future modeling and experimental efforts will be directed at elucidating a fuller understanding of the

nature of the coupling.

64
6 . NOMENCLATURE

a adsorption - diffusion constant, equation (15); activity

coefficient, equation (25)

a water activity coefficient within blister

a_ water activity coefficient in external solution

A flat area of blister on metal surface

b defined by (2n + 2 - x)

C dimensionless concentration, C./C. ; concentration of water,

equation (25)

C capacitance of charged electrodes

a

C. concentration of component i within coating

C. concentration of component i inside coating surface

10

C. critical value of C., equation (76)

d diameter of pigment particles

D diffusion coefficient or diffusivity

D. diffusion coefficient or diffusivity of component i

D. molecular (or Knudsen) diffusivity

E , E potential of anode and cathode, respectively

E^ 4 , E^„ initial potential of anode and cathode, respectively

OA OC

E , E Laplace transform of E and E , respectively

Ef final potential, equation (57)

H. Henry's Law constant for component i

i current

i initial value of current

o

65
k adsorption rate constant for component i ax

k , k anodic and cathode polarization rate constants, equations (51)

and (50) , respectively

k , k specific rate constants for the anode and cathode, respectively

k, desorption rate constant

d

k rate constant for osmotic pressure, equation (39)

P k , k , k rate constants for radius, area and volume, equations (36)

(37) and (38) , respectively

k' proportionality constant, equation (41)

k , k , k rate constants; equations (45), (46), (47) respectively

Z diffusion path length; thickness of corrosion products

L thickness of coating or film

L , L activation energia for anode and cathode, respectively A C

m molarity of impurity ion

m* saturation value of m

M mass of water in the blister

M amount of salt present at blister initiation site

s M solvent molecular weight

M salt molecular weight

s n number density of particles; summation index

N diffusive flux; number of occupied sites

N. diffusive flux of species i

l

N total number of sites

(p/v) volume ratio, pigment to vehicle

p. partial pressure of component i in external phase

xo

Q charge

Q initial value of Q

66
r characteristic dimension of blister

r, rate of dissolution based on surface

d

r. rate of adsorption of penetrant i per particle

R gas constant

R1 resistance of electrolyte solution

R„ resistance of corrosion products

(R ) ultimate or final value of R_

2 u 2

s Laplace transform variable

S surface area of blister mantle

S , S surface areas of anode and cathode, respectively

5 , S switches, figure 3

t time

t

1 dimensionless time

T absolute temperature

u transformation relation, equation (A-5)

u Laplace transform of u

V volume of blister

%

V solvent partial modal volume

x dimensionless distance, Z/L

Z depth into coating measured from free surface

GREEK LETTERS

a rate constant for film formation by Langmuir adsorption,

equation (58)

3, Y constants of proportionality, equation (59)

V gradient

6 characteristic film thickness for diffusion; potential difference

(E

A

- V

67
6 initial value of potential difference

o

e porosity of coating (volume fraction vehicle)

Att change in osmotic pressure

v number of ions formed from one molecule of impurity ion

(f> molar osmotic coeffient

p density of water within blister

p density of a pigment particle

P

p density of the vehicle (binder)

v

a constriction factor (equation 6)

T tortuosity factor (equation 6) ; time constant (equation 66)

T time constant based on final resistance; equation (69)

9 fraction of occupied sites, N/N

6 equilibrium surface coverage, equation (B-17)

68
7. REFERENCES

1. Perera, D. Y., and Van den Eynde, D. , "Alternation in Properties of

Organic Coatings for Buildings Due to the Leaching Process," Durability

of Building Materials and Components, ASTM STP691 , 698-710 (1980).

2. Kumms, C. A., "Physical Chemical Models for Organic Protective Coatings,"

Journal of Coatings Technology 52 (664), 40-53 (1980).

3. Ritter, J. J., and Kruger, J., "Studies on the Sub-Coating Environment of

Coated Iron Using Qualitative Ellipsometric and Electrochemical Techniques,"

Surface Sci . J?6, (1980).

4. Peterson, E. , Chemical Reaction Analysis , Prentice-Hall, New York, 1965.

5. Bruggeman, D. A., Ann. Physik 24, 636 (1935).

6. van der Meer - Lerk, L. A., and Heertjes, P. M. , "Blistering of Varnish

Films on Substrates Induced by Salts," J. Oil Col. Chem. Assoc. 58, 79-84

(1975).

7. van der Meer - Lerk, L. A., and Heertjes, P. M. , "Mathematical Model of

Growth of Blisters in Varnish Films on Different Substrates," J. Oil Col.

Chem. Assoc. 62, 256-263 (1979).

8. Robinson, J. M. , and Stokes, R. H. , Electrolyte Solutions , 2nd edition,

Butterworth, London, 1970.

9. Harned, H. S., and Owen, B. , The Physical Chemistry of Electrolytic

Solutions , 3rd edition, ACS Monograph No. 137, Reinhold Press, New York,

1958.

10. Bockris, J. O'M. , and Reddy A. K. N. , Modern Electrochemistry , Volume 2,

Plenum Press, New York, 1973.

11. Crank, J., and Park G. S., Diffusion in Polymers , Academic Press, New York,

1968.

12. Artamonova, R. V., et.al. , "Internal Stresses and Water Diffusion in

Polymers," (trans.) Vysokomol. soyed. A12, No. 2 (1970), pp. 336-342.

13. Uhlig, H. H. , Corrosion and Corrosion Control , 2nd edition, John Wiley

& Sons, Inc., New York, 1971.

14. Thirsk, H. R. , and Harrison, J. A., A Guide to the Study of Electrode

Kinetics , Academic Press, London, 1972.

15. Satterfield, C. N., Heterogeneous Catalysis in Practice , McGraw-Hill,

New York, 35-37, 1980.

69
APPENDIX A

SOLUTION TO MATHEMATICAL MODEL FOR

DIFFUSION OF WATER AND OXYGEN

THROUGH ORGANIC COATINGS

In this Appendix the analytical solutions to the mathematical models

developed in Section 3.1 are derived. These are the models which describe the

concentration of water and oxygen within the coating. The solutions are

summarized by equation number in Table 1 of Section 3.2, and are presented as

equations (14) through (24) in the text which follows. The solutions are

classified by time region of validity: short time, long time, or steadystate; by phenomena: with and without adsorption; and by location: within

the bulk of the coating and at the substrate-metal interface.

Equations (9) gives the partial differential equation which governs the

concentration profile of water and oxygen C. (Z,t) within the organic coating:

3C. 3 2 C. 6k.

E ir - D

i -r " -ir1 (1 - £) ci (9)

aL

Equations (10), (11), and (12) give the boundary and initial conditions:

C (0,t) - C. (10)

3C

(L,t) = (11)

3Z

C (Z,0) = (12)

70
The solution to Equation (9), subject to the conditions (10), (11), and (12),

gives the concentration profiles C. (Z,t) within the coating.

Introducing the dimensionless variables x = z/L, C = C./C. and

1 10

2 t' = tD./eL (equation 13), equations (9), (10), (11) and (12) become equations (A-l) , (A-2) , (A-3) , and (A-4) , respectively:

8C

3t'

9x

(A-l)

C (0,t') = 1 (A-2)

8C (l,f) = (A-3)

C (x,0) = (A-4)

The solution to Equation (A-l) subject to the conditions (A-2), (A-3) and

(A-4) gives the dimensionless concentration profile C (x,t') within the coating,

The parameter a is defined by Equation (15)

:

a = L

6k

a± (1-e)

dD

1/2

= L

6k

ai

dD (P/V) (Pv

/P

P

}

1/2

(15)

a is a dimensionless adsorption-diffusion constant which is fixed for any

coating system and environment. The paint films are presumed to contain a

uniform distribution of spherical pigment particles of diameter d. The assumption of particle sphericity is not critical to model development because the

concept of an equivalent shere (one having the same surface area as the non71
spherical particle) can be employed. The porosity £ is the volume fraction of

the coating occupied by the vehicle. Equation (2) is a volume balance made on

the coating:

£

1 + (p/v) (Pv

/P ) U)

Equation (2) is the basis for the equivalence of the second equality in

Equation (15)

.

A.l Long Time Model Solutions

The long time model solutions developed in this section are ones which

converge fastest at longer times. When the solution involves an infinite

number of terms fewer and fewer of these terms will be needed for convergence

as time increases.

By making the variable transformation from C (old dependent variable) to

u (new dependent variable) using the transformation relation:

2 , u = Ce

a fc (A-5)

equations (A-l) , (A-2) , (A-3) and (A-4) become equations (A-6) , (A-7) , (A-8)

and (A-9) , respectively.

2 IF = TT (A" 6)

3x

a

2 f u (0,t') = e

Z (A-7)

72
fj (l,f ) = (A-8)

u (x,0) = (A-9)

The boundary value problem expressed by these equations can either be solved

by either Fourier Series or Laplace Transform methods. The Laplace Transform

method is more general in that it can handle both long and short time solutions.

Thus it is employed here.

Taking the Laplace Transform of Equation (A-6) with respect to the time

variable t', the transformed equations become:

,2-

^-| - su = (A-10)

dx

u" (0,s) = 1/s-a

2 (A-ll)

^ (l,s) = (A-12)

QX

Here u is the transformed value of u. It is a function of x and s, where s in

the complex variable or transformed value of t'.

By standard methods the solution to Equation (A-10) is:

u (x,s) = A sinh /s~ (1 - x) + B cosh /s~ (1 - x) (A-13)

Evaluating the integration constants A and B using conditions (A-ll) and (A-12)

73
- , . cosh Js~ (1 - x)

. u (x,s) = ^ ~~ (A-14)

(s-a ) cosh /s

The inverse transform of Equation (A-14) can be found by evaluating it as

2 the product of two functions, f(s) = 1/s-a and g(s) = cosh Js~(1 - x)/cosh Ss~

,

and using the convolution theorem for the product of two functions of s:

u (x,t') = L

X [f(s) • g(s)] - / F (t'-u) • G(u) du (A-15)

Here the operator L indicates the inverse transform of the convolution

product f(s) times g(s)

.

2 , In Equation (A-15), F (t'-u) = e3

. G(t') can be evaluated from

g(s) using the method of poles and residues or from the Heavisides inversion

theorem:

oo 2 G(t') = 7T I (-l) n_1

(2n-l) exp [-

(2n~1)

7T 2 t'] cos [ (^f^) tt (1-x)] (A-16)

n=l

Performing the convolution integral in Equation (A-15) upon these functions:

co (_i)

n+1

( 2n-l) cos [ (!-*> (2n-l) ^ 2 ( -(2n-l)

2 / fc

, u(x,t) = tt I — ± ea

- e (A-17)

n=l 2 (2n-l) tt a H ;

Expressed in terms of the original dimensionless concentration C, equation (A-17)

takes the form:

74
2 -[a

2 + (2n-l)2 £-] t'

00 n+1

c (x t

. } = cosh a (1-x) _ ^ y (-1) (2n-l) e cos [(1-x) (2n-l)7T/2]

' cosh a ''£ 2 a + (2n-l) 7— (A-18)

4

Equation (A-18) is equation (14) of the main text of this paper.

This equation governs the concentration of water and oxygen within the

coating as a function of both time and position. Concentrations of the penetrants

water and oxygen, although they both are predicted to follow equation (A-8)

,

will differ because of their different values for the parameters C , D, and

k . This, in turn, through equation (15), will give different values of the

dimensionless parameter a. The additional parameters d, (p/v) , L and (p /p )

will vary with the nature of the coating system. Thus different coatings will

have different values of C and a and would be expected to perform differently

from one another in terms of their resistance to the penetration and permeation

of water and oxygen.

As mentioned previously, Equation (A-18) will take fewer terms to converge

at longer values of time and for this reason it is referred to as a "longtime" solution.

At very long dimensionless times t', Equation (A-18) reduces to:

c (Xj oo) = cosh a (1-x)

( cosh a

Equation (A-19) is Equation (23) of the main text of this paper. It governs

the steady-state concentration profiles within the coating. This relation can

also be derived more fundamentally by setting the accumulation term 8C/9t'

equal to zero in equation (A-9) and solving the resulting ordinary differential

equation.

75
The way in which the dimensionless concentration C varies with dimensionless time t* at the metal substrate interface (x=l) is given by:

2

oo (-l)

n+1

(2n-l) exp -[a

2 + (2n-l)

2 j- ] t

1 C (l,t») = sech a -

I

2 (A-20)

n=l 2 n 2 n

a + (2n-l) t—

Equation (A-20) is equation (17) of the main text. It governs the concentrationhistory at the metal substrate interface (x=l)

.

The steady-state concentration at the metal-substrate interface is given

by:

C (1,«) = sech a (A-21)

which is also equation (24).

In addition to the sub-models given here for the concentrations at the

metal substrate interface (x=l) , other sub-models can be derived for the case

where adsorption is insignificant (a=0) . Note that small a (tending towards

zero) is favored by small L, k , (p/v) and (p /p ) and large d or D. If

o. v p

k = then there is absolutely no affinity of the penetrant for the pigment

3. particles and a will be identically zero. For a = 0, equation (A-18) becomes:

00 2

C (x,t') = 1 - f I (-D

n+1

exp [-(2n-l)

2 |- t'] cos [(2n-l) \ (1-x)] (A-22)

n=l

Equation (A-22) is the same as equation (16)

.

Concentration histories at the metal substrate interface C (l,t') in the

presence of insignificant adsorption (a = 0) are given by:

76
°° n+1 2 2 c d.f) - i - 1 1 ^g-r- exP [-

(2n- 1

> n f] (a-23)

7T ^ , 2n-l

n=l

which agrees with equation (18) of the text.

A. 2 Short Time Model Solutions

The short time model solutions developed in this section are ones which

converge fastest at shorter dimensionless times t

1 . When the solution involves

an infinite number of terms, fewer and fewer of these terms will be needed for

convergence as time decreases. The short and long time solutions should give

the same numerical result since they are equivalent mathematically. At intermediate dimensionless times (t'~ 1) the series at short and long times should

be equally efficient in converging to a common numerical answer.

The solutions which converge fastest at the shortest dimensionless times

t' can also be obtained by Laplace Transform methods equation (A-14) can be

rewritten as:

-x •s -(2-x) /s u (x,s) = — (A-24)

. 2 -2/s-

(s-a ) (1 + e )

-2/s"

By dividing the numerator of Equation (A-24) by the term (1 + e )

:

u (x,s) =

I (-I)" exp [-(2n + x) 7s~ ] (-1) exp [-(2n + 2 - x) /s ] ,. ,_. n=0 2 +

2 U l ^> s-a s-a

The two separate terms in equation (A-25) , both of which are under the

summation sign, can be evaluated using the convolution integral, equation (A-15)

77
2 Letting f(s) = l/(s-a) and g(s) = exp [-b /s ], where b = (2n + x) in the

first summation and b = (2n + 2 - x) in the second summation, then it follows

a

2 b -b2 that, F(t') = e and G(t') = exp [-—p ]. Invoking the convolution

2 At'

integral, equation (A-15) , equation (A-5) and equation (A-25) are inverted to

give:

r (2n + 2 - x)

2

2 . i °° t exp - [- — + a u]

C (x,t') = — I (-l) n (2n + 2 - x) /

^

du

Ai\ n=0 u

'

., o r (2n + x)

2 2

t exp - [ T + a u] + (-l) n

(2n + x) /

^

du (A-26)

u

'

In equation (A-26) both terms are under the summation sign. Equation (A-26)

is identical to equation (19) . It is the general short time solution.

Equation (A-26) is a general solution to the diffusion problem governing the

dimensionless concentrations C within the organic coating as a function of

dimensionless distance x and time t'. It is mathematically equivalent to

equation (A-18) , or equation (14) . Together equations (A-18) and (A-26)

comprise a useful matched set of equations, one good for determining concentration profiles at long dimensionless times, (A-18) , and one good for determining concentration profiles at short dimensionless times, (A-26)

.

The way in which the dimensionless concentration C varies with dimensionless time t' at the metal substrate interface (x=l) is given by:

r (2n + l)

2

2 .

, °° t' exp - [-

— + a u]

C(l,t*)=—— I (-l) n

(2n + 1) / £». du (A-27)

y/lT n=0 u

78
Equation (A-27) can be shown to be identical to equation (21) . It governs the

concentration-history at the metal substrate interface (x=l) and is the short

time solution which complements equation (A-20) , the solution which converges

fastest at long times.

When adsorption of the diffusing species is insignificant, a = 0. Under

this condition the general short time solution, equation (A-26) reduces to:

00 t' 2 n t «-i\ 1 V t i\ n /o , o \ f (2n + 2 - x) du

C (x,t') = —: l (-1) (2n + 2 - x) J exp -

l - -> v— ~'j —r 4u 3 /2

/4tt n=0 u

t' 2

-L / n11 + /O _1_ \ f ( 2l1 + X ) du /A TON (-1) (2n + x) J exp —

'— -^j^ (A-28)

u

In equation (A-28) both terms are under the summation sign. Equation (A-28)

is identical to equation (20) . It governs the dimensionless concentration

profiles in the coating in the absence of significant adsorption onto the

pigment particles. Since it converges fastest at short dimensionless times

it complements equation (A-22) , which converges quickest at long dimensionless

times

.

Concentration histories at the metal substrate interface C (l,t') in the

presence of insignificant adsorption (a = 0) are given by:

C (l,t') = 2 £ (-l) n erfc ( 2n + 1

) (A-29)

n=0 2 7t~

The dimensionless concentration at the metal substrate interface C (l,t') is

expressed in terms of an infinite series of complementary error functions.

79
Equation (A-28) is identical to equation (22). It converges fastest at short

dimensionless times and thus complements equation (A-23) , the solution which

converges most rapidly at long dimensionless times.

80
APPENDIX B

SOLUTION TO THE MATHEMATICAL MODEL FOR

CORROSION PHENOMENA BENEATH AN ORGANIC COATING

In this Appendix the analytical solutions to the mathematical models

developed in Section 3.3 are derived. These are models which describe the

polarization which occurs when an electrical potential is dissipated in an

electrochemical cell. The models are the potential difference model (PDM) and

the equivalent circuit model (ECM). The equivalent circuit model has three

submodels depending on the nature of the resistance offered by the corrosion

products: constant resistance, Langmuir adsorption or film formation and

uniform accumulation of solid products.

B.l Potential Difference Model

In this model the driving force for corrosion is the difference in

potential 6 between the anode and cathode potentials, E and E , respectively.

Thus

sr - kc (ea

- V ' k

c

& (B-" V0) "

Eoc

and

^T " kA

(E

A

- V " k

A

6 CB-2>

E

A

( °> " E

0A

81
The initial conditions for equations (B-l) and (B-2) are E and E , respectively,

If the temperature within the electrochemical cell does not change then equations

(52) and (53) predict that the rate constant k and k will remain constant.

Under this condition equations (B-l) and (B-2) form a coupled set of linear

first order differential equations whose solution can be obtained by the

method of Laplace Transforms.

Taking the Laplace Transform of equations (B-l) and (B-2) and rearranging:

E

C

(S + k

C

) "

E 0C + k

C

E

A

<B"3)

and

E

A

(S + k

A> " E0A

+ k

A

E

C

(B -4>

Solving equations (B-3) and (B-4) simultaneously for the transformed potentials,

E

c

(s) and E (s) :

s + k k E

E c

•

s (s + r rn ^oc + tVtt1 (b "5) AC A

and

= E

0C

k

A

E

0C

(S + k

A + kC + E

OC

/E

OC> .. „

A s + k s (s + k ) (s + k + k ) l

'

Taking the inverse transform of equations (B-5) and (B-6)

82
lc + k -^

E

C

A CE

0C , kC

f

- E

0A ^ - (kA + V t v~ = k + k— +

k + k

( 1 ~ Y~ ) e (B " 7) E

0C

k

A

+ k

C

k

A

k

C

E

0C

and

k v _2A

E

A

A C E

0C

k

A , . E

0A . " (kA + V C

( 1 - == ) e " " (B-8)

E

0C

k

A

+ k

C

k

A + kC

E0C

Equations (B-7) and (B-8) are identical to equations (54) and (55) , respectively,

Since by convention, anode potentials are greater than cathode, E will decrease

exponentially and E will increase exponentially towards a final common potential E given by:

Ef

. ^

E

0C +

_

k

C

E

0A

(B _

9)

Equation (B-9) is identical to equation (57) . It is easily found from equations

(B-7) and (B-8) by letting t -> °°. E represents the steady state polarization

potential.

The driving force 6 = E - E is found by subtracting equation (B-7) from

equation (B-8) , when:

" (k

A +"^C* ' 6 " (E0A " E

0C'

6 (B-10)

Equation (B-10) is identical to equation (56) . The driving force 6 decreases

exponentially with time towards zero. Polarization is complete after a suitably

long time and the final potentials of the anode and cathode are equal to the

value predicted by equation (B-9).

83
B.2 Equivalent Circuit Models

The models derived in this section treat the polarization process in

terms of an equivalent electrical circuit [14] . Figure 3 is a schematic

representation of the circuit. The anode and cathode are considered as being

at opposite sides of a charged capacitor which has initial thermodynamic

potential 6 = E^, - E^ . In the presence of an electrolyte this potential is

o OA OC

dissipated across two resistances in series, R and R„ . Resistance R corresponds

to the resistance of the electrolyte, which is assumed to be constant.

Resistance R corresponds to the resistance offered by the corrosion products.

In the first submodel (constant resistance) R is taken constant. In the

other two models (Langmuir adsorption and uniform product accumulation) the

resistance R increases with time.

Consider the circuit depicted in Figure 3. At time zero switch S„ is

closed, switch S is opened and the circuit is complete. Application of

Kirchoff's second law gives:

i t (B. + R?

) + - / i dt = (60)

± Z L

q

or, since i = dQ/dt,

(R

i + V f

+ §

- ° (62)

The initial charge of the capacitor Q is given by:

Q = C 6 (63)

o

84
Equations (60) , (62) and (63) summarize the mathematical bases for the equivalent

circuit models developed for polarization.

B.2.1 Constant Product Resistance

When the products of corrosion have a constant product resistance R ,

equation (62) can be directly integrated to give:

^

" exp [ ( h

l\

2

) c 1 " ex

? [Z

f

1 ^ B-11>

The product (R.. + R„) C is the polarization time constant T. Equation (B-ll)

predicts that the charge on the capacitor Q will decay exponentially with time

towards zero with time constant T.

Since by definition i = dQ/dt, dif f erentation of equation (B-ll) yields:

j- = exp (-^ ) (B-12)

o

where the initial current flow i is given by:

6

'o

= (R

x +°R2

) (65)

Applying Ohm's law:

i 6 o o (B-13)

so that

85
2_ . i_ . |_ . exp <=£ , (B.14)

O O O

Equation (B-14) is identical to equation (64).

B.2.2 Langmuir Adsorption of Corrosion Products

The development of this submodel is based upon the formation of an adsorbed

film of reaction products at or in the vicinity of the electrode surface. The

products can accumulate as a monolayer of adsorbed ions or in a diffuse double

layer around the electrode. As the products build-up, their resistance R

increases towards an ultimate value R„ , corresponding to complete monolayer

2u

coverage.

The theory developed here is based on the classical Langmuir adsorption

theory for the physical adsorption of gases [15] . Consider the electrode

surface to contain a total number N of adsorption sites, and let N be the

t K

number of these sites which are occupied at any time t. Ions will adsorb onto

the surface at a rate which is proportional to the number of unoccupied sites

(N - N) and desorb at a rate proportional to the number of occupied sites N.

Writing the conservation law for ions arriving, leaving and being stored in

the film:

^7 = k (N - N) - k, N (B-15)

at a t d

where

k is the adsorption rate constant

ak is the desorption rate constant

d

Both rate constants have units of reciprocal time.

86
N

Let — " 9 be the fraction of the sites which are occupied. Then

N

t equation (B-15) can be rewritten as:

|f - k a - » - k e (B-i6)

The equilibrium surface coverage 6 can be found by setting d6/dt = in

equation (B-16)

6e " k-hr < B-17 > a d

where < 6 < 6 < 1 e Substituting equation (B-17) into (B-16) and integrating using the fact that

the surface is initially free of adsorbed ions:

6 "kfl t y- = 1 - exp [-f- ] = 1 - exp [ -(k + k ) t ] (B-18)

e e

Equation (B-18) predicts that the surface coverage will increase towards its

final or equilibrium value 6 . If the adsorption process is irreversible then

k, = 0, 6 =1 and equation (B-18) reduces to:

d e

9 1 - exp (-k t) (B-19)

The resistance attributable to the film will be in direct proportion to the

number of absorbed ions. Thus:

87
|- = 1 - exp (-at) (B-20)

R

u

Equation (B-20) is identical to equation (58) of the text.

We can now identify precisely the meanings of the constants R and a in

equation (58) by analogy to the Langmuir adsorption theory. Thus, the ultimate

resistance R corresponds to the equilibrium coverage 9 where the constant a. u ° e corresponds to the adsorption constant divided by the equilibrium coverage,

i.e., k /9 . a e Substituting equation (B-20) for R into Equation (62) a variables

separable differential equation results, which on integration becomes:

1

„ x R _ R ax 2- - f

- [a+ ^)e--^] » (B-21,

o o 1 1

T is the modified time constant C (R_, + R ). Equation (B-21) is identical

u 1 u

to equation (67). The first equality in equation (B-21) results from the

ratio of Q = C6 and Q = C6 . The capacitance of the electrochemical cell,

o o

C, is presumed constant.

Differentiating equation (B-21) and using the fact that i = dQ/dt:

R R ax

[ (1 + -S ) e at

- -H ] R R

%- = — (B-22)

Xo R . i + -±a- e_at)

R

l

Equation (B-22) is identical to equation (68)

88
Dividing equation (B-21) by equation (B-22) and using (B-20)

|- = ~ (1 +

^ ) (B-23)

o 1 Rn o o 1

Equation (B-23) is identical to equation (70).

Large values of the constant a correspond physically to rapid adsorption

and/or desorption of adsorbed ions at the electrode surface (i.e. large k + k,),

a d or to low equilibrium coverage (small 9 ) at high adsorption rates (large k )

Under these conditions equations (B-21) and (B-22) become equations (B-24) and

(B-25) , respectively.

n z R u

^o o 1

-u f-) R u

X o R

l

Equations (B-24) and (B-25) are identical to equations (71) and (72) . Equation

(B-25) follows directly from Equation (B-24) and i = dQ/dt. Equation (B-24)

can be found by the application of L'Hospital's rule to equation (B-21) as

a -> °°.

B.2.3 Accumulation of Solid Corrosion Products

If solid products of corrosion deposit or precipitate near or on the

electrode surface then the resistance R caused by the accumulation of these

products will increase with time. In some instances adsorbed films more than

several monolayers thick can behave in a similar manner to solid corrosion

products.

89
The resistance of the corrosion layer, if it remains homogeneous, will

be proportional to this thickness. The volume of corrosion products laid down

will be proportional to this thickness and by Faraday's law to the total

amount of current which has flowed through the electrical circuit to any time

t. Thus:

t t dQ

R = y / i dt = -y / — dt = y (Q - Q ) (B-26)

where

Y is a rate constant for the accumulation of solid reaction products.

If equation (B-26) is substituted into equation (62) , and the resulting

variable separable differential equation is integrated, then it follows that:

Q 6 CY (Q - Q) - t

Equation (B-27) is identical to equation (73) . It is a transcendental equation

in the variable Q and thus must be solved by trial and error procedures in

order to determine Q(t) and 6(t) . The corresponding formula for the current

ratio i/i is given by:

o

r-

= q q— (B - 28)

o C [R

± + YQQ " Y 3- ] o

Equation (B-28) is equation (74)

90
Once Q(t) is known, R~(t) can be found from equations (B-26) and (B-27).

R„(t) will in turn determine the relative thickness of the corrosion products

attached to the electrode.

91
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN 1150

2. Performing Organ. Report No 3. Publication Date

Sept. 1982

4. TITLE AND SUBTITLE

Mathematical Models for the Corrosion Protective Performance of Organic Coatings

5. AUTHOR(S)

James M. Pommersheim, Paul G. Campbell, and Mary E. McKnight

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Same as 6.

10. SUPPLEMENTARY NOTES

~2 Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less (actual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Mathematical models were developed for conceptual models describing

the principal phenomena that occur in the corrosion performance of polymeric coatings. These include models for water and oxygen permeability

through organic coatings, models for the growth of blisters beneath

coatings, and preliminary models for the polarization occurring at the

electrode surfaces. Results predicted by the models are discussed in

terms of the improvement of the protective function of the membrane.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

absorption; adhesion; adsorption; conceptual models; corrosion; mathematical

models; organic coating; osmosis; osmotic pressure; oxygen; permeability; pigment;

protective performance; substrate; vehicle; water.

13. AVAILABILITY

Q<[] Unlimited

| | For Official Distribution. Do Not Release to NTIS

|~X~~] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

^] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

99

15. Price

$5.50

USCOMM-DC 6043-P80

1>U.S. GOVERNMENT PRINTING OFFICE. 1982-360-997/2229




NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research of the National Bureau of Standards reports NBS research and develop- ment in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry,

engineering, mathematics, and computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement

methodology and the basic technology underlying standardization. Also included from time to lime are survey articles on topics

closely related to the Bureau's technical and scientific programs. As a special service to subscribers each issue contains complete

citations to all recent Bureau publications in both NBS and non- NBS media. Issued six times a year. Annual subscription, domestic

$18; foreign $22.50. Single copy, $4.25 domestic; $5.35 foreign.

NONPERIODICALS

Monographs—Major contributions to the technical literature on

various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with in- terested industries, professional organizations, and regulatory

bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special publications

appropriate to this grouping such as wall charts, pocket cards, and

bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and

studies of special interest to physicists, engineers, chemists,

biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide program coordinated by NBS under

the authority of the National Standard Data Act (Public Law

90-396).

NOTE; The principal publication outlet for the foregoing data is the Journal of Physical and Chemical Reference Data (JPCRD)

published quarterly for NBS by the American Chemical Society

(ACS) and the American Institute of Physics (AIP). Subscriptions,

reprints, and supplements available from ACS, 1 155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research results,

test methods, and performance criteria related to the structural and

environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of

the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing

organizations. Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the con- sumer. Easily understandable language and illustrations provide

useful background knowledge for shopping in today's technological marketplace. Order the above NBS publications from: Superintendent of Documents. Government Printing Office. Washington. DC 20402. Order the following NBS publications—FIPS and NBSlR's—from

the National Technical Information Services, Springfield. VA 22161

.

Federal Information Processing Standards Publications (FIPS

PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register

serves as the official source of information in the Federal Govern- ment regarding standards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amended,

Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or

final reports on work performed' by NBS for outside sponsors

(both government and non-government). In general, initial dis- tribution is handled by the sponsor; public distribution is by the National Technical Information Services, Springfield, VA 22161,

in paper copy or microfiche form.
U.S. Department of Commerce

National Bureau of Standards

Washington, D.C. 20234

Official Business

Penalty for Private Use $300

postage And fees paid

u s department of commerce

COM-215

THIRD CLASS

BULK RATE