Simulations of Nanoscale and macroscopic property changes on coatings with weathering.

Coatings are designed for and applied on a surface for both aesthetics and protection of the substrate. Many properties are measured to indicate performance, and eventual failure, of a coating under these two broad categories. Monte Carlo simulations have shown success in predicting trends in macroscopic properties during exposure. The Central Limit Theorem (CLT) is applicable because damage made to a coating can come from the accumulation of a vast number of very small damage events. Application of the CLT to property equations has generated additional equations for the prediction of properties of a coating with exposure, including measurable properties such as gloss, color, fracture toughness, and contact angle. These equations, when fitted to measured data, provide insight into the mechanisms of degradation processes, since the fitting parameters are physically based. They also offer a means to scale accelerated testing measurements to early field measurements of the property of interest for predicting lifetime in varied environments.

Keywords: Appearance, corrosion, durability, service life prediction, weatherability, molecular modeling, simulation, color

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The degradation of a coating is produced by the accumulation of a large number of random, small events caused by a large number of ultraviolet (UV) photons or aggressive molecules, e.g., water or acid pollution. For example, the terrestrial solar UV flux (1) is approximately equivalent to [10.sup.27] photons/[cm.sup.2]/year. After many small damage processes accumulate to produce the measurable damage, the coating features, e.g., surface height distribution or distribution of damage within the bulk, become normally distributed, according to the Central Limit Theorem (CLT). The CLT states that a variable or property resulting from the accumulation of numerous, random events is normally distributed. The CLT, as applied in counting statistics, also gives the equality between the variance of a measured change and the average number of changes. The variance of the surface topography or photochemical change, for example, will be used to relate the microscopic changes to large scale changes such as gloss or color. The CLT was applied throughout this study to predict measurable properties explicitly in algebraic equations. Information available from fitted values of parameters in these derived equations is descriptive of the degradation process, and is associated with a definable physical change such as an average size of polymer fragment released during photoscission. These parameter values, taken from the equations described below, allow one to predict behavior, extrapolate, interpolate, and compare different materials if there is an understanding of their process parameters. These equations offer the possibility of being able to discriminate when different mechanisms become dominant during the degradation process. Details of the derivations and assumptions are described elsewhere. (2)

Developing a physically based equation that describes a behavior or property of a coating allows one to interpolate, extrapolate, and characterize which aspects of the coating are important to a particular design property. In addition, such understanding may also delineate the conditions under which tracking one property can be used to follow the deterioration in another. Modeling is the Occam's razor of design; via simulation, one may find the minimum number of mechanisms necessary to accurately track and predict the behavior of interest. The process of modeling also helps quantify under what conditions a particular property may be expected to follow a prediction.

This article shows how a wide variety of measurable properties may be modeled through equations based on the CLT. Very simple equations were used here to compare with experimental data from other studies, in order to validate the reasonableness of their functional form. This demonstrates the broad applicability of the CLT to understand the kinetics of degradation and thus provide service life estimation. Explicitly predictive applications are left for future research. Monte Carlo simulations use the same rationale of applying random repeated degradation events in computer models. The Monte Carlo simulations are general and flexible in modeling, for example, in a situation arising before coating morphology becomes determined by the Normal (Gaussian) distribution, or where many processes compete for dominance (though the Monte Carlo often requires more input).

[FIGURE 1 OMITTED]

The property predictions were divided into two categories. The first included properties impacted mostly by changes on the surface, both chemical and topological. Second, the methodology surveyed processes that occur through the bulk of the coating, including well-known examples such as diffusion. Previous measurements taken at North Dakota State University for various studies and published in various forms were used to show how the equations represent experimental data, (3,4) including data from studies that employed alternating cycles of QUV[TM] and Prohesion accelerated weathering (ASTM 5894). The coatings used to test the equations and methods were a polyurethane unicoat (DEFT 03-GY-374, Navy TT-P-2756) and a glossy topcoat (polyurethane DEFT 03-GY-277, MIL-C-85285B Type I) over a primer (epoxy-polyamide DEFT 02-Y-040, MIL-P-23377G Type I Class C).

SURFACE DOMINATED PROPERTIES

The properties discussed in this section were dominated by surface changes that may have been due to both chemistry and topography. Surface roughness is a dominant influence of changes in gloss, (5) particularly in non-pigmented systems. Changes in contact angle with degradation have a component related to surface roughness (6,7) and a part due to changes in surface chemistry. Fracture toughness may be dominated by cracks originating in surface defects. (8-10)

A Monte Carlo degradation model has been developed that combines both topological and bulk changes. However, one of the limitations in the advance of the Monte Carlo approach is sufficient data for comparison of several concurrent properties for the same material, or details sufficient to reasonably generate the necessary input parameters. The use of the less general equations given below allows for fitting of parameters to generate the necessary input for use in more detailed modeling.

Contact Angle

The contact angle of water on the coatings was measured during exposure testing using an FTA 125 contact angle/surface energy analyzer (First Ten Angstroms, Portsmouth, VA). The change in contact angle was due to increased roughness and chemical changes on the surface. In deriving the equation below, it was assumed that the fraction of the surface that was oxidized reached a balance with the material ablated, and that the change in contact angle was dominated by the increase in roughness. Chemical changes or erosion that changed the material composition of the surface, i.e., significant change in pigment exposed, would require additional corrections or the use of a Monte Carlo simulation, which is being investigated. The effect of the surface roughness on contact angle is given by (6,7):

cos[theta] = r([[[gamma].sub.solid-vapor] - [[gamma].sub.solid-liquid]]/[[gamma].sub.liquid-vapor]) (1)

The [gamma]s are the surface-free energy between the subscripted phases. The entire term within the brackets consisting of the surface energies may be used as one fitting parameter for data manipulation. The contact angle is represented by [theta]. The roughness factor, r, actual surface area divided by projected area, is calculated in terms of the surface slopes, then applying [m.sup.2]=2[[sigma].sup.2]/[a.sup.2] (6) where m is the slope, [sigma] is the square root of the variance in height of the surface, and a is autocovariance of the surface. (5)

cos([theta](t)) = ([[[gamma].sub.solid-vapor] - [[gamma].sub.solid-liquid]]/[[gamma].sub.liquid-vapor])[square root of (1 + [4k't/[a.sup.2]])] (2)

The period of weathering, t, is assumed to be much longer than a cycle in the accelerated weathering protocol; k' is another parameter which must be obtained by fitting the data at this juncture. It is derived from the CLT and properties of the Gaussian distribution and describes how much the surface roughens per unit time. (2) Within the equation, k' is the parameter common to equations for the contact angle, gloss, and fracture toughness which are derived by assuming that increase in surface roughness caused by weathering degradation is the common, controlling influence. Figures 1 and 2 show how well equation (2) represents data. The fit for the polyurethane unicoat is reasonable within the scatter in the measurements. The fit for the gloss topcoat is poorer statistically, probably due to the larger scatter in the data resulting in a lower value of [R.sup.2]. If the contact angle is used to predict the end of useful service life, equation (2) represents an equation and method to predict the remaining protective life derived from meaningful physical mechanisms and statistical parameters; it is not an empirical construction for representing the data. The data is presently fit based only on topological roughness changes. Chemical changes can also be incorporated into equation (2) if assumptions are made about the relative fraction of the phases "as a function of time."

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Gloss

The gloss of a coating is frequently used as a nondestructive assessment of the progression of weathering degradation in the coatings industry. A Novo-Gloss[TM] meter (Rhopoint Instruments Ltd., East Sussex, UK) was used to measure gloss on the panels of the two coatings. Equation (3) was derived (2) based on gloss changes due to roughness increases. (5) Again, the assumption is that gloss is dominated by surface reflections.

[R.sub.s](t)/[R.sub.s](t = 0) = exp(-(4[pi]/[lambda])[.sup.2]k't) (3)

In this equation, [lambda] is the wavelength of light, [R.sub.s] is the scattered reflectance, and k' and t are the same quantities used in equation (2). Figure 3 shows the measured gloss and the fitting curves for the two coatings. The fits of the simple exponential in equation (3) to the data are quite satisfactory. Note that the fitting is done by applying a representative wavelength in equation (3), whereas a more complete fit would sum weighting factors over the glossmeter light source spectra and response. In this case the gloss is assumed to be dominated by specular reflectance.

Fracture Toughness

A crack in a protective coating produced by fracture or fatigue might provoke an adhesion failure or allow aggressive material to reach and possibly corrode the substrate. Predicting the relative fracture strength is useful for predicting lifetime in situations where cracking defines a major mode of failure resulting in the loss of the barrier properties of the coating. Here, deterioration in the relative fracture strength of two coatings was predicted based on values of parameters deduced from the measurements of changes in gloss with exposure. As an estimate of the reduction in coating strength, we used the Griffith failure criterion and assumed that the failure was initiated by a flaw or crack generated at the surface of the coating by the degradation processes, as is often found in polymer materials subject to weathering exposures. (8-10)

[[[sigma].sub.G](t)]/[[[sigma].sub.G](t = 0)] = (1 + k't/[[sigma].sub.initial.sup.2])[.sup.-1/4] (4)

[[sigma].sub.G] is the Griffith stress, which is the critical stress above which the crack propagates to failure, i.e., the fracture strength. Representing the fracture strength as a fraction of its initial value eliminates the need to know the other parameters in the Griffith failure criterion. These are Young's modulus and the fracture surface energy per unit area. The determining change in surface roughness is incorporated via the flaw size for a "penny-shaped" crack. The coating has failed when the flaws increase to the extent that it cannot withstand the stress state in which it finds itself. Changes in Young's modulus or surface energy, and thus absolute values of fracture strength, are not included at this stage of the modeling. The initial roughness of the unweathered coating, [[sigma].sub.initial], can be measured or, as here, combined with k' into a single constant for demonstrating the trend in the predicted fracture strength.

[FIGURE 4 OMITTED]

Figure 4 predicts the relative fracture toughness of the two coatings shown in the Contact Angle and Gloss sections of this article. No fracture toughness data was available for these coatings, but if measurements of the unweathered coatings could be made, the scaling factors from Figure 4 could be applied.

Equation (4) is based on the robust Griffith failure criterion and might be used to smooth experimental data (with the scatter typical of fracture data) to more accurately predict lifetime based on a relative fracture strength limit. Figure 5 shows the fit of measured fracture data of Nichols. (11) The fit allows for the extrapolation of the measurements to longer times in order to predict lifetime. The equation also allows for an estimate of the time behavior of the fracture toughness of a polymer based on the initial measurement of fracture toughness and an easily accessible measurement (gloss) for the field, uncontrolled exposure.

Correlation Between Surface Properties

Being able to correlate a nondestructive measurement to a crucial property is useful since it allows for not only the prediction of service life based on an a priori estimate of the service environment, but also for the ability to monitor and adapt the predicted service life based on the actual service environment without having to explicitly monitor that environment. Thus, the gloss or wetting angle (nondestructive tests) may be tracked with exposure and used to assess the progress in a critical property that might otherwise necessitate destructive testing, e.g., fracture strength. In circumstances where it is not practical or possible to track all the relevant environmental variables, the gloss can be measured and used to scale the deterioration rate and, thus, the severity of the environment based on a known or predicted deterioration rate. Presumably, if the severity of the environment can be assessed, then the rate of deterioration in other properties (besides gloss) also may be scaled appropriately in the environment in question.

As an example, the measured gloss from Figure 3 was used to get a fitted value of the variable k' to predict fracture strength in Figure 4. Any of the properties that can be predicted by the CLT and simple theory approximations [equations (2-4)] can be solved in terms of any other property. An example of an easily measured property that might be used to predict a less accessible property is using reflectance to predict fracture strength (assuming that they are both dependent on surface degradation). Since the algebraic equations are simple, in this case, the fracture toughness can be explicitly expressed as a function of reflectance. Thus, solving for k't in terms of reflectance in equation (3) and substituting it into equation (4) gives:

[FIGURE 5 OMITTED]

[[[sigma].sub.G](t)]/[[[sigma].sub.G](t=0)] = [1 + ([lambda]/[4[pi][[sigma].sub.initial]])[.sup.2] [ln[[R.sub.s](t = 0)]/[[R.sub.s](t)]]][.sup.-1/4] (5)

Further, in terms of the reflectance of a perfectly smooth surface of the same material, [R.sub.0]:

[[[sigma].sub.G](t)]/[[[sigma].sub.G](t = 0)] = (1 + ln[[R.sub.0]/[[R.sub.s](t = 0)]]ln[[[R.sub.s](t = 0)]/[[R.sub.s](t)]])[.sup.-1/4] (6)

In these circumstances, there would be no need to know the initial roughness explicitly to predict fracture strength from gloss. [R.sub.0] would be available from Fresnel's equations if we knew the refractive index of the material. Over a large portion of the reflectivity range the relative fracture toughness changes only slightly. This leaves only a very small exposure period early and late in coating life when the reflectivity (for a given illumination spectrum) is a sensitive indicator of fracture strength. More sensitivity might be gained if reflectance at various wavelengths could be used, a short wavelength early when the surface was relatively smooth and longer wavelengths as the surface roughens. Another use of the Monte Carlo approach may be to choose useful "pairs" of properties where the indication of one, or the other, is more sensitive to the extent of degradation.

BULK PROPERTIES

Bulk properties of a coating may change during exposure as the result of the accumulation of many individual events, just as the surface does. The CLT is the basis for such well known behaviors in bulk properties as Lambert-Beer's Law of optical attenuation and Fickian diffusion. The motion of the diffusing specie or the photon is consistent with a random walk problem, which is well represented by CLT. The additional step of chemical change due to diffusion of a reactant or photochemistry is another random process, usually with a low probability. The two coupled events, thus, frequently result in measurable properties that are linked to the CLT. We will produce equations based on this assumption and compare them to measured quantities. Processes relevant to changes in the bulk properties, such as color or resistivity, of a coating will be discussed in the following section.

Yellowing

A similar methodology can be applied to bulk processes as to surface phenomena. The number of photons passing through a material is huge, as is the number of possible absorbing sites. Attenuation of photons through bulk material can change various properties of the coating. A single photon may create or destroy a chromophore, or scission a polymer chain, or initiate other processes, or produce no effect. In addition, an initial photon may excite a bond which requires a second photon absorption of a given (perhaps different) energy to scission or become a chromophore. The yellowing of a polymer is an example modeled here as the absorption of a single photon. A similar approach would be relevant to modeling the changes in absorption with exposure measured by other forms of spectroscopy, e.g., infrared or UV absorption.

Yellowing, or some other change in color, or changes in optical density, are often related to the density of the chromophores present. In this article, we assumed that the yellowing was due to a chromophore produced by the photo-transition of a precursor molecule to a chromophore that can absorb energy in the visible spectra (see, for example, reference 12).

First, assume that there are a fixed number of initial sites, [N.sub.p0], of chromophore precursors per volume that can absorb a photon and be converted into a chromophore. The change in the number of precursors is then:

dN(x,t) = -N(x,t)[alpha][phi]dt (7)

The attenuation coefficient, [alpha], is the cross-section of chromophore precursors (an area). [phi] is the photon flux (in photons/[cm.sup.2]/sec), convolved with the quantum efficiency as a function of the wavelength. We assume for the first estimate that the cross-section of a chromophore precursor and the photon flux is constant with time. As we are interested in the total depth integrated chromophore concentration, the depth profile of the chromophore concentration can be easily calculated. Integrating to give the total number of chromophore precursors converted to chromophores as a function of exposure time, t, and integrating through the thickness:

[N.sub.p](t) = [N.sub.p0][e.sup.-[alpha][phi]t] (8)

[N.sub.p](t) is the number of chromophore precursors present at a chosen exposure time. The total number of chromophores produced may be calculated by subtracting the number of converted precursors from the initial total number of precursors. Thus the number of chromophores, N, increases as:

N(t) = [N.sub.p0](1 - [e.sup.-[alpha][phi]t]) (9)

The color change increases essentially linearly with time, while the number of susceptible sites is very large in comparison to the number of sites converted to chromophores. This is found by expanding the exponential in equation (9) and assumes the argument of the exponential is much less than 1. The approximate, linear relationship is given in equation (10).

N(t) = [N.sub.p0][alpha][phi]t (10)

[FIGURE 6 OMITTED]

Published data (1,13-15) (Wypych (1) p. 224; Rivaton (13) Figures 4, 5, 7, and 12; Davidson (14) Figure 27; and Das (15) Figures 9 to 12) appears well represented by equation (9). Several predictions can be made based on equations (9) and (10). For a fixed number of initial chromophore precursors, the color may saturate to a maximum yellowness with time (polystyrene, polyoxymethylene, and acrylonitrile-butadiene-styrene copolymer in figures 13.6 and 13.7 of Wypych (1) (additionally, Rivaton (13) pp. 127-143; Allen (16) p. 1; and Das (15) p. 11). A second case is where the number of chromophore precursors is very large in comparison to their absorption. In this case, the color increases roughly linearly with time, as in equation (10) (polyvinyl chloride and polymethylmethacrylate in figures 13.6 and 13.7 of Wypych (1)). The third case is where there are few chromophore precursors with a large absorption efficiency. In this case, the color changes nearly instantaneously to a small value that does not increase significantly with time (polyethylene in figures 13.6 and 13.7 of Wypych (1)). Changes in infrared absorption peaks during degradation show similar patterns of change. (17-19) Knowing which case a particular polymer follows explains much about the process dominant in that system.

[FIGURE 7 OMITTED]

Many complications are obvious. Diffusion can move chromophores, leaving the coating in an inhomogeneous state. Ablation or dissolution of the surface material may preferentially remove one type of material, impacting the original volume fraction of chromophores. Absorption of a photon may remove the chromophore precursor without producing a chromophore, which can be accounted for by attenuation and loss of chromophore precursors. Color change associated with the oxidation of a substrate under a clearcoat must have both color changes considered separately. Pigments add more complexity. These complications can be modeled if details of the behavior are known. Again, behavior that deviates from the prediction helps indicate that other processes of possible interest are present.

[FIGURE 8 OMITTED]

Undoubtedly, other contributions are of varying importance in different systems. Several aspects of reflectance color change are well known. Surface roughness affects the relative scattering efficiencies at different wavelengths, which in turn alters the color. Additionally, reflectance from a transparent coating is affected by the changes in color of the substrate, for example, due to corrosion of a metal substrate. Many aspects can be included in Monte Carlo simulations.

Equation (9), given here for the reduction in chromophore precursor concentration, could equally be applied to simple cases of crosslink scission by ultraviolet radiation or other aggressive environments. In this case, one would find an exponential decrease in crosslink density with exposure time that follows first order chemical kinetics. Such an exponential trend in crosslink density is well known (20,21) and has been modeled in a variety of approaches. (22,23) Again, if it is true that the chemical changes that are responsible for a color change are also responsible for a change in crosslink density, then the results from a nondestructive measurement may be applied to estimating the kinetics of polymer network breakdown.

[FIGURE 9 OMITTED]

Electrochemical Impedance Spectroscopy Response

Coatings, beyond their aesthetic designs, are often required to be protective of the underlying metal. Corrosion is a result of coating failure which can lead to structural failure of the substrate material. Therefore, understanding and predicting the time that a coating can protect a metal is both a cost and safety issue. The change in electrical impedance of a corrosion-protective coating is frequently used to predict time-to-failure, and even to define the failure condition. Combining diffusion of water into a coating with the dielectric properties of the constituent regions of the coating is used here to model its response. Simple Fickian diffusion is the result of penetrant molecules spreading through a material by random processes so that the kinetic variation of concentration, i.e., varying with the square root of time, is another manifestation of the CLT.

The behavior of a coating is often assessed via electrochemical impedance spectroscopy (EIS). (24,25) An alternating potential of various frequencies is applied to a solution over the coating and the current is detected at the metal substrate. The coating is described both by its resistance and capacitance while it remains intact. In this section we applied a simple and frequently employed model, the Randles equivalent circuit model, and examined the effect of diffusion of water into a coating and its effects on the various equivalent circuit elements. Measurements of the EIS data showed a great deal of variance. (26) The ability to derive a predictive equation and behavior can also assist in separating random sampling variances that may occur over the course of a test from failure modes connected with intrinsic flaws in the coating.

A Randles equivalent cell includes a relatively small solution resistance (representing the electrolyte external to the coating) in series with a capacitor that is in parallel with a resistor that represents the material properties of the coating. The impedance of the Randles circuit, neglecting the small solution resistance is:

Z = 1/[[1/R] + j[omega]C] = R(1 - j[omega]CR)/[1 + ([omega]CR)[.sup.2]] (11)

The resistance and capacitance of a material can be related to intrinsic material properties (relative dielectric constant, [[epsilon].sub.R], and resistivity, [rho]) and physical dimensions (area of EIS cell, A, and thickness of the region, d) through equations (12) and (13). The permittivity of free space (8.854 x [10.sup.-12] F/m) is [[epsilon].sub.0]. The relative dielectric constant and resistivity of materials are weak functions of frequency over the standard test spectrum (0.01-[10.sup.5] Hz), and are assumed not to vary significantly for homogeneous materials.

C = [[[epsilon].sub.0][[epsilon].sub.R]A]/d (12)

R = [rho]d/A (13)

Figure 6 describes the partitioning of the coating into distinct regions. The top region is the ablation zone. This may describe the material exposed to ultraviolet radiation, etc. that is physically eroded, solubilized, or pho-tooxidized and released from the bulk material. The thickness decrement due to these effects directly changes the resistance in proportion to the change in thickness [equation (13)]. The capacitance, applying equation (12), is proportionally increased by the thickness decrease. The net effect is to increase the spread in high frequency Bode plot curves with time. Ablation might be, to some extent, compensated by any swelling of the coating as it absorbs water. In this study, the ablation zone is neglected since many studies are concerned with coatings, e.g., primers, for which this may not be relevant.

A second region (Figure 6) is the diffusion-affected zone. This region is where the electrical properties change as coatings absorb water. A small volume fraction of water in a polymer reduces the resistivity by several orders of magnitude, depending on how the water is distributed physically and, thus, in terms of modeling, the effective media theory (EMT) is employed. In most models, and assumed here, the resistivity of the coating overtaken by the diffusion zone front is negligible when compared to that of the remaining pristine coating. An EMT of the Maxwell type (27-29) assumes that the secondary phase, water, clumps as unconnected spheres, and thus maximizes the resistivity of the diffusion-affected zone for a specific water concentration.

The third layer is the pristine coating. This is simply the original coating material, which still possesses its original electrical bulk properties.

The interfaces are assumed to be coplanar with the substrate. This assumption can be relaxed if detailed physical modeling in a finite element code, for example, suggests that aspects such as intrinsic partial defects, fillers, pigments, or polymer defects due to solvent escape paths need to be incorporated for accurate predictions.

An equivalent circuit is proposed as the starting point for the description of a degraded coating (Figure 7). It has two Randles cells in series; one describing the electrical properties of the diffusion affected material and the second Randles cell describing the resistance and capacitance of the shrinking pristine material. Other equivalent circuits may be developed after detailed investigation of the distribution of water in coatings.

In the low frequency regime, 0.01 Hz for example, the circuit impedance is dominated by the resistance. Equation (14) describes the impedance, with the insertion of equations (12) and (13) into equation (11).

log(|Z|) = log(R) - [1/2]log(1 + ([omega]RC)[.sup.2]) = log([rho]d/A) - [1/2]log(1 + ([omega][[rho]d/A][[[epsilon].sub.0][[epsilon].sub.R]A]/d)[.sup.2]) (14)

[FIGURE 10 OMITTED]

R is taken as the resistance of the remaining pristine coating, although the water-saturated resistance, which is generally negligible, could be included as the pristine coating becomes very thin. Here, [omega] is the angular frequency (2[pi].frequency) and C is the capacitance. The second term in equation (14) can be estimated for the low frequency of interest, 0.01 Hz, [[epsilon].sub.R] is the relative dielectric constant (~ 5 for many polymers) and resistivity [rho] (roughly 7 x [10.sup.9] [ohm]xm, taken from EIS spectra measured for the polyurethane unicoat topcoat). The second logarithmic term is 2 x [10.sup.-4] and, thus, negligible relative to the first log term.

The advance of the diffusion-affected zone interface may be controlled by one of two diffusion mechanisms. The first is Fickian diffusion, where the front is driven by the concentration gradient of the diffusing species (water) and the advance of the front is proportional to the square root of time. The second is the combination of plasticization and entropy, resulting in the movement of the water front being proportional to time, i.e., Type-II diffusion. The thickness of the pristine coating is given in equation (15) based on the diffusion layer (Fickian or Type-II) consuming the pristine coating.

d(t) = [d.sub.0] - ([K.sub.1]t + [square root of ([K.sub.2]t)]) (15)

[K.sub.1] is the proportionality constant for interface motion controlled by Type-II diffusion. [K.sub.2] is the proportionality constant for Fickian diffusion. Sometimes only [K.sub.1] or [K.sub.2] is used, with the other constant set to 0. Substituting equation (15) into equation (14) and neglecting the second log term of equation (16) gives:

log(Z) = log[[[[rho][d.sub.0]]/A](1 - ([K.sub.1]t + [square root of ([K.sub.2]t)]) / [d.sub.0]] (16)

For small [K.sub.1] or [K.sub.2]

log(Z) = log[[[rho][d.sub.0]]/A] - 1/ln(10)

([K.sub.1]t / [d.sub.0] + [square root of ([K.sub.2]t)] / [d.sub.0] + [K.sub.2]t / 2[d.sub.0.sup.2]) (17)

It has been found empirically (30,31) that the logarithmic plot of low frequency impedance as a function of exposure time shows a linear decrease and is explained by equation (17) for short times. The decrease in low frequency impedance accelerates after the initial linear rate of decrease (Figure 8). Empirically making a linear fit of the log (impedance) as a function of the exposure time shows how quickly the low frequency impedance drops with exposure, which may be useful if the coating failure is described by the impedance at low frequency. However, the linear approximation is not conservative. The coefficient from fitting the data more completely with equation (16) not only describes the low frequency impedance decrease with time, but it also gives information on the eventual failure rate. The fitting parameters [K.sub.1] or [K.sub.2] describe how fast the diffusion-affected zone interface advances.

FITTING TO EXPERIMENTAL DATA

The unknown parameters in equation (16) are physically based, not just empirical curve fitting coefficients. Equation (16) was used to fit the low frequency impedances for the two coating systems, the polyurethane unicoat and the high gloss topcoat, and to test for both Type-II diffusion and Fickian diffusion as the mechanism that advances the diffusion-affected zone.

[FIGURE 11 OMITTED]

The initial trend in impedance of the coating with thickness, [d.sub.0], is the variation that describes the resistance of the bulk material, without saturation. The second fitting parameter, [K.sub.1] or [K.sub.2], depending on the dominant diffusion mode, describes how the water advances as a fraction of total thickness per unit exposure time (weeks in this case).

EIS measurements were taken using a Gamry FS1 Potentiostat/Galvanostat/ZRA. Figure 8 shows the low frequency (0.01 Hz) impedance plot for the high gloss topcoat and polyurethane unicoat. The fit of the low frequency impedance data are reasonable. The fit to equation (16), using the [K.sub.1] parameter ([K.sub.2] set to 0) for Type-II diffusion, which has a linear advance of the saturation interface, is shown as the lighter dashed line. Using equation (16) and the [K.sub.2] parameter ([K.sub.1] set to 0) for Fickian advance of the saturation/water-affected zone is represented by the dotted line.

A consequence of the linear advance of the Type-II diffusion front is that when a coating begins to fail, the impedance decrease accelerates more quickly than for Fickian diffusion. The failure criterion can be defined either by predicting the time to a "failure" impedance or by finding the time to a preset deviation from the moving linear fit of the log (impedance) to log (exposure time) plot. Once the impedance is no longer well represented by a linear fit, the decrease in impedance (and thus protective ability) accelerates too fast for the coating to remain useful for a significant period.

Several other refinements to this model can be employed, and several material properties require further investigation. Another model might treat topcoat and primer separately when applicable; for example, when a topcoat is eroded by weathering but has a distinct behavior for water permeation. The EIS spectrum may depend on whether the coating is measured in the wet/hot/dry periods of the exposure cycle, although models may also be able to quantify this variation. The primer and a low gloss topcoat of the same study (2,30,31) are not well described by the present model and require other processes to be included in the model. However, modeling physical processes in calculating the resistance and capacitive components of coating impedance is a productive approach.

Weighting of the impedance data has a large effect on the quality of the fit at various frequencies. The EIS data here was fitted as the logarithm of the impedance by standard nonlinear least-squares techniques, thus weighting all values approximately equally (the values span 5 to 9). Otherwise, fitting with the impedance values gives much more weight to early (higher impedance) values. Uncertainty is likely greater at later exposure times, as seen by scatter in Figure 9, and perhaps the data at longer times should be treated to reflect this in some way.

MODELING THE ENTIRE EIS SPECTRUM

The low frequency component of the EIS spectrum is dominated by the resistance of the coating. The coating resistance is dominated by that thickness of the coating which does not have significant water infiltration. If water is present as a molecular solution in the coating or as discreet small Maxwell inclusions, its effect on the resistance will be less; but if it is present in channels that connect the substrate to the environment or in large inclusions, it may reduce the resistance of a coating layer by orders of magnitude.

Water has a high relative permittivity, [[epsilon].sub.R], of approximately 80 at normal ambient temperatures. Infiltration by water increases the relative dielectric constant of a polymeric coating and thus its capacitance, therefore reducing its impedance, but not by orders of magnitude. Resistance is much more sensitive to the presence of water than capacitance.

To model the entire frequency range normally measured in an EIS test requires the electrical properties of the diffusion-affected zone to be known over the broad frequency range of 0.01 Hz to [10.sup.5] Hz. EMTs describe the bulk properties of a material consisting of two or more constituents, depending on how the constituents are mixed. In the case of a coating, the water-affected region consists of the original coating that contains water with associated ions that might have changed the original coating material chemically. Models range from Maxwell type homogenizations, which treat the included water as isolated spheres distributed evenly through the diffusion-affected zone, to percolation models which assume channels of water progressing randomly toward the substrate (Figure 10). In this work, the diffusion of water into the coating has been treated as a continuous advancing front, described either by Fickian diffusion or Type-II diffusion, that reduces the resistance of the water-affected layer to zero.

The values of [K.sub.1] or [K.sub.2], derived from the low frequency impedance, may be used to predict how the rest of the impedance spectrum changes. Figure 9 shows the experimental EIS spectrum for the polyurethane unicoat topcoat measured periodically for 125 weeks. Predicted curves are similar and shown in Figure 11. The experimental impedance results do contain some scatter and do not decrease in a clean monotonic fashion. The spread in the high frequency impedance may follow an overall thickness decrease with time (Figure 12), although measurements of thickness with time were not taken at the time of the experiments.

It is noted that one of the discrepancies is due to the slope at high frequencies of the Bode plot (Figure 9) being lower than the model predictions (Figures 11 and 12). This slope should ideally be -1, since the high frequency response would be dominated by the capacitance of the coating. Actual measurements almost invariably show a slope less than -1, frequently between 0.8 and 0.9. Recent simulations of random combinations of capacitors and resistors exhibit this decreased slope behavior, with the change in slope proportional to the number fraction of resistors to capacitors. (32,33) Perhaps similar behavior could be expected with polymer, pigments, and water inclusions. In addition, if one includes the small solution resistance in the overall equivalent circuit, there is a reduction in this slope, but not enough using realistic values for the resistance of the electrolytes commonly used. Finite element simulation of a truly composite film with embedded high dielectric pigments in low dielectric polymer may be necessary to complete this approach. At reasonably high volume fractions of disparate materials the diffusion path is not likely to be homogeneous.

[FIGURE 12 OMITTED]

Nevertheless, the simple model duplicates the overall features of coatings tested by EIS. It predicts accelerated failure near end of service life, which is consistent with most published data and is consistent and supportive of the empirical observation that log (impedance) at low frequencies decreases linearly with time. (2,30,31)

CONCLUSIONS

The action of an aggressive environment is to change the structure of a coating by an enormous number of very small, random, repeated events. These events may be the action of ultraviolet photons, hydrolytic attack, or the ingress of water molecules into a coating, or other possibilities. Diffusion kinetics are the result of increasing numbers of penetrant molecules entering the coating and undergoing increasing numbers of random changes in position. Light absorption is similarly the result of many photons striking or missing the components of a coating at random.

The application of the CLT provides a method of incorporating the accumulated effect of these events into models that reproduce many aspects of measured coating properties including yellowing (or color), EIS response, contact angle, gloss, and fracture toughness. Many of these properties are also directly related to one another through either the surface topography or perhaps common features of the coating bulk, allowing one measured property to predict another. Results here confirm conventional practice for nondestructive testing during weathering studies which typically measures gloss as a function of exposure period and use that to correlate with compositional details that are technologically or commercially important.

Overall, the CLT-based approximations have given fitting equations of one or two variables based on simple physics that reproduce measured data in a reasonable way. The application of such equations allows for the evaluation of the variables based on physical properties of the degradation process and, thus, testing of the underlying mechanisms. If values of parameters deduced from the measurements via the equations are unphysical or if one variation with exposure fails to correlate with another then we gain the understanding that a mechanism is incorrect. The equations allow for the prediction of lifetime based on early measurements, or compositional variations, and understanding the degradation process parameters. Monte Carlo modeling can expand the range of application of the methodology at the expense of additional input parameters and less transparent correlations between properties. The process of trying to understand and predict coating behavior with exposure time and weathering defines new, relevant questions and suggests different approaches to measurement.

ACKNOWLEDGMENTS

The Air Force Office of Scientific Research funded this work. M. Nichols (Ford Motor Company) has kindly allowed us to use fracture data from one of their studies.

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B.R. Hinderliter and S.G. Croll--North Dakota State University*

Presented at the 82nd Annual Meeting of the Federation of Societies for Coatings Technology, October 27-29, 2004, in Chicago, IL.

* Dept. of Polymers and Coatings, P.O. Box 5376, Fargo, ND 58105.