Monte Carlo approach to estimating the photodegradation of polymer coatings.

The degradation of a polymer coating and predicting the coating lifetime based on physical properties and distribution within the coating of the polymer binder, pigments, and fillers are economically very important. As technologies advance and allow for control of coatings at the nanoscale level, methods such as Monte Carlo can be used not only to predict the behavior of a nanodesigned coating with time but also to design coatings, such as optimizing pigment particle distributions or optimum hard and soft phase distributions of the binders in multiphase systems for maintaining the desired property with time. Erosion of the coating surface was simulated using Monte Carlo techniques where terrestrial solar flux is the initiator for polymer segment cleavage and removal. The impact on the sensitivity of the polymer adjacent to the detached polymer segment can be increased or decreased in the model based on the chemistry and surface energy of the remaining polymer matrix. Multiple phases with varying sensitivity to degradation can be modeled. The Monte Carlo generates a statistically similar surface topography and chemistry of the coating. The results of the Monte Carlo model are compared to measurable properties such as gloss, fracture toughness, and wetting contact angle, using various published correlations of the property to the surface topology. The simulated properties change through the lifetime of the coating in ways that are consistent with observed behavior. Apparently, complicated changes in many properties can be described by the repeated application of simple, random processes.

Keywords: Polymer, degradation, simulation, Monte Carlo, gloss, fracture toughness, wetting angle

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The degradation of a polymer coating, and the related topic of predicting its lifetime based on important physical properties of the coating, have significant economic importance. Much effort is expended ranking coatings for various properties, predicting service lifetimes of coatings, and assessing the likely remaining service life of a coating.

We emulated the erosion of the surface of the coating using a Monte Carlo model using the solar spectrum photon flux as the initiator for polymer segment removal. The model was designed to predict behavior of coatings, though at this point, the generality of the model would allow the method to be applied to the surface erosion of any material due to a random process, such as rain or wind action, as the initiator to remove a piece of the surface material.

Whereas most fields of technology are relying heavily on computer design for new product development in areas such as automobile mechanical structure, airplane aerodynamics, nuclear reactor vessel metal embrittlement, fuel cell design, chemistry of new pharmaceuticals, and numerous others, coating design tends to be an experimental technology consisting of mixing based on experiential correlations of properties to constituent materials, and exposing the coating to accelerated testing. The ability to control the coating on a nanoscale level reinforces the need to model the behavior of a coating system through the life of the coating. The ability to control pigment size and distribution throughout a coating leads to the need to predict the best distribution of pigments within the coating to achieve the best combination of physical attributes not only initially, but also as a function of weathering.

The modeling consists of two steps: first, we used Monte Carlo to model temporal changes to surface topography and chemistry; and second, we took the new topography and chemistry and applied various published models or finite element models to it to predict macroscale (integrated over a spatial region) measurable properties such as gloss, fracture toughness, electrical impedance, UV visible spectroscopy, and wetting angle, which average over a spatial region.

MONTE CARLO SIMULATION OF PHOTODEGRADATION

The Monte Carlo method is a technique for modeling the evolution of a system whose final state is due to the repeated application of many small events. Each of these events, such as the freeing of a polymer segment, proceeds with an assignable probability, termed quantum efficiency for fragment removal. Our model consists of a coating layer divided into small boxes, and within any grid is only one material constituent. A single material is assigned to each grid location at the beginning of the computer simulation. Each material is assigned a probability, p, that an incident photon will remove that small grid of material. The computer chooses a photon at a random location on the surface of the coating. The computer then generates another random number. If that number is less than the assigned probability, p (the quantum efficiency), for which the material present at that grid will degrade, then the material in that grid is removed or modified (changed to another material). In addition, the material adjacent to the material in the photon struck grid can be modified to simulate the changes to the properties of the polymer chain that remain after the volatilization of a weak segment (Figure 1). The Monte Carlo model allows the random towers of unsupported polymer fragments to be lost, as they would be expected to be removed by processes such as rain or wind. General aspects of Monte Carlo modeling in physical systems can be found in reference 1.

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Figure 2 displays the evolution of a simple coating surface over time. The coating surface is assumed to consist of a single material. The material of the surface in Figure 2 is assumed to be unaffected by the degradation of the adjacent material. As the material surface erodes, the area of the surface affected and the depth of the pits both increase.

The goal of this model is to be able, with appropriate chemical data related to degradation products for the coating, to predict coating lifetimes based on total solar flux reaching the coating integrated over time. This is useful when nanotechnology allows for control of the detailed structure of a coating, and design needs also meet aging property requirements. Additionally, the ability to correlate measurable and nondestructive properties, such as reflectance with lifetime, may allow for the prediction of the remaining service life based on other requirements, for example, fracture toughness.

The reasonableness of our method is assessed in several ways. First, the surface topography is compared to AFM and optical profilometer measurements, not directly, but based on statistical surface attributes such as roughness and correlation length. A second method of testing is measuring the predicted integral properties that average over a statistically large surface area such as gloss or reflectance, and comparing them to the Monte Carlo predictions.

The most interesting aspect of the research so far is that the prediction of properties is functionally consistent with experimental results. These functional predictions are based on very general and simple assumptions of coating properties. The sigmoidal behavior of reflectance, for example, can be predicted by the repeated application of random processes in conjunction with a simple model of how reflectance is affected by surface roughness.

PREDICTING POLYMER COATING DEGRADATION

Several methods are available for the prediction of coating lifetime or to rank candidate coatings. The methods can be broken into two divisions: the first is experimental testing, including accelerated aging and extrapolation, and the second is predictions based on modeling the system.

One modeling method for predicting the lifetime of a coating is to apply a damage function to the independent variables that correlate with the degradation. Independent variables that correlate to the damage of coatings include such factors as solar spectrum and fluence (fluence is the integral of flux over time or the total number of photons per unit area), temperature, humidity, and water exposure. In this discussion we focus on solar fluence. The damage function is a way to weight the effect of the particular free variable, in our case, solar fluence, with a particular damage, for example, change in gloss. The damage measures frequently used to define the end of a coating's service life include loss of gloss, color change, fracture toughness, and erosion of substrate.

Focusing on solar irradiation as the initiator for coating degradation, an estimate for the damage as a function of solar irradiation is give by equation (1), (2,3) slightly modified to incorporate the variation with position and possibly material phase at that position.

D(x,t) = [[integral].sub.[[lambda].sub.min].sup.[[lambda].sub.max]][I.sub.0]([lambda],t,x)(1-[e.sup.-A([lambda],x,t)])[phi]([lambda],x,t)d[lambda] (1)

D is the instantaneous damage, or change in property in the coating at a particular material location in 3-space, x, and time, t. [lambda] is the wavelength of the incident light with the subscripts min and max referring to the highest and lowest energy of the light in the incident spectrum. A([lambda],x,t) is the absorbance of the light of wavelength [lambda] in the coating, since light must be absorbed in that material to produce the damage.

[phi]([lambda]) is the damage function and denotes the change in the measured physical property due to a unit fluence of photons at a given wavelength and material phase. The damage function is likely to be different for different properties, for example [[phi].sup.gloss]([lambda]) for gloss is not likely to be the same as [[phi].sup.toughness]([lambda]) for fracture toughness. A damage function is used to weight the flux to account for responses in ionizing radiation damage studies, though quantum yield or spectral efficiency are frequently used to describe the same process at lower photon energies.

The total damage over a length of time is the integral of the instantaneous damage over time [D.sub.total] = [integral]D(t)dt. Estimating the damage produced by a given flux requires a function that relates the effectiveness of the photons of each energy to produce that damage. Generating a damage function entails measuring experimentally the change in one or several properties over time during a test series. Actual service conditions are likely to vary substantially from the exposure conditions used in testing. When applied to actual service conditions, the testing may or may not accurately predict the ranking of coatings due to their damage susceptibility. Additionally, the prediction of the remaining service life of a coating based solely on the damage function requires the monitoring of the environmental conditions. A Monte Carlo model could help in correlating the nondestructively measurable properties that could be used to measure one property, and predict the remaining lifespan based on other properties. At a minimum, the Monte Carlo model results will allow more intelligent interpolation and extrapolation of damage functions.

Chemistry

A chemical model and chemical analysis are being developed in our group at NDSU (4) to predict the distribution of polymer segments that detach and are volatilized or would be washed away. The impact on the sensitivity of the polymer adjacent to the detached polymer segment can be increased or decreased in the Monte Carlo model based on the chemistry of the remaining polymer matrix. The effects of the modification of the material adjacent to a segment removed changes the predicted properties. In addition, the sensitivity and breakdown products of multiphase materials are being investigated.

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Photon Flux

The energy distribution and flux of photons incident on a particular surface of a coating is a function of many variables, including solar photon energy spectrum, angle of surface relative to position of the sun, atmospheric conditions, season, and time of day. The penetration of UV and visible light into polymers is fairly low; for example, in polyurethane, attenuation to 0.10 of initial flux is at a depth of 14.7 [micro]m at 340 nm, and a depth of 476 [micro]m at 280 nm. (5) Thus, most of the visible and UV radiation will be absorbed near the surface of the coating, and the attenuation is even stronger in the presence of pigments. In addition, many of the degradation pathways require oxygen or water. The diffusion of oxygen into a coating results in a significantly higher concentration near the surface. Additionally, reactions relatively far from the surface may result in degradation fragments that need to diffuse from their origin to a surface in order to escape before reacting back into the polymer chain. The loss of polymer fragments near the surface of the coating is less likely to result in recombination, compared to fragments diffusing out from deeper in the coating. Diffusion of oxygen in, diffusion of polymer segments out, and exponential attenuation of photon through a coating, lead to the general observation that degradation is predominately a surface effect. We will, as chemistry details become available, incorporate modification of subsurface chemistry into the model.

The photon flux can be as high as 1 kW/[m.sup.2] at midday in Florida. Using the solar spectrum, we calculate the integral photon flux as roughly [10.sup.18] [[photons]/[m.sup.2]sec]. The photon fluence over a year is immense, but the probability of any particular photon initiating a segment rupture is low. A low probability of quantum efficiency for coating polymer degradation and the very high number flux of photons makes Monte Carlo techniques well suited for modeling the coating degradation.

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Materials

The Monte Carlo model allows the photon impinging on a region to modify the material in such a way as to make it either more or less likely to be removed during subsequent photon strikes. Control can be set to remove a polymer segment impacted by a photon and the adjacent material modified to be more or less sensitive to future photon initiated erosion.

The sensitizing that occurs at the nanometer scale represents the increased chemical sensitivity resulting from polymer chain cleavage. At larger spatial scales, the increased sensitivity results from surface energy; increased surface area results in increased oxygen access and greater ease of polymer scission fragment escape before recombination. Decreased sensitivity may be due to the removal of fragments responsible for absorbing the photon quanta or the removal of bonds that are more easily broken.

The Monte Carlo model also allows the coating to be composed of multiple phases. These phases can either be randomly distributed or clumped, as in phase separation. The clumping presently consists of a generation of random spheres where the radius and standard deviation of the radius can be input. The spheres are allowed to overlap. At low densities of the secondary phase, the spheres are predominately single. At higher densities of the second phase, the secondary phase looks like the holes in a sponge.

The option to model two or many phases could be used to describe a system containing pigment or extender particles that do not erode, and just fall off when sufficiently free of surrounding matrix. Another multiphase system might describe base material containing a dispersed UV absorber that requires a certain number of hits by photons before the chemistry is assumed to break down and erode. Obviously, it may be used to explore the behavior of a polymer system that separates into two or more distinct phases that have different durabilities. (6)

MODEL RESULTS

The initial surface is assumed to be flat, without defects. In all models, softening refers to the material around an eroded region becoming more likely to be eroded when impacted by subsequent solar radiation than the base material. Conversely, hardening results in the material being more resistant to erosion, either by being less likely to absorb the solar quanta, or if the light quanta is absorbed, more likely to dissipate the energy without degradation. Thus, simple chemical changes during degradation can be readily modeled.

When the region adjacent to an eroded polymer segment is unaffected, the distribution of holes quickly becomes a Gaussian distribution (Figure 3). The distribution of pit depths, if the surrounding material is modified (by the erosion event), is not Gaussian initially. After long times, the distribution of surface depths becomes Gaussian due to the entire surface being modified.

The ratio of the average to standard (root mean squared) deviation in depth predicts how close the distribution of pits is to a Gaussian distribution. A Gaussian distribution has an average to standard deviation ratio of [square root of (2/[pi])] [congruent to] 0.80. The ratio of the average deviation to standard deviation is calculated by dividing the integrals of the weighted Gaussian distribution and is equal to:

[[integral].sub.-[infinity].sup.[infinity]][|x-[x.sub.average]|/[[sigma][square root of (2[pi])]]]exp(-(x-[x.sub.average])[.sup.2]/(2[[sigma].sup.2]))/[[[integral].sub.-[infinity].sup.[infinity]][[(x-[x.sub.average])[.sup.2]]/[[sigma][square root of (2[pi])]]]exp(-(x-[x.sub.average])[.sup.2]/(2[[sigma].sup.2]))][.sup.1/2] = [square root of (2/[pi])]

where x is the depth of the surface, [x.sub.average] is the average depth of the surface, and [sigma] is the standard deviation which is equal to the ([x.sub.average])[.sup.1/2] for a Gaussian system based on counting statistics.

Initial roughness of many smooth surfaces has an average to standard deviation of 0.1 to 0.2. (7) Thus, many surfaces start with surface profiles far from Gaussian and nearly smooth. This is the assumption presently applied for the initial condition for the model.

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In order to compare the model's prediction of behavior of a degrading system, a means to quantify the behavior is needed. Various theoretical and empirical equations are applied which measure the final physical state, in this case surface topology, and correlate it to another measurable property. An example is the use of the standard deviation of the surface profile, which can be measured by an atomic force microscope (AFM) or other profilometers and used to predict the reflectance based on the equation of Bennett and Porteus. (8) The Monte Carlo-generated surface may be parametrically compared to measurements of the true surface being modeled; for example, depth profile distribution.

This in turn can be used to predict integral properties, such as reflectance, wetting angle, and fracture toughness.

Fracture Toughness

A failure criterion of a coating may be crack propagation through to the base material. As the coating dries, it shrinks relative to the substrate resulting in planar, tensile stresses in the coating. By applying the Griffith crack criterion and using the maximum flaw size generated from the Monte Carlo model, an estimate of the relative strength of a coating can be predicted with time. We assume the coating is a nearly infinite plane. As an estimation of the reduction in coating strength we use [[sigma].sub.G] = [square root of (2E[gamma]/[pi]a)], where E is Young's modulus, [[gamma].sub.s] is the surface energy per unit area, and [alpha] is the maximum flaw size. The Griffith stress, [[sigma].sub.G], is the critical stress above which the crack continues to progress to fracture. The coating has then failed.

At each time step, the computer model surveys the surface for the deepest pit that has been generated to that point in time. This depth is taken as the flaw size to be used in the Griffith equation. Thus, the progress of fracture toughness can be monitored as the surface topography evolves. The decrease in fracture strength predicted by the Monte Carlo model for conditions in which the initial removal of a polymer segment results in a sensitization or hardening to erosion of adjacent material is shown in Figure 4. The fracture toughness of thin films is difficult to test quantitatively due to such confounding parameters as variations in the thin film thickness and initial defect distribution. The spread in the predicted decrease in fracture toughness over time is too large to precisely define the functional form the decrease in fracture toughness takes as a function of time. Trends in the decrease in fracture toughness, shown in Figure 4, show that the Monte Carlo model predicts or shows the same trend of an early accelerated decrease in the fracture toughness followed by a more gradual decrease as exemplified in published experimental data; for a very smooth initial surface the toughness shows a marked early decline. A slower initial decrease is seen in the system where weathering-produced defects need significant time to grow to exceed initial defects (in the Monte Carlo model case set to 5 nm). (9-17)

In an engineering system we would insert an initial distribution of defects consistent with the initial measurements of the coating prior to exposure and degradation. This initial defect distribution would decrease the initial steep slope seen in Figure 4. As expected, sensitizing the material adjacent to a removed segment reduces the fracture toughness more quickly, and vice versa. The coarse step decrease in the case in which the material adjacent to a removed segment is hardened is due to only a few defects being generated over the timescale used.

The change in fracture toughness for a two-phase system is given in Figure 5. The second phase in this example is 10% of the volume in all but the last case, where it is 40% of the volume, and consists of a secondary phase that is 100 times more sensitive than the base material. The dispersion of the second phase is either random, or taken as randomly dispersed spheres with a Gaussian distribution of a given radius in a number of grid cells and standard deviation of the radius. Thus, the second case consists of a base material containing 10% by volume of a second phase in the form of spheres of radius 24 and standard deviation of the radius of 1. The change in relative fracture toughness is least for a randomly dispersed secondary phase. The clumping of the sensitized phase results in quicker loss of toughness, as one should expect since the flaw size will be greater.

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As with all the correlations used in this article that relate surface topography to a physical property, other correlations exist. In this case, the simple Griffith crack criterion is applied, though other models are available and can easily be incorporated. Additionally, many finite element codes are available and could be used to predict fracture toughness based on the materials and surface profile supplied by the Monte Carlo program. Our analysis does not include the hardening effect of composites, particle bonding, or crack blunting.

Reflectance

Reflectance is a nondestructive way to measure the progression of coating damage. Correlation between the parameter used to define coating failure and change in reflectance could be used to estimate the remaining lifetime based on measuring the gloss. Gloss averages the reflectance over a spectrum of wavelengths. The specular reflectance calculated from the model is based on the model of Bennett and Porteus. (8)

Specular reflectance for normal incidence (7):

[R.sub.s] = [R.sub.0]exp[-(4[pi][sigma])[.sup.2]/[[lambda].sup.2]] (2)

where [R.sub.s] is the specular reflectance, [R.sub.0] is the reflectance for a perfectly smooth surface of the same material, [lambda] is the wavelength of the incident light, and [sigma] is the standard deviation of the surface from its mean level. The Monte Carlo results yield the standard deviation in the surface and how it progresses with time, and then equation (2) can be used to model how the reflectance changes with time. In the calculation presented here, [R.sub.0] is taken as unity and that the starting value of the standard deviation is zero. As pointed out in the previous section, realistic initial values of these quantities may easily be inserted into the modeling (as the data becomes available).

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The Monte Carlo model of surface erosion, as a function of solar spectral flux, predicts the sigmoidal decrease in reflectivity with exposure, which is consistent with experimental results (Figure 6). (18-22) The softening of material adjacent to a removed segment results in a faster degradation of the reflectance. The effect of a second phase, comprising spheres that are sensitized relative to the base material, is given in Figure 7. A higher concentration of secondary phase results in a faster degradation of gloss. Distribution of a secondary phase at random locations results in an accelerated degradation at early times, followed by a much slower degradation.

The reflectance may be able to predict the degree of damage, if the damage is plotted as a function of wavelength (Figure 8). Coating systems tend to be quite smooth, initially. With time, the roughness increases; first on a small dimension and then increasing the size scale of roughness with time. This may allow the reflectance as a function to be used as a spectrometer for damage. Gloss is a property often used to track the degradation of coatings over time. However, gloss meters are used with white light, or only one wavelength. Monte Carlo results and equation (2) show that reflectance decreases quickly for the short 300 nm wavelength, slower for the 500 nm, and slowest for the 800 nm wavelength (Figure 8). Tracking the reflectance for several wavelengths may allow for a nondestructive method of estimating the damage state of a coating. Additionally, pigmented systems have at least two distance scales: the sizes of the pigments and the pigment's average separation distance. Applying more advanced methods to predict reflectance in multicomponent systems could allow for the prediction of damage state, or be used to choose pigment particle size and distribution to optimize properties during the lifetime of the coating.

Accurately predicting the reflectance of a pigmented coating is outside the ability of equation (2), or even standard finite element models. An interesting prediction by the Monte Carlo model and correlation function from the surface roughness, equation (2), to reflectance is that the reflectance of low volume fraction of pigments decreases slower, though reaches a lower final reflectance. Controlling the pigment volume fraction distribution through a coating may allow optimization of the coatings reflectance with lifetime (Figure 9).

Wetting Contact Angle

The wetting angle changes with the roughness of the surface. At this point, we assume the chemistry of the surface does not change enough to alter the surface energy at each element of the coating. The change in contact angle is a modified version of Young's equation, which incorporates the ratio of the actual surface area to the projected surface. (23) The Monte Carlo simulation provides the projected surface area as a function of time. Clearly, the surface becomes rougher as degradation proceeds, and the surface area increases (Figure 2). We arbitrarily assume an initial wetting angle near 90[degrees], and find that at long times the wetting angle approaches 0[degrees], near complete wetting.

The wetting of a surface is a function of the physical profile of the surface and the surface chemistry. The assumption that the surface chemistry was unchanged during erosion, or, equivalently, the modified material has very similar surface energies between the solid surface and the vapor and liquid, can be relaxed later when chemical changes specific to a particular polymer are input. The defects introduced must be much smaller than the dimension of the liquid drop used to measure contact angle. An approximation for the effect of the surface roughness is (23):

r([[gamma].sub.solid-vapor] - [[gamma].sub.solid-liquid]) = [[gamma].sub.liquid-vapor] cos[theta] (3)

where r is the ratio of the actual versus projected area of solid contact with the solid phase. The [gamma]s are the surface free energy between the subscripted phases. [theta] is the wetting angle.

Assuming the wetting angle starts near 90[degrees] for a flat surface,

r([[[gamma].sub.solid-vapor] - [[gamma].sub.solid-liquid]]/[[gamma].sub.liquid-vapor]) = cos[theta] (4)

results in: [theta] = arccos(r([[[gamma].sub.solid-vapor] - [[gamma].sub.solid-liquid]]/[[gamma].sub.liquid-vapor])) (5)

We have assumed that the wetting angle of the initial surface before degradation is near 0[degrees]. Thus, ([[[gamma].sub.solid-vapor] - [[gamma].sub.solid-liquid]]/[[gamma].sub.liquid-vapor]) is chosen to be 0.1. Note that equation (5) cannot transition through a 90[degrees] angle since choosing ([[[gamma].sub.solid-vapor] - [[gamma].sub.solid-liquid]]/[[gamma].sub.liquid-vapor]) to be 0 results in equation (5) being independent of r.

The wetting angle is plotted in Figure 10. If a region adjacent to a removed segment becomes softened, the wetting angle changes more quickly. The plot of experimental data, (24) right side of Figure 10, shows great variation in the measured wetting angle, though the trends are consistent with the model's prediction. The wetting angle is not plotted for the two-phase system because the distribution of holes is not random when the second phase is clustered as spheres, and when the two phases may have potentially different wetting angles.

DISCUSSION

Degradation in Natural Exposure

Thus far in the discussion, we have focused on a qualitative agreement between modeled predicted behavior and experimental results in varying systems. We are presently collecting the required data, chemical degradation products, and quantum yields of polymer chain scission to make quantitative comparisons in urethane systems.

The times plotted on the graphs are in units of 1000 successful degradation events; thus, graphs with maximums of 1000 are giving a total of [10.sup.6] degradation events. Converting the time values listed on the preceding graphs to real times require several assumptions. Quantum efficiencies are predicted to vary from about [10.sup.-2] to [10.sup.-5] (25) for polymer segment removal. The quantum efficiency is measured for a simple polyurethane system using UV visible interference fringes to track coating thickness periodically as the coating is weathered in a Q-Panel model QUV/SE (QUV) accelerated exposure chamber. (3) The quantum efficiency is 4.3 X [10.sup.-5]/photon for a 1 n[m.sup.3] grid size, considering the QUV photon flux for wavelengths up to 340 nm. The quantum efficiency was calculated by taking the total thickness decrease of the polyurethane and dividing it by the total number flux of photons with wavelengths less than 340 nm and the time of the exposure that caused the thickness decrease in coating thickness. The quantum efficiency was taken over a 1 n[m.sup.2] surface area. This is on the lower end of the published quantum efficiencies but includes the probability not only of chain scission, but of multiple scissions that release a fragment and the probability of escape prior to recombination. It also includes the requirement that at least one complete chain is scissioned and usually two or more locations need to be scissioned to release the polymer segment.

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Applying the solar spectrum (26) and estimating that only photons of wavelengths less than 385 nm can cause degradation gives 1.3% of incident solar photon flux, which has enough energy to cause degradation. We also estimate a rough average monthly radiation level of 500 langleys (10) (0.0418 MJ/[m.sup.2] = 1 langley). Roughly [10.sup.27] photons with wavelengths less than 385 nm impact each square meter per year. From the Monte Carlo model side, a total fluence of [10.sup.6] to [10.sup.9] degradation events are modeled. The size of the model area is chosen as 256 grids square, equivalent to 0.256 [micro]m. This results in each eroded material piece being a 1.0 nm cube, corresponding to a polymer chain size of about 35 carbon atoms (taking the radius of a C[H.sub.2] bead to be about 1.9 A), which is a fragment that could conceivably be washed away or lost when broken into smaller volatile fragments. Assuming the index of refraction of the polymer coating is roughly 1.5 over the wavelengths of interest, the fraction that is transmitted into the polymer coating is roughly 96% from Fresnel's equations, with the balance being reflected. Including an estimate of the fraction of photons that interact in a 1.0 nm depth, we use the 0.068 [micro][m.sup.-1] as the extinction coefficient for a 280 nm photon (5) and from UV visible spectroscopy of our polyurethane; thus, about one percent of the photons incident on the surface is absorbed in our modeled surface grid. The Monte Carlo model thus simulates a fluence of 4 X [10.sup.27] photons/[m.sup.2] with 1000 time steps, which is a few times larger than the fluence estimated above for one year. Thus, the simulation might be compared to a system without UV stabilizers or pigments modeled over a few years time frame. It is encouraging that the simulations translate readily into reasonable estimates of degradation time frames.

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Correlation between Properties during Degradation

Being able to correlate a nondestructive measurement to a property of interest is useful since it allows not only the prediction of service life based on an a priori estimate of the service environment, but also the ability to monitor and adapt the predicted service life based on the actual service environment without having to explicitly monitor that environment. The results of the Monte Carlo simulations are used to predict physical property changes in the coating, both those that define failure and those that are measurable in a nondestructive way. These changes can be related to measurable properties and may be correlated to lifetime predictions of in-service items.

As an example, the fracture toughness is plotted as a function of gloss for a system where removal of a segment hardens or softens the adjacent material (Figure 11) or a two-phase system (Figure 12). Over a large portion of the reflectivity scale, the relative fracture toughness changes only slightly. This leaves only a very small time period early and late in coating life when the reflectivity is a good measure of toughness, for coatings that sensitize or harden adjacent material. A use for Monte Carlo may be to choose useful "pairs" of properties where the correlation is more sensitive to the extent of degradation.

A major codicil in the accurate prediction of coating properties is the ability to generate a correlation between a surface profile and the material distribution in a multiphase system with the property or damage being modeled. The description of the property need not be an equation; a finite element or ray tracing routine could also be used to tie the Monte Carlo modeled surface to the predicted property.

CONCLUSIONS

We have created a Monte Carlo model to predict surface coating degradation and use the results to predict change in the coating properties with lifetime. The model presently predicts qualitative changes of system properties based on various theoretical and empirical models which take surface profile properties and relates them to measurable system properties related to coating lifetime, such as gloss, relative fracture toughness, and wetting contact angle. As noted in the introduction, the Monte Carlo model is quite general in its application of a random process to a surface that removes a portion of the surface, and then the surface topology is used to predict physical properties or future behavior. This type of application could apply to surfaces randomly eroded by solvent or corrosive species, for example.

As nanotechnology advances allow for the control of multiphase coating and pigment distribution, methods need to be applied that predict the behavior of the system with time and not just the initial properties. For example, control of pigment distribution within a coating may generate a marvelous optical effect, but if weathering quickly changes the appearance drastically, the particular distribution may be of little value. Additionally, the ability to predict behavior may allow for the optimization of pigment particle distributions to maximize gloss through coating life.

Presently we can model a random dispersion of phases, or control the size distribution of a spherical secondary phase. This will be used to model the different phase domains in urethanes. The option is also used to model pigment particles that do not erode significantly, or simply fall off when sufficient supporting matrix material is removed. Ultraviolet absorbers might also be modeled as a phase, requiring a certain number of hits by solar radiation before the chemistry breaks down and the pixel is removed.

The Monte Carlo model uses an integral solar flux, through which the ability to use the solar spectrum as chemical data for specific coatings as a function of wavelength becomes available. Erosion is limited to only surface regions in this work. The option to modify subsurface material by photon initiated photo-chemical reaction (allowing easier erosion when that space becomes part of the surface) is available, contingent on data for the attenuation coefficient in the given coating for the specific wavelength.

The model qualitatively reproduces the lifetime behavior of wetting angle, reflectance, and fracture toughness for a single-phase polymer coating. We are also looking at diffusion to substrate and electrochemical impedance as a method to predict coating lifetime using Monte Carlo results. (27) In addition, the results scale to appropriate values of service life when reasonable estimates of the UV flux and damage are input. Further testing is presently underway to predict the degradation of a coating, including the chemistry required for qualitative modeling. We plan future testing of pigmented systems for comparison with the Monte Carlo models.

ACKNOWLEDGMENTS

The Air Force Office of Scientific Research under contract number F49620-99-1-0283 funded this work.

Presented at the 79th Annual Meeting of the Federation of Societies for Coatings Technology, October 28-November 1, 2002, in New Orleans, LA.

References

(1) Landau, D.P. and Binder, K., Monte Carlo Simulations on Statistical Physics, Cambridge University Press, 2000.

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(3) Martin, J.W., "Quantitative Characterization of Spectral Ultraviolet Radiation-Induced Photodegradation in Coating Systems Exposed in the Laboratory and the Field," Prog. Org. Coat., 23, 49-70 (1993).

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Brian Hinderliter and Stuart Croll -- North Dakota State University*

*Dept. of Polymers and Coatings, 1735 NDSU Research Park, Dr., Fargo, ND 58105.