2.1 :

Studied sample : white space paint

Samples that have been studied are white space paint samples. These coatings are composed of zinc oxide (ZnO) pigment particles, and a resin silicon binder (polydimethylsiloxane resin). The diameter distribution of the particles is evaluated from Scanning Electron Microscopy (SEM) observations, and is centered on 0,3 µm. The proportion in volume of the silicon resin in the paint is around 20 %. The total thickness of the paint is equal to 160 µm. The paint covers an aluminum substrate.

2.2 :

Experimental apparatus:

Samples have been irradiated by 45 keV-protons (doses from 4.10⁵ to 7.10⁸ Gy), using the SEMIRAMIS facility (Ref 1). The facility is composed by a set of vacuum chambers pumped out with cryogenic pumping units allowing to reach a vacuum level of the order of 2.10^-5 Pa. The system is connected to a solar UV simulator, two electron and proton Van de Graaf accelerators. Spectral reflectance of the material is measured for wavelengths between 250 and 2500 nm before and after irradiation. In order to avoid partial recovery effects due to air introduction in the chamber, these measurements were carried out in a secondary vacuum (better than 10^-4 Pa) with a Perkin Elmer spectrophotometer, equipped with a lateral sample integrating sphere. The measurement error is equal to 1%.

2.3 :

Study of the degradation and microscopy:

The evolution of the spectral reflectance is represented with the cumulated dose of 45 keV protons in (Figure 1).

Figure 1:

Spectral reflectance a the white paint irradiated by 45 keV protons

Two regions are degraded under irradiation: the first one appears in the infrared region of the spectrum. This degradation nearly completely disappears when the sample is put back in air after irradiation. As this degradation is reversible, we could not determine its origin because of a lack of in situ characterization tests.

On another hand, spectral reflectance shows the appearance of a large absorption band at 410 nm for the proton-irradiated sample. This degradation is irreversible and is correlated with the yellowing of the material (Ref 2).

Once the sample was put back in air, we observed its structure by Scanning Electron Microscopy (SEM). SEM images obtained before and after irradiation of the paint with 45 keV-protons are shown in (Figure 2). The study of the sample before irradiation shows that it is constituted of pigmented particles and vacuum. The particles are long-shaped and they are randomly oriented in the material. Their size distribution is centered on a diameter of 0.3 μm.

Figure 2:

SEM images of the paint before (left) and after (right) irradiation by 45 keV protons

After irradiation, no change in the shape or size of the grains is observed. Therefore, the degradation seems to be not related to some topological parameters of powder particles.

3.1 :

Mie theory for a multilayered sphere

We see from SEM that the resin is invisible on images. In fact, it constitutes a very thin coating (about 5 % of the total radius of the particles for a volumic proportion of 20 %) embedding ZnO pigments and leading to their cohesion. Then, the embedding medium of this pigment/binder complex is vacuum. As a consequence, we model the particles of the paint by two-layer spheres embedded in vacuum. Their absorption coefficient k and scattering coefficient s are determined by a modified Lorenz-Mie theory (Ref. 3).

The geometry corresponding to the scattering of a plane wave or a Gaussian beam by a multilayered dielectric sphere is depicted in Figure 3. In this configuration, m_j is the complex refractive index of the material in the j^th layer relative to the refractive index of surrounding medium (Ref 3). The j^th layer is characterized by a size parameter of x_j=2πr_j/λ, where r_j is the outer radius of the layer j and λ is the wavelength of the incident wave in the surrounding free space.

Figure 3:

Multilayered sphere geometry

The scattering coefficients a_n and b_n used in Lorenz-Mie theory can then be evaluated by use of a recursive computational scheme, taking the multilayered geometry into account (Ref 4). Once these coefficients are determined, we can deduce the different cross sections as a function of a_n and b_n : C_sca is the scattering cross section, C_abs is the absorption cross section and C_ext is the extinction cross section, defined as the sum of C_sca and C_abs.

This leads to determine the absorption coefficient k and the scattering coefficient s:

where N is the particles density.

We applied the model, varying the size of the resin coating from 1 % to 30 % of the total radius of the particle. The comparison between calculation and experiment shows that a size of 5 % gives the best result with an error less than 2 %. This result confirms our hypothesis of such a multilayered system.

3.2 :

Description of the four-flux model

During orbital missions, coatings used on spacecrafts are severely damaged by spatial environment. These missions being longer and longer, it becomes difficult to realize ground experiments in order to simulate optical degradation on coatings. In these conditions, predictive models are needed. They must allow us to predict optical property changes with irradiation dose. Moreover, they must be valid for the case of a dense material.

Multiple scattering theory, yet validated for coatings, allows calculating reflectance and transmittance values, and comparing them with experiment. It is interesting to obtain a simplification of this theory. Then, we have decided to use a four-flux model (Ref 6), derived from N-flux theories. This model is proved stricter than two-flux model of Kubelka-Munk (Ref 5), as it considers a specular component, contrary to Kubelka-Munk model. Moreover, this model permits to take nature of material into account, thanks to refractive indexes of matrix and pigmented particles. Particle size must be of the same order than incident light wavelength. Characteristics of the white paint studied here fit with four-flux model requirements. Therefore, this model has been chosen in order to study optical behavior of heterogeneous thermal control coatings (Ref 6, Ref 7).

We consider a slab (of infinite lateral extent) limited by planes (Z) and (O), which define a finite thickness, embedding discrete particles (Figure 4).

Figure 4:

Representation of the medium and the four flux

The incoming light on plane (Z) is constituted by a collimated beam I_c^Z and a semi-isotropic diffuse radiation I_d^Z. The incoming light on plane (0) is constituted by a collimated beam J_c⁰ and a semi-isotropic diffuse radiation J_d0. Then, the whole radiation field at location z is modeled as a sum of four parts:

An approximation of the radiative transfer equation leads to:

where k is the absorption coefficient, and s the scattering coefficient. ζ represents the forward scattering ratio and is equal to the energy scattered by a volume element located at z in the forward hemisphere over the total energy for a given incident light angular distribution. ε is the average crossing parameter: when we consider an element of medium with a thickness dz, the average path length which is traveled over by the light is εdz.

Different terms in (2-5) are explained as follows:

This equation system can be written under a matrix form:

The solution is:

where are the eigenvectors of matrix , and C₁, C₂, C₃ and C₄ have to be determined by the boundary conditions.

Each limiting plane of the slab is a separation between different media, possibly having different optical index. Therefore, there are reflections at the interfaces between layers. Then, the flux Ic(Z) inside the layer at the plane (Z) is the sum of the transmitted external incident collimated flux IcZ through the interface and of the reflected outgoing flux Jc(Z). This can be written as:

Similar relations are expressed for diffuse and collimated fluxes at plane (0) and (Z):

where r_c^Z, r_de^Z, r_di^Z, r_c⁰, r_de⁰, r_di⁰ are the different coefficients of reflection (Ref 7), I_d^Z is the incident diffuse flux on (Z), J_c⁰ is the incident collimated flux on (0) and J_d⁰ is the incident diffuse flux on (0) (see Figure 4).

These boundary conditions lead to determine constants C₁, C₂, C₃ and C₄. Knowing the incident collimated and diffuse flux, the outgoing fluxes I_c⁰, I_d⁰, J_c^Z and J_d^Z are written as:

These equations can be rewritten in terms of reflectances and transmittances:

where T_cc is the collimated-collimated transmittance, T_cd is the collimated-diffuse transmittance, T_dd is the diffuse-diffuse transmittance, R_cc is the collimated-collimated reflectance, R_cd is the collimated-diffuse reflectance and R_dd is the diffuse-diffuse reflectance. 0 and Z indicate each of the boundaries of the layer. Thanks to some algebra, these coefficients can be determined.

3.3 :

Extension to the four-flux model for a multilayer plane medium:

The case of multilayered material constitutes a generalization of the monolayered material case (Ref 7). We consider here that the material is composed of N layers of infinite lateral extent (Figure 5). In the layer i, outgoing flux is described versus the incident flux by:

Figure 5:

Relations between a layer and neighboring layers

where i+1 represents the neighboring layer in negative z direction and i-1 represents the neighboring layer in positive z direction. and are respectively incident collimated and diffuse fluxes on plane (Z). and are respectively incident collimated and diffuse fluxes on plane (O). T_cc is the collimated-collimated transmittance, T_cd the collimated-diffuse transmittance, T_dd the diffuse-diffuse transmittance, R_cc the collimated-collimated reflectance, R_cd the collimated-diffuse reflectance, and R_dd the diffuse-diffuse reflectance. O and Z indicate each of the two boundaries of the layer.

Outgoing flux of a layer enter the neighboring layer, then:

These relations lead to write system (13) under matrix form. Reflectance and transmittance coefficients are introduced in this matrix. Thanks to boundary conditions, fluxes at interfaces are determined from 4Nx4N-matrix inversion by Gauss transformation. In particular, the outgoing fluxes J_c^Z and J_d^Z are evaluated. Knowing the incident fluxes I_c^Z and I_d^Z, the total reflectance of the material is then given by:

4.1 :

Initial white paint

The four-flux model is applied considering a single layer medium. Figure 6 shows a comparison between theoretical and experimental reflectance of the white paint before irradiation. We observe a difference between the calculated reflectance and the measured one less than 2 %. This satisfactory result confirms the good behaviour of the model for this material.

Figure 6:

Comparison between theoretical and experimental reflectance of the white paint

4.2 :

Irradiated white paint

Experimental investigation showed that the irradiation by charged particles causes the yellowing of the paint and the apparition of an absorption band centred on 410 nm. We assume that this degradation can be theoretically modelled by a variation Δκ of the imaginary part of the refraction index of the zinc oxide pigment. This change appears in a thin layer near to the surface, which is defined by the penetration depth of the particles in the material and is equal to 0.8 μm. Consecutively, we have applied our model considering a two-layers system. The first layer is degraded, the second one is not affected by the irradiation and keeps its initial properties. The variation Δκ is calculated in the degraded layer with a dichotomy method. The exit condition is fixed by a difference between the calculated reflectance and the experimental one less than 10^-4. Figure 7 shows the variation Δκ of the imaginary part of the refractive index for several doses (4.10⁵, 7.10⁷ and 7.10⁸ Gy).

Figure 7:

Variation of the complex refractive index of zinc oxide in the degraded layer

We observe a strong change of Dk with the cumulated dose, with a maximum Δκ_max at 410 nm, corresponding to the maximum of the absorption band in reflectance measurements. From these results, the maximum variation of the complex refractive index Δκ_max was deduced as a function of the dose (Figure 8).

Figure 8:

Variation of the maximum of Dk with the dose

The difference between the degraded index and the initial one is very low for doses less than 10⁷ Gy and then fastly increases with the dose. This can be modelled by an exponential law, chosen in order to take a possible saturation of the degradation into account. Indeed, the density of defects created during irradiation and responsible for the optical degradation is not infinite and probably reaches a maximum. This has already been observed during long-term laboratory simulations and we can expect the same kind of saturation for our saturation. We could reproduce this thanks to the exponential law:

where Δκ₀ = -3.6, α = 10^-10 and d is the dose. The coefficients Δκ₀ and α are determined to optimally fit the experimental curve. Unfortunately, we didn’t observe the saturation during our experiment. This is why the law still has to be validated for doses greater than 10⁹ Gy.

The law was then introduced in a predictive code, which evaluates the degradation for doses up to 10⁹ Gy. We simulated the absorption band with a gaussian shape. The comparison between the calculation (Figure 9 a) and experiment (Figure 9 b) for three doses (4.10⁵, 7.10⁷ and 7.10⁸ Gy) was made in the spectral range 250-1200 nm. This corresponds to the region where the refractive index of ZnO is well known. In this range, the error in the calculation is less than 10 %.

Figure 9:

(a) Calculated reflectance (predictive code) and (b) experimental reflectance

This result is very promising and should be improved in the future by introducing the infrared degradation in the numerical code.