Handbook of Modern Coating Technologies

Monte Carlo simulation for photon diffusion

The arguments can be transformed to a model to evaluate the behavior of Iop and Isc, based on the Monte Carlo simulations [46(b)]. Fig. 6—5A and B illustrates the relationships of Nop and Nsc with the mean length of free path < r > of a photon using the graphs, respectively. As is rising, Nop first increases to a certain point and then decreases, and thus its mean length of optical path s is the longest one and its probability of encountering a pyrene
(B)
7000
NSC
FIGURE 6-5 Plot of Monte Carlo simulations. (A) NoP and (B) Nsc versus mean free path < r > of a photon in film of thickness d5 100 [39b]. 

FIGURE 6-6 Scanning electron micrographs of two-layer latex films, annealed for 30 min at (A) 25°C, (B) 120°C, (C) 140°C, and (D) 180°C temperatures [39b].

in the film is at the highest level for average < r > values. Hence, Nop hits the peak. On the other hand, for shorter < r >, the photons can simply flee from the front side, for longer < r > from the back side of the film. So, the number of the scattered photons Nsc from the back consistently decreases with rising values.
SEM results of two-layer latex films were obtained pre- and postannealing higher than Tg in a half hour. These powder film SEM was used twice, prior to annealing and when particle boundaries begin to disappear in the film due to 120° C annealing as presented in Fig. 6—6A and B, respectively. Fig. 6—6C illustrates such annealing with a micrograph of a film at 140° C where its part remained in relation to interparticle interfaces. When the temperature rises to at 180°C a transparent film is produced (see Fig. 6—6D). The fluorescence emission intensi¬ties (Iop) of the samples are shown in Fig. 6—7A corresponding to annealing temperature. The films in Fig. 6—6A and D exhibit the lowest Iop value while the film in Fig. 6—6C involves 
Annealing temperature (°С) Annealing temperature (°С)
FIGURE 6-7 Normalized (A) IoP and (B) Isc intensities versus annealing temperature for two-layer latex film samples a-d indicate the samples shown in Fig. 6-6A-D), respectively [39b].
the highest. The corresponding Isc intensities presented in Fig. 6-6 indicate a constant reduction as annealing temperature is rising (see Fig. 6—7B). The SEM results accord the photon diffusion model. It is adequately proved that the Iop variations depend on the mean length of optical path s of a photon travelling in the latex films [46(b)] and directly propor-tionally on its probability of encountering a pyrene molecule. Prior to annealing, the mean length of free path () is longer because of larger particle and interparticle voids. Following a couple of stages, the photon passing through the film reemerges from the front surface, which considerably shortens the mean length of optical path s. The voids are completely closed, followed by scattering mostly from the interparticle interfaces, and the mean length of free path increases or decreases depending on the size of reshaped par¬ticle. With the equal number of rescatterings, the film will expose to photons for a much greater time, which escalates the Iop values. The achievement of void closure allows for the interdiffusion of chains to remove these interfaces. It turns very transparent to the photon as the mean length of free path s diverges resulting in a proportional thickness of s. And then the Iop values lower while transparency of the film raises the Itr intensity. The film formation by hard latexes is illustrated in Fig. 6—8, including its relationships with the mean length of free path s and the mean length of optical path . The early stage of the process is also demonstrated in Fig. 6—8A. The close-packed particles produce a powder film partly with voids, yielding lower values of Iop and Isc due to smaller distances of and s, respec¬tively. Fig. 6—8B presents a film where interparticle voids begin to disappear because of annealing to elevate and the highest s values. In the film low Isc intensities are observed as well as the highest Iop. Finally Fig. 6-8C displays a fully transparent film where is the longest and s values are lower. Furthermore Iop and Isc were low. When consid¬ering photon diffusion theory and SEM together, it was seen that the maxima in direct fluo¬rescence intensities account for the healing processes at the interparticle interface while the film was being formed by high-T PMMA latex particles.
6.3.1.2 Polystyrene latex films
Surfactant-free emulsion polymerization produced P-labeled PS latex particles [64]. The equal amount of drops was placed on glass plates and further water evaporation to prepare 
(b)
Film with no void
(c)
/ \
Transparent film
(A)
FIGURE 6-8 Schematic illustration of (A) film formation from high T latex particles and (B) variation in mean free and optical paths ( and s) during film formation. I-III correspond to the film formation stages explained in the text [39b].
four latex films through particle dispersion. After that, samples were individually annealed higher than the Tg of PS (105°C) in the 5-, 10-, 20-, and 30-min periods at varying tempera-tures of 100°C-250°C. Film formation using these samples was studied with observations on the intensity of scattered excited light (Isc) from its surface and the fluorescence emission intensity (IoP) from a pyrene through SSF. Fluorescence measurements were made as explained in Section 6.3.1.1. Transmitted photon intensity Itr levels determined the develop-ment of transparency.
For the film treated with 10-min annealing, Itr, IoP, and Isc values corresponding to the annealing temperatures are graphed in Fig. 6-9. Following each procedure in annealing 

FIGURE 6-10 SEM of latex films, annealed for 10 min at (A) 120°C, (B) 130°C, (C) 150°C, and (D) 170°C temperatures. SEM, Scanning electron micrograph [64a].

process, the transmitted light intensity Itr started to escalate at higher than a preset tempera-ture, called minimum temperature of film formation T0. The scattered light intensity Isc jumped up just at the void closure temperature, Tv. Fluorescence intensity IoP first exhibits the behavior of increasing and hits the peak before decreasing as the annealing temperatures are raised. IoP reaches the highest level at the healing temperature Th.
The transparency of the annealed PS films accounts for the increased Itr at higher than T0. As mentioned in Section 6.3.1.1, the observations in this study demonstrated that the increased Itr values are very likely to agree to void closure, namely, PS first flows with film annealing and then the interparticle voids are covered. The sudden Isc increase which is identified at Tv overlaps with the inflection point on the Itr curve. At the lower degrees than Tv the roughness on the film surface leads to the isotropic light scattering. Annealing at Tv flats the film surface so that it becomes mirror-like. Thus, the reflecting light can be observed via the spectrometer with the photomultiplier detector. If annealing is carried on, the PS film becomes pellucid due to light and Isc falls to the bottom. Fig. 6—10 includes the SEM photo-graphs which confirms Fig. 6—10A where no particle deforms at 120°C. When temperature comes to 130°C the starting void closure enhances the growth of neck due to the viscous flow as illustrated in Fig. 6—10B. Then at Tv = 150°C this process completes where Isc is now at maximum (see Fig. 6—10C). For annealing at 150°C or more the film turns nearly pellucid as shown in Fig. 6—10D. However, it is assumed that the increased IoP at higher than T0 is consistent with the void closure up to the healing temperature Th (see Section 6.3.1.1). The interdiffusion between polymer chains may be responsible for the decreased IoP in the same condition. The behavior of IoP does not controvert the image in Fig. 6—8 available in Section 6.3.1.1).