Handbook of Modern Coating Technologies
Photon diffusion model for film formation
Photon diffusion theory bases the trip of a photon excited to or emitted from a dye molecule in a film formed by annealing latexes [141]. The overlapping probability (P) of a traveling photon with any distribution center in that film is calculated using the following equation:
where r represents the length between two successive centers the photon collides, and
Using Monte Carlo technique, potential collisions can be randomly simulated in a discre-tional p in between 0 and 1 (0 < p < 1). The value of r is estimated using Eq. (6.1) for a given < r > value following the completion of each collision. To compute the overall dis¬tance which a travelling photon has taken in the film, the z component is assigned based on those of the r values using the relation sz = П riz, where the index i represents the sequence of collision on the route of a photon. Out of the emerging photons from the film surfaces, both the back and the front, those which do not interact with a dye molecule such as pyrene can be identified under the specific constraints, respectively;
sz > d and sz < 0 (6.2)
where the front surface is the initial or null point, the emerging photons totally amount to Nsc which is assumingly proportional to the intensity Isc of the scattering light photons out of the latex film at the time of individual SSF measurements.
The association of a photon emitted from the latex film with the intensity (Iop)of fluores-cence was determined following after its probability of encountering a pyrene molecule was computed using the following expression:
q = 1 — exp( s/h) (6.3)
where s and h are the total and the mean trip length of the photon in the film (optical path), respectively, before the collision. For greater s, this probability is higher, or vice versa. The photon continues to travel even after being collided with a pyrene (Eq. 6.1). The occurrence of the photon emission from both surfaces of the film can be judged by comparison of the values of d to s, the z element of the total length (Eq. 6.2). Now the emerging photons from the front side totally amount to Nop, which is assumingly proportionate to IoP, the fluorescence intensity.
For Monte Carlo simulations the parameters of h and d were kept constant at 10 and 140, respectively. It is noted that h was assumingly proportional to the pyrene concentration of latex film as fixed throughout the formation process. For a given d, the mean distance of free pathway, < r >, remained within the range of 1 to 20, and the amount of incident photons during the simulations was identified as 3 X l04 for each < r > where the number of colli¬sions n was variable.
6.2.2 Void closure mechanism
Dillon et al. [142] are the first author who proposed the dry sintering of two interacted parti-cles by means of the polymer flow viscosity. The particles can be reshaped and the voids in between may be covered thanks to shearing stress from polymer-surface or -interfacial air tension. The necessary time for optical clarity and film formation is determined by the void
closure kinetics [38]. The following equality defines the relationship of the shrinkage of the voids spherical with radius r and the environmental viscosity n [4].
dr у 1
dt 2n p(r)
where the surface energy is represented by Y, the time by t, and the relative density by p(r). It is noted that the surface energy downsizes the voids when the term p(r) is variable depending on the quantity of voids, initial particle size, packing shape and other microstruc¬tural properties of the material. p(r) is the volume fraction of polymeric material to the voids, which is increasing up to an asymptote at the infinity as the limit of r is going to zero and decreasing for larger r values. Eq. (6.4) is very similar to the equation previously used to demonstrate why the lowest temperature varies by time in the film formation [3,39]. For this purpose, we keep the viscosity constant and take the integral of Eq. (6.4) as follows:
2n r
t = — p(r)dr (6.5)
Y r0
where t = 0, r0 corresponds to the initial void radius. The temperature of the melt polymer determines its viscosity because of the disability of macromolecular interaction, and the polymer chain segments skip from one to another equilibrium. Rising temperatures enable the free volume to enlarge adequately in the process, based on the elimination of potential barrier. The free energy of activation AG can be defined as higher barrier during viscous flow. According to Frenkel—Eyring model the temperature dependence of viscosity is repre¬sented as [143]
N0h (AG\ (6 6)
n =—exp kT (6-6)
where Avogadro's number represents N0, Planck's constant h, molar volume V, and Boltzmann's constant k. As reported, AG = AH — TAS, thus Eq. (6.6) can be rewritten as
n = Aexp AAH (6-7)
where the activation energy of viscous flow or the quantity of heat is AH, which is to be taken as a given value for one mole of the substance to create a jump during the viscous flow, and the entropy of activation of viscous flow AS. In the above expression, A represents a constant for the potential variables other than temperature. Eqs. (6.5) and (6.7) produce the following equation:
Now, we will interpret the fluorescence data to clarify the void closure mechanism.
