The UV photodissociation of the precursor only depends on the cross-section of the precursor molecule. In order to photolytically dissociate many precursor molecules, one has to look at the cross section in dependence of the wavelength and select a wavelength, where the cross section is high. This is the case when the photon energy meets the bond-dissociation energy of the radical and the precursor. In the modeling of the photodissociation process, a single photon photodissociation is assumed. For twoor more-photon absorption, Equations (1) and (2) have to be adopted, as for example described in [7-9].

The fluence can be described by the Beer-Lambert law. The fluence inside the precursor in a distance from the entrance window is described by

is the incoming fluence of the UV beam at the interface of the window into the reactor. From here on the UV photon flux will be referred to as the fluence (Joules per meter squared), regardless if the laser source is a pulsed one or emits a continuous wave. is the absorption coefficient of the precursor. It depends on the precursor density and the cross section for the UV wavelength,. The interface of the polymer and the precursor in z-direction is at. Inside the polymer, the fluence in dependence of is calculated to

is the absorption coefficient of the polymer.

To calculate the radical density generated at the height, the density of absorbed photons at that height has to be known. The density of absorbed photons can be calculated via the photons per meter squared, which is proportional to the fluence

is the photon energy in Joules. will be referred to as the photon fluence. The derivative with respect to of the photon fluence is the density of absorbed photons at the height

is negative in the case of absorption, as the photon density decreases.

In order to calculate the number of photons absorbed in the distance from to, one has to integrate the density of absorbed photons.

The relation between generated radicals and absorbed photons depends on the photolytic reaction, which is defined by the precursor molecule and the energy of the UV photons. This is also true for the radical density and the density of absorbed photons. In a simple case, each absorbed photon dissociates a precursor molecule generating a radical (or an ion, depending on the chemical polarity of the precursor molecule). In some cases where the photodissociation is not done by a single photon, the amount of generated radicals is not proportional to the photon flux [10]. On continuous wave (cw) sources, the flux does not change over time. For pulsed sources, the flux changes depending on the pulse shape. In the cases where two or more photons have to be absorbed by one and the same precursor molecule, two cases can occur. In the classical two-photon-absorption, two photons have to be absorbed at the same time. In the second case, the precursor molecule stays in a metastable excited state for a while after absorption of the first photon. In the latter case, the second photon needs to be absorbed while the molecule is in the excited state. Sometimes, both ways of photodissociation can occur [8]. The correlation between absorbed photons then strongly depends on the photon flux. This has extensively been described in literature [11].

The radical density can be generically described as

Here is the UV laser beam cross section and is the time constant in which the photons pass through the cross section. The function itself depends on the photolytical process and has to be derived individually for each precursor molecule in order to calculate the radical density.

In general, the highest radical density is generated where the fluence is the highest. Taking the described setup into account, the highest fluence is at the interface of the window to the precursor.

A high number of generated radicals is not enough for a high degree of halogenation. Also the distance between the radicals to the surface strongly affects the degree of fluorination. The radicals do have a limited lifetime, which sets their average travelling distance.

As radicals usually are highly reactive, it is assumed that the halogen radical will react with the first collision partner. For low irradiation doses, the radical density is much smaller than the precursor density. The radical density usually is a few orders of magnitudes smaller than the precursor density. Thus it is assumed that collisions between two radicals will take place only very rarely. In the model, this process is neglected. Only the two main processes are investigated; the first one, where the radical collides with a precursor molecule, and thus is no longer available for a reaction with the polymer, and the second one, where the radical collides with the polymer.

In order to calculate the number of halogens per unit area reaching the polymer surface, the probability of a radical to reach the surface in dependence of the distance and the precursor density is calculated. First the probability of the radical to travel a certain distance is calculated. The probability depends on the precursor density, the diameter of the precursor molecule and on the diameter of the radical. A distance, which radicals will reach, is called the mean free path. It is defined as

So the probability of a radical to reach a distance is

The radical can go into any direction from its origin. The probability to go into a certain direction is zero and cannot be used for further calculations, but a range of directions is the basis for the calculation of the probability of a radical to reach the polymer surface.

For calculating the probability of a radical to go into a range of directions, a conceptual model is used. All directions are represented by the surface of a sphere with its center point at the origin of the radical. The probability of the radical to go into a range of directions (pointing at a certain area of the sphere) is the surface of that area divided by the surface area of the whole sphere. In our case, where the probability of a radical to reach a surface (a layer) has to be calculated, the area of the sphere can be divided into many spherical segments. The spherical segments are parallel to the surface. The probability of the radical to pass through a certain segment is the surface area of the segment divided by the surface area of the sphere. The segments are defined by an angle, which is the angle between the surface normal and the direction of the radical and a second angle defining the range. The angular range is then from to. With an angle, the radical takes the shortest way possible, with an angle of, the radical travels parallel to the surface and with angles, the radical moves away from the surface. For the model consideration, the direction in x and y is not relevant (z is the surface normal). So the probability to reach the surface only depends on the height of the origin or the radical and the angle. Both defines the length of the path the radical has to overcome without a collision.

To calculate the overall probability of a radical with its origin in a distance of to the polymer surface travelling in a certain range of directions, the single probability of going into that range of directions and the probability to overcome the distance r without a collision has to be calculated. First, the probability to go into the direction is examined.

The probability of the radical to go into the direction of a certain spherical segment is

The surface area of a spherical segment is calculated via its radius and the height of the segment,. This leads to

The radius can be assumed as, the height h can be calculated via the two angles,

The probability has to be multiplied with the probability to reach a certain distance with .

In Equation (12), the probability of a radical to go into a certain direction and reaching the surface is presented. The probability strongly depends on. Note that we have to use the distance between the interface of the polymer to the precursor and the radical, which is. The general probability to reach the surface, independently of the direction is achieved via summing up the single probabilities for each direction.

The smaller is chosen, the more precise the result will be. With this formula, it is possible to calculate the probability to reach the surface of a radical with the origin at a height. To calculate the number of radicals reaching the surface from that height, the probability has to be multiplied with the generated radical density, the cross section of the laser beam and the height interval. The number of radicals reaching the polymer surface then is

To obtain the total number of radicals reaching the polymer surface from a volume between z and, has to be summed up.

Note that the number is with respect to the cross section of the laser beam.

With Equation (15), the number of radicals reaching the polymer surface can be calculated. In order to estimate the reaction speed, one has to look at several factors. In general, the reaction speed depends on the concentration of both reactants, in this case the concentration of the halogen and the hydrogen. The activation energy and the absolute temperature also play a major role in estimating the reaction speed. The activation energy itself depends on the ongoing reaction, which halogen radical is supposed to replace the hydrogen in which polymer. As the concentration of the hydrogen gets lower with an ongoing reaction, also the reaction speed will get lower. With the assumption, that the halogen radical concentration/density is independently of time, the concentration of the hydrogen atoms in the top layer of the polymer can be described by the following equation:

with as the reaction constant. The reaction constant includes the activation energy, the absolute temperature, the collision cross section of both reaction partners, their reduced mass, the Boltzmann constant and the Avogadro constant which all are integrated in Arrhenius’ equation.

with The equation cannot directly be applied to the reaction, as it only gives good results for reactions of ideal gases. The case of real molecules may be integrated via the insertion of the so called steric factor. This factor takes the deviation of the molecules shape from the ideal spherical shape into account. Usually the factor is measured by comparing the theory with the experiment.

As the concentration of one reaction partner will get lower with the time, the fluorine content of the surface will rise more slowly with ongoing time. The correlation can be expressed by the following equation:

The UV radiation also plays an important role in the reaction, which cannot be neglected. Depending on the polymer, the radiation is absorbed and activates certain bonds of the polymer. This lowers the activation energy of the reaction of the radical with the hydrogen at these bonds. Hence, the reaction of the halogen is more probable compared to unactivated bonds. To be accurate, this has to be taken into account when the reaction of the radicals with the polymer is being described.