2.3. Theoretical Modelling and Calculation of the Laser Fluency
The interaction of laser pulses with metallic surfaces has been studied extensively in relation to the photodesorption process [
19] and the damage threshold [
20]. Hereafter, we apply these well-established theories to the laser decontamination of a titanium implant surface. The reflectivity (
R) of the titanium surface is wavelength-dependent and is deduced from its complex refractive index (
n + i k) using the Fresnel equation:
As summarized in
Table 2, titanium reflectivity at 0.532 µm laser KTP (Potassium titanyl phosphate) laser is as low as 52% and increases to 90% at 10 µm (CO
2 laser), as expected from the improved screening of low-frequency electric fields by the metal electrons.
When the incidence angle departs from normal, the reflectivity becomes polarization-dependent, and the ‘averaged unpolarised’ reflectivity increases to reach 100% at grazing incidence. Upon reflection, the light is absorbed in a thin region underneath the metal surface corresponding to a penetration depth
(with
being the absorption coefficient) of a few tens of nanometers:
The rise in transient surface temperature can be determined by solving the heat diffusion equation:
where,
T is the local temperature,
A is the thermal diffusion coefficient (
),
is the heat capacity,
is the density, and
K is the heat conductivity of titanium.
S is the heat source corresponding to the local absorption of light. An analytical solution for the surface temperature rise (
) was derived by Bechtel et al. [
21] for a laser pulse (
) with a Gaussian temporal profile. The transient surface temperature (
) is given by:
where,
and
are pulse fluency and duration, respectively.
is the peak intensity of the laser pulse.
is a dimensionless function which can be derived from the parabolic cylinder functionsm [
21] and is tabulated in the inset of
Figure 1 which shows
.
From Equation (3), we deduce that the maximum temperature rise at the surface (
is inversely proportional to the square root of the laser pulse duration (
):
where,
is the maximum value of
η(x). This means that a Q-switch laser with a pulse duration of 6 ns can elevate the surface temperature with fluencies two orders of magnitude smaller than that of a laser working in relaxation mode with a pulse duration of 100 µs. The thickness of the metal layer, where most of the temperature gradient is localized, corresponds to the heat diffusion length (HDL). For the titanium substrate during the laser pulse, it is evaluated by quantity [
22]:
The HDL decreases with shorter pulses. This means that if the damage threshold is reached when
is higher than the titanium melting temperature, short pulses (<5 ns) will induce surface melting at a depth much smaller than the size of the implant surface microstructure. Therefore, despite surface melting, the shapes of the microstructure will be preserved. However, in the same condition, long laser pulses (>100 µs) will strongly modify the surface roughness (
Table 3).
Finally, Equation (3) indicates that the transient temperature at the interface scales with the laser pulse duration, independently from the laser pulse energy. No cumulative effect of heat will occur if the laser pulse separation in time is two orders of magnitude larger than the pulse duration.
The thermal desorption process of a single ad-molecule obeys the Arrhenius equation relating the rate of a chemical reaction to the absolute temperature (
T), a pre-exponential factor (
P) and the activation energy of the reaction (
Ea):
where,
C is the adsorbed molecule coverage,
is the desorption probability rate, and
Kb is the Boltzmann constant. When the complexity of the adsorbate increases and the interactions between ad-molecules must be taken into account, this rate equation must be adapted by defining, for example, the desorption order [
23]. The kinetics of the thermal desorption of organic contaminants or biofilms from titanium are not documented. The process is complex and includes several intermediate states such as the thermal degradation of the contaminants. For the sake of simplicity, however, we will model the decontamination of the titanium surface using the simplest form of the Arrhenius equation given by Equation (8).
From Equations (6)–(8), we can observe different trade-offs. When shortening the laser pulses (
), the pulse energy required to raise the surface temperature, as well as the HDL into the substrate, decreases with
. However, the transient temperature duration decreases with
which reduces the quantity of contaminant photo-desorption for the same value of
. This can be compensated by increasing
since the desorption rate increases exponentially with that quantity. To evaluate these trade-offs, we will adopt desorption kinetics parameters which are typical for organic molecules adsorbed on metals and titanium oxide surfaces [
24]. The pre-factor value is thus set to P = 10
14 Hz. The adsorption energy is set to
Ea = 30 kcal/mol (1.3 eV/molecule) corresponding to a strongly chemisorbed organic molecule which would desorb near 300 °C during a slow thermal programmed desorption ramp of 1 °K/s. We will assume that the laser operator will adjust the laser power to prevent damage to the implant surface occurring when the maximum temperature rise,
, reaches the titanium melting temperature (
= 1668 °C). Since
(with
= 25 °C), we will use, as setting parameter, the ratio:
This safety parameter (Sa) should be set at about 0.5–0.8 to preserve the implant surface topography. For the simulation, we further assume laser beam reflection coefficient of 61%. Figure 4 shows the remaining coverage of the contaminant after one laser shot as a function of the safety parameter (Sa) and for pulse durations of 0.5 ns, 6 ns, 100 µs and 100 ms.
Figure 2 shows the trade-off between the safety parameter and the pulse duration in order to achieve significant decontaminant desorption yield. Note that the safety parameter does not take into account the HDL which is much shorter in the case of a shorter pulse. This is in favor of short pulse durations, and this partly compensates, concerning the practical aspect of the safety, for the necessity to use a higher value of
Sa when using shorter pulses.
Table 4 summarizes the laser specifications required to achieve proper decontamination as a function of pulse duration. We assume that the operator selects a fluency to achieve 90% decontamination per shot. Therefore, about 10 shots are applied to ensure a decontamination level of 99.99999999%. The treatment of a complete implant, the surface of which is about one cm
2, requires about 2000 shots. Since the total optical energy impinging the implant, the implant volume (~5 × 10
−7 m
3), the titanium heat capacity and the reflectivity (61%) are known, the implant temperature rise (ITR) can be evaluated according to:
This modelling enables the prediction that a shorter pulse will improve the decontamination procedure. Very short pulses in the range of 500 ps appear advantageous. Further shortening of the laser pulse duration will not improve the decontamination process. Indeed, as observed for the measurement of metal surface damage thresholds [
25], the limited transfer rate of energy between the electronic and atomic vibration degrees of freedom in metals means that absorbed optical energy is thermalized in the substrate on a time scale of the order of 0.5 ns. Therefore, shorter laser pulses will not entail a shortening of the transient surface temperature.
The present modelling is certainly oversimplified. Indeed, it does not take into account either the absorption of the contaminant or the roughness degree of implant surfaces which may increase the reflectivity of the beam resulting in a reduction its efficiency. For those reasons, in our study, we decided to use a slightly higher energy density of 0.597 J/cm2 which is slightly higher than the theoretical one.