Experimental and theoretical investigations of free radical photopolymerization: Inhibition and termination reactions

In this work, the inhibition and termination reactions occurring throughout a free radical photopolymerization initiated by a type-I photoinitiator are studied by kinetic modeling. The role of the macroradicals as the main oxygen trapping agents during the inhibition time is identified, and the absence of primary radical consumption by oxygen can be related to a high initiation efficiency at early times. The ratio of the termination reactions reveals that bimolecular termination remains the principal pathway for the cessation of macromolecule growth, even at high polymer conversion. Moreover, the evolution of the termination ratio during the polymerization can be correlated to both the diffusional control of the polymerization reactions as the polymer network grows and the photoinitiator consumption. Finally, the effect of the incident light intensity and the initial photoinitiator concentration on the termination reactions is assessed, and the validity of the steady-state assumption applied to the macroradical concentration discussed.

The corresponding bimolecular termination rate constant kt,b is then proportional to the propagation rate constant kp with a reaction diffusion factor Rrd (Eq. 1). Reaction diffusion termination starts as soon as diffusional bimolecular termination is hindered by the increase of the medium viscosity.
kt,b = Rrdkp [C=C] (1) Better understanding of the complex FRP mechanism and its related effects is a recurrent topic in polymer science. For that purpose, several modeling methods have been developed, starting with the statistical approach developed by Flory in the 1940's [54][55][56][57] and followed by spatial [58][59][60][61][62] and kinetic modeling [14][15][16][17][28][29][30]38,43,44,46,47,. This last method appears especially interesting as it: i. enables direct comparison between experimental and simulated polymerization kinetics, ii. is relatively simple to implement, iii. gives access to the concentration of species which are difficult or even impossible to obtain from experiment (macroradicals, initiating radicals…). The one developed by Bowman's group is currently the most complete, considering the whole photopolymerization process from light absorption by a type-I PIS to termination reactions [14,46,47,[78][79][80][81]. It has been notably used to determine the effect of oxygen inhibition on the propagation and termination rate constants [82], as well as model layer-by-layer photopolymerization [83] by Taki et al.. Such a kinetic model has been recently adapted for the photoinitiation involving a type-II PIS combining isopropylthioxanthone and triazine derivatives, evidencing the crucial role of back electron transfer in the kinetics of three-dimensional polymer network formation [84]. However, there is still a lack of quantitative data concerning the amount and the competition between the termination reactions for such a simple system as a type-I photoinitator. Indeed, among the photopolymerization kinetic models available in the literature, most of them does not take into account simultaneously all the termination pathways [16,17,[29][30]38,43,44,46,47,[64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79][81][82][83] and/or does not address their evolution all along the photopolymerization process [16,17,38,[63][64][65][66][67][68][69][70][71][72][73][74][80][81][82][83]. 4 This paper aims to improve the understanding of the crucial oxygen inhibition and termination steps through the modeling of the whole FRP process initiated by a type-I photoinitiator. A kinetic model considering simultaneously all the termination pathways is thus optimized by comparison with experimental photopolymerization kinetics. It is subsequently modified to identify the main oxygen trapping agent in the case of a laminated photopolymerization reaction, and discuss the dual role of the initiating radicals between initiation and termination in function of their reactivity towards the monomer and the macroradicals as the diffusion control settles. For the first time, the evolution of each of the termination modes is discussed with numerical values all along the photopolymerization reaction in conjunction with the evolution of the polymerization rate constants and the concentration of the reactive species. This study is finally extended to different incident light intensities and photoinitiator initial concentrations, and a new view on the validity of the pseudo-steady-state assumption is provided in relation with the advancement of the polymerization reaction.

Experimental methods
The formulations for the experimental kinetic analyses were based on a mixture of SR349 (90 wt%) and DMSO (10 wt%), in which 0.5 to 1 wt% of TPO has been dissolved. They were stirred overnight and kept in the dark. The photopolymerization kinetics were experimentally followed with a real-time Fourier transform infrared (RT-FTIR) spectrometer Vertex 70 from Brucker Optics, equipped with a nitrogen liquid-cooled mercury cadmium telluride (MCT) detector [85]. It operated in rapid scan mode with a sampling interval of 0.12 s and 4 cm -1 resolution. The light irradiation was provided by a 395 nm LED (Roithner, LaserTechnik) whose incident intensity was checked with a calibrated fiber optic spectrometer (Ocean Optics, USB4000), on which a polypropylene (PP) film and a CaF2 pellet were placed to take into account light loss on the interfaces. Oxygen diffusion from the atmosphere was avoided by performing laminated experiments (the photosensitive resin was placed between two PP films and two CaF2 pellets). A 25 µm Teflon spacer was also used in order to adjust the thickness of the formulation.

5
The FRP kinetics were measured by following in real time the disappearance of the C=C bond stretching vibration band of the diacryclic monomer at 1637 cm -1 . The degree of conversion was calculated as: where (A1637)0 and (A1637)t are, respectively, the areas of the 1637 cm -1 vibration band before light irradiation and at an irradiation time t. The rate of polymerization Rp is derived from the degree of conversion by: with [C=C]0 the initial concentration of acrylic doubles bonds in the formulation. Each experiment was repeated three times to ensure a good reproducibility and the data presented here result from an average.

Computational methods
Kinetic modeling was implemented in Wolfram Mathematica 9 software. The rate constants were defined as functions of the fractional free volume f, and the reactions of the photopolymerization mechanism were expressed as a set of differential equations, which was numerically solved with the NDSolve function [84].

Kinetic model conception
The main kinetic rate constants are defined as functions of the fractional free volume f in order to take into account their progressive diffusional control throughout the photopolymerization process [14,46,47,[78][79][80][81]. f represents the fraction of unoccupied volume in the reaction medium. It decreases as the polymerization proceeds according to: 6 Subscripts M and P refer to monomer and polymer respectively, with  the thermal expansion coefficient (difference between liquid and glassy state expansion coefficients), Tg the glass transition temperature,  the volume fraction and  the volumetric mass density.
The expressions for the propagation (kp) and bimolecular termination (kt,b) rate constants are given in Eqs. 7 and 8 [14,46,47,[78][79][80][81]. The expression of kt,b includes both diffusional bimolecular termination and subsequent reaction diffusion processes. kp0 and kt,b0 correspond to the propagation and bimolecular termination intrinsic rate constants respectively, i.e. without any diffusional control. Ap and At,b are the parameters which govern the rate at which, respectively, kp and kt,b decrease with viscosity. Finally, f,cp and f,ct,b represent the critical fractional free volumes at which propagation and bimolecular termination become diffusionlimited respectively.
The initiation rate constant ki and the PRT rate constant kt,PRT have the same expression as the propagation rate kp (Eq. 7). The exponential factors A and the critical fractional free volume f,c coefficients for initiation and PRT are taken equal to Ap and f,cp, meaning that ki and kt,PRT start decreasing at the same moment and at the same rate as kp does [14]. The values of the initiation (ki0) and PRT (kt,PRT0) intrinsic rate constants depend on the nature of the primary initiating radical. As introduced in a previous paper [84], the oxygen inhibition and primary radicals (re)combination rate constants can be assimilated to the diffusion rate constant kdiff given by the Stokes-Einstein equation (Eq. 9), as it involves the mass transport of oxygen molecules or primary radicals. Following a proposition by Buback [43], the viscosity  exponentially increases with conversion from its initial value 0, with an exponential factor B optimized to 2 (Eq. 10) [84]. The global FRP mechanism involving TPO as a type-I PIS is given in Figure 1. Under light exposure in the UV, TPO dissociates to yield a phosphonyl (RA • ) and a benzoyl (RB • ) radicals (see Figure S2 in the ESI  Figure 1 is given in the ESI. The determination of the kinetic parameters for TPO-initiated photopolymerization has been fully described in [84] and their values are listed in the ESI. Among them, the initial dissolved oxygen concentration has been optimized to 3.210 -4 M by a trial-and-error fit to experimental data, in agreement with the experimental value obtained in [96] for a pure acrylic monomer.  Table 1. Figure 3 plots the FTIR spectra before and after light irradiation for 1 wt% of TPO (see Figure S3 in the ESI for 0.5 wt%).

Oxygen inhibition in laminated samples
As stated before, experiments and simulations were performed in the case of laminated samples. However, oxygen inhibition actually occurs until the O2 molecules initially dissolved in the resin are totally consumed. The production of inactive peroxy radicals from the reaction of oxygen on primary radicals (RA • and RB • ) and macroradicals is described by the following set of differential equations:

Termination reactions
Three different bimolecular termination reactions can occur during FRP: i. formation of a chemical bond between two macroradicals (combination), ii. hydrogen abstraction from a macroradical to a second one with formation of a double bond on the former (disproportion), iii. reaction between a primary radical and a macroradical (PRT). As they both involve a reaction between two macroradicals, combination and disproportion are lumped into a single mechanism called bimolecular termination (kt,b), while PRT by RA • and RB • are distinguished (kt,PRT RA• and kt,PRT RB• ). It is possible to model these termination reactions by the following set of differential equations: [bimol] represents the concentration of macroradicals terminated by bimolecular termination (combination or disproportion), either diffusional or through reaction diffusion (Eq. 14).
A deeper insight into the evolution of termination modes all along the photopolymerization is given in Figure 6, where the fractions of macroradicals and terminated species are   Figure 7). Despite its decrease (due to the diffusion control of pure bimolecular termination), the effective kt,b stays higher than 210 5 M -1 .s -1 and thus overcomes the other termination modes. As PRT is not yet significant, the fraction of macroradicals is consequently reduced. kt,b stays almost constant from 20 to 40 % of conversion because termination by reaction diffusion takes over. Indeed, kt,b becomes then proportional to kp (see Eq. 1). As can be seen in Figure 7, kp stays constant up to 40 % of conversion because propagation becomes diffusion-controlled at a higher conversion than bimolecular termination (small monomer molecules are more free to diffuse than macroradicals). As a consequence, the increase of bimolecular termination (mainly by reaction diffusion) carries on between 20 and 40 % of conversion but is slower (Figure 6), and the fraction of macroradicals decreases (PRT is still not yet efficient).
The decrease of bimolecular termination after 40 % of conversion is also associated to the proportionality between kt,b and kp. Indeed, propagation becomes diffusion-controlled at high conversion (here after 40 % of conversion) and effective kp falls (see Figure 7). The reaction diffusion rate constant being proportional to kp (Eq. 1), kt,b also diminishes and bimolecular termination then decreases. As PRT is also not significant in this range, the amount of macroradicals consequently raises. The predominance of bimolecular termination over PRT throughout the polymerization can be explained by its higher rate compared to that of PRT (see Figure S6 in the ESI). The evolution of kp and kt,b all along the photopolymerization is actually in agreement with what have been experimentally observed [43,48,49,79] and simulated [43,46,79]. The evolution of this ratio as a function of polymer conversion is given in Figure 8. It can be seen that the Rt,PRT/Ri ratio is close to zero for a large part of the process, only increasing after 50 % of conversion for both RB • and RA • . After a maximum at 82 % of conversion, the ratio finally falls for both primary radicals. However, it has to be noticed that the ratio for phosphonyl RA • radicals is largely lower than that for benzoyl RB • ones throughout the process. An initial low value of Rt,PRT/Ri means that initiation is largely more efficient than PRT for a large part of the FRP reaction. The following raise shows that competition of PRT with initiation becomes more efficient when the conversion increases. Indeed, the effective kt,PRT and ki rate constants decrease by means of the progressive diffusional control of the associated reactions during the process. In the model, the exponential factor A and critical free volume f,c parameters are identical for both kt,PRT and ki (equal to Ap and f,cp, vide supra), thus the reaction rate constants start decreasing at the same time and at the same rate. In parallel,  Figure 6. Indeed, most part of the initial TPO is consumed after 80 % of conversion, so the production of primary initiating radicals falls.
Initiation and PRT are then strongly reduced. As a consequence of the reduction of initiation, less macroradicals are produced (up to a complete cessation when TPO totally disappears).
However, bimolecular termination still occurs between the remaining macroradicals, so its fraction increases while that of macroradical decreases. This final enhancement of bimolecular termination explains the final decrease of [R(C=C)n • ] observed in Figure 8, causing that of the Rt,PRT/Ri ratio and thus the final decline of PRT fraction for both initiating radicals.

Effect of experimental parameters I0 s and [TPO]0
The incident light intensity on the surface sample I0 s and/or the initial photoinitiator concentration are generally tuned in order to optimize the photopolymerization process.
However, a modification of these experimental parameters will affect the initiation and termination reactions. Kinetic modeling represents an efficient way to quantify this influence with numerical values, especially that on the termination modes.

Influence of incident light intensity I0 s on termination reactions
The ability of the kinetic model to reproduce changes in the incident light intensity was confirmed with photopolymerization experiments made at 2 and 5 mW.cm -2 (see Figures S7 to S10 in the ESI). The effect of incident light intensity (from 2 to 50 mW.cm -2 ) on conversion and Rp curves is shown in Figure 9. Final conversion is almost not affected: only a slight lowering can be observed at higher I0 s . However, it is clear that the rate of free radical photopolymerization is enhanced and the inhibition time reduced. As expected, the shape of Rp vs. irradiation time curves is strongly modified when I0 s is raised: i. the inhibition time is reduced, ii. the Rp max position is switched to shorter times, iii. the Rp max values are greater, iv.
the curves are sharper, revealing both faster autoacceleration and autodeceleration processes. Increasing the incident intensity enhances the rate of primary radical production at early times, hence that of macroradical formation (see Figure S11  Increasing the incident intensity I0 s has a strong impact on termination proportions at final conversion, as shown in Figure 10. Indeed, both bimolecular termination and PRT fractions increase with I0 s , while occlusion falls. They all reach a plateau at higher incident intensities. It clearly shows that, contrary to what is generally suggested [14], PRT never becomes the major way of termination, even at high light intensities. The increase of bimolecular termination and the reduction of radical trapping by occlusion are surprising. Indeed, it is generally stated in the literature that increasing incident intensity causes an increase of both PRT, by means of a higher concentration of primary radicals [97,98], and occlusion at final conversion, due to a faster vitrification of the reaction medium [26][27][28]. In order to understand the influence of I0 s on the final termination ratio, the change in the fractions of terminated species and macroradicals for different light intensities is compared to that of TPO concentration (Figure 11). For a given light intensity, the successive steps in the change of bimolecular termination, PRT and macroradical fractions are identical to that discussed in the previous section. When I0 s is modified, change in these fractions is also identical before 60 % of conversion, but reveals interesting differences after. Indeed, the PRT fraction increases earlier and becomes higher when I0 s is enhanced for both RA • and RB • . It also decreases sooner at the end of the process. The final increase of bimolecular termination, as well as the corresponding decrease of macroradical fraction, is also shifted to shorter times when the incident intensity is increased. Finally, the TPO photolysis is faster when I0 s is enhanced, by means of a higher amount of photons per second being absorbed by the photoinitiator molecules. The earlier increase of PRT could be explained by a higher rate of macroradical production when I0 s is increased. This shifts the efficient competition between initiation and PRT reactions for primary radicals to a lower conversion, a fact supported by an earlier enhancement of the Rt,PRT/Ri ratio when I0 s is increased (see Figure S12 in the ESI). PRT has then more time to 22 macroradicals also stops sooner. Bimolecular termination has then more time to occur between the remaining macroradicals. The final fraction of bimolecular termination is then enhanced, while that of radical trapping by occlusion is consequently reduced. As TPO is consumed, PRT efficiency is also reduced because of the decrease of the primary radical formation. Its fraction then decreases earlier when I0 s is enhanced.
The influence of the incident light intensity on the temporal evolution of the macroradical concentration is especially interesting because the steady-state assumption is generally applied to [R(C=C)n • ] in order to derive relations between the rate of polymerization Rp and the incident light intensity [6,8,9,25]. This assumption notably implies that [R(C=C)n • ] is constant throughout the polymerization, hence d[R(C=C)n • ]/dt = 0. Figure 12 shows  This increase could be explained by the diffusional control of both the propagation and bimolecular termination steps (vide supra). Indeed, the macroradical concentration strongly increases when bimolecular termination is reduced. The final decrease is associated to the progressive photolysis of TPO and the associated cessation of initiation. As a consequence, macroradicals are not produced but termination reactions take place. Figure 12 then reveals that the steady-state assumption should be valid between 5 and 40 % of conversion, but not for higher values. Indeed, the macroradical concentration strongly evolves in this range. It confirms previous observations made in the literature with a simpler kinetic model [78] and then reconsiders the validity of the steady-state assumption still generally applied on [R(C=C)n • ] in photopolymerization kinetics.

Influence of initial photoinitiator concentration on termination reactions
The comparison of experimental and simulated data for 0.5 and 1 wt% TPO in Figure 2 proves the ability of the model to account for the effect of the initial TPO concentration on the kinetics. Figure   Increasing the initial concentration of TPO enhances the amount of primary radical produced per second, by means of a higher absorbance of the sample. Macroradicals are then formed faster (see Figure S13 in the ESI) and the total consumption of the initially dissolved  macroradicals, which explains the higher final fraction of macroradicals. The earlier efficiency of PRT could be explained by the higher concentration of primary radicals and their increased formation rate when [TPO]0 is raised. Indeed, the Rt,PRT/Ri ratio is higher for a same conversion when [TPO]0 increases, promoting PRT over initiation at a lower conversion (see Figure S14 in the ESI). As a consequence, this termination way finally represents a higher proportion of the macroradicals produced.

Conclusion
A kinetic model taking into account simultaneously all the possible termination pathways (bimolecular termination, primary radical termination and radical trapping by occlusion) was successfully applied to the photopolymerization kinetics initiated by a type-I photoinitiator.
Comparison with experiments reveals a good agreement both on conversion and rate of polymerization curves. The inhibition by the initially dissolved oxygen molecules in laminated systems was considered in order to take into account the experimental inhibition time. It points out for the first time that the macroradicals act as the principal O2 scavengers at the early stages of the process, whereas the primary initiating radicals are almost totally involved in initiation reactions with the monomer.
The model was then adapted to identify the relative contribution of the different termination ways throughout the photopolymerization process with numerical values. It was shown that bimolecular termination remains the major termination reaction during the whole process.
However, its ratio, as well as that of primary radical termination and macroradicals, actually evolves because of: i. the progressive diffusion control of the polymerization reactions as the polymer network grows, ii. the cessation of initiation when the photoinitiator is totally consumed. The impact of the incident light intensity and the initial photoinitiator concentration on the termination modes was also investigated. As expected, the photopolymerization process is faster and the inhibition time lower when both parameters are increased, but the final conversion only increases when [TPO]0 is raised. Increasing the incident light intensity enhances the fraction of bimolecular termination and that of PRT. However, it is interesting to notice that the former remains the major termination pathway even at high incident intensity.
When the initial photoinitiator concentration is raised, PRT becomes more efficient, but radical trapping by occlusion in the glassy 3D polymer network also surprisingly increases.
It will be very interesting to extend this approach to more complex photoinitiating systems (type-II or three-component photocyclic ones), as the underlying photochemistry will be impacted by the formation of the polymer network during the irradiation but will also influence back the polymerization process. This will be the subject of a forthcoming paper.