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We present a modeling study of photoinitiated polymerization in a thick polymer-absorbing medium using a focused UV laser. Transient profiles of the initiator concentration at various focusing conditions are analyzed to define the polymerization boundary. Furthermore, we demonstrate the optimal focusing conditions that yield more uniform polymerization over a larger volume than the collimated or non-optimal cases. Too much focusing with the focal length

UV-photoinitiated polymerization provides advantages over thermal-initiated polymerization, including fast and controllable reaction rates without a need for high temperatures or specific pH conditions [

The kinetics of photoinitiated polymerization have been studied by many researchers analytically, numerically, and experimentally [

We have recently developed semi-analytic modeling for photo-polymerization in a thick polymer (up to 10 mm) [

The above-described drawback exists for all photoinitiated systems that rely on illumination by a collimated laser beam, whose intensity decreases exponentially as a function of

To the best of our knowledge, this study provides the first presentation of a scaling law for optimal focusing, governed by the extinction coefficient and the initial concentration of the initiator. The focusing technique also provides a novel and unique means for uniform photo-polymerization (within a limited time of irradiation), which cannot be achieved by any other means.

As shown in _{0}

In Equation (1), _{0} =

Schematic of a focused UV laser propagating through an absorbing medium having thickness

For a thick polymerization system illuminated by a UV laser, the laser intensity, the photoinitiator and the photolysis product concentration, should be governed by a three-dimensional diffusion equation that can only be solved numerically. For a comprehensive analysis with an emphasis on the focusing features, we will ignore the diffusion effects such that the initiator profile may be described by a set of first-order differential equations.

The molar concentration of the photoinitiator _{0} is the initial value, _{0} = _{1}, with φ being the quantum yield and λ being the laser wavelength; and ε_{1} and ε_{2} are the molar extinction coefficient of the initiator and the photolysis product, respectively. In our calculations, the following units are used: ^{2}), λ in cm, _{j}^{ −1}. As with the conditions of references 9–12, we have ignored inhibition or self-focusing effects, which might be important in high light intensity case, but not in our low intensity case, 10–50 (mW/cm^{2}). Review of various kinetic conditions and different photosensitizers may be found in [

The coupled differential equations were solved, by finite element method, with the initial boundary conditions _{0} and _{0}. According to Equation (3), we can also obtain the additional conditions _{0} exp(_{0}_{0}_{1}_{0}_{2} = 0 and a collimated beam with _{2} ≠ 0, in which the photolysis product may still partially absorb the UV laser, the coupled differential Equations (3a) and (3b) become very difficult to solve analytically, and therefore, only numerical results (limited to the collimated case) have been reported thus far [

From our numerical results, shown later, the initiator concentration has a slowly varying spatial distribution, and thus, Equation (3) can be approximated to the first order as:

where _{(0)}(_{1} − ε_{2}) and _{2}. Substituting Equation (4a) to Equation(3a), we may easily find the integral expression for the first order solutions of the initiator concentration as follows:

The above equations provide an explicit formula for _{0}, and _{0}. Equations (4) and (5) show that the initiator concentration is a deceasing function of time (_{1}_{0}). Therefore, a higher concentration, shorter focusing, or a smaller _{1}_{0}). These two factors that compete in

If two active centers are produced upon defragmentation of the initiator, the local photoinitiation rate for the production of free radicals, _{1}

From the first order approximation shown by Equations (4) and (5), we readily see that the photoinitiation rate is proportional to ε_{1}_{0}, and the laser intensity is a competing, deceasing function of ε_{1}_{0}. Therefore, an optimal value of ε_{1}_{0} can be expected for a maximum photoinitiation rate derived from the balance between these two competing factors. The optimal condition will be shown in next section.

From Equation (5), for a given optimal focal length (_{1}_{0}*). This scaling prediction, based on our analytic formulas, will be quantitatively demonstrated later with numerical simulations.

Equation (3) will be solved using the finite element method. First, we will study the case of a collimated beam, where _{1} and _{0} on the transient profiles of

For a collimated beam, ^{−5} cm, _{2} = 0.075 (mM·cm)^{−1}. However, we have used a higher laser intensity, _{0} =20 mW, to shorten the time needed for the polymerization process.

The initiator concentration, _{0}

_{1}) on the profiles of the normalized _{0} = 2.0 mM at _{1}, which defines the coupling strength between the laser and the absorbing medium. In other words, larger coupling provides faster depletion of the initiator concentration, in which the depletion boundary always starts from the entrance plane (

Profiles of normalized initiator concentration _{0} _{1} = 0.1 to 0.5 (mM·cm)^{–1}, at an irradiation time

In Equation (2), the focusing function, _{0} for a fixed ε_{1} = 0.4 (mM·cm)^{−1} and various initial values of _{0}.

In _{0} = 2.0 mM. It is too focused for smaller _{0} < 2.0 and not focused enough for larger _{0} > 2.5. In other words, a shorter focal length is needed for a larger _{0}.

The data from _{1} and _{0}. This feature led us to search for a scaling law of _{1}_{0}) in the next section.

Profiles of normalized initiator concentration _{0} for a focused beam (with focal length _{1} = 0.4 (mM·cm)^{−1}, but for various initial concentrations _{0} = 1.0 to 3.0 mM.

As _{0} = 20 mM and various ε_{1} = 0.1 to 0.4 (mM·cm)^{−1}.

As shown by _{1} = 0.4 (mM·cm)^{−1}, _{0} between 1.0 and 4.0 mM. The optimal focal lengths (

We obtained _{0} = (1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0) mM, with ε_{1} =0.4 (mM·cm)^{−1} and _{1}_{0}). This scaling law is based on the numerical calculations and was also discussed based on our earlier analytic formulas, Equations (4) and (5). _{1}_{0} requires a tighter focusing, or smaller

Profiles (A to F) of normalized initiator concentration _{0} at ^{−1}. The initial concentration, _{0}, was varied between 1.0 and 4.0 mM (for _{0} = 1.0; (_{0} = 1.5; (_{0} = 2.0; (_{0} = 2.5; (_{0} = 3.0; (_{0} = 4.0 (mM).

Curve based on a scaling law. In addition, shown as dots are the data based on the calculated

We should note that the completion of the polymerization process at a given time may be defined by the amount of remaining _{0} > 0.3. The uncompleted polymerization is shown by the area with _{0} > 0.3.

We first show the collimated case. The time evolution of the polymerization boundary may be seen by the crossing positions of the horizontal red-line _{0} = 0.3 and the transient

Profiles of the initiator concentration under collimated laser polymerization, where the time evolution of the polymerization boundary is defined by the crossing positions of the horizontal red-line _{0} = 0.3. Curves 1 to 9 are profiles at _{0} =2.0 mM, and ε_{1} = 0.4 (mM·cm)^{−1}.

Same as

The above-discussed polymerization boundaries for various focusing conditions are further demonstrated by

For a collimated beam, the top portion (approximately 0.3 cm) of the medium is always polymerized starting from the surface (

As

Schematics of the time evolution of photo-polymerization via (1) a collimated beam, (2) a tightly focused beam (with

For the tightly focused case (with

The tightly focused case (2) in

We have presented comprehensive modeling for the kinetics of photoinitiated polymerization using a UV laser in thick polymer systems in which the photolysis product still partially absorbs the laser after polymerization. We have demonstrated that the focused beam at an optimal condition (_{1}_{0}) is derived numerically and shows that a larger extinction coefficient or a larger initial concentration of the initiator (larger value of (ε_{1}_{0})) require a tighter focusing or a smaller

This work was supported by the National Science Council of Taiwan under grants NSC 102-2221-E-039-005. It was also partially supported by the grant from Xiamen-200 program (Xiamen Science & Technology Bureau, China).

The authors declare no conflict of interest.