# Dual-Wavelength (UV and Blue) Controlled Photopolymerization Confinement for 3D-Printing: Modeling and Analysis of Measurements

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## Abstract

**:**

_{10}) and rate constant (k’) lead to higher conversion, as also predicted by analytic formulas, in which the total conversion rate (R

_{T}) is an increasing function of C

_{1}and k’R, which is proportional to k’[gB

_{1}C

_{1}]

^{0.5}. However, the coupling factor B

_{1}plays a different role that higher B

_{1}leads to higher conversion only in the transient regime; whereas higher B

_{1}leads to lower steady-state conversion. For a fixed initiator concentration C

_{10}, higher inhibitor concentration (C

_{20}) leads to lower conversion due to a stronger inhibition effect. However, same conversion reduction was found for the same H-factor defined by H

_{0}= [b

_{1}C

_{10}− b

_{2}C

_{20}]. Conversion of blue-only are much higher than that of UV-only and UV-blue combined, in which high C

_{20}results a strong reduction of blue-only-conversion, such that the UV-light serves as the turn-off (trigger) mechanism for the purpose of spatial confirmation within the overlap area of UV and blue light. For example, UV-light controlled methacrylate conversion of a glycidyl dimethacrylate resin is formulated with a tertiary amine co-initiator, and butyl nitrite. The system is subject to a continuous exposure of a blue light, but an on-off exposure of a UV-light. Finally, we developed a theoretical new finding for the criterion of a good material/candidate governed by a double ratio of light-intensity and concentration, [I

_{20}C

_{20}]/[I

_{10}C

_{10}].

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Photochemical Kinetic

_{1}, C

_{2}and C

_{3}for the ground state concentration of PA, PB and PC, respectively and [M] for the monomer, the kinetic equations for the dual-color and 5 radicals (R’, R, [N] and [X]) system are derived as follows [12,15,16]:

_{1}= b

_{1}I

_{1}(z,t), B

_{2}= b

_{2}I

_{2}(z,t)I

_{3}(z,t)/I

_{30}, B

_{3}= (b

_{3}/b

_{2})B

_{2}, g =1/(k

_{57}+ kC

_{3}), g’ = 1/(k

_{68}+ [M]), k

_{57}= (k

_{5}/k

_{7}), k

_{68}= (k

_{6}/k

_{8}) and k = (k

_{3}/k

_{7}). R

_{E}is the C

_{1}regeneration term given by R

_{E}= k

_{22}[N]R + 2k

_{T}R

^{2}. b

_{j}= 83.6a

_{j}q

_{j}w

_{j}; a

_{j}is the extinction coefficient for PA, PB and PC (with j = 1,2,3); the light wavelength (in cm), w

_{1}for the blue (at 470 nm) and w

_{2}for UV (at 365 nm) and light intensity I

_{j}(z,t) in mW/cm

^{2}; q

_{j}is the quantum yields of the PA triplet state and PB radical.

_{3}T*C

_{3}, with the steady-state T* = (g/k

_{7})B

_{1}C

_{1}. The UV light intensity, I

_{2}(z,t), is absorbed mainly by PB, in which the UV conversion of PA monomer is reduced by the presence of PB. On the other hand, the blue-conversion of PA could be significantly reduced by the UV-generated radical of PB, such that the inhibition depth may be controlled by the on-off of UV light (more will be discussed later). All the reaction rate constants are defined by the associated coupling terms. For examples, in Equation (7), k’ is for the reaction of monomer and radical R, which has a relaxation rate k

_{5}; k

_{12}is for the radical interaction of R’ and R and both have a bimolecular termination rate of k

_{T}. More detail derivation and definition of rate constants in g and g’ have been previously published [12,15].

_{2}C

_{2}/(k

_{22}R), [X] = g’B

_{2}C

_{2}/k

_{8}, and Equation (8) becomes:

_{T}is a total rate constant, which consists of two crosslink components attributed from the interaction of the monomer and [X] and R, respectively. Furthermore, the steady-state radicals, R’ and R, are given by:

_{12}R’. Equation (13) may be further approximated to R = (0.5H/k

_{T})

^{0.5}− k’[M](1 − d), with d = 0.5G

^{2}/(8k

_{T}H), for 2k

_{12}R’ << k’[M], which shows that R and efficacy, are increasing the function of H. The balance point of inhibition depth is defined by when R = 0, or 8k

_{T}H = 0 or gB

_{1}C

_{1}C

_{3}= B

_{2}C

_{2}, in which the PA initiated radical (R) is completely inhibited/consumed by the PB’s radical, [N]. We will have more discussion later.

_{1}) and UV (I

_{2}) are given by, when they are applied to the resin orthogonally and separately [13,15]:

_{j}is the extinction coefficients of PA (for j = 1) and PB (for j = 2) and their photolysis products, respectively; Q

_{j}is the absorption coefficient of the monomer at the blue and UV wavelength. Most previous modeling [8,9,10,11,12,13] assumed a constant C (z, t) in Equation (2.b). Our analytic formulas in this article will use a time-average of A (z, t) to count for the dynamic of light intensity due to PA and PB depletions. Accurate solutions of Equations (1) and (8) require numerical simulations (to be shown later). For analytic formulas, we will use approximated analytic formulas for the light intensity and the PI and PE concentration. The expressive closed forms of I

_{j}(z,t) and C

_{j}(z,t) allow us to solve for the first-order and second-order solutions of R, [M] and the conversion efficacy.

#### 2.2. Analytic Formulas for Efficacy

_{EFF}= 1 − [M]/[M]

_{0}= 1 − exp(−S), with [M]

_{0}being the initial monomer concentration, and the S-function is given by the time integral of the total rate factor R

_{T}given by Equation (1.f), d[M]dt = –R

_{T}[M], in which R

_{T}has three components defined by the coupling of the monomer [M] and the triplet-state, PA-radical and PB radical, respectively.

_{53}<< kC

_{3,}k

_{68}<< k”[M], gC

_{3}= g’[M] = 1, the solutions of Equations (1) to (3) are available by the approximated analytic formulas for I

_{j}(z,t) and C

_{j}(z,t), with j = 1,2,3, for PA, PB and PC, as follows [12,13]:

_{j}= b

_{j}I

_{j0}exp(−A

_{j}’z), A

_{j1}= 2.3(a

_{j}− b’

_{j})C

_{j0}I

_{j0}b

_{j}z

_{,}with A

_{j}’ is the time-averaged absorption given by Aj’ = 1.15(a

_{j}+ b’

_{j}) + 2.3Q

_{j}, bj’ is the extinction coefficient of the photolysis products. We note that the –A

_{j1}t term represents the decrease of A

_{j}’, or increase of light intensity due to concentration depletions of PA, PB and PC.

_{T}, we solved Equation (9) to obtain the total efficacy given by CEFF = 1−[M]/[M]

_{0}= 1−exp(−S), where S is the time integral of R

_{T}, which requires a numerical integration, in general. For analytic solutions, two cases were considered. For gB

_{j}C

_{j}<< k’R, case (i) H >> G, k’R = KH

^{0.5}, with K = 0.5k’/k

_{T}

^{0.5}; case (ii) H << G, k’R = k’H/G; where H

^{0.5}may be further reduced to H

^{0.5}= (B

_{1}C

_{1})

^{0.5}– 0.5(B

_{2}C

_{2})/(B

_{1}C

_{1})

^{0.5}, for (B

_{2}C

_{2}) << (B

_{1}C

_{1}) and gC

_{3}= 1, for k = k

_{3}/k

_{7}= 1.

_{EFF}= 1 − [M]/[M]

_{0}= 1−exp(−S), with S-function is given by:

_{j0}= B

_{j0}X

_{j},G

_{3j}= 0.5(B

_{j0}− A’

_{1j}), with B

_{j0}= b

_{j}I

_{j0}C

_{j0}, X

_{j}= exp(−A’

_{j}z)

_{,}A’

_{j}= 1.15(a

_{j}+ ${b}_{\mathrm{j}}$)C

_{j0}+ 2.3Q

_{j}, is a mean value of A

_{j}(z,t). We note that Equation (20) reduces to our previous formula for one-wavelength system with B

_{20}= 0 and H = B

_{1}C

_{1}.

_{T}

^{0.5}, Equation (9) becomes:

_{0,}or:

_{11}= 1/G

_{31}, E

_{12}= 1/(G

_{32}− G

_{31}), whereas transient state is given by E

_{11}= E

_{12}= t. Therefore, the inhibition effect given by the second term of Equation (21) is proportional to B

_{20}/(B

_{10})

^{0.5}/(G

_{32}− G

_{31}), with B

_{j0}= b

_{j}I

_{j0}C

_{j0}, for steady-state; and [tB

_{20}/(B

_{10})

^{0.5}] for transient state. Numerical data will be shown later. We also note that for a given B

_{1}C

_{1}, the radical R is a decreasing function of the ratio of R

_{AB}= (B

_{2}C

_{2})/(B

_{1}C

_{1})

^{0.5}. Therefore, the same R

_{AB}reaches the same efficacy. This feature will be numerically shown later.

#### 2.3. The Inhibition Depth and Time

_{H}) defined by the balance point of initiation and inhibition rate, or when R = 0, or H = 0. We find from Equation (14),

_{j0}= b

_{j}I

_{j0}, and C

_{j}(t) are the z-averaged function of C

_{j}(z,t). We note that Equation (8) defines an inhibition coefficient defined by β = (b

_{2}/b

_{1})[C

_{2}/(gC

_{1}C

_{3})], which depends on a multifactor and rate constants related by g = 1/(k

_{57}+ kC

_{3}). Our formula is more general than that of de Beer et al. [10], which is our special case when A

_{3}= 0, and C

_{2}= gC

_{1}C

_{3}, such that Equation (25) reduces to Equation (1) of de Beer et al. [10]: z

_{H}= (1/(A

_{2}– A

_{1})ln[βI

_{20/}/I

_{10}], with β(k37 + kC3 = [M])b

_{2}/b

_{1}. We note that b

_{j}= 83.6a

_{j}q

_{j}w

_{j}, which is defined by the extinction coefficient

_{for}PA, PB and PC (with j = 1,2,3); the light wavelength, w

_{1}for the blue (at 470 nm) and w

_{2}for UV (at 365 nm) and the quantum yields (q

_{j}). Moreover, in our more general formula, β is also proportional to 1/g = k

_{57}+ kC

_{3}, defined by the rate constants of k

_{57}and k = k

_{3}/

_{7}.

_{j}(z,t), Equation (25) in general is time-dependent, which was assumed as time-independent by de Beer et al. [10], when C

_{j}reaches a steady-state or remains as its initial value, in which the initiators depletion is ignored. To explore this dynamic feature, one may define an inhibition time (T

_{H}) given by when the radical R = H = 0. For a common situation that that C

_{1}(t) = C

_{10}exp(-B’t), with PA has a depletion rate, B’, much larger than that of PB and PC, such that C

_{2}(t) = C

_{20}, C

_{3}(t) = C

_{30}, both are much slowing decay function of time (to be shown by our numerical data later), we obtained an analytic formula:

_{2}/b

_{1})[C

_{20}/(gC

_{10}C

_{30})]. Equation (26) shows that T

_{H}is an increasing function of the depth (z), but a decreasing function of the concentration ratio C

_{20}/(C

_{10}C

_{30}), i.e., higher inhibitor concentration C

_{20}, results to a shorter inhibition time, which is desired for a faster on-off switching mechanism.

_{20}/I

_{10})

_{crit}defined by which initiation and inhibition rates are balanced to generate an inhibition depth, z

_{H}= 0 in Equation (25), and can be calculated by when Rmin = 1/$\mathsf{\beta}\prime $ = gC

_{10}C

_{30}/[(b

_{2}/b

_{1})C

_{20}], which is dependent on resin composition ratios and rate constants. de Beer et al. [10] reported $\mathsf{\beta}\prime =1$ in a TMPTA-based system.

_{T}, and defined by the S-function higher than a critical value, S > S

_{T}, or efficacy C

_{EFF}> C

_{T}, where S

_{T}= ln [1/(1 – C

_{T})], which can only be calculated numerically (to be shown later).

#### 2.4. Print Speed

_{1}= $\beta $B

_{2}, we obtain a similar formula:

_{2}b

_{20}/b

_{10})/(gC

_{1}C

_{3}), than the simplified function of de Beer et al. [10], with β =

_{20}/b

_{10}. A more accurate definition would be based on the S function, or time integral of Equation (10), rather than light dose given by Equation (20). However, S

_{max}needs numerical result integral of Equation (10), which is to be shown later.

#### 2.5. Curing Depth

_{10}t, is larger than a threshold value of E

_{TH}. Using the time integral of Equation (17) with neglected A

_{1}t, we obtain,

_{EFF}> C

_{T}, or when S > S

_{T}, with S

_{T}= ln [1/(1 − C

_{T})]. We obtain:

_{1}I

_{10}C

_{10}), and X’ = exp(−A’z

_{C}), with A’ = 1.15(a’

_{1}+ b’

_{2})C

_{10}+ 2.3Q

_{1}, is a mean value of A

_{1}(z,t).

## 3. Results and Discussion

_{2}= 0 (no UV light) for various concentration of the initiator, C

_{10}= (0, 0.5, 1.0, 3.0)%, coupling parameter b

_{1}, which is given by the absorption coefficient and blue-light intensity and also the role of the crosslink rate constant (k’), which gives the conversion in Equation (10). For simplicity, our modeling was limited to the surface layer of the resin, that is for z = 0. Spatial conversion profiles can be found in our previous study, which is limited to a single-wavelength system [13,14].

_{1}C

_{1}–B

_{2}C

_{2,}as suggested by our analytic formulas, Equation (6), our numerical input will be the initial values of B

_{j}or b

_{j}(with j = 1,2) rather than the light intensity or the absorption coefficients. We also showed that the IBE was strongly monomer-dependent, as reported by van der Laan et al. [11] by various reaction rate constants (k’, k

_{T}). The numerically produced temporal profiles were analyzed by our analytic formulas of Equation (20). Finally, our numerical profiles were fit to the measured data of de Beer et al. [10] and van der Laan et al. [11] in an on-off scheme.

#### 3.1. Efficacy Temporal Profiles

_{10}, led to higher conversion, as also predicted by our Equation (10), in which the total conversion rate (R

_{T}) was an increasing function of k’R, which was proportional to [gB

_{1}C

_{1}]

^{0.5}. However, the coupling factor b

_{1}played a different role that higher b

_{1}led to higher conversion only in the transient regime; whereas higher b

_{1}led to lower steady-state conversion (as shown by right Figure). This unique reverse effect in steady-state was also predicted by our analytic Equation (20).

_{T}and b

_{j}. Conversion of blue-only were much higher than that of UV-only and UV-blue combined, in which a high inhibitor concentration (C

_{20}) resulted in a strong reduction of blue-only-conversion, such that the UV-light served as the turn-off (trigger) mechanism for the purpose of spatial confirmation within the overlap area of UV and blue light. Figure 5 shows the conversion profiles under the same conditions as that of Figure 4, but for different resin formations, which were specified by our parameter k”. As reported by van der Laan et al. [11], different monomers have different C=C bond rate constants (K) under the exposure of blue, UV and blue + UV. For example, bisphenol ethoxylate diacrylate (BPAEDA) resins formulated with camphorquinone (CQ) and ethyl 4-(dimethylamino)benzoate (EDAB) had a maximum conversion rate constant Kmax= 0.675 (at blue + UV) for 0% butyl nitrite (BN), and reduced to 0.0106 (for 1%BN), a factor of 64 reduction, Therefore, it was a better candidate than trimethylolpropane triacrylate (TMPTA), which only had a three times reduction of Kmax.

_{20}), in which, for a fixed initiator concentration C

_{10}, higher C

_{20}led to lower conversion due to a stronger inhibition effect. However, as shown by Figure 7, same conversion reduction was found for the same H-factor of H

_{0}= [b

_{1}C

_{10}- b

_{2}C

_{20}]. This unique feature was also predicted by Equations (13) and (20).

#### 3.2. Analysis of Measured Data

_{1}, b

_{2}and rate constants (k’, k”, k

_{T}), but kept the same initiator concentrations (C

_{10}, C

_{20}, C

_{30}), as the measured data [10]. The monomer-dependence of the conversion for various resin formations is governed by the adjusted rate constants (in a relative amount), because the actual values are not available.

#### 3.3. The General Criterion for an Efficient UV-Inhibitor

_{OFF}and H

_{ON}for the H-value without and with-UV, respectively, an efficient candidate (or effective UV-inhibitor) requires two conditions: (i) high enough H

_{OFF}such that the conversion without UV-light larger than 50%; and (ii) high enough H

_{ON}such that the conversion of both blue and UV light is reduced to lower than 20%. This concept could be further described mathematically as follows. H = H

_{OFF}− H

_{ON}= H

_{OFF}(1 − H

_{ON}/H

_{OFF}). Therefore, a good candidate requires a large H

_{OFF}and also a high H-ratio, R

_{H}= H

_{ON}/H

_{OFF}. For example, for a fixed value of H

_{OFF}=10, a candidate with H

_{ON}= 6 leading to R

_{H}= 0.6, and H = 10 × (1 – 0.6) = 40, is not as good as a candidate having a higher H

_{ON}= 8 leading to R

_{H}= 0.8 and H = 80), which is four times lower, presenting a stronger inhibition triggered by the UV-light.

_{3}B

_{1}C

_{1}-B

_{2}C

_{2}, defines R

_{H}= B

_{2}C

_{2}/(gC

_{3}B

_{1}C

_{1}). We note that B

_{j}= b

_{j}I

_{j}, which is proportional to the light intensities (I

_{1}for blue-light and I

_{2}for UV-light) and the effective absorption constant (b

_{j}) governed by the quantum yield (q) and absorption coefficient at a specific wavelength. Therefore, a high H-ratio (R

_{H}) is determined not only by the material properties, but also the ratio of light intensity (UV/blue), and concentration ratio of the initiator and inhibitor, C

_{20}/C

_{10}. In addition, it is also rate-constant dependence, because the g-factor is given by g = 1/(k

_{57}+ kC

_{3}). Therefore, we concluded that the criterion for a good candidate is governed by collective factors, and at least by the double ratio of [I

_{20}C

_{20}.]/[I

_{10}C

_{10}]. The above criterion was an important new finding of our theoretical study, which requires further experimental study to confirm (to be discussed later).

#### 3.4. Role of Oxygen and Suggested Experiments

_{20}C

_{20}]/[I

_{10}C

_{10}] criterion could be experimentally justified by an experimental setup having adjustable light intensities and the initiator concentration. As shown by Figure 7, that same conversion reduction was found for the same H-factor of H

_{0}= [b

_{1}C

_{10}– b

_{2}C

_{20}]. This unique feature was also predicted by Equations (13) and (20). The experimental setup of Childress et al. [24], using various light red and UV light intensity (I

_{10}and I

_{20}), but fixed concentrations of C

_{10}and C

_{20}, could be easily extended for variable concentrations, such that our double ratio criterion could be justified.

## 4. Conclusions

_{10}) and rate constant (k’) led to higher conversion, governed by a scaling law of k[gB

_{1}C

_{1}]

^{0.5}. However, the coupling factor b

_{1}played a different role that higher b

_{1}led to higher conversion only in the transient regime; whereas higher b

_{1}led to lower steady-state conversion. For a fixed initiator concentration C

_{10}, higher C

_{20}led to lower conversion due to a stronger inhibition effect. However, the same conversion reduction was found for the same H

_{0}= [b

_{1}C

_{10}– b

_{2}C

_{20}]. Conversion of blue-only was much higher than that of UV-only and UV-blue combined, such that the UV-light served as the turn-off (trigger) mechanism for the purpose of PC. The UV-light initiated inhibition effect was strongly monomer-dependent and different monomers had different C=C bond rate constants and conversion efficacy.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematics of photochemical pathways of dual wavelength photopolymerization; in which crosslinkers are formed via two pathways, via the photoinitiator PA (under a blue light), and PB (under a UV light). The initiation radicals R and [X] crosslink with the monomer [M]; whereas the inhibition radicals [N] reduces the conversion efficacy by reducing the active radicals (R’ and R). Shown also is the co-initiator (PC), which reacts with the triplet state of PA (T*) forming an intermediate radical (R’). Bimolecular termination of R’ produces a propagating radical (R) which leads to crosslinks; terminations could be also resulted by the interaction of R and R’, and R and [N].

**Figure 2.**Conversion profiles of blue-light (without UV-light) for (left Figure) C

_{10}= (0.05, 0.1, 0.2, 0.4) %, for curve (1,2,3,4), for fixed b

_{1}= 0.1; and (right Figure) b

_{1}= (0.015, 0.05,0.15,0.5), for fixed C

_{10}= 0.2 %; for C

_{30}= 0.5 %, [M]

_{0}= 0.2 %, k’ = 1.0, k

_{T}= 0.5, k

_{57}= (k

_{5}/k

_{7}) = k

_{68}= (k

_{6}/k

_{8}) = k” = 35 (1/s).

**Figure 4.**Conversion profiles for blue-only (black curve—1), UV-only (blue curve—2) and both-light (red curve—3), for C

_{10}= 0.2%, C

_{30}= 3.0%, b

_{1}= 0.1, b

_{2}= 0.007, k” = 35; where solid color curves are calculated and bars are measured data of de Beer et al. [10].

**Figure 6.**The initiation radical (R, left) and conversion (right) profiles in the presence of UV light; for various inhibitor concentration, C

_{20}= (0, 1.0, 2.0, 3.0), for curves (1,2,3,4; red, green, blue, violet), for b

_{1}= b

_{3}= 0.1 (1/s/%), C

_{10}= 0.2 (%), C

_{30}= 0.5 (%), [M]

_{0}= 0.2 (%); k’ = 1.0 (1/s), k

_{48}= 1.0 (1/s), k

_{37}= 1.0 (1/s), k

_{57}= 0.01 (1/s).

**Figure 7.**The same as Figure 6, but for a fixed difference of [b

_{1}C

_{10}− b

_{2}C

_{20}] = 0.003, for C

_{20}= 0 (curve-1) and C

_{20}> 0, for curves (2,3,4) showing the overlapping of these three curves.

**Figure 8.**The initiation radical (left) and conversion (right) profiles for C

_{20}= (0, 0.5,1.0, 3.0), for curve (red, green, blue, violet), in the presence of both blue and UV light; for b

_{1}= 0.04, b

_{2}= 0.002, b

_{3}= 0.1 (1/s/%), k’ = 2.0 (1/s), k

_{48}=10 (1/s), k

_{37}= 20(1/s) and k

_{57}= 0.01 (1/s). In right figure, the background is measured data from van der Laan et al. [11].

**Figure 9.**Methacrylate conversion of a bisGMA/TEGDMA resin formulated with 0.2 wt% CQ/0.5 wt% EDAB/0/5 wt% BN and subject to a continuous exposure of a blue light, but an on-off exposure of a UV-light for 0.5 min, as indicated by the violet vertical areas; where black bars are measured data from van der Laan et al. [11] and red curve is our theoretical simulation.

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**MDPI and ACS Style**

Lin, J.-T.; Cheng, D.-C.; Chen, K.-T.; Liu, H.-W.
Dual-Wavelength (UV and Blue) Controlled Photopolymerization Confinement for 3D-Printing: Modeling and Analysis of Measurements. *Polymers* **2019**, *11*, 1819.
https://doi.org/10.3390/polym11111819

**AMA Style**

Lin J-T, Cheng D-C, Chen K-T, Liu H-W.
Dual-Wavelength (UV and Blue) Controlled Photopolymerization Confinement for 3D-Printing: Modeling and Analysis of Measurements. *Polymers*. 2019; 11(11):1819.
https://doi.org/10.3390/polym11111819

**Chicago/Turabian Style**

Lin, Jui-Teng, Da-Chuan Cheng, Kuo-Ti Chen, and Hsia-Wei Liu.
2019. "Dual-Wavelength (UV and Blue) Controlled Photopolymerization Confinement for 3D-Printing: Modeling and Analysis of Measurements" *Polymers* 11, no. 11: 1819.
https://doi.org/10.3390/polym11111819