Thiol–Ene Photopolymerization: Scaling Law and Analytical Formulas for Conversion Based on Kinetic Rate and Thiol–Ene Molar Ratio

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Article Thiol–Ene Photopolymerization: Scaling Law and Analytical Formulas for Conversion Based on Kinetic Rate and Thiol–Ene Molar Ratio

Kuo-Ti Chen 1, Da-Chuan Cheng 2, Jui-Teng Lin 3 and Hsia-Wei Liu 4,* 1 Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University, New Taipei City 24205, Taiwan; [email protected] 2 Department of Biomedical Imaging and Radiological Science, China Medical University, Taizhong 400, Taiwan; [email protected] 3 New Vision, Inc., New Taipei City 242, Taiwan; [email protected] 4 Department of Life Science, Fu Jen Catholic University, New Taipei City 24205, Taiwan * Correspondence: [email protected]; Tel.: +886-0910-688-968

Received: 22 September 2019; Accepted: 7 October 2019; Published: 10 October 2019

Abstract: Kinetics and analytical formulas for radical-mediated thiol–ene photopolymerization were developed in this paper. The conversion efficacy of thiol–ene systems was studied for various propagation to chain transfer kinetic rate-ratio (RK), and thiol–ene concentration molar-ratio (RC). Numerical data were analyzed using analytical formulas and compared with the experimental data. We demonstrated that our model for a thiol–acrylate system with homopolymerization effects, and for a thiol–norbornene system with viscosity effects, fit much better with the measured data than a previous model excluding these effects. The general features for the roles of RK and RC on the conversion efficacy of thiol (CT) and ene (CV) are: (i) for RK = 1, CV and CT have the same temporal profiles, but have a reversed dependence on RC; (ii) for RK >> 1, CT are almost independent of RC; (iii) for RK << 1, CV and CT have the same profiles and both are decreasing functions of the homopolymerization effects defined by kCV; (iv) viscosity does not affect the efficacy in the case of RK >> 1, but reduces the efficacy of CV for other values of RK. For a fixed light dose, higher light intensity has a higher transient efficacy but a lower steady-state conversion, resulting from a bimolecular termination. In contrast, in type II unimolecular termination, the conversion is mainly governed by the light dose rather than its intensity. For optically thick polymers, the light intensity increases with time due to photoinitiator depletion, and thus the assumption of constant photoinitiator concentration (as in most previous models) suffers an error of 5% to 20% (underestimated) of the crosslink depth and the efficacy. Scaling law for the overall reaction order, defined by [A]m[B]n and governed by the types of ene and the rate ratio is discussed herein. The dual ratio (RK and RC) for various binary functional groups (thiol–vinyl, thiol–acrylate, and thiol–norbornene) may be tailored to minimize side effects for maximal monomer conversion or tunable degree of crosslinking.

Keywords: thiol–ene functional group; kinetic model; photopolymerization; crosslinking; curing depth

1. Introduction Photopolymerization and crosslinking have been utilized in various medical and industrial applications [1–5]. Compared to thermal-initiated polymerization, photo-initiated polymerization provides the advantages of fast and controllable reaction rates and spatial and temporal control over the formation of the material without the need for high temperatures or harsh conditions [1,2]. Tissue engineering using scaﬀold-based procedures for chemical modiﬁcation of polymers has been reported

Polymers 2019, 11, 1640; doi:10.3390/polym11101640 www.mdpi.com/journal/polymers Polymers 2019, 11, 1640 2 of 17 to improve their mechanical properties by crosslinking or polymerization with UV or visible light to produce gels or high-molecular-weight polymers [3]. Industrial applications to date include the development of materials for thin films, 3D bio-printing, and microfabrication [5–7], in which the kinetics and mechanisms of photopolymerization have been extensively studied theoretically and experimentally [7–18]. The UV curing system is roughly categorized into three units: the radical system, cationic system, and anionic system. The radical system is mainly used for conventional UV curing systems. Thiol–ene photopolymerization (TEP) is based on the radical catalyzed addition of a thiol to a vinyl functional group, in which cross-linked polymer networks are formed via a two-step growth mechanism: (i) propagation of a thiyl radical through a vinyl functional group, (ii) followed by chain transfer from the resulting carbon radical to a thiol functional group, regenerating the thiyl radical [18–22]. In comparison, thiol–Michael photopolymerization (TMP) involves anion-mediated addition of a multifunctional thiol to an electron-deficient vinyl group of a multivinyl component [23–27]. Both TEP and TMP exhibit the advantages of typical photopolymerizations, including rapid reaction, optical clarity, negligible oxygen inhibition, solvent tolerance, high reaction yields, excellent mechanical properties, and they do not require solvents for processing [21–23]. In addition, TMP exhibits delayed gelation, and minimal byproduct formation and homopropagation effect of the vinyl group. Depending on the specific ene selected, TEP generally proceeds via a step-mode or mixed-mode chain-growth radical mechanism, and exhibits reaction kinetics strongly dependent on the electronic density of the ene (electron-rich vs. electron-poor) as well as the thiol and ene structures [18,19]. Both computational and experimental investigations have been performed to evaluate the role of solvents, monomers, and catalysts on this reaction mechanism [18–27]. The mechanism and kinetics of TEP utilizing a tetrafunctional thiol monomer copolymerized with acrylate, norbornene, vinyl ether, and vinyl-silazane-functionalized ene monomers were modeled and experimentally characterized by Cramers et al. [18] and Reddy et al. [20]. They demonstrated that reaction orders in TEP systems are controlled by the ratio of thiyl radical propagation (kP) to chain transfer (kCT) kinetic parameters, RK = kP/kCT, which is also related to the electron density of the vinyl group and the carbon radical stability, which could provide spatiotemporal control over the course of the reaction. The scaling law for the functional group concentration of thiol, [A], and ene, [B], given by [A]m[B]n, with m and n ranging from 0 to 2, is related to the type of ene and the rate ratio. For example, for RK >> 1, polymerization rates are first order in ene concentration (or n = 1.0) and nearly independent of the thiol concentration (or m = 0); in contrast, m = 1.0 and n = 0 for RK << 1. For RK values near unity, polymerization rates are approximately 0.5 order in both the thiol and ene functional group concentrations (m = n = 0.5). However, a scaling law of m = 0.4 and n = 0.6 has been found in an acrylate system (with RK = 13) due to contributions from homopolymerization [18]. In general, competitive propagation/chain transfer process must occur, with one of the elementary reaction steps often becoming rate-limiting. These processes also determine the rate-limiting step and time to reach the gel-point conversion [24]. The temporal profiles of thiol and ene functional groups were successfully modeled and compared with measured data by Cramers et al. [18] with various rate-ratio RK and concentration-ratio RC, except for the thiol–norbornene system which has a strong viscosity effect. Therefore, their model excluding the viscosity effect predicted a higher efficacy than their measured data. Although Cramers et al. [18] presented the temporal profiles of the functional groups based on numerical solutions, however, no analytical formulas were reported. Most of the previous models [18–27] have been based on oversimplified assumptions of constant photoinitiator (PI) or photosensitizer (PS) concentration (without depletion), and thus the light intensity in these models follows the conventional Beer–Lambert law (BLL), which is only valid for optically thin polymers and will suffer huge errors in optically thick polymers [15,16]. This study will focus, for the first time, on the analytical formulas for the conversion efficacy for various kinetic rate ratios and functional group concentration molar ratios. A set of more general kinetic rate equations was derived, including viscosity and homopolymerization effects. Numerical data were Polymers 2019, 11, x FOR PEER REVIEW 3 of 17

Cramers et al. [18]. We also demonstrated that our model for a thiol–acrylate system including homopolymerization effects, and for a thiol–norbornene system including viscosity effects, fit much better with the measured data than the model of Cramers et al. [18]. Furthermore, the dynamic light Polymersintensity2019 resulted, 11, 1640 by the PI depletion has been discussed, in which the optically thin assumption3 of 17in previous models [18–27] caused errors of 5% to 20% on crosslink depth and efficacy. We have also discussed the scaling law for the overall reaction order, defined by [A]m[B]n, and governed by the analyzedtypes of usingene ourand analytic the rate formulas ratios. and This compared study withprovides the experimental useful guidance data of Cramersfor free et al.radical [18]. We also demonstrated that our model for a thiol–acrylate system including homopolymerization photopolymerization via the roles of the dual ratios (RK and RC) in various binary functional groups, eincludingffects, and thiol–vinyl, for a thiol–norbornene thiol–acrylate, system and includingthiol–norbornene. viscosity effects, fit much better with the measured data than the model of Cramers et al. [18]. Furthermore, the dynamic light intensity resulted by the PI2. Materials depletion and has beenMethods discussed, in which the optically thin assumption in previous models [18–27] caused errors of 5% to 20% on crosslink depth and efficacy. We have also discussed the scaling law for2.1. thePhotochemical overall reaction Kinetics order, defined by [A]m[B]n, and governed by the types of ene and the rate ratios. This study provides useful guidance for free radical photopolymerization via the roles of the As shown in Figure 1, a two-monomer system [A] and [B] involves three crosslinking pathways: dual ratios (R and R ) in various binary functional groups, including thiol–vinyl, thiol–acrylate, two radical-mediatedK C(or electron transfer) pathways (1 and 2), and one oxygen-mediated (or energy and thiol–norbornene. transfer) pathway (3). The ground state photosensitizer (PI) is excited to its triplet excited state T* by 2.a quantum Materials yield and ( Methodsq). In a type I process, T* interacts directly with [A] and [B] to form the intermediate radicals R’ and S’, which then produce the reactive R and S which could interact with oxygen 3O2, 2.1.[A], Photochemical[B], or bimolecular Kinetics terminations R2 and S2. For a type II (or oxygen-mediated) process, T* interacts 3 1 3 withAs O2 shown to form in a Figure singlet1, aoxygen two-monomer O2, which system could [A] interact and [B] with involves [A], [B], three or crosslinkingbe relaxed to pathways: O2. Both twotype radical-mediated I and type II reactions (or electron can occur transfer) simultaneous pathwaysly (1in andphotopolymerization, 2), and one oxygen-mediated and the ratio (or between energy transfer)these processes pathway depends (3). The groundon the state type photosensitizer of photosensitizer (PI) is excited(PS) or to photoinitiator its triplet excited (PI) state used, T* bythe a quantumconcentrations yield (ofq). PS In or a typePI, substrate I process, monomers T* interacts and directly oxygen, with the [A]kinetic and rates [B] to involved form the in intermediate the process, and the light intensity, dose, PI depletion rate, etc. Greater details of these kinetics were published3 in radicals R’ and S’, which then produce the reactive R and S which could interact with oxygen O2, [A], [B],our orearlier bimolecular work [13–15]. terminations In a thiol–ene R2 and S polymerization2. For a type II (or system, oxygen-mediated) the functional process, groups T* are interacts insensitive with 3to oxygen inhibition; therefore,1 it can be treated as a type-I-dominant system. 3 O2 to form a singlet oxygen O2, which could interact with [A], [B], or be relaxed to O2. Both type I and typeFor a II thiol–ene reactions system, can occur as shown simultaneously in Figure in 2, photopolymerization,the photochemical pathways and the are ratio more between simple these and processescan be shown depends by two-step on the type growth of photosensitizer mechanism, whic (PS)h orinvolves photoinitiator (i) propagation (PI) used, of the a thiyl concentrations radical (R) ofthrough PS or PI,an substrateene functional monomers group and [B]oxygen, to form thea carbon kinetic radical rates involved (S), and in(ii) the chain process, transfer and from the light the intensity,resulting dose,carbon PI radical depletion (S) to rate, a thiol etc. Greaterfunctional details group of [A], these regenerating kinetics were the published thiyl radical in our (R) earlierwhich workreacts [ 13with–15 [A]]. In to a thiol–eneform the reaction polymerization cycle, where system, R and the functionalS could be groupscoupled are and insensitive also terminated to oxygen by inhibition;bimolecular therefore, recombination, it can be or treated react with as a type-I-dominant[A] and [B] in general. system.

FigureFigure 1.1. Schematics of three photochemical pathways in a two-monomer system, A and B, inin thethe 3 presencepresence ofof groundground statestate oxygenoxygen 3OO22, for radical-mediated pathways (1 and 2),2), andand oxygen-mediatedoxygen-mediated pathwaypathway (3).(3). PS is the ground state photosensitizer (PS), having excited and triplet states PS* and T*; terminationsterminations includeinclude unimolecular,unimolecular, bimolecularbimolecular recombination,recombination, andand inter-radicalinter-radical coupling coupling [ 15[15,17].,17].

For a thiol–ene system, as shown in Figure2, the photochemical pathways are more simple and can be shown by two-step growth mechanism, which involves (i) propagation of a thiyl radical (R) through an ene functional group [B] to form a carbon radical (S), and (ii) chain transfer from the resulting carbon radical (S) to a thiol functional group [A], regenerating the thiyl radical (R) which Polymers 2019, 11, x FOR PEER REVIEW 4 of 17

The thiol–ene pathways of Figure 2 are a special situation of the more general pathways of Figure 1, where the functional groups are insensitive to oxygen inhibition and can be treated as a type-I- dominant system, neglecting oxygen-mediated type II reactions. In addition, thiol monomer, [B], do not couple directly with thiyl radical (R), but with the carbon radical (S) to regenerate R to form the Polymersreaction2019 cycle., 11, 1640Therefore, the rate equations for thiol–ene system (Figure 2) are the reduced form4 of of 17 the two-monomer system (Figure 1). For simplicity, the intermediate radicals in Figure 2 (R’ and S’) are not shown in the kinetic scheme, but they have been included in our kinetic equations and reacts with [A] to form the reaction cycle, where R and S could be coupled and also terminated by calculations. bimolecular recombination, or react with [A] and [B] in general.

Figure 2. Schematics of UV-light initiated photopolymerization kinetics of thiol [A] and ene [B] Figure 2. Schematics of UV-light initiated photopolymerization kinetics of thiol [A] and ene [B] functional groups, in which the thiyl radical R reacts with [B] to form a carbon radical (S), which reacts functional groups, in which the thiyl radical R reacts with [B] to form a carbon radical (S), which reacts with the thiol and regenerates R to form the reaction cycle; R and S could interact with each other or be with the thiol and regenerates R to form the reaction cycle; R and S could interact with each other or terminated by bimolecular recombinations, S2 and R2. be terminated by bimolecular recombinations, S2 and R2. The thiol–ene pathways of Figure2 are a special situation of the more general pathways of FigureUsing1, where the following the functional short-hand groups notation: are insensitive C for the to oxygenconcentration inhibition of PI and ground can be state treated (with as an a type-I-dominantinitial value C0); [A] system, and neglecting[B] for the oxygen-mediatedconcentrations of the type thiol II reactions. and ene Infunctional addition, groups; thiol monomer, [R’] and [B],[S’] for do notthe coupleintermediate directly radicals, with thiyl and radical [R] and (R), [S] but for withthe reactive the carbon radicals, radical we (S) obtained to regenerate a set of R general to form therate reaction equations cycle. associated Therefore, with the the rate two-monomer equations for system thiol–ene shown system by (FigureFigure 22,) areas follows the reduced [13,17]. form of the two-monomer system (Figure∂C1). For simplicity, the intermediate radicals in Figure2 (R’ and S’) are not shown in the kinetic scheme, but = they−bIg( have[A] +[B]+kg′C been included[[O in])C our + kinetic R equations and calculations.(1) ∂t Using the following short-hand notation: C for the concentration of PI ground state (with an initial value C0); [A] and [B] for the concentrationsdI(z, t) of the thiol and ene functional groups; [R’] and [S’] = −A(z, t)I(z, t) (2) for the intermediate radicals, and [R] and [S]dz for the reactive radicals, we obtained a set of general rate equations associated with the two-monomer system shown by Figure2, as follows [13,17].

∂[R] = 2∂kC[𝑅′] −(2k[𝑅] + k[R][S] + 𝑘"[R][𝑂]) − 𝐺 (3) ∂t = bIg([A] + [B] + kg0C[[O ])C + R (1) ∂t − 2 E

∂[S] dI(z, t) = 2k[𝑆′] −(2k[=𝑆] +A (kz, t)[RI(][z,S t])+k"[S][𝑂]) + 𝐺 (4)(2) ∂t dz −

∂[R] 2 2 = 2k∂[O[R]0] (2k [R] + k [R][S] + k00 [R][O ]) G (3) ∂t T1 =− −[bIg𝐶T1 + k"([R]12 + [S])][O ]+P 2 − (5) ∂t ∂[S] 2 h 2 i = 2kT2[S0] (2kT2 S] + k12[R][S] + k00 [S][O2 ) + G (4) ∂t − ∂[A] = − 𝑅 [A] (6) ∂[O2] ∂t = [bIgC + k00 ([R] + [S])][O ] + P (5) ∂t − 2 ∂[B] ∂[A] = − 𝑅 [B] (7) ∂t = R [A] (6) ∂t − T ∂[B] = R [B] (7) ∂t − T

RT = bICgg0[O2] + k0([R] + [S]) (8) G = k [R][A] k [S][B] (9) 41 − 42 Polymers 2019, 11, 1640 5 of 17

A(z, t) = 2.3[( a0 b0)C + b0C + Q] (10) − 0 where b = aqI(z,t); q is the quantum yield of the PI excited state; a = 83.6wa’; w is the wavelength of light (in cm); and I(z,t) is the light intensity with a unit of mW/cm2. a’ and b’ are the molar extinction coefficients (in 1/mM/%) of the initiator and the photolysis product, respectively; and Q is the absorption coefficient of the monomer and the polymer repeat unit. 2 2 RE is the regeneration of PI ground state given by RE = (k91[R] + k92[S])[O2] + 2kT1R + 2kT2S ; g = 1/(k57 + k37[O2] + [A] + [B]), g’ = 1/(k68+ k48C + [A] + [B]), k = (k11/k8); k37 = k3/k7, k57 = k5/k7; k68 = k6/k8, k48 = k4/k8, where k5 and k3 are the relaxation rate of the PI triplet state (T*) and the coupling rate of T* and oxygen; k71 and k72 are the coupling rate of T* and monomers [A] and [B]; k88, k81, and k82 are the coupling rates of singlet oxygen and monomers; k41, k42, k51, and k52 are the coupling of radicals R and S and monomers. Greater detail and derivation of the above equations may be found in References [13,15], which also show the kinetic equations for the triplet state (T*) and singlet oxygen [X]. Therefore, Equation (1) to (10) of this article describe the quasi-steady-state form of Equation (1) from Reference [13] under the quasi-steady-state condition dT*/dt = d[X]/dt = 0 = d[R’]/dt = d[S’]/dt = 0, which also defines the steady-state value of radicals [R’] and [S’]. Substituting these values into Equations (3) and (4), we obtain the steady-state reactive radicals [R] and [S]:

2k [R]2 + k[R][O ] + G bICg[A] = 0 (11) T1 2 − h 2 i 2k S] + k0[S][O G bICg[B] = 0 (12) T2 2 − − For a thiol–ene system, which has the advantages of being relatively uninhibited by oxygen. Therefore, we may ignore the oxygen related terms k” [R][O2] and k” [S][O2], in Equations (1) to (10). Without oxygen inhibition, k57 << [A]+[B], g([A]+[B]) = 1 for k72 = k71, and g’[O2] = 0. We further assumed, as proposed by Cramer et al. [10], that the consumption rates of the thiol and ene functional groups could be set as equal, d[A]/dt = d[B]/dt, which allowed us to solve for Equations (6) and (7) for [S], thus solving Equation (11) for [R]. Substituting these steady-state values, [S] and [R], into Equation (8), we obtained the simpliﬁed rate equations of monomers [A] and [B] as follows.

d[A] = Rp (13) dt − d[B] = Rp(1 + HM) (14) dt − p Rp = K 0.5bIC/G (15)

2 2 1 G = (FkP[B])− + (kCT[A])− + (FkPkCT[B][A])− (16)

HM = (kCV/ kCT)([B]/(D[A]) (17) where HM is the revised term for the homopolymerization effect, with D = 1 + kCT[B]/(FkP[A]). 0.5 K = kP/kT is an effective rate constant; k52 = kCT, kT1 = kT2 = kT, are, respectively, the rate constants for chain transfer, propagation, and termination; and k52 = kCV is the rate of the homopolymerization effect, HM, which is given by the k52S[B] term of Equation (9); however, we ignored the k51S[A] term, since [A] (thiol) has weaker homopolymerization than [B] (ene). In Equation (16), we also revised the propagation rate constant kP, by a reduction factor (F) for the viscosity effect on the ene, given by F = 1 d’C , where d’ is a fit reduction rate and C is the − EFF EFF conversion efficacy of the ene group given by C = 1 exp( S), where S is given by the time integral EFF − − of [A]/[A]0, or the solution of Equation (13). The viscosity effect reduces the available free volume or decreases kP in the diffusion-controlled region [11]. Without the extra two consumption terms of the ene group viscosity and homopolymerization effects, HM = 0 (or kCV = 0) and F = 1, Equations (13) and (14) have the same total reaction rate function, RP, which is symmetric with regards to [A] Polymers 2019, 11, 1640 6 of 17 and [B] such that their conversion efficacies are also symmetrically related. These features are further demonstrated later in the paper by our numerical data. The above revised kinetic equations can be reduced to those of Cramer et al. [18] under the following three assumptions: (i) homopolymerization and viscosity effects are neglected (with HM = 0 and F = 1); (ii) the PI concentration is a constant, or Equation (1) dC/dt = 0 (or bIt << 1, for small doses), and (iii) light intensity is a constant, or Equation (2), dI/dt = dI/dz = 0, which is valid only for a short exposure time, or an optically thin polymer. Accurate solutions of Equations (13) to (17) require numerical simulations. For analytical formulas, we used approximated analytic formulas for the light intensity and the PI concentration, such that we did not need to solve for Equation (1), and the expressive closed forms of I(z,t) and C(z,t) allowed us to solve analytically for the first-order solutions of Equations (13) and (14) for the chain and propagation limited cases. 2.2. Analytical Formulas for Efficacy The monomer conversion efficacy for a bimolecular termination process is given by C = 1 EFF − [M]/[M] = 1 exp( S) with [M] = [A] or [B], and the S function is given by the solution of Equations 0 − − (13) and (14) as follows. Analytical solutions are available for the two limiting cases defined by the ratio of chain (kP) and propagation rate (kCT), defined as RK = kP /kCT: in Case i, kP << kCT (chain limited), and in Case ii, kP >> kCT (propagation limited). For Case i, kP[B] << kTC[A], Equation (15) becomes

Rp = K √0.5bIC Fkp[B] (18)

Using the above RP and solving for the first-order solution of Equation (14), for the case of F = 1 (viscosity effect is neglected) and F’ = 0 (or homopolymerization effect is neglected), we obtained the first-order solution of [A] and [B], which allowed us to calculate F’ and the conversion efficacy of the ene, C = 1 [B]/[B] , V − 0 CV = 1 exp( S) (19) − − Zt S = KFkp(1 + HM) √0.5bICdt (20) 0 which also gives [B] = [B] exp( S), and solve for Equation (13); thus, we obtained the conversion 0 − efficacy of the thiol, defined by C = 1 [A]/[A] , T − 0

CT = 1 ([B] /[A] )ST (21) − 0 0 Zt ST = KFkp √0.5bIC exp( S)dt (22) − 0

For Case ii, kP[B] >> kTC[A], Equation (15) becomes

Rp = K √0.5bIC kCT[A] (23)

Similarly, solving for Equations (13) and (14), we obtained the second-order conversion eﬃcacy for thiol (CT) and ene (CV), as follows.

CT = 1 exp( S0) (24) − − Zt S0 = KkCT √0.5bICdt (25) 0 Polymers 2019, 11, 1640 7 of 17

Polymers 2019, 11, x FOR PEER REVIEW CV = 1 ([A] /[B] )SV 9 of(26) 17 − 0 0 t Note that Equation (38) is a nonlinearZ function of ZC; therefore, there is no analytical formula for SV = Kk (1 + HM) √0.5bIC exp( S0)dt (27) ZC vs. TC. However, this formula can beCT found by plotting the curve− of TC vs. ZC, then rotating the axis to show the curve of ZC vs. TC.0 Numerical results based on Equation (38) are shown later in the paper. 2.3. The Second-Order Solution

3. ResultsEquations and Discussion (19) to (27) are the first-order solutions for the limiting cases of kP[B] >> kTC[A] and kP[B] << kTC[A]. To include the high-order terms for the case of comparable kP and kCT with kP[B] > 3.1.k [A],Efficacy Equation Spatial (22) Profiles was revised to (1 d) S, with d = (k /k ) [B’]/[A’], which is a decreasing function TC − P CT of theAs first-order shown by solution Equation product (31), the [B’] S /function[A’]. Therefore, also defines the conversion the monomer efficacy conversion of [B] is efficacy an increasing given byfunction CEFF = of1 − the exp( concentration−S), and S has ratio a transient [A]0/[B]0 .state Similarly, function for kE(Pz,t),[B] < governedkTC[A], Equation by the dynamic (25) was profiles revised of to (k /k ) (1 1/d) S’, which is a decreasing function of [A’]/[B’]. Therefore, the conversion efficacy of [A] theCT lightP intensity− I(z,t) and PI concentration, C(z,t), given by Equations (28) and (29). Based on Equationsis a decreasing (19) and function (31), ofthe [A] spatia0/[B]0l .profiles This feature of the of conversion the conversion efficacies efficacy of the of [A] monomer and [B] [B] can (or be seenene) areby theshown symmetrical in Figure formulas, 3; for a fixed as shown light intensity, by Equations efficacy (21) andis an (26). increasing This feature function is consistent of exposure with time the (t),measured but a decreasing data shown function in Figure 3of of the Cramer depth et (z). al. [Similarly,18] for a thiol–vinyl Figure 4 shows ether system that the with efficacy kP = 1.2k is CTan. increasingThe common function feature of is light that intensity for kP = k(forCT ,a the fixed conversions exposure oftime). [A] andHowever, [B] have for symmetrical very large exposure formats times,and are with proportional an S function to the approaching ratios [A]0/[B] its0 andsteady [B] 0state,/[A]0 ,with respectively. E(z,t) = 1 This in Equation symmetry (32), is broken the efficacy when becomeskP and k CTan increasingare not equal, function or when of z, thereas reported are other by our consumption previous studies factors [15,16] of [B], which such asalso viscosity discussed or thehomopolymerization scaling laws of the e Sff ects.function Greater in greater details details. are discussed later.

Figure 3. ConversionConversion efficacy eﬃcacy vs. vs. depth depth (z) (z) for for t t== (10,(10, 20, 20, 30, 30, 40) 40) s, s, given given by by curves curves 1, 1, 2, 2, 3, 3, and and 4, 4, 22 respectively, forfor aa ﬁxedfixed lightlight intensity intensity I 0I0= = 33 mWmW/cm/cm ,C, C00 = 0.1%,0.1%, b = 0.001,0.001, a’ a’ == 130130 (1/cm/mol), (1/cm/mol), K = 5.0,5.0, b’ = QQ == 0,0, B’ B’ == 0.0.

2.4. Eﬀects of Viscosity and Homopolymerization

To include the effects of homopolymerization given by HM, the first-order solutions of [A] and [B] can be substituted into Equation (20) for the HM function; we thus obtained the second-order conversion efficacy defined by a revised S-function of Equation (20), which resulted a higher efficacy for [B] and [A], as predicted by Equation (20), for the case of kP[B] << kTC[A]. In comparison, when kP[B] >> kTC[A], HM does not affect the first-order solution of [A], as shown by Equation (25), but it reduces the efficacy of [B], as shown by Equation (27). We confirmed these features by numerical solution, as detailed later in the paper. To include the viscosity effects, or when F < 1 in Equation (16), the free volume was reduced when crosslink efficacy increased. The reduction factor (F) only affected the propagation rate constant FkP in Equation (17), but not that in Equation (23). Therefore, it only reduced the efficacy of [B]. We propose F = 1 [1 exp( S’)], with S’ proportional to the efficacy and given by a fit function − − − S’ = 2[1 exp( mb’I t)]/(mb’I )0.5, where m and b are fit parameters to be found later by comparing − − 0 0

Figure 4. Conversion efficacy vs. depth (z) for I0 = (1, 2, 3, 5) mW/cm2, given by curves 1, 2, 3, and 4, respectively, for t = 20 s.

Polymers 2019, 11, 1640 8 of 17 with the measured data. The reduction factor is proportional to the available free volume, or a decreasing function of the conversion eﬃcacy, which is proportional to S’. To include both the homopolymerization and viscosity eﬀects, F and F’ are needed to solve for Equation (3). Numerical solutions, shown later, demonstrated the above features predicted by our analytical formulas.

2.5. Dynamic Light Intensity Solving Equations (1) and (2) for the light intensity, I(z,t), and PI concentration, C(z,t), concentration, we numerically found S and S’ and then the conversion functions, CV and CT. We further derived the analytical form of conversion eﬃcacy, which required closed forms of I(z,t) and C(z,t), as follows. Using our previously developed approximated analytical formulas [13,15]

I(z, t) = I exp[ A0z] (28) 0 −

C0(z, t) = C exp[ Bt] (29) 0 − A0(z, t) = 2.3(a0C + Q) B0t (30) 0 − where B = bI exp( A”z), B’ = 2.3(a’ b’)C I bz, with A” as the averaged absorption given by A” 0 − − 0 0 = 1.15(a’ + b’) C0 + 2.3Q. Note that the –B’t term represents the decrease of A’, or increase of light intensity due to PI depletion. Equation (29) is the approximated solution of Equation (1) with oxygen ignored, or k88[X] = 0 and g[A] + g’[B] = 1. Using Equations (28) and (29), we obtained the analytical form of Equation (20) for the cases of F = 1 and F’ = 0: p S = K 0.5bXI0C0 E (z, t) (31)

E(z, t) = [1 exp( B00 t)]/B00 (32) − − where B” = 0.5(B B’) and X = exp [ 2.3(a’C + Q) z]. Equation (31) gives the close form of Equation − − 0 (20) for CV, and also allowed us to numerically integrate Equation (6) to obtain CT for Case i. Similarly, for Case ii and Equation (25), S’ = (kCT/kP) S, which gave a close form of CV, and CT from Equation (27). Note that Equation (30) deﬁnes the dynamic feature of the light intensity, which is an increasing function of time due to the depletion of the PI concentration. It also provides the nonlinear depth (z) dependence of A’z, given by B’. The above analytical formulas provide useful information with which to analyze and predict the critical roles of each of the inﬂuencing factors without numerically solving the coupled equations.

2.6. Gelation Time The critical (gelation point) conversion, C , may be deﬁned by the classical Flory Stockmayer CT − equation [10]: q C = 1/ R ( fT 1)( fV 1) (33) CT C − − where fT and fV are the functionalities of thiol and ene monomers, respectively; and RC = [A]0/[B]0 is the thiol–ene stoichiometric initial molar ratio. For the case of neglected induction time, we found that the gelation time can be given by solving the exposure time of Equations (31) and (6) under the critical condition of Equation (33). For example, for the case of kP[B] << kTC[A], we solved for Equations (31) and (33) to obtain the gelation time (TGEL) for the ene:

T = (1/B00 ) ln[1/(1 B )] (34) GEL − 1 B1 = B00 L/ K √0.5b (35) L = ln[1/(1 C )] (36) − CT Polymers 2019, 11, x FOR PEER REVIEW 9 of 17 Polymers 2019, 11, 1640 9 of 17 Note that Equation (38) is a nonlinear function of ZC; therefore, there is no analytical formula for ZC vs. TC. However, this formula can be found by plotting the curve of TC vs. ZC, then rotating the axis toSimilarly, show the we curve obtained of ZC the vs. gelation TC. Numerical time for results the case based of k onP[B] Equation>> kTC[A] (38) given are shown by Equation later in (20), the givenpaper. by the same formula as Equation (34), but Equation (35) was revised to: √ 3. Results and Discussion B1 = B00 L/[kTCK 0.5b] (37)

3.1. EfficacyHowever, Spatial there Profiles are no analytic formulas for CT for the case of kP[B] << kTC[A], or CV for the case of kP[B] >> kTC[A] due to the complex integrations of Equations (20) and (25). As shown by Equation (31), the S function also defines the monomer conversion efficacy given 2.7.by C CrosslinkEFF = 1 − exp( Depth−S), and S has a transient state function E(z,t), governed by the dynamic profiles of the light intensity I(z,t) and PI concentration, C(z,t), given by Equations (28) and (29). Based on A crosslink depth (ZC) is deﬁned by when the conversion eﬃcacy is higher than a critical value, Equations (19) and (31), the spatial profiles of the conversion efficacies of the monomer [B] (or ene) CT > CCT, or when S > SCT with SCT = ln [1/(1 CCT)]. Using Equation (31), and with S = SCT = 2 (or are shown in Figure 3; for a fixed light intensity,− efficacy is an increasing function of exposure time CCT = 0.86), ZC is related to the crosslink time (TC) by: (t), but a decreasing function of the depth (z). Similarly, Figure 4 shows that the efficacy is an increasing function of light intensity (for1 a hfixed exposurep time). However,i for very large exposure TC = ln 2B00 / K 0.5bXI0C0 1 (38) times, with an S function approachingB00 its steady state, with E(z,t)− = 1 in Equation (32), the efficacy becomes an increasing function of z, as reported by our previous studies [15,16] which also discussed Note that Equation (38) is a nonlinear function of Z ; therefore, there is no analytical formula the scaling laws of the S function in greater details. C for ZC vs. TC. However, this formula can be found by plotting the curve of TC vs. ZC, then rotating the axis to show the curve of ZC vs. TC. Numerical results based on Equation (38) are shown later in the paper.

3. Results and Discussion

3.1. Efficacy Spatial Profiles As shown by Equation (31), the S function also defines the monomer conversion efficacy given by C = 1 exp( S), and S has a transient state function E(z,t), governed by the dynamic profiles EFF − − of the light intensity I(z,t) and PI concentration, C(z,t), given by Equations (28) and (29). Based on Equations (19) and (31), the spatial profiles of the conversion efficacies of the monomer [B] (or ene) are shown in Figure3; for a fixed light intensity, e fficacy is an increasing function of exposure time (t), but a decreasing function of the depth (z). Similarly, Figure4 shows that the e fficacy is an increasing function of light intensity (for a fixed exposure time). However, for very large exposure times, with an S functionFigure 3. approachingConversion efficacy its steady vs. depth state, (z) with for t E(z,t) = (10, =20,1 in30, Equation40) s, given (32), by thecurves effi 1,cacy 2, 3, becomes and 4, an increasingrespectively, function for ofa fixed z, as light reported intensity by I our0 = 3 previous mW/cm2, studiesC0 = 0.1%, [15 b ,=16 0.001,] which a’ = also130 (1/cm/mol), discussed K the = 5.0, scaling lawsb’ of = the Q = S 0, function B’ = 0. in greater details.

Figure 4. Conversion eﬃcacy vs. depth (z) for I = (1, 2, 3, 5) mW/cm2, given by curves 1, 2, 3, and 4, Figure 4. Conversion efficacy vs. depth (z) for I00 = (1, 2, 3, 5) mW/cm2, given by curves 1, 2, 3, and 4, respectively,respectively, forfor tt= = 20 s.s.

PolymersPolymers2019 2019, ,11 11,, 1640 x FOR PEER REVIEW 10 ofof 1717

3.2. Dynamic Profiles of the Light Intensity 3.2. Dynamic Profiles of the Light Intensity Figure 5 shows the dynamic profiles of the PI-normalized concentration and the increase of light intensityFigure obtained5 shows by the the dynamic numerical profiles solution of the of PI-normalized Equation (1). Depending concentration on andthe coupling the increase parameters of light intensityb and A, obtainedas shown by by the Equation numerical (30), solution the depletion of Equation of C(z,t) (1). causes Depending the increasing on the coupling I(z,t), defined parameters by an b and A, as shown by Equation (30), the depletion of C(z,t) causes the increasing I(z,t), defined by increasing percentage dI = [I(z,t) − I0]/I0, which is also a decreasing function of the depth (z), per the an increasing percentage dI = [I(z,t) I ]/I , which is also a decreasing function of the depth (z), per Beer–Lambert law. Figure 6 shows an− increase0 0 of 6% to 10% light intensity (at z = 100 µm) for b = 0.05 the Beer–Lambert law. Figure6 shows an increase of 6% to 10% light intensity (at z = 100 µm) for to 0.2, a’ = 200 (1/%/mM), b’ = 100 (1/cm/%), Q = 50 (1/cm), and I0 = 5 mW/cm2. Figure 6 shows that 2 blarger= 0.05 b tovalues 0.2, a’had= a200 faster (1/% PI/mM), depletionb’ = and100 (1hence/cm/ %)a larger,Q = light50 (1 intensity/cm), and increase. I0 = 5 mW For/ cmthick. polymers Figure6 shows that larger b values had a faster PI depletion and hence a larger light intensity increase. For (>100 µm), the conversion efficacies of the thiol (CT) and ene (CV) functional groups are also depth µ thick(z)-dependent, polymers ( >as100 wellm), as thetime-dependent. conversion effi caciesThe ex ofample the thiol shown (CT) andin Figure ene (C 5V )demonstrates functional groups that arethe alsoassumption depth (z)-dependent, of constant light as wellintensity as time-dependent. and PI concentration The example is valid shownonly for in optically Figure5 demonstratesthin (<100 µm) thatpolymers the assumption under a small of constant dose (<1.0 light mJ/cm intensity2), i.e., and under PI concentrationthe condition of is validEquation only (9c) for with optically B’ = 2.3(a’ thin (<100 µm) polymers under a small dose (<1.0 mJ/cm2), i.e., under the condition of Equation (9c) with − b’) bC0I0tz < 0.1. For optically thick polymers under a larger dose, with bI0tz = 0.1 to 0.2, the thin- B’ = 2.3(a’ b’) bC I tz < 0.1. For optically thick polymers under a larger dose, with bI tz = 0.1 to polymer model− assumption0 0 will cause an error of 5% to 20%, depending on the depth0 of light 0.2,propagation the thin-polymer or polymer model thickness, assumption and its will absorption cause an coefficient. error of 5% The to 20%,influence depending of B’ on on the the crosslink depth ofdepth light and propagation efficacy oris polymershown later thickness, in the andpaper. its absorptionNote also coethat,ffi cient.as shown The influenceby Equation of B’ (3), on thethe crosslink depth and efficacy is shown later in the paper. Note also that, as shown by Equation (3), photoinitiation rate function (RP) is proportional to the product of I(z,t) and C(z,t), which are two thecompeting photoinitiation parameters, rate function as shown (R byP) isFigure proportional 5. Therefore, to the there product are opti of I(z,t)mal andvalues C(z,t), of I(z,t) which and are C(z,t), two competingas shown earlier parameters, in Figures as shown 3 and by 4. Figure 5. Therefore, there are optimal values of I(z,t) and C(z,t), as shown earlier in Figures3 and4.

Figure 5. Dynamic proﬁles of normalized photoinitiator (PI) concentration (blue curves) and light Figure 5. Dynamic profiles of normalized photoinitiator (PI) concentration (blue curves) and light Polymersintensity 2019, 11 increases, x FOR PEER dI (redREVIEW curves) for b = (0.05, 0.1, 0.2), for curves 1,2,3. 11 of 17 intensity increases dI (red curves) for b = (0.05, 0.1, 0.2), for curves 1,2,3.

3.3. Crosslink Depth Profiles Using the analytical formula Equation (38), we were able to investigate the roles of PI, oxygen concentration, light intensity, and exposure time in crosslink depth. Figure 6 shows that crosslink depth (ZC) (for the case of B’ = 0) is an increasing function of light intensity and exposure time. Figure 7 shows the influence of dynamic light intensity due to PI depletion given by B’ in Equation (30): that the assumption of B’ = 0 suffers an error of 10% to 20% (underestimated) of ZC for depths of 300 to 500 µm. Therefore, the assumption is valid only for optically thin polymer thinner than 200 µm for the case of absorption constant a’ = 130, and thinner than 100 µm for a stronger absorption of 260 (1/cm/%), or when C0 = 0.2%, as shown by Equation (9), B’ = 2.3(a’ − b’) C0I0bz. The influence of B’ on the efficacy profiles is shown in Figure 8, namely that neglecting the B’ factor also results in an underestimation of the efficacy. Figure 6. Crosslink depth (Z ) vs. light intensity (I ) for t = (15, 20, 25, 30) s, given by curves 1, 2, 3, Figure 6. Crosslink depth (ZCC) vs. light intensity (I0)0 for t = (15, 20, 25, 30) s, given by curves 1, 2, 3, and 4, respectively, for C = 0.1%, based on Equation (14), for B’ = 0. and 4, respectively, for C0 = 0.1%, based on Equation (14), for B’ = 0.

Figure 7. Crosslink depth (ZC) vs. light intensity (I0), for B’ = 0 (green curve), and B’ = 0.06I0z > 0 (red

curve) for C0 = 0.1%, t = 20 s.

Figure 8. Efficacy vs. depth (z) based on Equation (14) for I0 = (1, 2, 3, 5) mW/cm2, given by curves 1,

2, 3, and 4, respectively, at t = 20 s, C0 = 0.1%; for B’ = 0 (solid curves), and B’ = 0.06I0z (dashed curves).

Polymers 2019, 11, x FOR PEER REVIEW 11 of 17 Polymers 2019, 11, x FOR PEER REVIEW 11 of 17

Polymers 2019, 11, 1640 11 of 17

3.3. Crosslink Depth Profiles Using the analytical formula Equation (38), we were able to investigate the roles of PI, oxygen concentration, light intensity, and exposure time in crosslink depth. Figure6 shows that crosslink depth (ZC) (for the case of B’ = 0) is an increasing function of light intensity and exposure time. Figure7 shows the influence of dynamic light intensity due to PI depletion given by B’ in Equation (30): that the assumption of B’ = 0 suffers an error of 10% to 20% (underestimated) of ZC for depths of 300 to 500 µm. Therefore, the assumption is valid only for optically thin polymer thinner than 200 µm for the case of absorption constant a’ = 130, and thinner than 100 µm for a stronger absorption of 260 (1/cm/%), or whenFigure C0 = 0.2%,6. Crosslink as shown depth by (Z EquationC) vs. light (9), intensity B’ = 2.3(a’ (I0) for b’)t = (15, C0I 020,bz. 25, The 30) influence s, given by of curves B’ on the1, 2, e 3,ffi cacy − profilesandFigure is 4, shownrespectively, 6. Crosslink in Figure fordepth C80, = namely(Z 0.1%,C) vs. based light that neglectingonintensity Equation (I0 the) (14),for B’ t for= factor (15, B’ =20, 0. also 25, results 30) s, given in an by underestimation curves 1, 2, 3, of the effiandcacy. 4, respectively, for C0 = 0.1%, based on Equation (14), for B’ = 0.

Figure 7. Crosslink depth (ZC) vs. light intensity (I0), for B’ = 0 (green curve), and B’ = 0.06I0z > 0 (red Figure 7. Crosslink depth (ZC) vs. light intensity (I0), for B’ = 0 (green curve), and B’ = 0.06I0z > 0 (red curve)Figure for7. Crosslink C0 = 0.1%, depth t = 20 (Zs. C) vs. light intensity (I0), for B’ = 0 (green curve), and B’ = 0.06I0z > 0 (red curve) for C0 = 0.1%, t = 20 s. curve) for C0 = 0.1%, t = 20 s.

2 Figure 8. Eﬃcacy vs. depth (z) based on Equation (14) for I0 = (1, 2, 3, 5) mW/cm , given by curves 1, 2, Figure 8. Efficacy vs. depth (z) based on Equation (14) for I0 = (1, 2, 3, 5) mW/cm2, given by curves 1, 3, and 4, respectively, at t = 20 s, C0 = 0.1%; for B’ = 0 (solid curves), and B’ = 0.06I0z (dashed curves). 2 2,Figure 3, and 8. 4, Efficacy respectively, vs. depth at t =(z) 20 based s, C0 = on 0.1%; Equation for B’ =(14) 0 (solid for I0 curves),= (1, 2, 3, and 5) mW/cmB’ = 0.06I,0 givenz (dashed by curves curves). 1, 3.4. Numerical2, 3, and 4, Resultsrespectively, of Conversion at t = 20 s, C0 = 0.1%; for B’ = 0 (solid curves), and B’ = 0.06I0z (dashed curves).

This section shows the numerical results of the conversion eﬃcacy of thiol and various ene functional groups by solving Equation (1) to (10). These numerical data were analyzed by our analytic formulas and compared with the experimental data of Cramers et al. [18]. We also demonstrated that our modeling for a thiol–acrylate system including homopolymerization eﬀects, and for a thiol–norbornene

Polymers 2019, 11, x FOR PEER REVIEW 12 of 17

3.4. Numerical Results of Conversion This section shows the numerical results of the conversion efficacy of thiol and various ene functional groups by solving Equation (1) to (10). These numerical data were analyzed by our analytic formulas and compared with the experimental data of Cramers et al. [18]. We also demonstrated that our modeling for a thiol–acrylate system including homopolymerization effects, and for a thiol– norbornene system including viscosity effects, fit much better with the measured data than the model of Cramers et al. [18] for the high efficacy region. Figure 9 shows the conversion efficacy of ene, [B], for a fixed z = 100 µm, with a rate ratio RK = kP/kCT = 0.2, for various light intensity of I0 = (1, 5, 25) mW/cm2, which demonstrates that, for a fixed light dose, higher light intensity has a higher transient efficacy, but a lower steady-state conversion. These numerically obtained features were also predicted by Equation (10), and may be compared Polymerswith Figure2019, 115., 1640 12 of 17 We next explored the roles of the rate ratio RK = kP/kCT for various concentration ratios RC = [A]0/[B]0. For kCV = 0 and F = 1, the G function of Equation (16) was the same for Equation (13) and system including viscosity effects, fit much better with the measured data than the model of Cramers (14), which are symmetric equations. Therefore, [A] and [B] had identical solutions, except that their et al. [18] for the high efficacy region. dependence on RC is reversed, i.e., [B] was governed by [B]0/[A]0 and [A] by [A]0/[B]0. These opposite Figure9 shows the conversion e fficacy of ene, [B], for a fixed z = 100 µm, with a rate ratio dependences on the concentration ratio RC lead to the reversed curves of 1 and 3 in Figure 10 for [B] R = k /k = 0.2, for various light intensity of I = (1, 5, 25) mW/cm2, which demonstrates that, andK [A].P CTThese features seem unexpected numerically,0 but they were clearly predicted by our for a fixed light dose, higher light intensity has a higher transient efficacy, but a lower steady-state analytical formulas, Equations (24) and (26), for the case that kCV = 0 and F = 1, or when the conversion. These numerically obtained features were also predicted by Equation (10), and may be homopolymerization and viscosity effects were neglected. compared with Figure5.

2 Figure 9. Conversion eﬃcacy of ene (CV) vs. time at various light intensity of I0 = (1, 5, 25) mW/cm , Figure 9. Conversion efficacy of ene (CV) vs. time at various light intensity of I0 = (1, 5, 25) mW/cm2, given by curves 1, 2, and 3, respectively, for b = 0.05, kp = 0.1 105, k = 2 105 (with a rate ratio 5 CT 5 given by curves 1, 2, and 3, respectively,6 for b = 0.05, kp = 0.1 × 10× , kCT = 2 × 10× (with a rate ratio RK = RK = kp/kCT = 0.2), kT = 1 10 (L/mol.s), and [A]0 = [B]0 = 5.0 mol/L, neglecting the viscosity and kp/kCT = 0.2), kT = 1 × 10×6 (L/mol.s), and [A]0 = [B]0 = 5.0 mol/L, neglecting the viscosity and homopolymerization eﬀects (or kCV = 0 and F = 1). homopolymerization effects (or kCV = 0 and F = 1).

We next explored the roles of the rate ratio RK = kP/kCT for various concentration ratios

RC = [A]0/[B]0. For kCV = 0 and F = 1, the G function of Equation (16) was the same for Equations (13) and (14), which are symmetric equations. Therefore, [A] and [B] had identical solutions, except that their dependence on RC is reversed, i.e., [B] was governed by [B]0/[A]0 and [A] by [A]0/[B]0. These opposite dependences on the concentration ratio RC lead to the reversed curves of 1 and 3 in Figure 10 for [B] and [A]. These features seem unexpected numerically, but they were clearly predicted by our analytical formulas, Equations (24) and (26), for the case that kCV = 0 and F = 1, or when the homopolymerizationPolymers 2019, 11, x FOR PEER and viscosityREVIEW eﬀects were neglected. 13 of 17

Figure 10. Conversion eﬃcacy of ene (left) and thiol (right) for R = 1.0 and various concentration Figure 10. Conversion efficacy of ene (left) and thiol (right) for KRK = 1.0 and various concentration ratios R = [A] /[B] = (0.5, 1.0, 2.0), given by curves 1, 2, and 3, respectively, neglecting the viscosity ratios RC C = [A]00/[B]00 = (0.5, 1.0, 2.0), given by curves 1, 2, and 3, respectively, neglecting the viscosity and homopolymerization eﬀects (k = 0 and F = 1). and homopolymerization effects (kCVCV = 0 and F = 1).

3.5. Analysis of Measured Data Figure 11 shows the conversion in a thiol–vinyl system, using the same parameters used in Figure 4 from Cramers et al. [18], with RK = kP/kCT = 0.2 and kCV = 0, for various RC = (0.5., 1.0, 2.0). Our calculated curves fit very well with the measured data of Cramers et al. (with fit b = 0.01). Figure 12 shows the effects of homopolymerization (with kCV = 0.25 × 105) in a thiol–acrylate system, using the same parameters used in Figure 13 from Cramers et al. [18] with kp/kCT = 13, but also including the dynamic light intensity and PI depletion. The modeling data of Cramers et al. were consistent with ours for specific values of b and z. However, for different PI concentration profiles (or different b), or polymer thicknesses (z), the conversion efficacies of thiol and ene were changed accordingly. As predicted by Equations (24) and (25), the conversion efficacy of thiol [A] did not affected by HM or kCV. In contrast, as predicted by Equation (27), the efficacy of acrylate [B] increased when kCV increased. This feature was also shown by our numerical data presented in Figure 11, in which our model fit very well the measured data of Cramer et al. [18], except for the transient region in which we neglected the induction time (or lag period) due to the initial interaction of the functional groups and the impurities or oxygen. Therefore, a further improved model including induction time would be needed for strong induction systems as discussed by Claudino et al. [22].

Figure 11. Calculated conversion efficacy of vinyl (left) and thiol (right), for kP = kCP = 1.0 × 105, (or RK

= 1.0), kT = 1 × 106 (L/mol.s), and various concentration ratios RC = [A]0/[B]0 = (0.5, 1.0, 2.0), given by curves 1, 2, and 3, respectively; the background gray circles are the measured data of Cramer et al. [18].

Polymers 2019, 11, x FOR PEER REVIEW 13 of 17

Figure 10. Conversion efficacy of ene (left) and thiol (right) for RK = 1.0 and various concentration

ratios RC = [A]0/[B]0 = (0.5, 1.0, 2.0), given by curves 1, 2, and 3, respectively, neglecting the viscosity

and homopolymerization effects (kCV = 0 and F = 1).

3.5. Analysis of Measured Data Figure 11 shows the conversion in a thiol–vinyl system, using the same parameters used in Figure 4 from Cramers et al. [18], with RK = kP/kCT = 0.2 and kCV = 0, for various RC = (0.5., 1.0, 2.0). Our calculated curves fit very well with the measured data of Cramers et al. (with fit b = 0.01). Figure 12 shows the effects of homopolymerization (with kCV = 0.25 × 105) in a thiol–acrylate system, using the same parameters used in Figure 13 from Cramers et al. [18] with kp/kCT = 13, but also including the dynamic light intensity and PI depletion. The modeling data of Cramers et al. were consistent with ours for specific values of b and z. However, for different PI concentration profiles (or different b), or polymer thicknesses (z), the conversion efficacies of thiol and ene were changed accordingly.Polymers 2019 , As11, 1640predicted by Equations (24) and (25), the conversion efficacy of thiol [A] did13 not of 17 affected by HM or kCV. In contrast, as predicted by Equation (27), the efficacy of acrylate [B] increased when kCV increased. This feature was also shown by our numerical data presented in Figure 11, in which3.5. Analysis our model of Measured fit very well Data the measured data of Cramer et al. [18], except for the transient region in whichFigure we neglected 11 shows the the induction conversion time in (or a lag thiol–vinyl period) due system, to the using initial the interaction same parameters of the functional used in groupsFigure and4 from the Cramersimpurities et or al. oxygen. [ 18], with Therefore, RK = kP /ak CTfurther= 0.2 improved and kCV = model0, for variousincluding RC induction= (0.5., 1.0, time 2.0). wouldOur calculated be needed curves for strong ﬁt very induction well with system the measureds as discussed data by of Claudino Cramers etet al.al. (with[22]. ﬁt b = 0.01).

5 FigureFigure 11. 11. CalculatedCalculated conversion conversion efficacy efficacy of of vi vinylnyl (left) (left) and and thiol thiol (right), for kP = kkCPCP == 1.01.0 × 10105, (or, (or R RK K 6 × = 1.0), kT = 1 106 (L/mol.s), and various concentration ratios R = [A] /[B] = (0.5, 1.0, 2.0), given by = 1.0), kT = 1 × 10× (L/mol.s), and various concentration ratios RC C= [A]0/[B]0 0 =0 (0.5, 1.0, 2.0), given by curvescurves 1, 1, 2, 2, and and 3, 3, respectively; respectively; the background graygray circlescircles are are the the measured measured data data of of Cramer Cramer et al.et [al.18 ]. [18]. Figure 12 shows the effects of homopolymerization (with k = 0.25 105) in a thiol–acrylate CV × system, using the same parameters used in Figure 13 from Cramers et al. [18] with kp/kCT = 13, but also including the dynamic light intensity and PI depletion. The modeling data of Cramers et al. were consistent with ours for specific values of b and z. However, for different PI concentration profiles (or different b), or polymer thicknesses (z), the conversion efficacies of thiol and ene were changed accordingly. As predicted by Equations (24) and (25), the conversion efficacy of thiol [A] did not affected by HM or kCV. In contrast, as predicted by Equation (27), the efficacy of acrylate [B] increased when kCV increased. This feature was also shown by our numerical data presented in Figure 11, in which our model fit very well the measured data of Cramer et al. [18], except for the transient region in which we neglected the induction time (or lag period) due to the initial interaction of the functional groups and the impurities or oxygen. Therefore, a further improved model including induction time wouldPolymers be 2019 needed, 11, x FOR for PEER strong REVIEW induction systems as discussed by Claudino et al. [22]. 14 of 17

Figure 12. Calculated conversion eﬃcacy of acrylate (left) and thiol (right), for k = 2.2 105, k P 5 CP Figure 12. Calculated5 conversion efficacy of acrylate (left) and thiol (right), for kP = 2.2 × 10×, kCP = 0.17 = 0.17 10 (or RK = 13), and various concentration ratios RC = [A]0/[B]0 = (0.5, 1.0, 2.0), including 5 × 10 ×(or RK = 13), and various concentration 5ratios RC = [A]0/[B]0 = (0.5, 1.0, 2.0), including homopolymerization eﬀects (with kCV = 0.25 10 ). The background gray circles are the measured homopolymerization effects (with kCV = 0.25 × 105). The background gray circles are the measured data of Cramer et al. [18]. data of Cramer et al. [18].

Figure 13. Conversion efficacy vs. depth (z) for C0 = (0.05, 0.1, 0.2, 0.4) %, given by curves 1, 2, 3, and

4, respectively, for I0 = 3 mW/cm2 at t = 20 s.

Figure 14 shows the viscosity effect on the conversion efficacy using the same parameters shown in Figure 2 from Cramers et al. [18], with kP = kCP = 1.0 × 105 (or RK = 1.0) for a thiol–norbornene system. Note that the model-predicted conversions of Cramers et al. (shown by black curves) were higher than their measured data, due to their assumption of a constant reaction rate (kP). In our revised model, we included a reduction factor (F < 1) in Equation (22) for the viscosity effect, which reduces the available free volume specifically for the high efficacy region. We propose that F = 1 − [1− exp(−S’)], with S’ proportional to the efficacy and given by a fit function S’ = 2[1 − exp(−mb’I0t)]/(mb’I0)0.5, with m = 0.5 and b’ = 0.03 fit to the measured data, as in Figure 2 of Cramers et al. [18]. The reduction factor is proportional to the available free volume, or a decreasing function of the conversion efficacy, which is proportional to S’. As shown in Figure 14, a much better fit to the measured curves was found in our revised model (shown by red curves). Note that without the viscosity effect (or kCV = 0, F = 1), Figure 14 reduces to Figure 11, in which the symmetrical feature of CV and CT becomes asymmetrical, resulting from the viscosity effect, which mainly affects the propagation parameter of norbornene (kP). This reduction factor (F) affects both CV and CT specifically for the case of kP = kCT (or RK = 1.0). Note that the modeling curves in Figure 2 of Cramers et al. [18] are the same as in our Figure 11 (with kCV = 0, F = 1).

Polymers 2019, 11, x FOR PEER REVIEW 14 of 17

Figure 12. Calculated conversion efficacy of acrylate (left) and thiol (right), for kP = 2.2 × 105, kCP = 0.17

× 105 (or RK = 13), and various concentration ratios RC = [A]0/[B]0 = (0.5, 1.0, 2.0), including Polymershomopolymerization2019, 11, 1640 effects (with kCV = 0.25 × 105). The background gray circles are the measured14 of 17 data of Cramer et al. [18].

Figure 13. Conversion eefficacyﬃcacy vs.vs. depthdepth (z)(z) forfor C C00= = (0.05,(0.05, 0.1,0.1, 0.2,0.2, 0.4) 0.4) %, %, given given by by curves curves 1, 1, 2, 2, 3, 3, and and 4, 2 respectively,4, respectively, for for I0 =I0 3= mW3 mW/cm/cm at2 at t =t =20 20 s. s.

Figure 14 shows the viscosity effect effect on the conv conversionersion efficacy efficacy using the same parameters shown in Figure2 from Cramers et al. [ 18], with k = k = 1.0 105 (or R = 1.0) for a thiol–norbornene in Figure 2 from Cramers et al. [18], with kP =P kCP =CP 1.0 × 105 ×(or RK = 1.0)K for a thiol–norbornene system. system.Note that Note the thatmodel-predicted the model-predicted conversions conversions of Cramer ofs Cramers et al. (shown et al. (shownby black by curves) black curves)were higher were higher than their measured data, due to their assumption of a constant reaction rate (k ). In our revised than their measured data, due to their assumption of a constant reaction rate (kPP). In our revised model, wewe includedincluded aa reduction reduction factor factor (F (F< <1) 1) in in Equation Equation (22) (22) for for the the viscosity viscosity eff ect,effect, which which reduces reduces the available free volume specifically for the high efficacy region. We propose that F = 1 [1 exp( S’)], the available free volume specifically for the high efficacy region. We propose that− F− = 1 − [1− with S’ proportional to the efficacy and given by a fit function S’ = 2[1 exp( mb’I t)]/(mb’I )0.5, with exp(−S’)], with S’ proportional to the efficacy and given by− a fit− function0 S’ 0= 2[1 − m = 0.5 and b’ = 0.03 fit to the measured data, as in Figure2 of Cramers et al. [ 18]. The reduction factor exp(−mb’I0t)]/(mb’I0)0.5, with m = 0.5 and b’ = 0.03 fit to the measured data, as in Figure 2 of Cramers iset proportionalal. [18]. The reduction to the available factor freeis proportional volume, or ato decreasing the available function free volume, of the conversion or a decreasing efficacy, function which isof proportionalthe conversion to efficacy, S’. As shown which in is Figure proportional 14, a much to S’. better As shown fit to in the Figure measured 14, a much curves better was foundfit to the in ourmeasured revised curves model was (shown found by in red our curves). revised Note model that (s withouthown by the red viscosity curves). eff Noteect (or that kCV without= 0, F = the1), Figure 14 reduces to Figure 11, in which the symmetrical feature of C and C becomes asymmetrical, viscosity effect (or kCV = 0, F = 1), Figure 14 reduces to Figure 11, in whichV theT symmetrical feature of resulting from the viscosity effect, which mainly affects the propagation parameter of norbornene CV and CT becomes asymmetrical, resulting from the viscosity effect, which mainly affects the (k ). This reduction factor (F) affects both C and C specifically for the case of k = k (or R = 1.0). propagationP parameter of norbornene (kP). ThisV reductionT factor (F) affects both CPV andCT CT specificallyK Note that the modeling curves in Figure2 of Cramers et al. [ 18] are the same as in our Figure 11 (with Polymersfor the 2019case, 11of, xk FORP = k PEERCT (or REVIEW RK = 1.0). Note that the modeling curves in Figure 2 of Cramers et 15al. of[18] 17 k = 0, F = 1). areCV the same as in our Figure 11 (with kCV = 0, F = 1).

Figure 14. Calculated conversion eﬃcacy of norbornene (left) and thiol (right), including the viscosity

Figureeffects (with14. Calculated a reduction conversion factor, F efficacy< 1), for of various norbornene concentration (left) and ratios thiol (right), RC = [A] including0/[B]0 = (0.5,the viscosity 1.0, 2.0), effectsfor curves (with 1,2,3; a reduction with k = factor,k = 1.0F < 1),10 for5 (R various= 1.0). concentration The calculated ratios curves RC without = [A]0/[B] (in0 black= (0.5, dash) 1.0, 2.0), and P CT × K forwith curves (in red) 1,2,3; the with viscosity kP = k effCTects = 1.0 are × also105 (R shown.K = 1.0). The calculated curves without (in black dash) and with (in red) the viscosity effects are also shown. 3.6. General Features of Conversion Efficacy 3.6. General Features of Conversion Efficacy Our numerical results for the conversion efficacies CT (for thiol [A]) and CV (for ene [B]) showed that theOur roles numerical of the reactionresults for rate the ratio, conversion RK = k Pefficacies/kCT, and C theT (for concentration thiol [A]) and ratio, CV (for RC = ene[A] [B])0/[B] showed0, were that the roles of the reaction rate ratio, RK = kP/kCT, and the concentration ratio, RC = [A]0/[B]0, were consistent with our predicted results based on analytical formulas which provided more general features for the roles of RK and RC, summarized as follows: Without viscosity (with F = 1) or homopolymerization (or kCV = 0) effects, [A] and [B] have an equal overall polymerization rate (RP); CV (CT) is an increasing (decreasing) function of the ratio RC. For RK = 1 (or kp = kCT), CV and CT have the same temporal profiles, but have a reversed dependence on RC, as shown by Figure 10. For RK << 1, [B] and CV are almost independent from RC, with a second-order correction proportional to RC having asymmetrical dependence on RC, given by (1 − d), with d = RK/RC, as shown by Figure 11. For RK >> 1, [A] and CT are almost independent from RC, but the second-order correction is inversely proportional to RC, an opposite trend to that of CV. As predicted by Equation (25), the first- order solutions (with neglected d = 0) of CT and CV were independent from RC. For RK << 1 and with the homopolymerization effect considered (with kCV > 0, F’ > 0), a revised S function of Equation (21) and Equation (27) predicts that CV and CT would have the same profiles and both be decreasing functions of kCV. In comparison, for the case of RK >> 1, kCV does not affect [A], as shown by Equation (26), which predicts that CV is an increasing function of kCV, but CT is a slightly decreasing function of kCV, due to its second-order correction. With the presence of viscosity effect, or when F < 1, in Equation (22), the free-volume is reduced when crosslink efficacy increases. The reduction factor (F) only affects the propagation rate constant Fkp in Equations (20) and (22), but not in Equation (25). Therefore, the viscosity effect does not affect the efficacy for the case of RK >> 1, and affects the efficacy for other ratios of kP and kCT, where the viscosity effect reduces the efficacy of [B], as predicted by Equation (22). As predicted by the analytical formula of Equation (31) and the numerical data of Figure 9, for a fixed light dose, higher light intensity has a higher transient efficacy, but a lower steady-state conversion. This unique and unusual feature of the light intensity dependence is the result of a bimolecular termination. In contrast, in type II unimolecular termination, the conversion is mainly governed by the light dose, rather than its intensity. For optically thick polymers, the influence of dynamic light intensity due to PI depletion is given by B’ in Equation (30), which predicts that the assumption of B’ = 0 (as in most previous models) suffers an error of 5% to 20% (underestimated) for a crosslink depth (ZC) ranging 300 to 500 µm, and also underestimates the efficacy, as shown by Figures 7 and 8.

Polymers 2019, 11, 1640 15 of 17 consistent with our predicted results based on analytical formulas which provided more general features for the roles of RK and RC, summarized as follows: Without viscosity (with F = 1) or homopolymerization (or kCV = 0) effects, [A] and [B] have an equal overall polymerization rate (RP); CV (CT) is an increasing (decreasing) function of the ratio RC. For RK = 1 (or kp = kCT), CV and CT have the same temporal profiles, but have a reversed dependence on RC, as shown by Figure 10. For RK << 1, [B] and CV are almost independent from RC, with a second-order correction proportional to R having asymmetrical dependence on R , given by (1 d), with d = R /R , as C C − K C shown by Figure 11. For RK >> 1, [A] and CT are almost independent from RC, but the second-order correction is inversely proportional to RC, an opposite trend to that of CV. As predicted by Equation (25), the first-order solutions (with neglected d = 0) of CT and CV were independent from RC. For RK << 1 and with the homopolymerization effect considered (with kCV > 0, F’ > 0), a revised S function of Equation (21) and Equation (27) predicts that CV and CT would have the same profiles and both be decreasing functions of kCV. In comparison, for the case of RK >> 1, kCV does not affect [A], as shown by Equation (26), which predicts that CV is an increasing function of kCV, but CT is a slightly decreasing function of kCV, due to its second-order correction. With the presence of viscosity effect, or when F < 1, in Equation (22), the free-volume is reduced when crosslink efficacy increases. The reduction factor (F) only affects the propagation rate constant Fkp in Equations (20) and (22), but not in Equation (25). Therefore, the viscosity effect does not affect the efficacy for the case of RK >> 1, and affects the efficacy for other ratios of kP and kCT, where the viscosity effect reduces the efficacy of [B], as predicted by Equation (22). As predicted by the analytical formula of Equation (31) and the numerical data of Figure9, for a fixed light dose, higher light intensity has a higher transient efficacy, but a lower steady-state conversion. This unique and unusual feature of the light intensity dependence is the result of a bimolecular termination. In contrast, in type II unimolecular termination, the conversion is mainly governed by the light dose, rather than its intensity. For optically thick polymers, the influence of dynamic light intensity due to PI depletion is given by B’ in Equation (30), which predicts that the assumption of B’ = 0 (as in most previous models) suffers an error of 5% to 20% (underestimated) for a crosslink depth (ZC) ranging 300 to 500 µm, and also underestimates the efficacy, as shown by Figures7 and8. Scaling laws for the functional group concentration of thiol, [A], and ene, [B] are given by [A]m[B]n. For RK >> 1, the polymerization rates are first order in the ene concentration (or n = 1.0) and nearly independent of the thiol concentration (or m = 0); in contrast, m = 1.0 and n = 0 for RK << 1. For RK values near unity, polymerization rates are approximately 0.5 order in both thiol and ene functional group concentrations (m = n = 0.5). However, a scaling law of m = 0.4 and n = 0.6 was found in an acrylate system (with RK = 13), due to contributions from homopolymerization. Based on the above-described general features for thiol–ene polymer systems, it is possible to tailor the two ratios, RK and RC, and choose the appropriate ene functional group to minimize side effects such as viscosity and homopolymerization for maximal monomer conversion or tunable degree of crosslinking. The binary thiol–vinyl system used in this study may be expended for multiple-component systems, such as the ternary thiol–ene–ene and thiol–ene–acrylate systems reported by Reddy et al. [20]. Furthermore, the model and formulas developed for the free-radical-mediated thiol–ene system may be extended (to be published elsewhere) for anionic chain process such as base-catalyzed, thiol-Michael addition reactions [26]. Further information on the monomer properties discussed in this article and the factors influencing photopolymerization kinetics and optimal materials with low molecular weight, low viscosity, and in situ polymerization mechanisms have been reported in earlier publications [28–30]. Polymers 2019, 11, 1640 16 of 17

4. Conclusions We demonstrated that our model for a thiol–acrylate system including homopolymerization effects, and for a thiol–norbornene system including viscosity effects, fit much better with the measured data than the previous model. Furthermore, we found that the dynamic light intensity due to photoinitiator depletion cannot be neglected in optically thick polymers. The efficacies of the thiols and enes depend on both RK and RC, which may be tailored together with the choice of ene functional group to minimize side effects such as viscosity and homopolymerization for maximal conversion efficacy.

Author Contributions: Conceptualization, J.-T.L.; Data curation, K.-T.C. and D.-C.C.; Formal analysis, J.-T.L. and D.-C.C.; Funding acquisition, H.-W.L. and D.-C.C.; Methodology, J.-T.L.; Software, K.-T.C. and D.-C.C.; Supervision, J.-T.L. and H.-W.L.; Validation, J.-T.L. and K.-T.C.; Visualization, J.-T.L. and H.-W.L.; Writing—original draft, J.-T.L.; Writing—review & editing, J.-T.L. Funding: This research received no external funding, and the APC was funded by China Medical University, Taiwan, ROC. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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