# Simulation of models for multifunctional photopolymerization kinetics.

INTRODUCTIONFree radical chain photopolymerizations of multifunctional monomers are light-induced reactions that convert a liquid monomer into a solid polymer. The use of light provides many advantage, such as the ability to cure rapidly in the absence of a solvent under ambient conditions, flexible monomer chemistry, spatial and temporal control, and the formation of a highly crosslinked network, i.e., a polymer which is insoluble in any organic solvent [1-8]. Photopolymerization of multifunctional monomers such as tetraacrylates, triacrylates, and diacrylates produces densely crosslinked networks that are useful in applications such as decorative and protective coatings, replication of optical disks and aspherical lenses, biomaterial, drug delivery systems, and dental restoration [9].

Photopolymerization with spatial and temporal control can be carried out at ambient temperatures with and without oxygen through the use of appropriate photo-masks, which focuses illuminating light on the surface of resin. Such control on the reaction process provides a unique opportunity to produce wealth of photopolymers for use in special purposes in a customized manner [10-14]. Therefore, any affords in understanding the kinetics and the chemistry of the photo-curing process is very important. There are many studies involving the kinetics of photopolymerization, chemistry and modeling [2-6, 9].

The gelation point is the point where a transition from a liquid monomeric resin to a highly crosslinked gel is accomplished. A more direct measurement of mechanical properties during photo-induced free radical polymerization is preferable. Microrheological techniques can easily and directly detect changes in mechanical properties including the drastic increase in viscosity at the gelation point during photopolymerization, in contrast to traditional rheology techniques where bulk properties are measured and the sample is subjected to an externally imposed shear strain [9, 15].

In free radical polymerization, mass transfer limitations of various species become highly important since most of the elementary reactions become diffusion-controlled. Reactions that are influenced by diffusion phenomena include chemical initiation reactions, propagation of growing chains, and termination of these "live" macroradicals [16].

In this work, a kinetic and stochastic modeling of the photopolymerization process was perfonned that takes into account the molecular weight and crosslink density change. Such improvements in the theoretical models would greatly aid the process improvements where higher throughput, better spatial resolution, and improved surface finish are important, i.e., stereolithography [15], The objective of the models developed in this study is to model the critical degree of polymerization that results in gelation. In addition, the gelation time refers to the time where the critical degree of polymerization was measured.

The kinetic model of photopolymerization process involved solving a system of coupled first order ordinary differential equations (ODE) describing the change of the concentrations of photoinitiator, monomer, and live and dead polymeric radicals as a function of reaction time. In this model, the radical termination is assumed to be radical by combination and rate constants of propagation and termination were taken to be diffusion controlled.

In the stochastic Monte Carlo model, which is based on a probabilistic approach, was employed to repeat the simulations of the experiments that measured the dependence of the gelation time on the initiator loading concentration when the resin did not contain diffused oxygen. Furthermore, the potential of the model is validated using FTIR and DSC data that can be used to model complex reaction mechanisms.

Free Radical Photopolymerization Kinetics

Initiation, propagation, and termination are the three primary reaction mechanisms in the photopolymerization process.

Initiation

The initiation step starts when photoinitiator molecules absorb photons to form photoinitiator free radicals, I*. The fragmentation of a photoinitiator molecule into its free radicals is represented by the following reaction.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Here, S represents the photoinitiator molecule and [k.sub.i] is the initiation rate constant for the dissociation of photoinitiator molecules via photon absorption. Photoinitiator free radicals then attack monomers to form primary radicals. This reaction step is represented by Eq. 2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where M represents a monomer molecule, [R.sup.*.sub.1] represents the primary radical and [k.sup.l] represents the kinetic rate constant for the initiation step.

Propagation

During propagation, primary radical molecules, [R.sup.*.sub.1], formed at the initiation step continue to add monomer molecules in a chain-like fashion to form macro-radicals of different molecular weight. The propagation reaction between a free radical and a monomer molecule can be represented by the following equation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The resulting polymer radical, [R.sup.*.sub.2], propagates by Eq. 4 and this process continues as indicated in Eq. 5

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [R.sup.*.sub.n] represents a growing polymeric radical chain of n monomer units and [k.sub.p] is the kinetic rate constant for propagation. The rate of propagation depends on the functionality of the monomers. The photopolymerization of monomers with one double bond results in linear chains while the photopolymerization of multifunctional monomers produces highly crosslinked polymeric networks. During the propagation step, multifunctional monomers are added successively into polymer chains as units containing pendent double bonds. As the propagation reaction continues, these pendent double bonds are attacked by radical sites either on the same chain or on different chains. A propagation reaction, which takes place between a pendent double bond and a radical site on the same growing polymer radical, is called a primary cyclization reaction [2, 17, 18].

Termination

Termination reaction can take place by coupling of two growing polymer radicals to form a single dead polymer of chain length equal to the sum of the chain lengths of the combining radicals. This reaction is represented by Eq. 6 and is commonly called termination by combination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [k.sub.tc] represents the kinetic rate constant for termination by combination [8]. Radical termination may also occur as a result of an atom transfer from one growing polymer radical to another. This type of termination reaction is called termination by disproportionation and leads to two different types of dead polymer molecules, as shown in Eq. 7

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (?)

where [k.sub.td] represents the kinetic rate constant for termination by disproportionation. Termination can also take place by the reaction of an initiator radical with a growing polymer radical as shown in Eq. 8

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [k.sub.tp] represents the kinetic rate constant for termination by photoinitiator radicals.

EXPERIMENTAL STUDIES

Materials

In this study, monomers having different number of functional groups were selected and were: 2(2-ethoxyethoxy) ethyl acrylate, triethylene glycol diacrylate, trimethylolpropane triacrylate, and ethoxylated pentaerythritol tetraacrylate, which are known by the acronyms of SR256, SR272, SR351, and SR494, respectively. The photoinitiator Irgacure 651 (DMPA) was used. All the monomers used in the experimental studies were purchased from Sartomer, and the photoinitiator was purchased from Ciba. The micron-sized, inert fluorescent silica-based tracer particles used in the microrheology experiments were provided as a courtesy by a research group at the University of Twente, the Netherlands.

Experimental Measurements

To probe the rheological properties of the resins as it polymerizes, microrheological techniques have been used. The experimental setup and data acquisition have been comprehensively explained by the previous publication of the authors [9]. The microrheology measurements were done for four photopolymerizing resins for different photoinitiator loading concentrations.

To determine the double bond conversion of the resin as a function of the photoinitiator loading concentration a Bruker brand FTIR spectrometer was used. For this purpose, a specific sample chamber, which is disposable and is compatible with the sample holder of the spectroscope that allowed transmittance of both IR and UV wavelengths into the resin was designed and fabricated. The fabrication began by grinding the FTIR spectrograde powder potassium bromide (KBr) salt purchased from Crystal Labs. A Graseby Specac Press was used to make the KBr pellets. Two teflon spacers, separated by a 1 mm gap, were placed in parallel on top of the KBr disc. The other KBr disc was then centered and placed over the gap between the Teflon pieces. The stack of materials was then fastened with metal clips. The gap of the chamber could hold ~2-3 [micro]L of sample. The sample was loaded into the chamber using a micropipette and then the chamber was sealed with vacuum grease when necessary to prevent evaporation. The samples were loaded into the sample chamber and then placed with the sample holder in the FTIR spectrometer. The sample was then positioned so that the IR laser beam directly illuminated the center of the KBr disc. The data was transferred to the computer and recorded by Opus software. After a blank spectrum of KBr disc was recorded, spectra for the uncured samples and cured samples were recorded one after another and the results were compared and correlated with each other. After the data acquisition, Opus programs were used to analyze the data further. FTIR experiments were done with SR256 resin using three different photoinitiator loading concentrations for a range of UV light exposure times (curing times).

The DSC experiments were conducted isothermally to study the double bond conversion as a function of temperature. For this purpose, ~1.16 mg of SR494 resin with 0.02% Irgacure 651 photoinitiator by weight percentage was put in an aluminum sample pan using a micropipette. A TA Instruments DSC Q1000 with photocalorimetric accessory was used to monitor the photopolymerization of the resin. An Exfo Photonic Solutions Novacure 2100 light source, which had source power of 0.06 mW, was used with a filter to produce 365 nm UV light. A continuous flow of nitrogen gas was passed through the DSC apparatus to prevent oxygen from inhibiting the photopolymerization process. In these experiments the normalized heat flow signal obtained from the DSC apparatus as a function of the reaction time was integrated to get the total heat generated by the photopolymerization reaction at four different reaction temperatures, which were 303, 343, 383, and 403 K.

MODEL DEVELOPMENT

A Deterministic Model Incorporating Chain Length Effects

The main motivation to search for other models for the simulation of the photopolymerization process was to see the effects of autoacceleration, autodeceleration, volume shrinkage, and reaction diffusion terms on the system besides the basic photopolymerization reaction mechanisms of initiation, propagation, and termination. This model was further improved to include the chain-length dependence in the termination reaction constant [19]. The kinetic equations describe the time variation of the species concentrations such as the photoinitiator, monomer, polymeric radical, and dead polymer concentrations in the reaction volume.

Applying the concept of conservation of mass to the reaction volume under these elementary reaction mechanisms leads to the reaction rate equations below for the photoinitiator molecule concentration, the double-bond-weighted monomer concentration, and the concentrations of polymeric radicals and dead polymer molecules of every possible length. Thus, the rate equation for the photoinitiator concentration, [S], becomes

d[S]/dt = -2.303[epsilon][S][A.sub.in]z (9)

where [A.sub.in] represents the amount of radiation on the surface of the resin at t = 0, e is the molar absorptivity coefficient in units of [m.sup.3]/mol m, and z is the depth of the resin. For the monomer concentration, [M], the following rate equation can be written:

d[M]/dt = -[k.sub.p][M][[R*].sub.tot] (10)

where [[R*].sub.tot] represents the total concentration of polymeric radicals of all possible lengths in the reaction volume at any time, and [k.sub.p] is the propagation constant. The rate equation for the concentration of primary radicals, [[R.sup.*.sub.1]], is then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [k.sub.fp] represents the polymer chain transfer kinetic constant, [D[P.sub.j]] is the concentration of dead polymer chain length of j, and [k.sub.t,i,j] is the chain-length-dependent termination kinetic constant for the termination of a polymer radical of length i by a polymer radical of length j. The mathematical expression for chain-length dependent [k.sub.t,i,j] is given below [20].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

For the concentration of polymeric radicals of length i, [[R.sup.*.sub.i]], the rate equation can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The rate equation for the concentration of dead polymers of length i, [D[P.sub.i]], is given as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The ODE model described here is comprised of the coupled ordinary differential equations, which are solved numerically. By applying the ODEs Eqs. (9-14) to the SR494 resin composed of four functional monomers, it is assumed that the primary radical concentration, [[R.sup.*.sub.1]], has one monomer, three double bonds, and one radical. The growing of primary radicals is assumed to take place in a chain-like fashion through the addition of monomers where only a single double bond on each added monomer is broken. The model assumes live and dead polymer molecules of every possible length are present in the reaction volume. The dead polymer concentration acts like a chain-transfer agent and helps the crosslinking of the growing polymeric molecules. Since the dead polymer concentration is used as a source for regenerating live polymers of every length, this mimics the crosslinking between polymer molecules. The essential kinetic parameters and the activation energies used in this model are given in Table 1.

A Stochastic Model

To simulate the experimental data, a probabilistic approach, which is based on the stochastic Monte Carlo model (SMCM) was used. SMCM is a powerful computational method developed by Gillespie, which takes into explicit account the fact that the time evolution of a spatially homogenous (well-stirred) chemical system is a discrete and stochastic process [21, 22]. Gillespie named this method the Stochastic Simulation Algorithm (SSA) that probabilistically answers the questions "When will the next reaction occur?" and "What kind of reaction will it be?" SMCM model has been comprehensively explained by the authors previously [9].

The SSA, describes the time evolution of an isothermal, well-mixed system of chemically reacting molecules contained in a fixed reaction volume of V. SSA can be applied to a variety of physical phenomena such as chemical reactions, diffusion, and radioactive decay. The common element in all these systems is their stochasticity, which arises from the inability to predict either the order of events or which events will occur in a given time. In these systems, the variables describing the populations of different species as a function of time are random variables, the successive values of which are not independent from each other.

In brief and concise description of SSA, the reaction volume is assumed to contain molecules of N distinct chemical species [S.sub.i] (i = 1, 2, ..., N) interacting via M reaction channels [R.sub.[mu]] ([mu] = 1, 2, ..., M). The number of the species [S.sub.i], at any given time, t, in the reaction volume will be denoted by X,{t), thus, the main goal of SSA is to calculate the [X.sub.i](t) values from their initial values [X.sub.i]([t.sub.0]) at initial time [t.sub.0]. A well description of each reaction channels should be defined according to the process [9].

We make the following simplifying assumptions: that the intensity of radiation is uniform throughout the sample, that there is no substantial change in volume of the sample, that the system is maintained at constant temperature. With the incorporation of the effect of cyclization and chain length, the model correctly predicts the CLDT behavior observed in multivinyl free-radical photopolymerizations.

Comparison of Experimental Results with the Deterministic and Stochastic Simulation

Figures 1-4 compares both experimental and simulation results for the deoxygenated photopolymerization of SR494, SR351, SR272, and SR256 resins used in this study, respectively. As can be seen from Figs. 1-4, gelation times for all of the resins decline with increasing photoinitiator concentration, since the yield of initator-derived radicals would increase with the concentration of DMPA. Monomers with multiple reaction sites require shorter times to reach the point of gelation where the determination of the gelation point (critical conversion) values of the multifunctional monomers were predicted by using Flory criteria given in Eq. 15 was used [23],

[[alpha].sub.c] = 1/f-1 (15)

By using Eq. 15 the gelation point values of SR494, SR351, SR272, and SR256 resins are calculated as 0.14, 0.2, 0.33, and 1, respectively.

Also, Figs. 1-4 show further that the nonlinearity in the curve depicting the dependence of gelation time as a function of the photoinitiator loading concentration increases as the functionality of the monomers increases for the experimental data. However, the dependence of the gelation times on the photoinitiator loading concentrations predicted by the deterministic ODE model appears to be linear in this logarithmic scale. On the other hand, both sets of SMCM simulations predict at least the qualitatively correct behavior for this dependence. As can be seen, the stochastic model agrees far better with experiment than the other deterministic model for SR494, SR351, SR272, and SR256 resins (Figs. 1-4).

As can be seen from Figs. 1-4, for all resins investigated in this study as the photoinitiator loading concentration increases, the stochastic model's predictions agree with the trend of the experimental measurements. In contrast, with the deterministic ODE model the agreement worsens as the photoinitiator loading concentration increases.

The stochastic rate constants are determined by dividing the corresponding deterministic rate constants by the reaction volume; it is difficult, however, to measure the reaction volume accurately. This may account for some of the discrepancies between the current stochastic model's predictions and the experimental results.

The overall qualitative agreement between experiment and the deterministic ODE model simulation results is much better for SR351, SR272, and SR256 resins than for SR494 at lower photoinitiator loading concentrations (Figs. 1-4). The nonlinear dependence of the gelation time on the photoinitiator loading concentration somewhat decreases as the functionality of monomers decreases. This explains the increase in the qualitative agreement of the simulations and the experimental measurements as a function of decreasing monomer functionality. However, as can be seen in Figs. 1-4, the stochastic models do capture the nonlinear character of the photopolymerization process in every case.

For SR351, SR272, and SR256 resins the gelation time values predicted by the deterministic ODE model decrease with the increasing photoinitiator loading concentrations up to 4% by weight and then level off and stay almost constant regardless of the increase in the photoinitiator loading concentration.

Thus, the deterministic ODE model compares well with the experimental results for the three lowest photoinitiator loading concentrations, but fails to follow the experimental profile for higher concentrations. The SMCM simulation with chain-length dependent rate constants and primary cyclization reactions follow the experimental profile closely; it performs particularly well for lower photoinitiator loading concentrations, and qualitatively follows the experimental trend even at the highest photoinitiator loading concentrations.

The nonlinear dependence of gelation time on the photoinitiator loading concentration for SR256 resin is not as prominent as for higher multifunctional monomers; it occurs only for those reactions with the highest photoinitiator loading concentrations (Fig. 4). Because of the decreasing nonlinearity in the experimental data for SR256, the curves for the linear deterministic ODE simulations agree reasonably with the experimental curve until the highest photoinitiator loading concentration values; and, even for the high photoinitiator loading concentration values, the deviations between the ODE simulations and experiment are relatively small.

Although the deterministic ODE models developed in this study do not accurately predict the nonlinear dependence of the gelation time on both the photoinitiator loading concentration and the functionality of the monomers in the absence of oxygen, the literature is full of successful applications of the deterministic approach to the chemical kinetics of polymerization of monomers with one or two double bonds [1-4, 16, 24], As can be seen from Fig. 4, the deterministic ODE model predicted the experimental behavior of the photopolymerization of SR256 resin as accurately as the stochastic model. Thus, it may be concluded that the deterministic approach is particularly useful and productive for simulating the photopolymerization of monomers with fewer double bonds. The deterministic ODE model simulations for the photopolymerization of SR256 resin composed of monomers with one double bond shown in Fig. 4 confirm this conclusion.

Validation of Stochastic Mode!

To further validate the SMCM, FTIR, and DSC measurements were also conducted. For this purpose, SR256 and SR494 resins were chosen for FTIR and DSC measurements, respectively.

Validation by FTIR Measurements

The degree of conversion of a photopolymerization process can be determined from FTIR measurement results [25], In this study, FTIR experiments were performed to measure the double bond conversion during the photopolymerization of the SR256 resin. For this set of experiments, photoinitiator loading concentration of 1%, 5%, and 10% by weight percentage were used. As mentioned previously, each SR256 monomer has one carbon-carbon double bond in its vinyl group, which has a stretching frequency of 1640 [cm.sup.-1]. As an example, Fig. 5 shows three different FTIR spectra of SR256 resin with the photoinitiator loading concentration of 5% by weight percentage. The highest curve in Fig. 5 represents the FTIR spectrum of the uncured resin and the lower two curves represent the FTIR spectra taken at the end of 4 and 15 s curing times, respectively.

As can be seen from, Fig. 5 the absorption peak from the C=C stretching mode at 1640 cm 1 has almost disappeared from the lowest curve corresponding to measurements done after curing for 15 s, indicating that almost all of the C=C bonds have been broken during the photo-polymerization process after 15 s of curing. This set of experiments was repeated at eight different UV light exposure times.

At the end of each experiment, the area under the C=C stretching peak at 1640 cm 1 was calculated automatically by the FTIR spectrometer using the OPUS software package. The relationship between the monomer conversion and the peak area (PA) is:

Monomer Conversion = 1 - PA (t)/PA(t=0) (16)

Using Eq. 16, the monomer conversions were calculated for the set of experiments conducted at the required times. Figure 6 compares the experimental and predicted double bond conversions as a function of the exposure time where the photoinitiator loading concentration was 1% by weight. In this figure, points marked with diamonds and squares represent the experimental measurements and the predictions of the SMCM, respectively.

The other two sets of experiments were done with photoinitiator loading concentrations of 5% and 10% by weight. Figures 7 and 8 show the results from FTIR experiments along with the results from the corresponding SMCM simulations for photoinitiator loading concentrations of 5% and 10%, respectively. As can be seen from Figs. 6 and 7, the agreement between experimental results and the model predictions for the 1% and 5% photoinitiator loading concentrations are quite remarkable given the complexity of the process.

For the 10% photoinitiator loading concentration, the model predictions for the conversion times at the highest double conversion values (or longer conversion times) are somewhat smaller than those experimentally measured (Fig. 8). The overall behavior of the simulation and experimental curves, however, are still remarkably similar. For longer curing times, larger amounts of heat are produced in the reaction volume by the photopolymerization process; the resulting increase in temperature is not included in the determination of the reaction probabilities in the simulations. This might be the reason for these discrepancies. These figures also clearly show the sharp increase in the conversion rate--that is, the slope of the double bond conversion curves in these graphs--as the photoinitiator loading concentration increases.

The gelation times obtained from the FTIR measurements were also compared with the passive microrheology results published in the previous study [9]. The gelation time results acquired from FTIR experiments were found to be very close to the gelation times determined from the passive microrheology experiment data [9]. For example, the gelation times obtained from the FTIR and passive microrheology experiments conducted for SR256 resin with 1% photoinitiator loading concentration in the absence of oxygen gave 15 and 16.2 s, respectively. For the 5% photoinitiator loading concentration the same measurement gave 6 and 4.8 s, respectively. It should be noted that Flory predicts 1 as the critical conversion for the photopolymerization of monomers with one double bond, which indicates 100% conversion. In reality, there is never 100% conversion in a finite amount of time as seen from FTIR experiments. Thus, the gelation times determined by the SMCM simulations for SR256 resin correspond to 95% conversion rather than 100% conversion [25].

The SMCM correctly predicts the behavior of conversion as a function of reaction time with different photoinitiator loading concentrations in the photopolymerizing resin. The success of the predictions of the SMCM based on the SSA seen in these last three figures holds promise for its potential to treat a variety of interesting chemical problems with complex reaction mechanisms.

Validation by DSC Measurements

In DSC experiments, the normalized heat flow signal obtained from the DSC apparatus as a function of the reaction time was integrated to get the total heat generated by the photopolymerization reaction at four different reaction temperatures. The heat flow signals from the DSC as a function of reaction time at 303, 343, 383, and 403 K are shown in Fig. 9. The double bond conversion values were then calculated by dividing the area under the heat flow signal curves by the total heat of the photopolymerization process, which is assumed to be the amount of heat generated when all the monomer double bonds have been broken via photopolymerization [26, 27].

The SMCM simulation results obtained for the photopolymerization of SR494 resin with photoinitiator loading concentration of 0.02% by weight at reaction temperatures of 303, 343, 383, and 403 K are compared with the corresponding experimentally measured conversions values in Figs. 10-13.

SR494 thermally polymerizes at temperatures of 413 K and above; all the temperatures in these DSC experiments were less than this critical temperature. Therefore, the increase in the conversion values as a function of temperature is apparent in these curves. Yet, an increase can be seen in the conversion as a function of reaction time. This may be due to the thermal energy distribution, which makes it possible for some resin molecules to possess sufficient thermal energy to excite and eventually become polymeric radicals. As can be seen in Figs. 10-13, the simulations also predict the same behavior and show a clear rise in the conversion as a function of temperature.

As can be seen from these figures the agreement between the SMCM simulations and the experiments are very good at all temperatures. The temperature effect in the simulations is represented by multiplying the reaction propensities by an Arrhenius factor. These results further validate the stochastic Monte Carlo model as a reliable method to make predictions about the photopolymerization of resins composed of multifunctional monomers.

CONCLUSION

In this work, the gelation times for resins composed of monomers with different functionalities, which were measured using passive microrheology technique, were compared with the results calculated from two different models. It was found that the results from the mathematical models based on deterministic approach (ODE) agree quite well for lower multifunctional monomers but fail to track the behavior of the nonlinear photopolymerization kinetics. The second model, SMCM, that used the stochastic approach was successful in capturing the nonlinear behavior of photopolymerization process for all the monomers used in this study. Also, further validation of the SMCM was done using two known techniques; FTIR and DSC. Thus, it is concluded that SMCM is a reliable method to make predictions about the photopolymerization of resins from multifunctional monomers.

ACKNOWLEDGMENTS

The first author wants to thank Dr. Bredveeld and Dr. Slopek for helpful scientific and technical discussions and Amy Tsui for her contribution in the technical reading of this article.

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Gokcen A. Altun-Ciftcioglu, (1) Aysegul Ersoy-Mericboyu, (2) Clifford L. Henderson (3)

(1) Department of Chemical Engineering, Marmara University, Goztepe, 34722, Istanbul, Turkey

(2) Department of Chemical Engineering, Istanbul Technical University Maslak, 34469, Istanbul, Turkey

(3) School of Chemical and Biomolecular Engineering, Georgia Institute of Technology Atlanta, GA, 30332, US

Correspondence to: G.A. Altun-Ciftcioglu; e-mail: gciftcioglu@marmara.edu.tr

Contract grant sponsor: Marmara University, Scientific Research Fund (BAPKO), Istanbul, Turkey; contract grant number: FEN-D-0804100095.

DOI10.1002/pen.23713

Published online in Wiley Online Library (wileyonlinelibrary.com).

TABLE 1. Kinetic parameters used in ODE model. Parameters Symbols SR494 Model Parameter for [A.sub.p] 6.1 parameters propagation rate Parameter for [A.sub.t] 6.4 termination rate Pre-exponential factor [A.sub.Ep] 28.4 Pre-exponential factor [A.sub.Et] 8916 Activation energy [E.sub.p] 1627 for propogation Activation energy [E.sub.t] 2103 for termination Reaction diffusion [R.sub.rd] 0.013 constant Absorptivity (initiator) [epsilon] 19.9 Initiation quantum yield [[phi].sub.i] 0.6 Parameters SR351 SR272 Model Parameter for 4 2 parameters propagation rate Parameter for 4 2 termination rate Pre-exponential factor 28.376 1600 Pre-exponential factor 8916 3600 Activation energy 1626 18230 for propogation Activation energy 2102 2940 for termination Reaction diffusion 0.0011 0.002 constant Absorptivity (initiator) 0.15 0.15 Initiation quantum yield 0.6 0.6 Parameters SR256 Units Model Parameter for 0.66 -- parameters propagation rate Parameter for 1.2 -- termination rate Pre-exponential factor 1600 [m.sup.3]/mol s Pre-exponential factor 3600 [m.sup.3]/mol s Activation energy 18230 J/mol for propogation Activation energy 2940 J/mol for termination Reaction diffusion 0.002 [m.sup.3]/mol constant Absorptivity (initiator) 0.15 [m.sup.3]/mol m Initiation quantum yield 0.6 -

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Author: | Altun-Ciftcioglu, Gokcen A.; Ersoy-Mericboyu, Aysegul; Henderson, Clifford L. |
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Publication: | Polymer Engineering and Science |

Article Type: | Report |

Date: | Aug 1, 2014 |

Words: | 5358 |

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