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Degree of Cure of Epoxy/Acrylic Photopolymers: Characterization With Raman Spectroscopy and a Modified Phenomenological Model.

INTRODUCTION

Photopolymers possess unique capabilities including ambient temperature curing and control over where and when the polymer cures. These capabilities have prompted a variety of products like dental materials, contact lenses, coatings, microfluidic device fabrication, tissue engineering matrices, and photolithography. Recent developments in photopolymer technology include monomer mixtures and co-polymers. These applications are limited by our understanding of the photopolymerization process.

Stereolitography (SLA) is an additive manufacturing that employs photopolymers. SLA processes liquid photopolymers by exposing thin liquid layers of them to a raster scan of laser light for curing. The curing layer adheres to the layer below it and between/around the raster scan paths as the polymer chain reactions proceed. After the pattern, has been traced, the stereolithography machine's elevator platform descends the thickness of a single layer (specified by the operator), typically 0.05 Correspondence to: T. A. Osswald; e-mail: tosswald@wisc.edu 0.15 mm (0.002" to 0.006"). Then, a blade, sometimes resin emitting, usually sweeps across the cross section of the part, recoating it with fresh material. On this new liquid surface, the subsequent layer pattern is scanned, joining the previous layer. The sliced layers combine to form a solid 3D model. After the building process, parts are usually cleaned of excess resin and are subsequently cured in an ultraviolet oven.

Most manufacturing problems faced by stereolithography users are correlated to the behavior of photo-polymers during processing. Some of these include warpage, the presence of unreacted and potentially ex tractable monomers, and unknown green strengths of stereolithography parts. The final part strength is known to vary when using different operating conditions including layer thickness [1-5], over-cure depth [3, 6], light intensity [1], scanning velocity [1, 2], and post curing time [6]. The effects of these strength variations were found through extensive testing of materials without any modeling or in situ material analysis.

An in situ monitoring of the cure process in combination with distinguished data would provide a better understanding of the cure process by enabling more accurate cure modeling of these materials with strength predictions and process monitoring with instant parameter adjustment feedback.

The photopolymerization reaction can be achieved using two different mechanisms, free-radical and cationic. The first development was the acrylate chemistry free-radical photopolymerization, which is limited by oxygen inhibition [2] and shrinkage. An example of a simplified free radical cure reaction can be seen in Fig. 1. Free radical photopolymerization occurs when initiator molecules are converted to free radicals by a light source with wavelengths in the UV spectrum in most cases. These free radicals react with and attach to a monomer, initiating a polymer chain that self-propagates until it reacts with another free or attached radical [8], Oxygen inhibition occurs in acrylics when the free radicals at the surface are scavenged by the air's oxygen to form a peroxy radical rather than react with the monomers. This inhibits the propagation and termination reactions leading to partially cured products [9]. The second development in resins was the epoxy's cationic photopolymerization process. An example of a simplified cationic cure reaction is shown in Fig. 2. Cationic photo-polymerization is different from free-radical in that it uses photo-initiators to open one bond in the monomer rings rather than converting one carbon double bond to two single bonds. The main drawbacks with these reactions are inhibition by water vapor [1, 11, 12] typically in industrial environments [2], shrinkage, and a slower reaction timeframe relative to acrylics. Inhibition by water occurs in a similar manner to acrylic's inhibition by oxygen but requires water vapor or high humidity to occur.

A photopolymer cure profile is similar to a cake or epoxy cement. The same analysis is conducted with similar types of kinetics analysis of heat and/or mix activated thermosets. The first difficulty lies in the need for an adapter to conduct the UV light exposure experiments in a customary testing apparatus [13].

Differential scanning calorimetry (DSC) works well for heat and mix activated thermosets by monitoring the heat released/ absorbed during the cure process. A hybrid DSC system called photo-differential scanning calorimetry (PDSC) is adapted with a UV light source for activation. It is used to obtain the rate of cure (rate of photopolymerization) dc/dt, as well as the degree of cure c [14-20]. This method records the heat generated from the chemical reaction over time and corresponding light intensity I. Drawbacks of this method include that it is based on conduction. Fast curing and minimal thermal change systems like photopolymers are less accurately captured by the thermocouple's relative slow speed of response. In addition, PDSC only measures the bulk conversion of the entire sample and it only measures an approximate conversion of different components when the exothermal peaks do not overlap [21, 22]. Two peaks in a DSC analysis still contain information from both acrylic and epoxy conversions. In the field of heat activated thermosets, Cruz et al. showed how DSC underestimates cure percentages as a result of measuring the bulk thermal changes rather than the chemical reactions [23]. The measurement is a correlation to the heat of conversion rather than the chemical reaction. More cons include the inability to measure in situ, lack of thickness consistency, and the need to adapt a UV-light source to the apparatus [13].

One solution addressing this has been the developments in spectroscopy for polymer characterization. These include real time infrared (RTIR) [22, 24], real-time near infrared (NIR) [25], and Raman [20] spectroscopies. These methods monitor the changes in frequency of photons in monochromatic light upon interaction with a sample. We define these changes in quantities between the different Raman measurements over time as the chemical conversions and consequently change in the degree of cure occurring in the materials.

The real time infrared spectroscopy (RTIR) has been used on photopolymers with modified thin chambers. Ye et al. demonstrated how the glass transition temperature [T.sub.g] varies and correlates to different resultant crosslinked polymers when photopolymerized under different temperatures. The testing required a chamber that was fully enclosed and sensitive to sample thickness [26]. Another complication of the RTIR process is the possible need of sample dilution to prevent saturation [27].

The NIR is similar to RTIR except it requires a thick film and only gives a bulk conversion observation [28]. This study targets the stereolithography and other additive manufacturing process like selective laser sintering and selective laser melting which cures in thin layers at the surface. Because the surface-and not the bulk--layer that is affected by the atmosphere and layered manufacturing system is of interest, this method is less feasible.

Raman spectroscopy works without the restrictions of dilution, sample thickness restrictions, and can measure small areas. Raman spectroscopy uses vibrational spectroscopy based on the "Raman" effect which involves the inelastic scattering of photons [29, 30]. The intensity and energy of gathered, scattered photons--Raman spectral intensity--provide information on the existence of certain bonds and their quantity in the test volume. In a Raman spectrum graph each bond is correlated to a specific energy, which is represented by a wavelength in the x-axis (see Fig. 3a). Monitoring the testing material with the Raman-Spectroscope allows the detection of changes in the intensity of these peaks over time (see Fig. 3b).

The Tornado Raman spectroscopy system was chosen for its prediction capabilities, non-destructive measurements, and in situ monitoring capability. The cure profile of this epoxy/acrylic photopolymer was inferred from the Raman spectrum evolution and later analyzed to profile the rate of cure.

EXPERIMENTAL

The photopolymer of study was Accura[R] 60 by 3D systems[R]. This resin is used in SLA additive manufacturing (3D printing) systems. Accura[R] 60 is an epoxy/acrylate (cycloaliphatic diepoxide/aliphatic tetra-acrylate) resin containing reactive diluents (see Table 1). This photopolymer includes both photo-cationic initiators and photo-radical initiators to activate the epoxy/acrylate monomers respectively.

The material samples were prepared on 3.175 mm (0.125 in) 6061 T6 aluminum plates that are precleaned with acetone. A 14.287 mm (0.5625 in) circle was scratched on the aluminum plate and one drop of photopolymer was applied to the circle. The thickness of the sample was reduced as close to the thickness cured by the laser in the Viper SI2 SLA[R] system as possible by spearing it over the plate. The sample being thin also reduced any shrinkage effects and allows generated heat to conduct out through the aluminum substrate. The aluminum plate was placed in an enclosure with a cover to block the fluorescent lighting.

The acrylic monomer exposed to oxygen inhibits the cure, which can slow down and sometimes postpone the cure cycle. The tests here were done in ambient air to match the stereolighography's and post cure UV oven's operating conditions with the oxygen.

To gather significant and consistent Raman peak heights, Raman camera exposure times of at least 0.5 s are needed. This study proposes a stepwise technique to quantify the degree of cure. As the curing of the photopolymer is rapid (significant cure within 1 s) in comparison with the camera exposure time needed for satisfactory resolution, the tests were run with curing exposure time intervals of 0.2 s between each Raman measurement. The samples were then tested at different light intensities. Experiments were also run at 0.4 s and the differences were minimal.

One spectrum of Accura[R] 60 was recorded at the liquid uncured state. The UV cure light was activated for 0.2 s. A 15 s pause was done before the 30 s (2.5 s averaged over 12 measurements) Raman measurement was taken. The pause was included to allow the sample to cool to ambient temperature and allow the degree of cure to reach equilibrium. This cycle was repeated for a minimum of 45 exposures. The UV cure light time was increased and the cycle was repeated until the polymer has been exposed to a minimum of 252 s (cumulative) of UV cure light. Figure 4 summarizes the stepwise method.

Tests done with longer pauses before each measurement reduced the influence and noise of the epoxy's post cure reactions, but due to resource limitations, they were kept at 30s increments.

The apparatus used was a Tornado Spectral System's remote probe Raman spectrometer. It utilized an RFH-400 Optical Head combined with the RFT-10 Non-contact Objective Lens Assembly focused at 50 mm. The camera used was a Tomado Hyperflux U1 with a 785 nm Raman Stokes configuration (see Fig. 5).

The operating conditions were chosen to compare with the 3D Systems[R] Viper SI2 SLA[R] 3D printer manufacturing process. The UV cure light wavelengths were filtered to 550 nm [+ or -] 250 nm. The UV light intensity was chosen at 0.08,0.10, and 0.13 W.

The temperature change was measured during the cure cycle and followed the room's temperature within the range of 0.6[degrees]C for the 0.08 W and 0.10 W tests. The temperature change due to the exothermic reaction was considered to be negligible and absorbed by the aluminum substrate.

CHARACTERIZATION CURVES

The peaks of the spectra were analyzed by the KnowItAll software by BioRad Systems, Hercules, CA, in order to characterize the associated bonds. Table 2 summarizes the chemical bond relations to Raman peaks for Accura[R] 60.

To determine the relevant spectrum from a Raman measurement, the Baseline Estimation and Denoising with Sparsity algorithm BEADS developed by Ning et al. was implemented [40]. This algorithm assumes, that the measured signal y consists of the information of the spectrum x, the underlying baseline b, and white Gaussian noise w as seen in Eq. I.

y = x + b + w (1)

The algorithm formulates a minimization problem, which contrasts the shape of the spectrum x with the noise w, while minimizing the impact of negative values for x and identifying the baseline b and noise w by low and high pass filtering, respectively. A matching set of algorithm parameters was evaluated for this particular application [41]. Every Raman measurement was evaluated separately.

The strong peaks that changed during the cure in the Raman spectrum were correlated to the estimated chemical bonds in Accura[R] 60 (see Fig. 3). All the spectrums x(t) are normalized by a reference peak to compensate for changes of the intensity due to temperature history or other external influences. The reference peak 1,445 [cm.sup.-1] (C[H.sub.2] and RCH=C[H.sub.2]) was chosen for being the most stable peak of interest.

The normalization of a value of the spectrum x is defined in Eq. 2, with i the wavenumber and t the time.

[bar.x](i,t) = x(i,t)/x(1,445cm-1,t) (2)

The peaks that were chosen to be analyzed were peak 1,634 [cm.sup.-1] commonly correlated with the acrylic C=C bond change due to reticulation, and peak 1,262 [cm.sup.-1] that correlates to epoxy rings converting to ether groups with C--O and C--C stretching [7, 42-44]. Equation 3 represents the definition of degree of cure in the Raman analysis, with [A.sub.0] and [A.sub.[infinity] being the normalized peak intensity of the uncured and fully cured material respectively [45, 46].

C(t) = [A.sub.0] - A(t)/[A.sub.0] - [A.sub.[infinity]] (23)

It showed to be more sufficient to use a range of intensity values around the peak instead of just the peak intensity value. Figures 6-9 show the influence of light exposure power on the degree and rate of cure.

The rate of cure dc/dt was calculated by dividing the change in the degree of cure by the change in time.

The rate of cure results shown in Figs. 7 and 9 exhibit large pause before curing for the low light intensity values of 0.08 and 0.1 W. The slopes were however consistent. These tests were done in an oxygen free atmosphere (nitrogen) and the start times were then found to be consistent indicating that the oxygen was the cause. The stereolithography process is done in ambient air so these tests were continued with the oxygen to closer mimic the manufacturing process on the part's surface.

The oscillation in the data was a result of two primary sources. The first source was the data itself with oxygen inhibition, manual measurement timescales, and short delays between the exposure and measurements. The second was the BEADS algorithm, which is very sensitive to the definition of 0 and 100% cure as well as automated optimizing parameters that can vary how much the area under the Raman peaks influences the normalization process. Future work includes testing in the nitrogen environment setup to understand the light intensity's direct influence on the cure process without oxygen inhibition.

PHOTOPOLYMER CURE RATE MODELS

The following mathematical models focused on the characterization of photopolymers as a holographic recording material. Table 3 attempts to summarize the key variables of the models restricted to the rate of photopolymerization (rate of cure). The present study on the other hand focuses on a rate of cure model, dc/dt which can be implemented in a numerical analysis of residual stress build-up and warpage of SLA processes. We restrict our analysis to a phenomenological model rather than a mechanistic model that included the species balance and neglected the influence of diffusion mechanisms.

Phenomenological Models for the Degree of Cure c

Models that incorporate the actual operating parameters and chemical reactions are inherently more versatile and flexible. The two primary variables in the cure of photopolymers include light intensity I and exposure time t. The influence of UV light intensity on the rate of cure of photopolymers was shown to be correlated by Kwon et al. in 1999 [47], and Moreau et al. in 2003 [38, 39], Kwon's model incorporated the light intensity input 1 into a rate of cure model that used a constant exponential of 0.5 (see Eq. 4).

[dc.sup.m]/dt = [K.sub.p]C [[[phi]I(x)(1-T)/[K.sub.t]].sup.0.5] (4)

Moreau et al. incorporated light intensity dependence [I.sup.[gamma]] into a linear shrinkage model--[gamma] being equal to 0.5. Linear shrinkage was then assumed linearly related to the degree of cure c. In these holographic properties tests, I was considered to be modified from the normal light exposure and including a fringe visibility V influence on the light intensity (see Eqs. 5-7). The fringe visibility V becomes zero for a uniform field of illumination. When testing photopolymers exposed to stereolithography laser or UV-lamp light on the surface, the fringe visibility is zero. This makes I equal to [I.sub.0].

[mathematical expression not reproducible] (5)

K(t)= [k.sub.o][1 - exp(-t/[tau])] (6)

I = [I.sub.0] * [1 + [V.sub.cos]([K.sub.g]x)] (7)

In 2003, Neipp et al. proposed the use of a model including the effects of polymerization rate k(t) light intensity I, and monomer concentration f([c.sup.m]) as seen in Eqs. 8 and 9.

[dc.sup.p]/dt = k(t) x I[(x).sup.[gamma]] x f([c.sup.m](x,t)) (8)

k(t) = [k.sub.0][e.sup.(-t/[tau])] (9)

In this case [k.sub.0] and [tau] are constants and t is the time of the reaction [47]. This equation is inherently closer to the actual cure process because it incorporates the influence of the variable's light intensity I and degree of cure c. Neipp et al. equated the [gamma] value to zero--linear intensity response--to simplify the mathematical analysis.

In 2005, Gallego et al. proposed an improvement in the field I of modeling photopolymers which added a polymerization rate with the Trommsdorff effect [48] (see Eqs. 10 and 11)

[dc.sup.p]/dt = k(t) x I[(x).sup.[gamma]] x f([c.sup.m](x,t)) (10)

k(t) = [k.sub.0][e.sup.(-tI/[tau])] (11)

This changes the time variable of the analysis to the cumulative sum of the UV light intensity 1 over time. It also defines the [tau] constant as a correlation to the model's influence of the energy input ([J.sup.-1]). The model also used the inclusion of the light intensity I with a y power value similar to Neipp et al. (see Eqs. 8 and 9).

In all of these cases the equations that utilized [gamma] were simplified by either assuming a [gamma] value of 1 [48, 49] or a constant polymerization rate [50-53].

The use of this Arrhenius like polymerization rate k(t) and incorporation of [I.sup.[gamma]] matches the effects of a photopolymer's cure profiles as well as the data collected in this study more accurately.

The influence of [gamma] in the light intensity I was later analyzed by Jallapuram et al. in 2007 using Raman measurements of acrylamide photopolymers [44]. In a simplified model that did not include the reactivity and concentration of monomers, the values obtained from [gamma] differed from the assumed value of 0.5 (see Eq. 12) [39].

[d.sub.c]/dt = [t.sub.0][I.sup.[gamma]] (12)

Model Variables

The present data analysis uses an analogy with autocatalytic models from heat/mix activated thermosets adapted to photopolymers, with a variable value for [gamma]. Equations 13-16 summarize the proposed model.

[mathematical expression not reproducible] (13)

f(c) = ([k.sub.1] + [k.sub.2][c.sup.m])[(1 - c).sup.n] (14)

[mathematical expression not reproducible] (15)

[mathematical expression not reproducible] (16)

Using the test data the models were matched to the actual rate of cure. This determines model variables and can compare the model effectiveness with an [R.sup.2] value. This study used the Levenberg-Marquardt and sequential quadratic programming algorithm methods [54-57]. IBM SPSS software was used to analyze the model with the measured data to obtain the corresponding fitting parameters. Tables 4 and 5 show the fitting parameters for both the acrylic and epoxy components. To understand the dependency of I in the Arrhenius like polymerization rate, two approaches, a simplified model with [[gamma].sub.2] equal to 1 and [[gamma].sub.2] as a fitting parameter were investigated. There was a minimum difference between the fitting in the two approaches and in this work the simplified model was used to generate the results. Figures 10 and 11 show a comparison of the modeled data versus the actual measurements.

The combined degree of cure of the material (see Figs. 12 and 13) was obtained without averaging over a large area. A part's cure can be inconsistent and uncured from lack of sufficient UV exposure, oxygen and water inhibition, physical distance between initiator and monomer chain propagation. As a result, even post-cured parts will have some material in the part that is not fully cured and averaged in the results.

CONCLUSIONS

The use of the Raman spectrometer worked to track the cure cycle through the elimination of the acrylic monomer's C=C bonds and the epoxy monomer's ring opening C--O and C--C stretching changes. The Raman measurements were performed without the thickness restriction, specialized sample chamber, or dilution to measure the actual chemical conversion rates of both acrylic and epoxy within the samples. The proposed phenomenological model has more versatility and usefulness in being able to predict the rate of cure that can potentially be implemented in a numerical simulation to predict part warpage. The Accura60's individual epoxy and acrylic components were identified independently with concentrations at different stages in the cure cycle. With this information and the monomer concentration of the photopolymer, the full cure characteristics were obtained. One can now model the full resin cure percentage and rate with this technique.

ACKNOWLEDGMENT

The authors thank Tornado Spectral Systems for allowing usage of their system.

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Blair Martin (ID), John Puentes, Lorenz Wruck. Tim A. Osswald

Polymer Engineering Center, Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706-1691

Contract grant sponsor: The German Academic Exchange Program (DAAD).

DOI 10.1002/pen.24550

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

Caption: FIG. 1. Example of a simplified radical (a) initiation of a monomer and (b) the monomer propagation reaction [7].

Caption: FIG. 2. Example of a simplified cationic/epoxy monomer (cyclohexene oxide) cure (a) activation and (b) propagation with ACE (Active Chain End) mechanisms [10].

Caption: FIG. 3. Raman spectrum of Accura 60 for (a) uncured and (b) uncured with cured stages.

Caption: FIG. 4. Gantt chart of the stepwise method.

Caption: FIG. 5. Raman spectrometer setup showing (1) Raman spectrometer probe and (2) UV cure light collimator.

Caption: FIG. 6. Degree of cure c of acrylic component in Accura 60 at various light intensities. Analysis based on peak 1,634 [cm.sup.-1] correlated with the acrylic C=C bond change due to reticulation.

Caption: FIG. 7. Rate of cure dc/dt of acrylic component in Accura[R] 60 at various light intensities. Analysis based on peak 1,634 [cm.sup.-1] correlated with the acrylic C=C bond change due to reticulation.

Caption: FIG. 8. Degree of cure c of epoxy component in Accura[R] 60 at various light intensities. Analysis based on peak 1,262 [cm.sup.-1] correlated with the epoxy rings converted to ether groups. The two figures correspond to 8 s (a) and 250 s (b) time scales of the same measurements.

Caption: FIG. 9. Degree of cure c of epoxy component in Accura[R] 60 at various light intensities. Analysis based on peak 1,262 [cm.sup.-1] correlated with the epoxy rings converted to ether groups. The two figures correspond 8 s (a) and 250 s (b) time scales of the same measurements.

Caption: FIG. 10. Rate of cure dc/dt of epoxy component in Accura 60 at various light intensities. Analysis based on peak 1,262 [cm.sup.-1] correlated with the epoxy rings converted to ether groups.

Caption: FIG. 11. Rate of cure dc/dt of acrylic content in Accura 60 at various light intensities compared to phenomenological model. Analysis based on peak 1,634 [cm.sup.-1] correlated with the acrylic C=C bond change due to reticulation.

Caption: FIG. 12. Modeled rate of cure dc/dt of Accura[R] 60 at various light intensities. Analysis combining epoxy/acrylic with the concentration of the monomer epoxy/acrylic at 75%/25%. The two figures correspond to 8 s (a) and 250 s (b) time scales of the same measurements.

Caption: FIG. 13. Modeled rate of cure dc/dt of Accura[R] 60 at various light intensities. Analysis combining epoxy/acrylic monomers.
TABLE 1. Chemical composition of Accura[R] 60 from MSDS [31].

Description        Components                    Weight percent

Diepoxide          3,4-Epoxycyclohexylmethyl,        40-60
monomers           3,4-epoxycyclohexane
(cycloaliphatic)   carboxylate [32-36]
                   Approximate base structure.

Tetra-acrylate     Ethoxylated pentaerythritol       12-22
monomers           tetraacrylate [7, 37]
(aliphatic)

UV initiators      Triarylsulfonium salts            0.5-4

Toughness          Propylene carbonate               0.5-4
additives

Description            Chemical structure

Diepoxide          [formula not reproducible]
monomers
(cycloaliphatic)

Tetra-acrylate     [formula not reproducible]
monomers
(aliphatic)

UV initiators      [formula not reproducible]

Toughness          [formula not reproducible]
additives

TABLE 2. Chemical bond relations to Raman peaks [28, 38, 39].

Peaks ([cm.sup.-1])    Group or bond representation

3,450                  Hydroxyl stretching
2,850 and 2,925        CH3 group stretching
1,723 and 1,730        CO (carbonyl groups)
1,620 and 1,634 (a)    Acrylic C=C bond
1,407 and l,445 (b)    Stable C[H.sub.2] and RCH=C[H.sub.2]
l,262 (c)              Epoxy rings converted to ether groups
1,261 (c)              Epoxy (increasing)C--0 and C--C stretching
1,234/40/50 (c)        Epoxy rings converted to ether groups
l,214 (b)              Stable aromatic ether stretch
1,173/1,186            CO stretching
1,106 (l,180) (b)      Stable para-disubstituted phenyl band
914 (c)                Epoxy rings converted to ether groups
788.6/760 (c)          Epoxy COC and C[H.sub.2] angular deformations
601 (b)                Stable nonreactive methacrylate CCO COO group

(a) Corresponds to Acrylic bonds.

(b) Corresponds to stable peaks.

(c) Corresponds to Epoxy bonds.

TABLE 3. Contributions to the photopolymer rate of cure model.

Author           Publication   Added to modeling of
                    year       photopolymers

Kwon et al.         1999       Added I into a dc/dt model

Moreau et al.       2002       Incorporated light
                               intensity [I.sup.0.5]
                               into a linear shrinkage
                               model. Linear shrinkage was then
                               assumed linearly related to c.

Neipp et al.        2003       Added P to the cure rate model
                               (Simplified to [gamma] = 1).

Gallego et al.      2005       Incorporated I and [gamma] into a
                               dc/dt model from Neipp et al.
                               in the Arrhenius like equation
                               (Simplified with [gamma] = 1)

Jallapuram          2007       Included [I.sup.[gamma]]
et al.                         to the dc/dt model

Author           Rate of photopolymerization
                 (cure) dc/dt models

Kwon et al.      [dc.sup.m]/dt = [K.sub.p]C                      (4)
                 [[[phi]I(x)(1-T)/[K.sub.t].sup.0.5]

                 [c.sup.m] = Monomer concentration
                 [K.sub.p] = Propagation rate constant
                 [K.sub.t] = Termination rate constant
                 [phi] = Constant quantum yield of
                 production of radical
                 I(*) = Light intensity as a
                 function of depth
                 T = Transmittance of film

Moreau et al.    [mathematical expression not reproducible]      (5)
                 k(t) = [k.sub.o] [l-exp (-t/[tau])]             (6)
                 [[epsilon].sub.l] = Linear shrinkage
                 [mathematical expression not reproducible] =
                 Linear shrinkage of fully
                 cured sample
                 [k.sub.o] = Polymerization constant
                 [tau] = Time constant

Neipp et al.     [dc.sup.p]/dt = k(t) *                          (7)
                 I[(x).sup.[gamma]] * f([c.sup.m](x, t))         (8)
                 k(t) = [k.sub.o][e.sup.(-t/[tau])
                 [c.sup.p] = Polymer concentration
                 k(t)= Arrhenius like polymeri-
                 zation rate
                 [k.sup.o] = Polymerization Constant
                 [tau] = Time constant
                 I(x)= Light intensity as a func-
                 tion of depth
                 [gamma] = Light intensity fitting
                 parameter
                 [c.sup.m] = Monomer concentration

Gallego et al.   k(t) = [k.sub.o][e.sup.(-tI/[tau])]             (9)
                 t = model's influence of the
                 energy input ([J.sup.-1])
                 See Neipp et al. for the
                 description of other variables.

Jallapuram       dc/dt = [t.sub.o][I.sup.[gamma]]                (10)
et al.           [t.sub.o] = Polymerization time
                 constant
                 I = Light intensity
                 [gamma] = Light intensity fitting
                 parameter

TABLE 4. Model parameters for the epoxy component.

Parameter         Estimate with            Units
                   RSS = 1.063

[a.sub.2]           143.3204      [mathematical expression
                                     not reproducible]

1/[tau]              6.9430       [mathematical expression
                                     not reproducible]

[[gamma].sub.1]      0.1980                  --

[a.sub.1]            0.0001                  --

m                    2.0633                  --

n                    2.7214                  --

[[gamma].sub.2]        = 1                   --

Parameter         Estimate with            Units
                   RSS = 1.044

[a.sub.2]          207.006172     [mathematical expression
                                     not reproducible]

1/[tau]             8.626373      [mathematical expression
                                     not reproducible]

[[gamma].sub.1]     0.628486                 --

[a.sub.1]           0.000130                 --

m                   2.019168                 --

n                   2.718495                 --

[[gamma].sub.2]     1.199393                 --

Estimates have 1.044 residual sums of the squares.

TABLE 5. Model parameter for the acrylic component
(RSS stands for the residual sums of the squares).

Parameter         Estimate with            Units
                   RSS = 0.626

[a.sub.2]            11.9935      [mathematical expression
                                     not reproducible]

1/[tau]              2.9561       [mathematical expression
                                     not reproducible]

[[gamma].sub.1]      0.6201                  --

[a.sub.1]            -0.0052                 --

m                    0.9643                  --

n                    0.6019                  --

[[gamma].sub.2]        = 1                   --

Parameter         Estimate with            Units
                   RSS = 0.332

[a.sub.2]            4.9483       [mathematical expression
                                     not reproducible]

1/[tau]              0.0009       [mathematical expression
                                     not reproducible]

[[gamma].sub.1]      0.1912                  --

[a.sub.1]            -0.0123                 --

m                    0.8886                  --

n                    1.0106                  --

[[gamma].sub.2]      -2.2597                 --
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Author:Martin, Blair; Puentes, John; Wruck, Lorenz; Osswald, Tim A.
Publication:Polymer Engineering and Science
Article Type:Report
Date:Feb 1, 2018
Words:5950
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