A computer model for a laser induced photopolymerization process has been established
which simulates stereolithography. The model couples irradiation, chemical reaction, and heat
transfer equations to provide insights into rate processes occurring in the volume element
contacted by the laser beam. Quantities predicted include the spatial variation in conversion of
monomer to polymer, depletion of photoinitiator, and local variations in temperature in and
around the spot contacted by the laser. This allows predictions to be made about the laser dwell
time, depth penetration and uniformity of the photopolymer formed in the process.Mechanical Engineerin

Chartoff, Richard P.

Flach, Lawrance

English

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A COMPUTER MODEL FOR LASER PHOTOPOLYMERIZAnONLAWRANCE FLACHDepartment of Chemical and Materials EngineeringandRICHARD P. CHARTOFFRapid Prototype Development LaboratoryandCenter for Basic and Applied Polymer ResearchUniversity of DaytonDayton, Ohio, U.S.A.AbstractA computer model for a laser induced photopolymerization process has been establishedwhich simulates stereolithography. The model couples irradiation, chemical reaction, and heattransfer equations to provide insights into rate processes occurring in the volume elementcontacted by the laser beam. Quantities predicted include the spatial variation in conversion ofmonomer to polymer, depletion of photoinitiator, and local variations in temperature in andaround the spot contacted by the laser. This allows predictions to be made about the laser dwelltime, depth penetration and uniformity of the photopolymer formed in the process.IntroductionOne aspect of the stereolithography process that has not received much attention duringinitial development of the overall procedure concerns the physical and chemical changes thatoccur in the small zone of monomer/polymer mixture directly exposed to laser irradiation, andthe rates at which these changes occur. The work described here addresses this aspect of theprocess.Mathematical models have been developed for the photopolymerization process whichallow computation of light intensity, absorbed light, photoinitiator concentration, monomerconcentration, and temperature profiles, as functions of time, for the exposed reaction zone andsurrounding monomer/polymer mixture. The model discussed here·.deals with· the situation wherethe laser light spot remains stationary on the surface of the resin vat. A second model, onecapable of handling the situation where the light spo!mOvesacross the surface of the resin vatin a series of small discrete steps, is being developed. Some preliminary results obtained fromthe moving spot model are presented to illustrate some of its capabilities.155Mathematical.•• ModelThe mathematical model for the stationary light spot situation was developed byconsidering a cylindrical region ofex.posedmaterial as shown in Figure 1. For this situation,where the light spotdoes<not move with time, and the resin is spatb~llyhomogeneous, cylindricalsymmetry· can be assumed. This allows .light intensities,concenttations and temperatures to befunctions of the two spatial coordinates (r and z) and time (t).The incident laser light is assumed to have a Gaussian intensity distribution-2 (..!) 2I(r,z,t) == I o e R at z==O, r~O , t~O (1)The beam radius, R, being the value of r at which light intensity drops to e-z of the peak: value.The decrease in light intensity with depth into the resin due to absorption by photoinitiatoris given byaI == -eSIaz (2)with the light intensity at the surface (z=O) given by eqn.(l). The light absorbed by photoinitiatoris thus given byIII. (r , z, t) == eSI • (3)Since photoinitiator is consumed by the photochemical initiation process, the rate ofchange of photoinitiator concentration can be calculated fromas == -~I (4)at awith S(r,z,t) = So at t=O, r~, z~.Monomer is consumed by the polymerization reaction, so the rate of change of monomerconcentration can be calculated from(5 )with M(r,z,t) =Mo at t=O, r~, z~.156According to Odian [1], a rate expression of the following fonn can be used for aphotochemical polymerization process:[cl>I ]0.5R;p = k;p M k ea ( 6)If an expression of this fonn is valid then eqn.(5) becomesaM = -k M [ cI>.Ia ]0,5 .at ;p k t;(7)The temperature field in the vicinity of the exposed region is given by a transient heatconduction equation with heat source tenn (heat generated by polymerization) [2](8 )withT(X,z,t) = To at t=O, X:I?;O , Z:I?;OaT = 0 at X=O, Z:I?;O, t:l?;OaxT = To at x .....oo, Z:I?;O, t:l?;OT = To at z .....oo, X:I?;O , t:l?;OaT = 0 at z=O, X:I?;O, t:l?;OazThe partial differential equations presented above together with the appropriate boundaryand initial conditions constitute the mathematical model. It should be realized that this is amodel and is an approximation to the real situation. In addition, a number of assumptions areimplicit in the model fonnulation as presented here. Briefly these are as follows:1. No flow of material (convection or diffusion) in any direction.2. No optical effects (scattering, diffraction, refraction or reflection).3. Heat generation due to heat of polymerization only.4. Negligible heat loss from surface of monomer vat.5. All physical and thennodynamic parameters constant.6. Photoinitiator fragmentation products non-light-absorbing.When considering a mathematical model for the moving spot situation, certainmodifications to themodelaspresentedabove, are required. Since the moving spot situation nolonger possesses cylindrical symmetry, a change to rectangular coordinates is necessary, and a157transient heat conduction equation in three spatial dimensions, x, y, and z, is required. Motionof the laser light spot is achieved simply by changing the coordinates of the light spot center onthe surface of the monomer vat.The.mathematicallllodel·thusconsists of a .systelll of.couplodpartial.• differential••equationsof different types, the most complex of .which is the· parabolic, second order·heat conductionequation. In order to set up numerical solution. techniques for the set of equations, the infinityboundary conditions for the heat conduction equation were approximated by specifying thetemperature at some fmite r and z values. Ithas been found that values of r=5R and z=2D areadequate for the situation used to test model operation (see Figure 2). This approximation is onlyreally valid for short times «0.5 seconds) as for longer times the boundary conditions start toinfluence the computed temperature values. Both Runge-Kuttaand Alternating Direction Implicitmethods 4] were used to numerically solve the equations.Results and DiscussionThe mathematical model presented above is general in that it can be used to model anyresin system provided the appropriate physical, thermodynamic, and kinetic parameters aresupplied by the user. The model was tested using physical and thermodynamic properties for aHDDA monomer, and a 15 mW laser light source. This information together with thedimensions of the exposed region, etc., is summarized in Table 1. Polymerization rateinformation for HDDA was obtained from a calorimetric study by Tryson and Shultz [5]. Thesedata were used because they represented the most comprehensive available data set containingtemperature dependent rate information. The data and kinetic parameters were derivedexperimentally by curve fitting to the 1/2...order kinetic expression of eqn.(6). Thus, in order tocalculate consistent results it was necessary to use the 1/2-order rate expression in our treatment.It should be noted that mechanistically an expression that is 1st-order with respect to lightintensitY is most appropriate for photopolymerization of multifunctional monomers [6]. Thecomputer model is completely general, however, and will accommodate any rate expression withappropriate rate constants.Figures 3 and 4 are contour plots prepared from data obtained from the stationary spotmodel. Figure 3 illustrates how conversion of monomer to polymer varies with position anexposure time of 0.3 seconds. The contours clearly illustrate how conical or "bullet-shaped" test-pieces would be formed by short.;term exposure with a stationary laser beam. Figure 4 showsthe corresponding temperature field in the vicinity of the exposed segment.Figures 5 and 6 represent some preliminary results obtained from the moving spot modelcurrently under development. Figure 5 shows temperature profiles along the y-axis, and howthese change with time as the light spot moves in the positive y...direction. It is interesting tonote that for short times «0.3 seconds) the center of the laser spot precedes the peak in thetemperature distribution, whereas for longer times (>0.4 seconds) the center of the spot lagsbehind the temperature peak. The fact that the resin is heated prior to exposure may be animportant factor which could affect the nature of the polymer formed during the process. Figure 6158shows the temperature contours on the surface of the monomer vat for an exposure time of0.3 seconds.ConclusionA mathematical model for the stereolithography process has been developed and testedfor a HDDA monomer reaction system and stationary laser light source. The model can be usedto investigate process dynamics in slices of various thickness, with different laser light intensities,exposure times, and physical and chemical parameters. Results obtained allow prediction of laserdwell time, light depth penetration, and uniformity of polymer formed during the process. Modelpredictions are reasonable, but experimental verification has yet to be performed. A model forthe moving spot situation is currently under development,and preliminary results obtained lookpromising.AcknowledgementsThe authors gratefully. acknowledge the financial support of the State of Ohio EdisonMaterials Technology Center (EMTEC), the United States Air Force, and an industrial consortiumof 14 major companies.NomenclatureCp heat capacity (J g-l K-l)D slice thickness (cm)I light intensity (E cm-z sec-I)Ia absorbed light (E cm-3 sec-I)Ie, peak light intensity (E cm-z sec-I)k thermal conductivity (Wem-l K-l)~ propagation rate constant (1 mo}"l sec-I)~ termination rate constant (l mo}"l sec-I)M monomer concentration (mol 1"1)Mc, initial monomer concentration (mol 1"1)r radial position (cm)R nominal radius of laser beam (cm)~ polymerization rate (molcm-3 sec-I)S photoinitiator concentration (mol 1"1)So initial photoinitiator concentration (moll-I)t time (sec)T temperature eC)z depth of light penetration (cm)~ heat of polymerization (J mo!"l) .e molar absorptivity of photoinitiator (J morl)c1> quantum yield for initiationp density (g cm-3)159References1. Odian, G., "Principles of Polymerization", 2nd Edition, John Wiley & Sons, Inc., NewYork (1981).2. Bird, R.B.,W.E.Stewartand E.N.Lightfoot, "TransportPhenomena", John Wiley & Sons,Inc., New York (1969).3. Camahan, B., H.A. Luther and J.O. Wilkes, "Applied Numerical Methods", John Wiley& Sons, Inc., New York (1969).4. Ferziger, J.H., "Numerical Methods for Engineering Application", John Wiley & Sons,Inc., New York (1981).5. Tryson, G.R. and A.R. Schultz, "A Calorimetric Study of Acrylic Photopolymerization",J. Poly. Sci.: Poly. Phys. Ed., vol. 17, p. 2059 (1979).6. Decker, C., "Laser-Induced Photopolymerization", Materials for Microlithography, ACSSymposium Series, p.207 (1984).Table 1. Information and data used for test case.Value UsedLaser beam radius (R)Slice thickness (D)Peak light intensity (10)Initial photoinitiator concentration (So)Initial monomer concentration (Mo)Initial temperature (To)Molar absorptivity of photoinitiator (e)Quantum yield for initiation (<I»Density (p)Thermal conductivity (k)Heat capacity (Cp)Heat of polymerization (M\)Polymerization rateLaser step size (moving spot)Laser step period (moving spot)1600.0125 cm0.05 cm4.81xl0's E cm'z sec'l0.15 mol r l3.0 mol r l35°C200 I morl cm'l0.11.15 g cm'32.0xl0'3 W cm'l K'l1.70 J g,t K'l1.66xlOS J morlData for HDDA [5]0.00125 cm0.025 sec-tr.lH:x:~-LASERLIGHT\ U/\,. - .,MONOMER VATz=O (SURFACE)IIIII/- -- ....I .. z=DIIFigure 1. Cylindrical region used for development of mathematical model, showingcoordinate system origin.aTr=O r=R= 0az r=5RaTar= 0Z=Oz=Dz=2DT=ToT=ToFigure 2. Finite region used for numerical solution ofheat conduction equation.1610.0250.0150.05 a.....-......J--....I-_..I.I.-"""'-..............L--'--'--e....-u.-_J--......u_--'-0.025 -0.01 5 -0.005 0.005RADIAL POSITION (em)0.010.04_0.02E~~I.I.lCl 0.03Figure 3. Contour plot showing % conversion of monomer as a function of position.Time of exposure =0.3 seconds. Contour interval =6%.0.0250.0150.05 ~......_..J.I-~........,.;~~.........-.....:;......lI'.--'--l-'_"'""-...-J---J.--'-0.025 -0.01 5 -0.005 0.005RADIAL POSITION (em)0.010.04_ 0.02E~~I.I.lCl 0.03Figure 4. Contour plot showing temperature as a function of position.Time of exposure =0.3 seconds. Contour interval =2.6°C.16290 .......-.....,.--...,..---r---.,....-......,~-....,.--...,..--.,....--,---....,8070EllJct::::::>~ 60llJa..::::llLlJI- 50a 0.5 1.0aeeLaser position I I I I I I I I I I I0.150.120.06 0.09V-POSITION (em)0.0330 L.-_--L__-J..__-l-__..I---_---''---_-.J.__--'-__-'--__J..-_--I0.00Figure 5. Temperature profiles along y-axis as obtained from moving spot model.Plot time increment = 0.1 seconds.0.0625 .......-.....,.----,---...,---.,...---r-----,.--...,..--.,....--,-----,0.05000.03750.0250Eu 0.0125.......zoE 0.0000(/)&:I -0.0125x-0.0250-0.0375-0.05000.200.160.04. 0.08 . 0.12V-POSITION (em)Contour plot showing. temperature distribution on surface of monomer vatas obtained from the moving spot model. Time of exposure = 0.3 seconds.Contour interval =4.00C.-0.0625 1....-_---"__--'-__.......__"'--_---' --'-__,........__........_-'---_--'0.00Figure 6.163

A Computer Model for Laser Photopolymerization

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A Computer Model for Laser Photopolymerization

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