Behavior in the vicinity of a moving exposed region.

INTRODUCTION

Stereolithography involves generation of a three-dimensional plastic part by photopolymerization. A lamina (layer or "slice") of solid is formed at the surface of a tank of liquid monomer by applying a UV laser light spot by vectored scanning in a computer generated graphic pattern. The laser light spot is typically 10 mils in diameter and has a Gaussian intensity profile. Individual thin slices of polymer (approximately 10 mils in thickness) are formed at the surface, each representing an adjacent cross section of the part being fabricated. The first (or bottom) slice of the part is formed on an elevator platform, which, after completion of the first layer, is lowered to allow new liquid monomer to flow onto the working surface. This process is repeated, with each layer adhering to the previously exposed section of the part now submerged in the monomer vat. The thickness of each slice is determined by the diminution of light intensity with depth into the resin and by the gel point of the mixture. This "cure depth" is typically determined before part building by an experimental procedure on the SLA. The process has been described more comprehensively in a companion publication (1), which presents the basis for a mathematical model that simulates important aspects of the process.

We present here an extension of this initial work. The previously described model dealt with the situation of photosensitive material exposed to a stationary source of laser light. In this paper the basic model is modified to allow computer simulation of a dynamic situation where the laser scans across the surface in a straight line. The case of the moving light source more closely simulates the actual stereolithography process where a computer generated pattern is traced out on the vat surface. This pattern is constructed from a series of interwoven straight line segments or vectors.

The modified mathematical model allows computations of conversion and temperature information along the path of laser motion.

BACKGROUND

Since the mathematical model for case II, the dynamic situation where the laser scans across the resin surface, is based on case I, the stationary laser model, the latter will be summarized briefly here. The reader is referred to the first paper of this series (1) for additional details on the basic model for case I. The essential elements of the basic model are as follows.

Incident light intensity:

I(r, z, t) = [I.sub.0][e.sup.-2[(r/[W.sub.0]).sup.2]] at z = 0, r [greater than or equal to] 0, t [greater than or equal to] 0 (1)

(see Nomenclature for definitions).

Decrease in light intensity with depth:

[Delta]I/[Delta]z = -[Epsilon]SI. (2)

Light absorbed by photoinitiator:

[I.sub.a](r, z, t) = [Epsilon]SI. (3)

Rate of change of photoinitiator concentration:

[Delta]S/[Delta]t = -[Phi][I.sub.a] (4)

with S(r, z, t) = [S.sub.0] at t = 0, r [greater than or equal to] 0, z [less than or equal to] D.

Rate of change of monomer concentration:

[Delta]M/[Delta]t = -[[Kappa].sub.p]M[[[Phi][I.sub.a]/[[Kappa].sub.t]].sup.0.5] (5)

with M(r, z, t) = [M.sub.0] at t = 0, r [greater than or equal to] 0, z [less than or equal to] D.

Temperature:

[Rho][C.sub.p][Delta]T/[Delta]t = [Kappa][1/r [Delta]/[Delta]r(r [Delta]T/[Delta]r) + [[Delta].sup.2]T/[Delta][z.sup.2]] + [Delta] [H.sub.p][R.sub.p] (6)

with

T(r, z, t) = [T.sub.0] at t = 0, r [greater than or equal to] 0, z [greater than or equal to] 0

[Delta]T/[Delta]r = 0 at r = 0, z [greater than or equal to] 0, t [greater than or equal to] 0

T = [T.sub.0] at r [approaches] [infinity], z [greater than or equal to] 0, t [greater than or equal to] 0

T = [T.sub.0] at z [approaches] [infinity], r [greater than or equal to] 0, t [greater than or equal to] 0

[Delta]T/[Delta]z = 0 at z = 0, r [greater than or equal to] 0, t [greater than or equal to] 0

For a finite solution region (r [less than or equal to] 5[W.sub.0], z [less than or equal to] 2D) the temperature equation initial and boundary conditions become:

T(r, z, t) = [T.sub.0] at t = 0, 0 [less than or equal to] r [less than or equal to] 5[W.sub.0], 0 [less than or equal to] z [less than or equal to] 2D

[Delta]T/[Delta]r = 0 at r = 0, 0 [less than or equal to] z [less than or equal to] 2D, t [greater than or equal to] 0

T = [T.sub.0] at r = 5[W.sub.0], 0 [less than or equal to] z [less than or equal to] 2D, t [greater than or equal to] 0

T = [T.sub.0] at z = 2D, 0 [less than or equal to] r [less than or equal to] 5[W.sub.0], t [greater than or equal to] 0

[Delta]T/[Delta]z = 0 at z = 0, 0 [less than or equal to] r [less than or equal to] 5[W.sub.0], t [greater than or equal to] 0

A number of assumptions and approximations were made during the development of this model in order to simplify the analysis, but it is believed that these in no way affect the significance and general usefulness of the calculated results. The reader is referred again to Part I of this two-paper series (1) for a more detailed discussion of these assumptions and approximations.

MATHEMATICAL MODEL

Although many of the equations used are the same as for the previously developed stationary laser light model, the dynamic model can accommodate the light spot moving across the surface of the resin vat in a series of small discrete steps. Laser motion in the actual stereolithography apparatus also occurs in such a discrete fashion, although to the human eye motion appears to be continuous. Motion of the light spot results in a loss of cylindrical symmetry, and a change to rectangular cartesian coordinates is required. All dependent variables thus now become functions of x, y, z, and t.

The moving laser light model was developed in two stages. Initially, two spatial dimensions were considered, viz., the x and y directions at the surface of the monomer vat (z = 0). After some reasonable preliminary results were obtained, the model was expanded to include a third spatial dimension (the z direction). For the sake of brevity, only the three-dimensional model is described here. It should be noted, however, that the two-dimensional model can be easily obtained from the three-dimensional model by setting z equal to a constant value of zero and by eliminating the third spatial dimensional from the equations.

The mathematical model was developed by considering a region of exposed material as shown in Fig. 1, with the laser light spot moving across the surface in the positive y-direction. Analysis is only required for half the indicated region since the y and z axes lie on a plane of symmetry.

Equation 1 for the incident light intensity is still valid provided the value of r (distance from center of light beam) is calculated as follows:

r = [[(x - [x.sub.0]).sup.2] + [(y - [y.sub.0]).sup.2].sup.0.5] (7)

where ([x.sub.0], [y.sub.0]) = x and y coordinates of the center of the laser beam, and motion is accomplished simply by changing [x.sub.0] and/or [y.sub.0].

Equations 2 through 5 remain essentially unaltered apart from the coordinate system change, and Eq. 6 becomes the three-dimensional heat conduction equation as follows:

[Rho][C.sub.p][Delta]T/[Delta]t = [Kappa][[[Delta].sup.2]T/[Delta][x.sup.2] + [[Delta].sup.2]T/[Delta][y.sup.2] + [[Delta].sup.2]T/[Delta][z.sup.2]] + [Delta] [H.sub.p][R.sub.p] (8)

with

T = [T.sub.0] at t = 0, all x, y, and z

[Delta]T/[Delta]x = 0 at x = 0, t [greater than or equal to] 0, all y and z

T = [T.sub.0] at x = X, t [greater than or equal to] 0, all y and z

T = [T.sub.0] at y = 0, t [greater than or equal to] 0, all x and z

T = [T.sub.0] at y = Y, t [greater than or equal to] 0, all x and z

[Delta]T/[Delta]z = 0 at z = 0, t [greater than or equal to] 0, all x and y

T = [T.sub.0] at z = Z, t [greater than or equal to] 0, all x and y

Inspection of Eq. 5 reveals that the polymerization reaction would stop entirely when the light source is extinguished (when I and [I.sub.a] become zero). It is, however, well known that polymerization continues after extinction of the light source in photopolymer systems such as those used in stereolithography. The models can be modified to accommodate this "dark reaction" by introduction of an intermediate species ("radicals"), which form during exposure and are then consumed during the dark period. Polymerization thus continues after exposure. The following set of equations would then replace Eq. 5 above:

[Delta]R/[Delta]t = [Phi][I.sub.a] - [[Kappa].sub.1][R.sup.2] (9)

[Delta]M/[Delta]t = [[Kappa].sub.2]MR (10)

with R(x, y, z, t) = 0 at t = 0 and all x, y, and z, and where R = "radical" concentration (mol [1.sup.-1]) and [k.sub.1] and [k.sub.2] are empirical rate parameters that characterize the kinetics of the light and dark reaction phases. It should be realized that this representation of the light and dark phases does not use a quasi-steady-state approximation for radical concentration during the light period. The light and dark period equations are obtained simply by setting [I.sub.a] equal to the calculated value during the light period and equal to zero during the dark period. The kinetic parameters would typically be determined from laboratory kinetic studies where conversion vs. time information can be obtained from an experiment where the UV source is extinguished after a short period of time (pulse experiment). Conversion of monomer to polymer in an experiment of this nature may be monitored by a high-speed technique such as real-time infrared spectroscopy (2).

NUMERICAL METHODS

The three-dimensional moving light source model's equation set is very similar to that of the stationary spot model. As a result the numerical methods used for approximate solution of the equations are almost identical. The incorporation of a third spatial dimension in the heat conduction equation, however, requires use of a three-dimensional Alternating Direction Implicit (ADI) method. This technique is described by Carnahan, Luther, and Wilkes (3).

The solution domain (0 [less than or equal to] x [less than or equal to] X, 0 [less than or equal to] y [less than or equal to] Y, 0 [less than or equal to] z [less than or equal to] Z) is discretized by setting up a three-dimensional mesh, and the process variables are computed at each node of the mesh using the following procedure:

1. Compute light intensity distribution at surface (z = 0) using Eq. 1.

2. Compute light intensity at all nodes using Eq. 2 (4th order Runge-Kutta).

3. Advance photoinitiator and monomer concentrations by one time increment using Eqs. 4 and 5 (4th order Runge-Kutta).

4. Advance temperature distribution by one time increment using Eq. 8 (Alternating Direction Implicit).

5. Move light spot (if necessary). Check for maximum integration time termination. Return to step 1.

Of some concern here was the size of the finite solution region used to approximate the "infinite" resin vat, and the fact that the fixed boundary conditions used along these boundaries could significantly influence the computed temperature and concentration profiles. Some experimentation was performed during which the size of the solution region was modified and computed profiles compared. The influence of the boundaries was found to be limited to the immediate vicinity of the boundaries, and in all cases an appropriate region could be chosen based on the length of the line of plastic being formed.

RESULTS AND DISCUSSIONS

Presented here are the results from a simulation of a photo-polymerization reaction using a 0.25-mm-diameter UV beam and a slice thickness of 0.5 mm. Material properties and reaction rates are based on those for a hexanedioldiacrylate (HDDA) monomer, and the light source is a 15 mW HeCd 325 nm laser. Model parameter values that were used in the simulation are summarized in Table 1. These are fairly typical values for a HDDA monomer system, and the polymerization rate information used takes both temperature and conversion effects into account.

Values for the kinetic parameters used in Eq. 5 were obtained from the work of Tryson and Shultz (4). The ratio [k.sub.p] to [Mathematical Expression Omitted] was fit to an Arrhenius expression of the following form:

[Mathematical Expression Omitted]

In this expression both [k.sub.0] and [E.sub.a] were allowed to vary as functions of conversion and fit to simple exponential expressions. By doing this both the temperature dependence of the reaction rate as well as its dependence on conversion could be accommodated. It should be realized that fitting the kinetic parameters to expressions of this type is purely empirical (no mechanistic significance), and is used simply to obtain numerical values for the polymerization reaction rate.

Figures 2a and b are contour plots showing the temperature distribution on the surface of the resin vat at exposure times of 300 ms and 800 ms, respectively. As mentioned previously, and as can be seen in Fig. 2b, the imposed conditions along the boundaries of the finite solution region influence the outer temperature contours only and have minimal effect on those in the direct vicinity of the exposed region.

[TABULAR DATA FOR TABLE 1 OMITTED]

Figure 3 illustrates how the temperature profile develops along the line of plastic, i.e., along the y axis. It can be seen that after a short time (typically the time taken to traverse 5-6 beam diameters) the temperature distribution no longer changes significantly and exhibits pseudo-steady-state behavior. It is interesting to note that the peak temperature actually precedes the center of the laser beam. This is due to radial conduction and the chemical reaction occurring in the leading edge of the Gaussian light intensity distribution. At higher reaction rates it is possible for a peak to occur in the temperature distribution at short times. This is shown in Fig. 4, where the reaction rate is increased by a factor of 5. The temperature peak results from rapid heat generation due to the polymerization reaction and the heat release rate exceeding the rate of heat conduction away from the exposed region.

The transient temperature behavior at a fixed position (the midpoint of the line) before and after exposure is shown in Fig. 5. The temperature rises rapidly prior to the passage of the beam and then decays more gradually back to ambient conditions after the beam has passed that position. The temperature at which the majority of the polymerization occurs may significantly affect the nature and properties of the polymer formed. The center of the beam passes over the midpoint of the line of plastic at 1.5 s, so peak temperatures once again actually occur prior to the passage of the beam.

An assumption made during initial model development was that no dark or residual polymerization occurred (1), i.e., that polymerization ceased as soon as the light was extinguished, This assumption was made in order to use kinetic data available in the literature. It is, however, well known that dark polymerization does occur, although the effect has not been well quantified. Model modifications as discussed above allow polymerization to continue after termination of exposure. A typical model predicted conversion vs. time curve for polymerization with dark effects is shown in Fig. 6. An experimental program to quantify the dark effects is necessary, however, in order to effectively use this model modification.

Model-predicted conversion profiles at different laser scan rates, together with the known gel point of the polymer mixture, can be used to compute stereolithography apparatus "working curves." These working curves represent the relationships between cure depth, cure width, and laser scan rate. Working curves are used to set actual machine operating parameters and are typically determined experimentally. Figure 7 is an example of a set of computer-predicted working curves. Computer predictions of this type could conceivably be used in next-generation stereolithography machines for on-line modification of operating parameters such as the laser scan rate and laser light intensity.

CONCLUSIONS

The development of mathematical models for laser photopolymerization as applied to stereolithography has gone through a number stages. Initially, simple one-spatial-dimension models were developed for the stationary reaction zone situation. Additional spatial dimensions were then added, and the models modified to allow motion of the irradiated zone. The final model allows computation of all relevant time-dependent variable values in a three-dimensional region around a moving irradiated region. This closely simulates the actual stereolithography process and as such reveals considerable detail concerning the transient and steady-state behavior in and around the exposed region.

1. A pseudo-steady-state temperature profile develops along the line of plastic. The peak temperature occurs slightly in advance of the center of the laser beam.

2. The temperature at any fixed point along the path of exposure rises rapidly prior to the passage of the beam, and then cools gradually by conduction after exposure.

3. Dark polymerization can be accommodated by inclusion of a species formed during UV light exposure and then subsequently consumed.

4. A knowledge of the polymer gel point, together with conversion profiles, allows computation of stereolithography working curves.

Results of this type have resulted in an enhanced understanding of a relatively new process, and have the potential to be utilized in next-generation stereolithography apparatus control systems.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support of the State of Ohio Edison Materials Technology Center (EMTEC), the United States Air Force, and an industrial consortium of twelve major companies.

NOMENCLATURE

[C.sub.p] = Heat capacity (J [g.sup.-1] [K.sub.-1]).

D = Slice thickness (cm).

I = Light intensity (E [cm.sup.-2] [s.sup.-1]).

[I.sub.a] = Absorbed light (E [cm.sup.-2] [s.sub.-1]).

[I.sub.0] = Peak light intensity (E [cm.sup.-2] [s.sup.-1]).

k = Thermal conductivity (W [cm.sup.-1] [K.sub.-1]).

[k.sub.1] = Rate constant 1-dark reaction (1 [mol.sup.-1] [s.sup.-1]).

[k.sub.2] = Rate constant 2-dark reaction (1 [mol.sup.-1] [s.sup.-1]).

[k.sub.p] = Propagation rate constant (1 [mol.sup.-1] [s.sup.-1]).

[k.sub.t] = Termination rate constant (1 [mol.sup.-1] [s.sup.-1]).

M = Monomer concentration (mol [1.sup.-1]).

[M.sub.0] = Initial monomer concentration (mol [1.sup.-1]).

r = Distance from center of beam (cm).

R = "Radical" concentration (mol [1.sup.-1]).

[R.sub.p] = Polymerization rate (mol [cm.sup.-3] [s.sup.-1]).

S = Photoinitiator concentration (mol [1.sup.-1]).

[S.sub.0] = Initial photoinitiator concentration (mol [1.sup.-1]).

t = Time (s).

T = Temperature ([degrees] C).

[W.sub.0] = Nominal radius of laser beam (cm).

x = x-coordinates (cm).

X = Maximum value of x (cm).

y = y-coordinate (cm).

Y = Maximum value of y (cm).

z = z-coordinate (cm).

Z = Maximum value of z (cm).

[Delta] [H.sub.p] = Heat of polymerization (J [mol.sup.-1]).

[Epsilon] = Molar absorptivity of photoinitiator (1 [mol.sup.-1]).

[Phi] = Quantum yield for initiation.

[Rho] = Density (g [cm.sup.-1]).

REFERENCES

1. L. Flach, and R. P. Chartoff, Polym. Eng. Sci., this issue.

2. C. Decker, and K. Moussa, Am. Chem. Soc., Div. Polym. Chem. Polym. Prepr., 29, 1, 516 (1988).

3. B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods, John Wiley & Sons, Inc., New York (1969).

4. G. R. Tryson, and A. R. Shultz, J. Polym. Sci.: Polym. Phys. Ed., 17, 2059 (1979).