New Simulation Tool Optimizes SL Systems
A thermal-kinetic model using the finite element method to study, simulate and optimize stereolithography.
Stereolithography is one of the most popular rapid prototyping processes. It involves the curing or solidification of a liquid photosensitive polymer by a laser beam scanned across its surface.1,2 The laser supplies energy that induces a chemical reaction, bonding large numbers of small molecules and forming a highly cross-linked polymer. The first layer of the object to be built is formed on an elevator platform, which is then lowered to allow new liquid material to flow onto the working surface.1,2 This process is repeated, with each new layer adhering to the previous one. Finally, a post-cure operation is needed to complete the cure process.1,2
The use of stereolithography is still in its infancy. Much of the technology and operating procedures are based on empirical correlations and work experience and very little is known about the physical and chemical changes occurring in the material due to the irradiation. A good understanding of the curing process is an important factor to improve both the precision and quality of the models and to develop well-adapted polymeric systems.
A new simulation tool used to study and optimize stereolithographic systems has been developed, which enables the control of several kinds of information including:3-7
- The progress of the curing reaction for estimating the time required to produce a model.
- The spatial solidification profile for controlling the precision of the process.
- The temperature variations, to know if the thermal effects are compatible with the process.
Simulation Model
To simulate the solidification process of stereolithographic resins, a mathematical model was developed3-7 based on the understanding of the fundamental physical and chemical phenomena that govern the behavior of a thermosetting material in stereolithographic applications.This simulation model enables a better understanding of the mechanism of cure and the influence of different parameters such as the operating conditions, the material properties and the temperature distribution in the sample. It also enables the modeling of the heat flow while curing takes place, assuming that, by irradiating and/or heating the material, the affected volume absorbs energy, causing a phase change in the material (liquid to solid transformation).
According to Model 1, the temperature field in the exposed region is expressed by the two-dimensional heat conduction equations in cylindrical coordinates3-7 where r is the density, C is the specific heat, r is the radial direction, z is the axial coordinate, T is the temperature, t is the time, kr and kz are thermal conductivities, H is the total heat released during the chemical reaction and da/dt is the rate of gel formation given by an appropriated phenomenological equation.3-5
The solution of this equation requires both the knowledge of the initial temperature and the initial value of the amount of solid material formed, as well as appropriate boundary conditions. In addition, light intensity is assumed to have a Gaussian intensity distribution and the decrease in light intensity with depth is supposed to obey the Beer-Lambert law.
The mathematical model is based on computer software named STLG-FEM (Stereo-Thermal and LithoGraphic processes through the Finite Element Method), which uses the finite element method (FEM) and linear isoparametric rectangular elements (see Figure 1).
Numerical Examples
This simulation software was used to predict the necessary time to achieve a complete cure for a layer of unsaturated polyester resin containing 0.5 wt% of photo-initiator, which was irradiated at room temperature using an UV lamp. A resin layer of thickness 0.03 cm and length 3 cm was considered. In addition, both the center of the layer and the center of the light beam with a Gaussian distribution profile (see Figure 2) were assumed to be coincident.Consequently, due to the geometry and light distribution symmetry of the problem, only one half of the resin layer was considered for simulation purposes. A single layer of 30 rectangular elements was used to model the problem, where each element has a horizontal dimension of 0.05 cm and a vertical one of 0.03 cm. The variation of the UV light along the resin thickness was assumed to follow the Beer-Lambert law. This variation is indicated in Figure 3. Finally, the transient analysis was performed considering 400 increments of 0.05 min.
The variation of the fractional conversion obtained by simulation as a function of position and time (see Figure 4 a and b) shows that conversion typically decreases by increasing the distance from the center of the beam due to the decrease of light intensity, but increases with the irradiation time. This graph was obtained by considering the nodes at the surface of the resin.
Figure 5 a and b are contour plots of fractional conversion for an exposure time of, respectively, 1.6 minutes (the time at which the maximum temperature is obtained) and 20 minutes. These contours clearly indicate a conical variation of conversion, which could be an important factor in terms of the quality of the obtained models.
The computational tools proposed here also can be used to predict the optimal fabrication conditions. Figure 6 shows the working curves of the resin layer being studied under different light intensities. Fractional conversions indicated in this figure are the average conversion values of the entire layer. As observed, light intensity increases the amount of fractional conversion and reduces the necessary exposure time.
Conclusions
A new and more accurate computational tool for the study of stereolithography and related processes describes the major events occurring during the chemical process of photo-solidification, enabling control and optimization of the most important para-meters of stereolithographic applications.
References
¹M. Burns, Automated Fabrication - Improving Productivity in Manufacturing. Prentice Hall, New Jersey, 1993.²P.F. Jacobs, Rapid Prototyping and Manufacturing: Fundamentals of Stereolithography. SME, Dearborn, (1992).
3P.J. Bartolo; G. Mitchell; Proceedings of International Conference on Mechanics of Structures, Materials and Systems, Wollongong, M.N.S. Hadi; L.C. Schmidt; A. Basue Eds; ISBN 0 864186568, 2001, 145.
4P.J. Bartolo; G. Mitchell; Proceedings of the 7th European Conference on Advanced Materials and Processes, Associazione Italiana di Metallurgia; ISBN 88-85298-39-7, 2001.
5P.J. Bartolo; G. Mitchell Europhys. Conf. Abst. 2000, 241, 141.
6P.J. Bartolo; G. Mitchell; Proceedings of the 8th International Conference on Rapid Prototyping, Tokyo, T. Nakagawa; Y. Marutani; M. Imamura; M. Agarwala; A. Lightman; D. Klosterman; R. Chartoff Eds., 2000, 80.
7P.J. Bartolo; G. Mitchell; Proceedings of the 9th European Conference on Rapid Prototyping and Manufacturing, Athens, R.I. Campbell Ed. 2000, 11.
