Stereolitografideki fotopolimerizasyon sürecinin matematiksel modellenmesi ve simulasyonu

Mathematical modeling and simulation of photopolymerization process in stereolithography

Özet

Ultraviyole (UV) ışın etkisi ile bir sıvı reçinenin istenilen geometriye sahip katı cisimlere dönüşümünü sağlayan stereolitografi tekniğinin temel süreci olan fotopolimerizasyon sürecinin anlaşılması, stereolitografi tekniğinin kullanıldığı üretimler için kritik öneme sahiptir. Çalışma koşullarının fotopolimerizasyon sürecine olan etkilerinin belirlenmesi, stereolitografide istenilen boyutta ve kalitede ürünün elde edilmesini sağlamaktadır. Bu çalışmada, stereolitografi cihazında farklı koşullarda gerçekleştirilen fotopolimerizasyon sürecinin simulasyonu için deterministik yaklaşımı temel alan bir matematiksel model geliştirilmiştir. Geliştirilen matematiksel modeli oluşturan iki boyutlu kısmi diferansiyel denklemlerin çözümü COMSOL paket programı yardımıyla sonlu farklar yöntemi uygulanarak belirli sınır koşulları için çözülmüştür. Bu modelde, fotopolimerizasyon sırasında gerçekleşen kimyasal tepkimelere (başlama, yayılma, sonlanma ve yavaşlama) ait kinetik eşitliklerin yanısıra ısı ve kütle aktarım etkilerini temsil eden eşitlikler de göz önünde bulundurulmuştur. Matematiksel modelin, stereolitografide fotopolimerizasyon yoluyla katı cisim oluşma sürecini başarılı bir şekilde tanımlayabildiği görülmüştür. Simulasyonlar UV ışın kaynağının üç farklı tarama hızı (2.72x$10^ {-1}$ m/s, 2.72x$10^ {-2}$ m/s ve 1.18x$10^ {-2}$ m/s) için yapılmıştır. Simulasyon sonuçları, UV ışın kaynağının tarama hızındaki değişimin tepkime dönüşümüne olan etkisini ve buna bağlı olarak katı cismin boyutlarında meydana gelen değişimi de açıkça ortaya koymuştur. Yapılan simulasyonlar sonucunda UV ışınının reçine içindeki derinliğine bağlı olarak tepkime dönüşüm eğrileri elde edilmiş ve bu eğrilerin yardımıyla fotopolimerizasyon sonucu oluşacak katı cismin boyutları belirlenmiştir.

Abstract

Stereolithography (SL) is one of the most widely used and cost-effective method for creating threedimensional objects from thin layers of hardened liquid polymers. Generally, an intense ultraviolet (UV) light source is used to solidify these liquid polymers, which are also known as resins, from a series of consecutive two-dimensional (2-D) cross sections. Often data from computer-aided design software is used to control the precise movements of the UV light source as it builds the object. The resulting product may serve as a prototype for engineering designs before its mass production and for low-volume manufacturing applications. Photopolymerization, which is the underlying basic reaction mechanism of SL has a wide range of applications such as: creating decorative and protective coatings, fabricating biomedical prostheses, contact lenses, dental restorations, manufacturing electronic components paints or printing inks, composite materials, and making fiber optic coatings. In SL technique a liquid resin is converted to a solid object of desired geometry by photopolymerization process. Thus, the modeling and simulations of the photopolymerization process that takes place in SL is very important for products manufactured by this technique. The determination of the variation of photopolymerization reaction conversion and the gelation time under different conditions is important for manufacturing products with desired qualities. The absorption of the UV light by the photoinitiator molecules mixed into the resin creates highly reactive radicals, and these radicals interact with the functional groups of monomers that compose the resin. This, in turn, converts the monomers into radicals and starts a chain reaction, which causes a large percentage of the monomers in the resin to ultimately become entangled in a highly cross-linked polymeric network. The key advantages of using light-induced photopolymerization are that such processes tend to be less damaging to the environment; because they generally use smaller amounts of solvents or solvent-free formulations, very high reaction rates at room temperature, spatial control of the process, low energy input; thus being generally economic, and chemical versatility since a wide variety of polymers can be polymerized photochemically. Over the last few decades, a considerable literature has accumulated for the purpose of understanding the kinetics, photoinitiators, polymerization systems, and applications of the photopolymerization process. The most important parameters which govern the photopolymerization process are the temperature, the UV light equipment, the UV light penetration depth, the UV light properties (wavelength and intensity), the functionality and reactivity of the monomer, and initial loading concentration and reactivity of the photoinitiator. The kinetics studies mainly measured and simulated double bond conversion and determined the effect of the parameters just mentioned above on the overall double bond conversion. The effect of these parameters on the cure depth of the sample and photoinitiator loading concentration, in contrast, has not been nearly as well studied. In this study, the effect of the scanning speed of the UV light source on the photopolymerization process in SL was simulated. For this purpose, a mathematical model consisting of 2-D Partial Differential Equations (PDE) and based on deterministic approach was developed. This model was solved by using COMSOL package software by application of finite element method. Besides the kinetic equations representing the process reactions, the equations that include the heat and the mass transfer effects in the reaction volume were also considered in the model. The resin chosen for the simulation composed of acrylate monomer, which has four functional units, and the photoinitiator, which has high absorption coefficient at the frequency of the UV light used in the simulations. Simulations were performed for three different UV light scanning speeds of 0.272 m/s, 0.0272 m/s and 0.018 m/s. These speeds chosen so that the resulting polymerized solid objects would have different dimensions. The dimensions of these solid objects were calculated and compared by using the model described above. The simulation results showed that the effect of oxygen inhibition reactions are more dramatic at higher UV light scanning speeds. Thus, the dimension of the parts obtained at lower UV light scanning speed of 0.018 m/s were found to be higher than the parts obtained at UV light scanning speed of 0.0272 m/s. In addition, due to the oxygen inhibition effect any polymerized solid parts were not obtained at the highest UV light scanning speed of 0.272 m/s from the simulation of the model.