A 3D printer is basically a very specialized chemical reactor. It has inputs of light and energy, and it outputs waste and valuable products. The big difference between a 3D printer and a more typical reactor is that in most chemical reactors the information input is a combination of concentration, mass, momentum, and temperature gradients, which along with the reactor geometry, ultimately dictate the final product. Unlike most chemical reactors a 3D printer is capable of precisely controlling when and where energy is applied in the system, making it more controlled in some sense.
The implication of this approach is that 3D printers can readily be understood with simple chemical engineering models. This Instructable explains the most pertinent of these models to printer settings development. Ultimately, these models are much simpler than reality, and the rules devised from these should be treated as heuristics, i.e. they will get you 80% of the way there, but will give erroneous results sometimes. This is not a thesis, this is just a collection of simple rules of thumb for starting to understand DLP/SLA printers.
Step 1: Model 1: a Basic Model for Resin Curing
The most important model for understanding a 3D printer is the "working curve" (or work function) that describes the thickness of the polymer that forms as a result of light absorption, as seen in the above figure. This model come from the combination of two other simple models. First, as light passes thru an absorbing media its intensity decays in an exponential fashion described by the Beer-Lambert law. Second, when a photopolymer resin absorbs enough light (i.e. energy) it undergoes a very rapid transition from a free flowing liquid to a solid (albeit a very fragile gel).
The consequence of these two models is that as a photopolymer is illuminated nothing happens until the critical energy is reached and polymer first forms at the the window surface. Subsequently, the layer then slowly glows in thickness away from the light source.
Mathematically this result is obtained by re-writing the Beer-Lambert law in terms of energy (rather than light intensity), and then solving for the thickness:
layer thickness = dp * ln ( Dose / Ecrit),
where the penetration depth (dp) describes the penetration of light into the resin (dp = 1 / sum(extinction coefficient * concentration) for all of the resin components), and critical energy (Ecrit) describes how much energy the resin takes to cure. Both of these constants are very important, as measuring and understanding them is key to getting any resin to work, and we'll get back to later in a later instructable,
A less hand waving derivation of the above equation can be found in Jacob's Rapid prototyping and Manufacturing: Fundamentals of Stereolithography or in Gong et.al.
Step 2: One Slight Hitch
The working curve model is a pretty good description of the cure, except for one small detail, oxygen. The most common chemistry used in SLA/DLP printers is the radical polymerization of (meth)acrylate monomers. For (meth)acrylates to polymerize the oxygen concentration must be below ~10^-6 M, whereas the oxygen concentration in most monomers is about 10^-3 M ( the oxygen concentration in air is ~10^-2 M, and water ~10^-4 M ). For a polymerizaton to start the resin must first consume all of the oxygen dissolve in it, and then it must continue to consumer any oxygen that is diffusing into it.
DLP printers using PDMS and amorphous teflon windows rely on oxygen inhibition for their separation mechanisms to work. Right at the window and resin interface is a thin layer of unreacted monomer that is rich in oxygen because the window is acting as an oxygen source. For PDMS printers this can be any where from 1- 50 microns thick. Unfortunately no simple model to describe the thickness of this layer, but it can be a calculated using very sophisticated ones (coming soon). The top figure that shows the acrylate double bond concentration as a function of time and depth in conditions relevant to a 3D printer. Away from the window the double bonds are rapidly depleted, and if the x-axis is made logarithmic the contour lines become linear as the working curve model would suggest. However, near the window a thin layer of unreacted monomer exist and is only slowly depleted.
The take away is that the growth of a layer is slightly more complex than one might imagine at first. As seen in the right graphic after some amount of illumination a thin layer 1st forms between the previous printed layer and the window. As time progresses the layer rapidly grows away from the window towards the previous layer, and more slowly towards the window. Eventually the new layer links up with the old one and adheres to it. It then continues to slowly grow towards the window. In time the layer will grow all the way to the window surface, and if this lubricating layer is eliminated the printer may "jam."
Step 3: Model 2: Stefan Adhesion,why Just Pulling the Print Up Is Difficult
The working curve model explained how energy (light) is applied in a DLP/SLA printer. However, mass (resin and a print) must also be manipulated in a 3D printer. In an SLA printer the slowest step tends be the laser writing, as it takes awhile to get all the energy into the correct locations in the layer being written. However, because DLP based 3D printers can apply a lot energy rapidly over a large area these printers tend to be limited by how fast resin can be manipulated rather than light, and it is important to understand how mass flow limits printer speed.
It is instructive to 1st consider why most printers do no operate like Carbon's M1, that is just pulling the print up out of the vat. Nice and simple, right?
If you have two rigid parallel plates separated by a thin gap (relative to the diameter of the plates) and a lubricating liquid in between the plates the separation force is described by the Stefan equation shown in the graphic above. This force is dictated by the viscosity of the liquid, the size of the gap, the separation speed, and most critically the radius of the smaller plate. What is critical is that the separation force depends on the 4th power of the radius. This means that doubling the radius of the plate increase the separation force by a factor of 16! Sure you can make a more rigid printer, but at some point the force will tear your print apart, so one has to slow down to decrease the force. As the force is proportional to the separation speed. One, has to slow down alot.
Note: a detailed derivation of the Stefan adhesion equation is provided in Prieve's Advanced Fluid mechanics with Vector Field Theory.
Step 4: Implications of Fluid Flow: DLP/SLA Separation Mechanisms
Because the Stefan adhesive force can quickly get out of hand as objects get bigger most printers find ways to work around it. Printers like Ember and the B9 Creator using a sliding mechanism. Like a sweating glass of water on a hot day, it much easier to slide a print around on a surface, than to pull it straight off the window. This is because the separation force is now controlled by a Coutte type flow and the
Force = viscosity * Area * speed / gap
Thus instead of pulling striaght up the printer can then slide of the PDMS to an area which a much bigger gap and then pull up with a much lower Stefan adhesive force.
Another approach is to replace the rigid window with a more flexible one. This results in the concentration of the separation force on a point or line that depends on the angle of the two surfaces. The Form1, Asiga, and many EnvisionTec printers take advantage of this phenomena. A short description of the force analysis is found here.
Step 5: Next Steps
This basic models provide a quick description of what is going on in a 3D printer. The next step to creating settings is applying these models. The following Instructables will guide you thru the process.
Measuring Light Intensity - this is the 1st step to know what your printer is actually doing
Creating a working curve - Understanding how a resin cures and what does is needed
Creating Settings files - Once you know the dose a resins needs you need to create a settings file
1) How to tune Embers Settings for New Resins This Instructable walks you thru most of Embers settings and describes settings development. Starting with the working curve considerably speeds up settings development.
1) Modeling layer growth using a Differential Equation based model with Temperature and Diffusion dependent rate constants (see attached document)
4) Estimating the inhibition layer thickness using Ember (coming someday).
6) Using Pattern Mode - How to control the printer on a pixel elvel