This simulation predicts how deep a layer of photopolymer resin cures when it is exposed to UV light in a digital light processing (DLP) 3D printer. Light enters the top of a 1 mm deep resin sample and is attenuated by a photoabsorber as it travels down; resin cures wherever the local exposure exceeds a critical value.

  1. Use the irradiance slider (below the light intensity profile) to set the light intensity I0 at the resin surface and the exposure time slider to set how long the light is on. Their product is the incident exposure E0, shown below these sliders.
  2. Use the photoabsorber slider (below the sample cross-section) to change the photoabsorber concentration [A]. More photoabsorber absorbs light faster, which decreases the penetration depth Dp and the cure depth.
  3. Use the O2 inhibition slider to include the effect of dissolved oxygen, which scavenges radicals and reduces the cure depth. At "none" the standard Jacobs working-curve equation is used (n = 1); moving right decreases n to as low as 0.68.
  4. The left figure shows a cross-section of the 1 mm sample: cured polymer (gold) forms at the top where light enters, liquid resin (gray) remains below, and the black line (marked by the arrow) is the cure boundary at the cure depth Cd. If Cd exceeds 1 mm the whole sample is cured and is marked as over-cured (hatched).
  5. The right figure shows the irradiance as a percentage of the surface irradiance versus depth z, with the cure depth marked by a dashed line, the penetration depth marked by a dot, and the percentage of the surface irradiance that reaches the bottom of the sample (1000 μm) labeled at the end of the curve.
  6. A message appears below the resin sliders if the resin does not cure or if the entire sample is over-cured.
  7. Open Details to see the equations with the current values substituted.
Light attenuation (Beer–Lambert law)

UV light at the resin surface (z = 0) has irradiance \(I_0\). The photoabsorber attenuates the light exponentially with depth:

$$I(z) = I_0\,\exp\!\left(-\varepsilon[A]\,z\right)$$

The penetration depth is the depth at which the intensity falls to \(1/e\) (about 37%) of \(I_0\):

$$D_p = \frac{1}{\varepsilon[A]}$$
Exposure

The incident exposure (energy dose per unit area at the surface) is

$$E_0 = I_0\,t$$

and the exposure at depth z is \(E(z) = E_0\exp(-z/D_p)\).

Cure depth (Jacobs working curve)

Resin cures where the local exposure exceeds the critical exposure \(E_c\). Setting \(E(C_d) = E_c\) gives the cure depth:

$$C_d = D_p \ln\!\left(\frac{E_0}{E_c}\right)$$

No curing takes place (\(C_d = 0\)) when \(E_0 \le E_c\).

Oxygen inhibition

Dissolved oxygen scavenges radicals, so the cure depth is lower than predicted by the Jacobs equation. This is modeled with an empirical exponent \(n\) on the irradiance:

$$C_d = \frac{D_p}{n}\ln\!\left(\frac{I_0^{\,n}\,t}{E_c}\right)$$

where \(0.68 \le n \le 1\). The inhibition slider value \(s\) (0 to 1) sets \(n = 1 - 0.32\,s\); \(n = 1\) recovers the Jacobs equation.

Sample quantities
$$\frac{I(L)}{I_0} = \exp\!\left(-\frac{L}{D_p}\right) \qquad \text{fraction cured} = \min\!\left(\frac{C_d}{L},\,1\right)$$
Definitions
\(I_0\)irradiance at the resin surface (mW/cm2)
\(I(z)\)irradiance at depth z (mW/cm2)
\(t\)exposure time (s)
\(E_0\)incident exposure (mJ/cm2)
\(E_c\)critical exposure for curing = 10 mJ/cm2 (fixed)
\(\varepsilon\)absorptivity of the photoabsorber (natural-log basis) = 1000 M−1 cm−1 (fixed)
\([A]\)photoabsorber concentration (M)
\(z\)depth below the resin surface (μm)
\(D_p\)penetration depth (μm)
\(C_d\)cure depth (μm)
\(n\)oxygen-inhibition exponent (dimensionless)
\(L\)sample depth = 1 mm = 1000 μm

The values of \(\varepsilon\) and \(E_c\) are typical of common acrylate photoinitiator systems.

Current values

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by Alexander Osterbaan under the direction of Professor Christopher N. Bowman. Address any questions or comments to LearnChemE@gmail.com.

Sample cross-section
Light intensity profile