Computed Tomography-Based Volumetric Additive Manufacturing: Development of a Model Based on Resin Properties and Part Size Interrelationship—Part I

Abstract

This study presents an analytical description of the computed tomography-based volumetric additive manufacturing (VAM) process, with an emphasis on the impact of resin properties on product dimensions. The main issue addressed in this study is the assessment of the dimensional limitation of the objects produced using the VAM process, which is usually reported to be of the order of one centimeter. An analytical model is introduced to predict the product size based on the resin property (penetration depth—D_p), vial size (radius), and the duration of part formation, and the results indicate significant correlations among these parameters. A method of D_p measurement and analysis that is appropriate for the VAM process is also introduced. Mathematical justification is provided along with experimental verification for the effects of the main governing factor, D_p, on the maximum possible product size. Multiple criteria are also introduced for selecting the appropriate size of the resin container (vial) based on the desired object size and the value of D_p. It was found that the D_p is a crucial factor in analysis and experimentation in the VAM process, and its value is fundamentally different from the one obtained in the conventional polymerization AM methods. The product dimension based on the resin property, vial size, and time for the formation of the part is introduced by the analytical model. This model provides valuable insights into the complex interplay of factors influencing VAM outcomes and can facilitate informed decision-making in material selection and process design.

1. Introduction

Volumetric additive manufacturing (VAM) is a novel layer-less fabrication technique enabling the rapid conversion of photoreactive polymer resins into intricate 3D structures in a matter of seconds [1,2]. This technique also facilitates the production of overhanging and spanning geometries without requiring support materials and achieves the rapid printing of intricate designs [3,4].

A number of VAM processes have been investigated, but [5,6,7] the “Tomographic VAM” process has been generally adopted due to its higher resolution, simplicity of setup, and less sophisticated material chemistry. In this process, a 3D model undergoes conversion into a sequence of 2D images, termed “sinograms” through the Radon transform, a mathematical algorithm. These sinograms are generated by capturing multiple images of the 3D model from different angles, thereby offering varied perspectives of the model in a 2D format. Subsequently, mathematical operations such as the Fourier Transform (FT) are performed, and before going to the inverse transform, the function is multiplied by the frequency parameter (called “high-pass filtered”) to improve the targeted image sharpness. This computation yields a series of light patterns, which, when delivered by the projectors onto the photosensitive resin, result in the fabrication of the final part [6]. The stereolithography process depends upon the interaction between the incident optical illumination and resin photochemistry. The model is built upon the core assumption that light absorption within the photopolymer resin follows the Lambert exponential attenuation law. This law can be expressed in terms of the penetration depth, D_p, a parameter specific to each resin, as in Equation (1) [8,9]:

$$I_{(x,y,z)}=I_{\left(x,y,0\right)}{e}^{{\displaystyle \frac{-z}{D_p}}}.$$

(1)

This equation expresses the irradiance of light intensity at a specific point within a 3D resin volume I_(x,y,z) compared to its intensity at the surface of the resin in the lateral plane (x, y, 0) where the light beam enters. It is assumed that a critical energy dose (E_c) is required at any point within the resin volume for it to be cured. The intensity of the light is always at a maximum at the surface of the resin, and the relationship between the depth of cure (C_d) (cured layer thickness) [10], the exposure dose (E_s), critical energy (E_c) in J/cm² required for curing, and the D_p can be written as shown in Equation (2) [9]:

$$C_d=D_p.\mathrm{l}\mathrm{n}\left({\displaystyle \frac{E_s}{E_c}}\right).$$

(2)

It should be emphasized that E_s is the amount of energy that is required by the light source (before entering the resin). It is then attenuated through the resin, depending on the resin property, D_p, down to E_c, at the depth of C_d. Beyond this point, the light energy is lower than E_c; thus, no further curing is expected. The energy E_s is, then, the product of intensity $I_{\left(x,y,0\right)}$ by the time of exposure, say, t.

Studies on the VAM process are mostly focused on refining processes to enhance printing efficiency, accuracy, and material versatility. Researchers have explored various aspects, including optimizing light projection methods, understanding photopolymerization kinetics, and developing novel materials tailored for this method [1,4,11,12,13]. Rackson et al. introduced a study on optimizing digital models in tomographic VAM by adjusting projection images and intensities to mitigate volume errors [14]. Bhattacharya et al. [3] propose an optimization method for the tomographic VAM by enhancing the computation accuracy of energy dosage using gradient descent optimization. A penalty minimization loss function was introduced to identify insufficiently polymerized regions [3]. Chen et al. [15] discovered the Maximum Likelihood Expectation Maximization (ML-EM) method. This method starts with an initial guess of light patterns and iteratively refines them based on the difference between expected and actual energy dosages, leading to photopolymerization in the resin. The process continues until the difference is minimized, resulting in optimized light patterns being projected into the resin. This method outperforms previous approaches, especially in large-scale, strong absorption, and high-complexity scenarios [15]. Salajeghe et al. [16] address sedimentation issues in VAM, which occur when solidified sections sink while the rest continue to solidify. They pinpoint factors contributing to this, including the geometry-to-container ratio, curing time, container rotation speed, part orientation, and material properties. Sedimentation can affect accuracy and hinder low-viscosity resin use. They propose a computational approach and develop a numerical model to simulate sedimentation during the VAM process [16]. Orth et al. [17] devised a method to optimize energy transmission in VAM, enabling real-time reconstruction and the display of the part during printing. Their work emphasizes the importance of knowing and accounting for the optimal exposure time, crucial for producing consistent and precise printing outcomes [17].

As a novel additive manufacturing technique, the VAM process presents various ambiguities still to be uncovered and several parameters awaiting optimization. One limitation, which is prevalent in the literature, is the size of the fabricated object, which is usually around 10 mm (or one-centimeter scale) when the conventional tomographic VAM process is used [18]. The main cause is the high attenuation of the light through the resin in the container. It is mainly governed by the resin D_p, and the distance the light travels within the resin, in the vial. One could think that for a larger object size, there is a need for a larger container size; however, in a larger container, light must travel through a greater distance before reaching the center of the container or the designated printing area. A longer traveling distance results in a higher attenuation of the light, potentially leading to insufficient energy carried by the light, which can lead to underexposure in the inner regions of the print. Hence, a comprehensive analysis is required to understand and determine this limitation by performing an analytical assessment of the relationship among the processing parameters (e.g., D_p), container size, and the printed object size. Therefore, an analytical model is introduced in this study to predict the largest possible object size for a given resin, a given vial size (radius), and also the time of part formation. A refined mathematical expression has been developed and presented in this manuscript to define this relationship. The analytical results have been evaluated using a set of experiments and a customized experimental setup has been designed for this study. At this stage of study, for the sake of simplification in mathematical operations, a simple part design, namely a cylindrical shape, was considered. Additionally, the projected rectangular images were uniform in light distribution. This method can be extended to further complex arrangements.

2. Analytical Approach

2.1. Obtaining D_p for the VAM Process

In this study of the VAM process, the size of objects is in the centimeter scale, which means that the light can travel up to this distance to cure the resin in the vial center region, without curing the outer region, and more specifically, the resin along the inner surface of the vial. The distance of penetration can be represented by the “D_p” value, which is a resin property. The higher the value of D_p, the higher the depth of curing. However, this property is normally reported for the polymerization methods of SLA and DLP.

It is very important to emphasize that the method for obtaining D_p for the VAM process is fundamentally different from the method used in the SLA/DLP process. In a traditional polymerization method, the resin is cured in a vat by exposing it to the light from the top or bottom. The critical point to note is that when the time of exposure is increased, the additional cured layer is formed by the light that passes through the already cured layers. The attenuation of light can be very different in the cured resin than in the uncured resin. Therefore, D_p, which is inversely related to attenuation, will be largely different for the cured resin from that of the liquid resin. The widely practiced and accepted method of D_p of the resin, which simulates the standard SLA/DLP processes, is to expose the working resin to different energy doses, typically controlled by varying the exposure time in a vat. As shown in Figure 1, various samples are then produced, and their thicknesses are measured and plotted against the corresponding exposure energy. It is to be noted that the exposure energy is calculated by the multiplication of time to the light intensity of the projector; the light intensity should also be measured utilizing the proper device, prior to the calculation. The slope of the resulting working curve represents D_p, where the energy axis is plotted in a logarithmic scale [8,10].

For most working resins that are used in DLP or SLA methods, which may also be used in the VAM process [19,20], the value of D_p is less than 1 mm, where the measurement method simulates the DLP or SLA process [21,22,23]. This means that the amount of light intensity will be reduced to 37% at 1 mm of penetration from the surface, and to almost zero at a depth of 5 mm from the surface. If the same value is applied for the VAM process, it will imply that it is impossible to have any resin curing in the central zone when the vial diameter is larger than 10 mm (or a radius of 5 mm), while the reality shows otherwise. The source of the difference can be sought in the fact that the conventional method for D_p measurement (for SLA/DLP methods) does not simulate the VAM method. The proposition is that the measurement method for D_p should be different to simulate the corresponding process.

The proposed explanation is that the derivation of D_p for the VAM process is fundamentally different from DLP or SLA processes, since the light penetrates through the uncured “liquid-state” resin to reach the regions near the central zone, as the vial rotates. And because of the significant difference in the attenuation within the solid state (as it proceeded in SLA/DLP) and that of liquid (as in VAM), the value of D_p can be vastly different. Figure 2 exhibits the relative displacement of the vial vs. the light projector, showing that the light beam penetrates into the resin to reach the center. In this process, the vial containing the liquid resin rotates while a stationary light projector directs energy towards it. The light beam penetrates into the resin, reaching the center of the vial as it rotates. As the vial rotates, the light exposure varies across different sections of the resin. The center region of the resin consistently receives energy, resulting in the photopolymerization of the resin in this zone and its transition into a solid (cured) state, represented by the orange region.

The outer zones of the resin, which are exposed to the light for shorter durations due to the vial’s rotation, do not receive sufficient energy to initiate the curing process and thus remain in a liquid state (depicted in yellow). The intensity of light exposure is highest at the center and decreases progressively as the resin moves away from the light source due to the rotation of the vial. This results in a gradient of curing from the center towards the outer regions. While the targeted center zone receives energy in all relative positions, the other zone remains in a liquid state due to the lower reception of energy.

Therefore, the value of D_p obtained from SLA/DLP methods will not simulate, and thus not apply to, the VAM process, and this value must be uniquely measured utilizing a similar method as that of the VAM process. We have tested and validated this hypothesis later in this manuscript.

A VAM setup, based on resin contained in a rotating vial, is used for measuring D_p. An object of a small size, say a circular image of 1 mm in diameter, is projected at the central region of the rotating vial, and the curing time is measured. It is hypothesized that the size of the projected object does not affect the time for the initiation of curing, although a small object is selected for this evaluation. In this setup, the light will pass through the outer regions of the zone of interest without initiating curing, reach the center of the vial, and start curing it. Note that due to the rotation of the vial, the exposure time is greatest at the center/axis of rotation and starts to decrease as the distance away from the axis increases. Therefore, the regions closer to the center receive sufficient accumulated energy for curing and solidifying faster compared to regions farther away from the center. By varying the vial size and measuring the initiation time of curing at the center, the working curve for the VAM process can be obtained.

Determining the curing time, including process initiation and termination times of the resin at any location, is very useful. It is worth pointing out that in order to measure D_p with respect to time, information about the light intensity, I_s, of the projector being used is not required, noting that I_s is required to measure critical exposure energy (E_c). E_c is a fundamental property of the working material and is independent of the source intensity. However, it will be shown that another parameter, namely, critical exposure time, t_c, can be introduced, and using it, the curing time at any location and stage of the process can be derived. The depth of the cure in a resin can be expressed based on the Beer–Lambert law expressed in Equation (2).

The parameter D_p is a critical characteristic of the working material that governs the curing thickness (in SLA/DLP methods), and its action is opposite to the attenuation of the light through the medium. For the VAM process, this parameter can be interpreted as an indicator of the possibility of creating an object inside the vial before curing occurs on the vial’s inner surface. In other words, it can be surmised that a lower value of D_p causes surface curing before the formation of the object in the central working space of the vial.

The required energy, E_s, given by the light source, is the product of its projected light intensity, I_s (or the light intensity before entering the resin), and the exposure time t. On the other hand, E_c is the critical energy to initiate curing at the desired point at the depth of C_d, Therefore, E_s will be the required energy from the light source to cure a point at the depth of $Cd$. We propose that E_c be replaced by another term, t_c $,$ which is the critical exposure time. This means that time, $tc$, is required for the source light intensity, I_s, to yield the critical energy, E_c (or ${\mathrm{E}}_{\mathrm{c}}={\mathrm{I}}_{\mathrm{s}}.{\mathrm{t}}_{\mathrm{c}}$). In other words, if the source light is directly projected with a set intensity, I_s, onto a thin layer of the working resin, it takes time, t_c, to cure that tiny layer. Or, in the VAM process, if the vial is not rotated while being exposed to the projected light, it takes time t_c to cure the tiny layer of resin at the interface of the vial (inner) surface and the resin where the light enters the vial. This parameter is not an intrinsic property of the working material, as it depends on I_s or the light source. Therefore, it will differ from one projector to another, which may have a different light intensity or power. It is true that if we know E_c, we can determine the time required for curing the resin. However, knowledge of E_c is not needed if t_c is known, which can then be used to find t at any location directly. We can rearrange Equation (2), based on the above-discussed parameters, to find a revised equation relating C_d with time:

$$E_c=I_s. t_c (\mathrm{or} t_c={\displaystyle \frac{E_c}{I_s}}),$$

(3)

$$C_d=D_p\mathrm{l}\mathrm{n}\left({\displaystyle \frac{I_s. t}{I_s. t_c}}\right)=D_p\mathrm{l}\mathrm{n}\left({\displaystyle \frac{t}{ t_c}}\right),$$

(4)

or

$$C_d=D_p\mathrm{l}\mathrm{n}\left(t\right)-D_p\mathrm{l}\mathrm{n}\left(t_c\right).$$

(5)

For the VAM process, where the vial is rotating, the parameter C_d is the distance from the inner surface of the vial to the center, namely, the inner radius of the vial, i.e., when the curing starts at the center, this time is recorded. It is to be noted that the projected image should be as small as possible, to ensure that the curing begins at the center and to mitigate the chances of initiating curing at the undesirable regions.

This formulation indicates that for the VAM process, if the relationship for C_d (or R, the vial radius) is plotted against the logarithmic time t, its slope will represent D_p, and this is the same as plotting C_d against the logarithmic E. The critical exposure time, t_c, can also be computed from the intercept term.

2.2. VAM Analysis—Uniform Light Intensity

In the introductory analysis, a circular light source of uniform intensity with radius r₀ is projected into a cylindrical resin-filled container with a radius R. For simplicity, the light intensity is assumed to be uniform, and the projected object is a cylindrical shape with radius r₀ and an arbitrary height. The objective is to calculate the initiation time at any point Q and understand the relationship between the resin properties, processing parameters, and the setup geometries that lead to the creation of a robust object. One important outcome will be to compare the cure time at the vial’s central zone with that of the vial’s inner surface. It should be mentioned that one of the major dilemmas in the VAM process is when the inner region adjacent to the vial surface is cured before the formation of the object in the central region.

Assuming a vial with radius R and a light source of radius r₀ with uniform intensity, I_s, the amount of absorbed energy at an arbitrary point, Q (Figure 3), can be calculated as follows.

In the figure, “s” represents the distance from the vial’s inner surface of an arbitrary point, Q, located at a radius of r, from the Beer–Lambert law, and the light intensity at that location can be calculated as follows:

$$I_Q=I_s{e}^{-{\displaystyle \frac{s}{D_p}}} \mathrm{with} I_s\left(r\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.$$

(6)

The absorption covers an angle of 2α, which can be derived from the geometry as follows:

$$\sin \left(\alpha \right)={\displaystyle \frac{r_0}{r}} \mathrm{or} \alpha ={\sin }^{-1}({\displaystyle \frac{r_0}{r}}).$$

(7)

The distance, s, shown in Figure 3, represents the event when point Q absorbs energy at the zone nearer to the light source. Due to the vial rotation, point Q will not receive light after angle, α, counterclockwise, until it reaches a point with the angle of π−α, and continues to receive the light until it reaches the angle of π + α, representing the farther zone; this zone receives the more attenuated light with respect to the nearer zone. The distance s for the near and far zones can be calculated as follows:

$$s^N=R\sqrt{1-{\left({\displaystyle \frac{r}{R}}\right)}^2{\sin }^2\theta }-r\cos \theta$$

(8)

$$s^F=R\sqrt{1-{\left({\displaystyle \frac{r}{R}}\right)}^2{\sin }^2\theta }+r\cos \theta .$$

(9)

The letters N and F denote “near” and “far” zones, respectively. The intensity, I_s, has the unit of W/m². To compute the absorbed energy at each location per unit of surface, the intensity value has to be multiplied by the time. Since the intensity at any location varies as the vial rotates, the total energy absorbed can be determined by the integration of all incremental energies impinging at this location during rotation, noting that this consists of two sections, the “near” section where it is closer to the light source, E^N, and the “far” or opposite section, which is on the opposite side of the source, E^F (Figure 3):

$$dt={\displaystyle \frac{d\theta }{\omega }}$$

(10)

where t is the exposure time and θ is the angle traversed during this time.

$$E^N={\int }_{-\alpha }^{+\alpha }{I}_{Q}dt={\int }_{-\alpha }^{+\alpha }{I}_s{e}^{-{\displaystyle \frac{s^N}{D_p}}}{\displaystyle \frac{d\theta }{\omega }} ={\displaystyle \frac{2I_s}{\omega }}{\int }_0^{+\alpha }{e}^{-{\displaystyle \frac{s^N}{D_p}}}d\theta .$$

(11)

and similarly,

$$E^F={\displaystyle \frac{2I_s}{\omega }}{\int }_0^{+\alpha }{e}^{-{\displaystyle \frac{s^F}{D_p}}}d\theta .$$

(12)

Then,

$$E^t=E^N+E^F={\displaystyle \frac{2I_s}{\omega }}\left[{\int }_0^{+\alpha }\left\{e^{-{\displaystyle \frac{R\sqrt{1-{\left(\frac{r}{R}\right)}^2{\sin }^2\theta }-r\cos \theta }{D_p}}}+e^{-{\displaystyle \frac{R\sqrt{1-{\left(\frac{r}{R}\right)}^2{\sin }^2\theta }+r\cos \theta }{D_p}}}\right\}d\theta \right].$$

(13)

It must be noted that the above energy is based on “one single rotation”. The integration can be easily solved numerically for any arbitrary point.

Some special cases are discussed below:

Center Exposure: Very close to the center, the radius, $r$, will tend to 0, and the angle of exposure will be α = π/2; thus, the equation will be simplified as below:

$$E^t\left(r=0\right)={\displaystyle \frac{2I_s}{\omega }}\left[{\int }_0^{+{\displaystyle \frac{\pi }{2}}}\left\{e^{-{\displaystyle \frac{R}{D_p}}}+e^{-{\displaystyle \frac{R}{D_p}}}\right\}d\theta \right]={\displaystyle \frac{2\pi I_s}{\omega }}{e}^{-{\displaystyle \frac{R}{D_p}}},\mathrm{for\; a\; single\; rotation}.$$

(14)

The term ${\displaystyle \frac{2\pi }{\omega }}$ represents the time of vial rotation, T_v, and $E^t$ is energy exposure for a single rotation. It is assumed that the point completes at least one rotation; for the sake of more accuracy, the exposure time is the integer of this time, T_v.

$$E^t\left(r=0\right)=T_v{I_{s}e}^{-{\displaystyle \frac{R}{D_p}}},\mathrm{for\; a\; single\; rotation}.$$

(15)

This value should be multiplied by the number of rotations when revolving for time t, that is,

$$N={\displaystyle \frac{t}{T_v}}.$$

(16)

Therefore, the total absorbed energy will be as follows:

$$E_{tot}\left(r=0\right)={\displaystyle \frac{t}{T_v}}.T_v{{.I}_s.e}^{-{\displaystyle \frac{R}{D_p}}}=t.{I_s.e}^{-{\displaystyle \frac{R}{D_p}}}.$$

(17)

The amount of energy should reach the value of E_c to initiate curing. The ratio of the critical exposure energy to the amount of absorbed energy in a single rotation will determine the number of required rotations for curing:

$$N={\displaystyle \frac{E_c}{E^t }}.$$

(18)

But earlier, we defined an alternative definition for critical exposure energy in terms of critical exposure time, t_c ($E_c=I_s.t_c$). Substituting this, we can obtain the total time for curing as follows:

$$t_{tot}\left(r=0\right)=N.T_v={\displaystyle \frac{I_s. t_c}{T_v{I_{s}e}^{-\frac{R}{D_p}}}}.T_v={\displaystyle \frac{ t_c}{e^{-\frac{R}{D_p}}}} =t_c{. e}^{+{\displaystyle \frac{R}{D_p}}}.$$

(19)

From the above simple equation, one can estimate the time of cure initiation at the central zone. It depends exponentially on the ratio of the vial radius R, divided by the D_p. It is also evident that the time is independent of rotational speed ω, and dependent on the critical exposure time t_c. Therefore, one can estimate the time of curing at any point without knowing E_c, as long as t_c for the given setup is known.

In general, for each arbitrary point, the time for curing is as follows:

$$t_{tot}\left(r\right)={\displaystyle \frac{\pi t_c}{\left[{\int }_0^{+\alpha }\left\{e^{-\frac{R\sqrt{1-{\left(\frac{r}{R}\right)}^2{\sin }^2\theta }-r\cos \theta }{D_p}}+e^{-\frac{R\sqrt{1-{\left(\frac{r}{R}\right)}^2{\sin }^2\theta }+r\cos \theta }{D_p}}\right\}d\theta \right]}}.$$

(20)

For the inner surface curing, the time will be calculated as below:

$$t_{tot}\left(R\right)={\displaystyle \frac{\pi t_c}{\left[{\int }_0^{+\mathsf{\beta }}\left\{e^{-\frac{R\sqrt{1-{\sin }^2\theta }-R\cos \theta }{D_p}}+e^{-\frac{R\sqrt{1-{\sin }^2\theta }+R\cos \theta }{D_p}}\right\}d\theta \right]}}$$

(21)

$$t_{tot}\left(R\right)={\displaystyle \frac{\pi t_c}{\left[{\int }_0^{+\mathsf{\beta }}\left\{1+e^{-\frac{2R\cos \theta }{D_p}}\right\}d\theta \right]}}$$

(22)

where

$$\sin \left(\beta \right)={\displaystyle \frac{r_0}{R}} \mathrm{or} \beta ={\sin }^{-1}\left({\displaystyle \frac{r_0}{R}}\right).$$

(23)

An essential criterion for ensuring a sound part formation is that the time of the cure at the center is less than that of the vial surface. Otherwise, the surface will be cured before part formation, and the light attenuation through the solidified skin on the vial surface will diminish the conditions for object formation. We can compare the cure times at the center and the surface to determine the following necessary condition:

$$t_{tot}\left(r=0\right) \le { t}_{tot}\left(r=R\right),$$

(24)

$$t_c{. e}^{+{\displaystyle \frac{R}{D_p}}}\le {\displaystyle \frac{\pi t_c}{\left[{\int }_0^{+\mathsf{\beta }}\left\{1+e^{-\frac{2R\cos \theta }{D_p}}\right\}d\theta \right]}},$$

(25)

or

$$e^{+{\displaystyle \frac{R}{D_p}}} .\left[{\int }_0^{+\mathsf{\beta }}\left\{1+e^{-{\displaystyle \frac{2R\cos \theta }{D_p}}}\right\}d\theta \right]-\pi \le 0.$$

(26)

We can now determine the relationship between the D_p/R ratio with angle β, which is the ratio of the light radius to the radius of the vial, i.e., r₀/R. The numerical solution of Equation (26) yields the relationship between the two ratios and is plotted in Figure 4.

Note that the region above the curve satisfies the inequality. It can be observed that as the full projection of the light source increases, specifically when the radius r₀ approaches the vial’s radius R (r₀→R), the required penetration depth D_p of the resin becomes significantly larger, which may not be easily attainable. This analysis indicates that the largest object dimension should be kept below some fraction of the vial diameter, e.g., about half of the vial radius, as it may exceed the practical limits of the resin’s ability to cure efficiently, especially at greater depths.

We can also find the relationship between the projection radius and the required vial radius. Non-dimensionally, they can be represented as r₀/D_p and R/D_p ratios. Therefore, by knowing the D_p of the resin, we can determine the radius of the required vial for an object that has to be fabricated. This relationship is plotted in Figure 5, where the region below the curve satisfies the following inequality:

This graph highlights the relationship between the object’s radial size and the ability of the resin to undergo sufficient curing under a uniform light projection. As the light beam is projected onto the resin, the penetration depth D_p dictates how deeply the light can reach before the resin solidifies. The inequality graph provides insights into the maximum achievable radial dimensions of the object by identifying the relationship between these dimensions and the material’s characteristics, such as its ability to allow light penetration. For the uniform light projection, the maximum attainable projection radius is as follows:

r₀ (max) = 0.92 D_p,

(27)

with a given vial radius of

R = 1.43 D_p,

meaning that

r₀(max) = 0.64 R.

(28)

This is the main finding of this research work, which mathematically explains the limitation of the object size in the VAM process. As it can be seen, there are two solutions for any given r₀ that determine two boundaries. The boundary at the far left is too constraining, as it implies that the radius of the vial is almost the same as that of the projection ${\left({\displaystyle \frac{\mathrm{R}}{{\mathrm{D}}_{\mathrm{p}}}}\right)}_{\mathrm{m}\mathrm{i}\mathrm{n}}$; this cannot be a good practice to select the vial radius of the same size as the dimension of the object being produced. Consequently, the right side of the curve is more important for design consideration, and it means that for a given radial dimension r₀ of the object being produced, the vial radius, R, must not be greater than the corresponding value, ${\left({\displaystyle \frac{\mathrm{R}}{{\mathrm{D}}_{\mathrm{p}}}}\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$. The inequality can then be approximated by assuming that R is, in general, greater than D_p, also favoring the right side of the curve. This is a justified assumption, since the value of D_p is, in general, low, and also, one can consider selecting a larger vial for a given projection radius, r₀. Then, by neglecting the exponential part of the integration, the inequality will be reduced to the following:

$$e^{+{\displaystyle \frac{R}{D_p}}} .\left[{\int }_0^{+\mathsf{\beta }}d\theta \right]-\pi \le 0 or {\beta e}^{+{\displaystyle \frac{R}{D_p}}}-\pi \le 0.$$

(29)

By rearranging the above inequality and substituting $\beta ={\sin }^{-1}\left({\displaystyle \frac{r_0}{R}}\right)$, we obtain the following inequality:

$${\displaystyle \frac{r_0}{D_p}}\le {\displaystyle \frac{R}{D_p}} \sin \left({\displaystyle \frac{\pi }{{ e}^{+\frac{R}{D_p}}}}\right).$$

(30)

The graph for this approximation is also shown, which indicates a good fit with the original inequality, especially at the values of R/D_p > 2, where the fit is almost exact. The main interpretation of this approximation is that the light exposure at the far zone ($E^F)$, during rotation, can be ignored. This is justifiable, as the light attenuation in this region is too high to be able to contribute additional energy.

If, for instance, we consider the D_p from the current work (5.70 mm), then the maximum object radius (according to the numerical solution of the full formulation of the inequality) will be as follows:

$$r_0\left(max\right)=0.92D_p\cong 5.2 \mathrm{m}\mathrm{m}.$$

The vial radius is as follows:

$$R=1.43D_p\cong 8 \mathrm{m}\mathrm{m}(\mathrm{or} \mathrm{the} \mathrm{vial} \mathrm{diameter} \cong 16 \mathrm{m}\mathrm{m}).$$

Another example is when the desired projection radius is selected, e.g., 4 mm, namely r₀/D_p = 0.70. Then, the lower and upper limits for R/D_p will be 0.8 and 2.3, respectively (see Figure 5). The numbers correspond to the vial radii of 4.6 and 13 mm, respectively (note that D_p = 5.70 mm). Therefore, a vial radius between these two values can be selected for producing the given object. In practice, a larger diameter is generally favored due to reasons such as the curvature of the vial; a smaller radius means a larger surface curvature, which adversely affects the projection accuracy as a result of refraction. On the other hand, a vial radius greater than 13 mm will not produce a sound product, as the attenuation of the light through the resin will be too large, so the surface curing will be favored before the start of curing in the inner regions. It is interesting to note that if a vial of a different radius (say, radius R of 20 mm or R/D_p$\cong 3.5$) is selected, then the maximum dimension (radius) of the object is expected to be about 1.9 mm as r₀/D_p $\cong$ 0.33; this is a very large reduction (more than 50% from 4 mm to 1.9 mm) for a feasible part size. The larger size of the vial (above the limit) results in a reduction in the size of the part that can be produced due to the attenuation effect it exerts.

To simplify the criterion, three approaches are introduced for the selection of vial diameter: (1) Single Value, (2) Mid-Point, and (3) Max-Limit.

Single Value: As illustrated in Figure 5, the value of R/D_p at the peak for r₀/D_p is applicable across all other r₀/D_p values. Consequently, one can conveniently utilize this value (i.e., 1.35) to designate the vial size for manufacturing objects of any size. This embodies an elegantly simple approach to determining the suitable vial size.

$${\displaystyle \frac{R}{D_p}}\cong 1.35 \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} \mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{s}.$$

(31)

Mid-Point: While Figure 5 represents a comprehensive criterion for the relationship and boundaries among the governing parameters of projection radius, vial radius, and D_p, for simplicity and as a rule of thumb, we introduce the middle value (exhibited in Figure 6) for the relationship, as represented by the “Middle Vial Diameter ($R_{Mid})\mathrm{”}$. When sketching the curve representing the middle points at various ${\displaystyle \frac{r_0}{D_p}}$ values, we may introduce a linear fit with a simple formulation as below:

$${\displaystyle \frac{R_{Mid}}{D_p}}\cong -{\displaystyle \frac{r_0}{D_p}}+2.3$$

(32)

or

$${\displaystyle \frac{r_0}{D_p}}\cong -{\displaystyle \frac{R_{Mid}}{D_p}}+2.3.$$

(33)

Max-Limit: Since the choice of vial diameter is favored toward larger sizes rather than smaller sizes (to mitigate light deflection and its compensation), the right wing of the curve can be considered as the main criterion to relate the governing parameters. The right wing can be very closely fit with Equation (33) depicted in Figure 7, which maintains the exponential relationship ($y=a.e^{b.x}),$ signifying the nature of the process (light attenuation):

$${\left({\displaystyle \frac{r_0}{D_p}}\right)}_{Max}\cong \pi e^{-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\mathrm{R}}{D_p}}}.$$

(34)

And re-arranging it to find the vial radius based on the object size, we attain the following:

$${\left({\displaystyle \frac{\mathrm{R}}{D_p}}\right)}_{Max}\cong -1.5\ln \left({\displaystyle \frac{r_0}{\pi D_p}}\right).$$

(35)

2.3. Numerical Analysis—Object Growth

The numerical analysis was conducted on the VAM process of uniform light projection at various projection radius r₀ values, for the vial radius of 12.5 mm (or inner diameter of 25 mm). Considering the working resin with D_p and t_c of 5.7 mm and 4.8 s, the time of curing was sketched vs. the (cured) object size. This considers the part growth beyond the projection radius (r₀). The analysis is shown in Figure 8.

The figure indicates that as the projection radius increases, the occurrence of surface curing to precede inner region curing becomes more probable (the curve end point tends towards down where the curing time becomes closer to that of the central points). Therefore, according to the theoretical evaluation to achieve a larger part (than the maximum possible of about 4 mm in radius) with this experimental resin (with D_p = 5.7) faces the curing at the vial surface.

2.4. A Consideration on Rotational Speed

As we noticed in the relationship, there was no explicit dependency of the cure time on the rotational speed, ω. This is understandable, as the higher speed causes less time of exposure at every revolution; thus, more revolution is needed to reach the critical exposure for curing. This means that the total time is the dominating factor and not the speed of rotation. However, an extreme limitation can be defined. In the given analysis, it was assumed that point Q receives the light at both near and far zones from the projected light ($E^N$ and $E^F$). Therefore, at least one rotation has to occur before part formation. On the other hand, the exposure time at the vial surface (where it receives the maximum light intensity, I_s), and at one rotation, should be less than the critical exposure time, t_c. The time of exposure of the surface can be derived as below, based on the exposed angle, β:

$$T_v.{\displaystyle \frac{4\beta }{2\pi }}\ll t_c,$$

(36)

that is,

$$T_v\ll t_c.{\displaystyle \frac{\pi }{2\beta }} \mathrm{or} N\gg {\displaystyle \frac{2\beta }{\pi t_c}} \mathrm{rev}/\mathrm{\text{s}}.$$

(37)

The lowest time can be assumed when the angle β is π/2. Then, we will have the following:

$$T_v\ll t_c \mathrm{or} N\gg {\displaystyle \frac{1}{t_c}} \mathrm{rev}/\mathrm{\text{s}}$$

(38)

or

$$\omega \gg {\displaystyle \frac{2\pi }{t_c}} \mathrm{rad}/\mathrm{\text{s}}.$$

Regarding this research work, where t_c was obtained as 4.8 s, the limit for the rotational speed is as follows:

$$\mathsf{\omega }\gg 1.3 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s} \mathrm{or} N\gg 0.21 \mathrm{r}\mathrm{e}\mathrm{v}/\mathrm{s} \mathrm{or} T_v\ll 4.8 \mathrm{s}.$$

For a typical angle of π/6 (exposed area at half vial radius) for the angle β, the corresponding values will be as follows:

$$T_v\ll 14.4 \mathrm{s}, \mathsf{\omega }\gg 0.44 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}, N\gg 0.07 \mathrm{r}\mathrm{e}\mathrm{v}/\mathrm{s}.$$

According to the above, it is recommended that the rotational speed is adjusted much higher than the proposed limit to cover many full revolutions; it is surmised that the part formation will be more uniform. In this current curve, the period of 3 s was adjusted for the revolution.

3. Experimental Materials and Preparation

An anycubic transparent plant-based resin (soybean-based) was used as a photosensitive polymer resin. Phenylbis (2,4,6-trimethylbenzoyl) phosphine oxide with a molecular weight of 418.46 g/mol and 97% purity obtained from Sigma-Aldrich, Oakville, Canada, was used as a photoinitiator. Mineral oil by Life Brand was employed for light refraction compensation.

The resin and initiator mixture was prepared by adding 0.2% initiator in the resin and stirring at 100 rpm for 1 h. This mixture, after preparation, was stored under a dark cover to avoid the pre-curing of the resin.

3.1. Experimental Setup

The setup used for VAM is exhibited in Figure 9.

This includes a video projector emitting a UV light of a specific wavelength. The brightness of the projector was adjusted to 40%. A glass vial holding the photosensitive material and initiator mixture was attached with a lab-built apparatus, with a rotating head, to rotate the vial at a speed of 3 s per revolution. Three different vial diameters were used for measuring D_p; ODs are 14 mm, 22 mm, and 27 mm, and IDs are 12 mm, 20 mm, and 25 mm. The glass vial was submerged in a transparent container holding mineral oil. The measured refractive index of mineral oil was 1.49 ± 0.02 and 1.47 ± 0.02 for resin, respectively, indicating that the mineral oil is a sound refraction compensation media for the working resin utilized in this study. This measurement was performed through a Snell’s law experiment in the lab. A double convex lens with a focal length of 150 mm was placed in between the projector and the glass vial to focus and direct light beams precisely onto the build area.

3.2. Experimental Design

Measurement of D_p: To measure the D_p using the DLP printer, pre-defined squares with different exposure times were printed, and the thickness of these squares was measured. The results were plotted using the Beer–Lambert law of absorption, as described by other researchers [22,24].

To measure D_p on VAM, an image of the small circle of 1 mm diameter (as shown in Figure 10a) was projected on the rotating vial holding the resin–initiator mixture (exhibited in Figure 10b,c). The projected image was passed through a lens and mineral oil, as shown in Figure 9. The initiation time for curing was recorded, and the experiment was repeated for the three vial sizes with 12, 20, and 25 mm inner diameters (or inner radii of 6, 10, and 12.5 mm).

Limit curve: The credibility of the limit curve given in Figure 5 was assessed via carrying out experiments at points with different object radii and vial diameters. In this study, two vial diameters were utilized: 12 and 25 mm. The object radius was chosen as 3, 4, and 5 mm. According to Figure 5, the points tested are given in

Table 1. Experiments conducted for validating the limit curve.

r0/Dp [r0 (mm)]	R/Dp [R (mm)]
0.53 [3] 0.70 [4] 0.88 [5] (outside the limit)	1.05 [6] 2.2 [12.5]

:

Object growth: For the evaluation of analytical modeling, the experiments were conducted for making a cylinder of varying diameters, from 2 mm to 10 mm. The image of a rectangle was projected on the rotating vial, and the test was conducted for varying times from 20 s to 180 s, depending on the projected diameter.

Table 2. List of experiments conducted for the evaluation of analytical modeling.

Exp#	Projected Diameter (mm)	Test Time (s)
1	2	60
2	2	90
3	2	120
4	2	150
5	2	180
6	4	50
7	4	70
8	4	90
9	6	50
10	6	60

shows the list of experiments conducted.

4. Results and Discussion

4.1. Depth of Penetration, D_p

Figure 11 shows the results obtained from both the VAM and DLP methods to calculate D_p. As it can be seen, there is a large difference (larger than one order of magnitude) between the two data. For the VAM method, the vial radius indicates the curing at the center of the vial that is distant from the vial surface by the vial radius; this represents the depth of cure. Therefore, the given data for the VAM method include the time when the curing starts at the center of the vial. A concluding remark is that the D_p, utilized in the analysis of the VAM process must be obtained from a similar method; thus, employing data that are obtained from the DLP method will be misleading.

From the above data and the trendline equation, one can obtain D_p and t_c for the VAM process as below:

$${\mathrm{D}}_{\mathrm{p}}=5.70 \mathrm{m}\mathrm{m}$$

$${\mathrm{t}}_{\mathrm{c}}=\exp \left({\displaystyle \frac{8.96}{5.70}}\right)=4.8 \mathrm{s}$$

4.2. Experimental Verification

Limit curve verification:

The corresponding points tested for the verification of the limit curve (Figure 5) are depicted in Figure 12:

The tests were run to produce these objects to verify the limit curve. Interestingly, the results indicated the validation of the theoretical evaluation for the limit curve; points 1, 2, and 3 were readily produced with no indication of a surface cure. However, point 5 could not be obtained, as this condition resulted in excessive gel formation at locations beyond the targeted geometry, including at the vial surface. In fact, the model formulation predicts this as the undesirable region where gel formation is initiated at the vial surface before the object completion occurs, as the critical exposure threshold is exceeded, and it receives enough energy for gel formation. This transition in the resin bulk from the liquid phase to the gel phase prevents further controlled solidification [25].

Thus, the results reinforce the assumption that reaching the critical exposure energy is equivalent to the onset of solidification, highlighting the importance of controlling the exposure to avoid premature gelation that interferes with the desired object formation. The produced objects are shown for the corresponding points in

Table 3. Produced parts for the purpose of limit curve verification according to Figure 12.

Vial Internal Diameter	Projected Diameter	Curing Inside Vial	Produced Object
12 mm	6 mm (r0 = 3 mm) (Point 1)	At 16 s	6.1 mm Dia
	8 mm (r0 = 4 mm) (Point 3)	At 16 s	8.3 mm Dia
25 mm	6 mm (r0 = 3 mm) (Point 2)	At 50 s	6.3 mm Dia
	8 mm (r0 = 4 mm) (Point 4)	At 50 s	8.5 mm Dia with gel
	10 mm (r0 = 5 mm) (Point 5)	At 45 s	No object was produced due to a lot of surface curing

.

It is to be noted that for point 4, where it is almost at the boundary, the test showed that, although the object was completely produced, the part finish coincided with surface curing. As mentioned above, the tests were conducted with unmodified light irradiation; as a result, the product features’ accuracy was low, with some irregularity, especially at the edges. Additionally, for some runs, the issue of sedimentation was also observed in the 25 mm vial.

• Object Growth:

Experimental data for the projection radii of 1, 2, and 3 mm were carried out so that after the creation of the desired object (that appeared almost at once), the exposure was continued to investigate the object growth and compare the result with the theoretical prediction. The results are added to Figure 13, as shown by markers for various object sizes. It is seen that the experimental results closely agree with those of theoretical values. The further growth is hindered by the initiation of vial surface curing (where the light passes through the solid resin at the surface), which prevents an authentic VAM process (where the light passes through the working resin in a liquid state). It is seen that the time of stoppage is closely relevant to the initiation time of curing at the surface (far right endpoints).

The results also predict the full part formation, almost at once, as the initiation times of curing up to the desired radius are almost the same. This is in full agreement with the observation of part formation in the VAM process. In fact, after a certain time, the whole part appears in a narrow time window. After this, the part growth is slow due to fractional exposure for the region at the larger radius. Although it is not considered in the predictions given in Figure 8, one other reason is the blockage of light by the part formation at the center to penetrate to the far segment of the light exposure during rotation. As explained in the theoretical section, the resin at the rotation receives light in two segments, the near region and the far region. The far region receives less light; however, it is more favored for the central region than for the surface region, due to the difference in distances. Hence, when the part formation occurs, then the growth will be more hindered at the object location (central); thus, the part size becomes smaller. When the part formation slows down, then the surface curing will be further promoted. This modification was carried out on the numerical evaluation and is presented in Figure 13.

The modification appeared to be more accurately matching the experimental data and thus should be considered as one effective factor in the VAM process. It is to be emphasized that this modification does not affect the curing time at the surface for this vial radius (12.5 mm).

Table 4. Growth of object inside vial with an internal diameter = 25 mm when different diameter cylinders are projected and final object produced.

Projected Diameter	Curing Inside Vial and Part Produced
2 mm (r0 = 1 mm)	At 60 s 2.9 mm Dia	At 150 s 3.8 mm Dia
4 mm (r0 = 2 mm)	At 70 s 4.4 mm Dia	At 90 s 4.7 mm Dia
6 mm (r0 = 3 mm)	At 50 s 6.3 mm Dia

displays the progress during the process and the object produced.

5. Conclusions

An analytical model has been developed for the VAM process to understand the interrelationship between the geometric parameters and material properties, and it has been verified experimentally. The main objective as to determine the limitation of the part size in the VAM process and explore the factors that affect it. At this stage of this continuing research work, solid cylindrical objects of various sizes were produced using a uniform light distribution. Experiments were designed to validate the analytical results, and a thorough study was conducted to measure the D_p in the VAM process. The following contributions and conclusions can be deduced from this work:

• The D_p, is a crucial factor in the analysis and experimentation in the VAM process. Its action and value are fundamentally different from the one obtained by the conventional AM polymerization methods, such as SLA and DLP. Thus, a method for its measurement and analysis is introduced for the VAM process. The value was found to be about one order of magnitude larger than those obtained in DLP.
• Analytical results reveal the region for the plausible object size formation in accordance with the D_p and the vial size, bounded by a bell-shaped curve. A particular point of interest is the extremum point on the curve, indicating the maximum possible size of the part that can be produced, regardless of container size, that solely depends on D_p. For uniform light distribution, the ratio of the maximum part size to D_p is 0.92. This is based on the competition for curing between the central regions and the regions adjacent to the inner surface of the vial.
• According to the analysis and the limit curve, for any desirable size of the object (r₀), there is a wide selection of vial sizes (R) that can be selected. The smaller the object size, the wider the selection spectrum for the vial size. The actual sizes are governed by the value of D_p, which is the principal criterion for the design of the process.
• A limit graph for the vial diameter showing lower and upper boundaries is introduced by the analytical results. The upper boundary is more important for the selection of the vial diameter for a desired part size. It highlights three criteria for the selection of vial diameter—Single Value, Mid-Point, and Max-Limit.
• A set of experiments was conducted to evaluate the analytical results. The experimental results strongly support the experimental outcomes, affirming the reliability of the analytical evaluation.