Multiphysics modelling of the mechanical properties in polymers obtained via photo-induced polymerization

Photopolymerization is a chemical-physical reaction where a UV light triggers free radical polymerization [26]. At the first step of the reaction, the liquid resin monomer M, whose concentration in space and time is indicated with C_M(X, t), is irradiated by the UV light. The light irradiation allows photo-initiator molecules Ph_I, initially placed inside the resin monomer with a concentration C_I(X, t), to be converted into free radicals R^•, whose concentration is herein indicated with C_R(X, t). During the second step, free radicals react with monomer molecules, providing the activation of the functional groups P^•. From a physical viewpoint, this functional group simply represents a polymer chain which behaves like a snake, growing into the medium by reacting with other monomer molecules. The propagation proceeds until a termination stage is achieved: this occurs when either the polymeric chain meets a free radical (\( {P}^{\bullet }+{R}^{\bullet}\overset{k_t}{\to }{P}_{dead} \)) or it binds to another chain encountered along its growth path \( \left({P}^{\bullet }+{P}^{\bullet}\overset{k_t}{\to }{P}_{dead}\right) \). A sketch of the photopolymerization process, along with the reaction scheme, is reported in Fig. 2, where β, k_p, and k_t are the rate constants characterizing this chemical process.

As the chemical reaction proceeds, the monomers in the liquid solution are gradually consumed by combining with radicals and forming polymer chains, whose density progressively grows providing a stiffness increase of the material. A parameter typically used in the kinetic description of photopolymerization is the degree of cure, defined as [27]: which quantifies the amount of the monomer molecules converted into polymer chains. In the initial stage of the process (we refer here to the reference configuration of the domain whose points are identified by the position vector X), where the resin is in a liquid state, the degree of cure is ϱ(X, t = 0) = 0, since the concentration of the monomer molecules is equal to the initial one. As the reaction proceeds, ϱ(X, t) increases in time because C_M(X, t) < C_M(X, t = 0), i.e., the number of monomer molecules reduces due to the polymer chains growth (Fig. 2), and tends to 1 when the chain formation process is almost complete. The degree of cure is of crucial importance since it can be related to the concentration of active chains c_a(X, t) (number of chains per unit volume), i.e., chains capable of transmitting a force being connected at both ends to other neighboring chains. The elastic parameters of the polymer network, and consequently its stiffness, depend on such a concentration. Indeed, as shown in [27] and other previous works [31], the concentration of the active chains after the gelation point, i.e., when the material can be considered to become stiffer and stiffer as the photopolymerization proceeds, can be related to the degree of cure by means of an exponential relation, defined as [27]: where μ is the shear modulus, E_c, E_d, and s are fitting parameters, to be determined under the assumption of incompressible behavior. The concentration of the active chains within the medium is assumed to depend point-by-point by the degree of cure in the continuum, which is also point-wise dependent on the UV light intensity I(X, t) and its duration.

In the specific process of photopolymerization, light is irradiated over the resin surface and diffused within the layer thickness, allowing photo-initiators to be decomposed into active species. The evolution of the photo-initiators (which are consumed in order to provide activation of free radicals according to \( {Ph}_I\overset{\beta }{\to }2{R}^{\bullet } \)) can be modelled by defining the time rate concentration: where \( \dot{\blacksquare} \) stands for the time derivative of the quantity ■, i.e., \( \dot{\blacksquare}=\frac{\partial \blacksquare }{\partial t} \), while β represents the photodecomposition rate, which is a property of the photo-initiators. After such an activation, free radicals evolves triggering the initiation, propagation and termination of polymer chains [27]. The evolution of the free radicals can be expressed as: where m is the number of radicals generated in the photodecomposition (for instance, it can be assumed m = 2). k_t is the termination rate, expressed as a function of curing in the form k_t(t) = k_t0 · g(ϱ), where k_t0 is the initial termination rate constant and g is the curing-dependent term related to the diffusion controlled termination. The resulting problem is nonlinear. As it can be noticed from Eq. (4), the concentration of radicals increases due to photo-initiator decomposition while it reduces because of the termination mechanisms of propagation. Additional mechanisms, such as oxygen inhibition and other related chemical problems [32‐34] , are here neglected for the sake of simplicity, but could be taken into account by slightly modifying Eq.(4) (see section 4.4).

As the combination with radicals proceeds, the monomers in the solution are gradually consumed, allowing the polymer chains initiation and propagation to occur (\( {R}^{\bullet }+M\overset{k_p}{\to }{P}^{\bullet } \) and \( {P}^{\bullet }+M\overset{k_p}{\to }{P}^{\bullet } \)). Such a mechanism can be described by the following differential equation: where k_p is the propagation rate constant that depends, analogously to k_t, on the curing evolution itself [27].

The system represented by Eqs. (3)–(5) can be solved by assigning the proper initial conditions known from the liquid resin properties, i.e., C_I(X, t = 0), C_R(X, t = 0) and C_M(X, t = 0), together with the knowledge of the rate constants (both the initial values and evolution equations are needed to solve the problem), and with the light intensity I(X, t) within the continuum (see Sections. 2.2 and 3.1). As discussed above, the main aim of kinetic models is the evaluation of ϱ(X, t), in order to subsequently compute the concentration of active chains c_a(X, t) which is responsible for the mechanical properties of the polymer at the mesoscale (see Section. 2.3).

Interestingly, at a given position in the continuum, the problem of defining the evolution of the mechanical properties of the material can be directly related to the increase of active chains’ concentration caused by the curing process. By applying the time derivative to both members of Eq. (1), we get\( \dot{\varrho}\left(\boldsymbol{x},t\right)=-\frac{{\dot{C}}_M\left(\boldsymbol{x},t\right)\ {C}_{M0}}{{\left[{C}_{M0}\right]}^2} \)(being C_M0 = C_M(X, t = 0)) and finally, by exploiting the monomer molecules evolution reported in Eq. (5), we obtain:

The concentration of the active chains evolves according to an exponential function, see Eq. (2), which can be reformulated in this form: where it has been assumed that ϱ_gel ≅ 0, while \( \overline{\mu} \) is the shear modulus of the fully cured material (i.e., for an ideal curing time t → ∞). We assume that a small number c_a0 of active chains exists when the material is in the initial (nearly) liquid state, i.e., c_a(X, t = 0) = c_a0. The knowledge of the current value of the degree of cure, obtained by solving the differential Eqs. (3)–(6), suffices to evaluate the corresponding current chain concentration value by using Eq. (7). The concentration of active chains at the time t_c (curing time) can be evaluated as \( {c}_a\left(\boldsymbol{X},{t}_c\right)={c}_{a0}\left(\boldsymbol{X}\right)+{\int}_0^{t_c}\dot{c_a}\left(\boldsymbol{X},t\right)\ dt \). Further, according to the previous assumption, the time derivative of the chain concentration can be expressed as \( {\dot{c}}_a\left(\boldsymbol{X},t\right)=\alpha {c}_a\left(\boldsymbol{X},t\right)\dot{\varrho}\left(\boldsymbol{X},t\right) \), where α is a model parameter. The chain concentration at the end of curing can be finally obtained as follows: which emphasizes how the final concentration of the polymer chains can be controlled by properly driving the time rate of the degree of cure during the photopolymerization process.

As shown In Section 2.1, the evolution of the chemical species involved in the photopolymerization process is triggered by a light radiation provided by a UV-light moving source. The light radiation is absorbed by photoinitiators, photoabsorbers, and by other light reactive species present in the matrix. Light diffusion within the material is attenuated and the variation of its intensity is a function of the concentration of the absorbing species and the molar absorptivity. In order to quantitatively describe how the light intensity depends on the spatial position in the material, the well-known Beer-Lambert law can be used [29, 35].

Let us consider the large deformation regime for which, in the reference undeformed configuration (having domain and boundary Ω₀, ∂Ω₀, respectively), the generic point is indicated with X, while in the current deformed configuration (having domain and boundary Ω, ∂Ω, respectively) the corresponding point is identified by x. In a one-dimensional setting, where the light beam is assumed to impact perpendicularly the surface of the material (along the vertical Z axis), the diffusion equation can be easily obtained. At a given time instant, it can be assumed that the radiant flux I(Z) of the light outgoing from a slice of material lying perpendicularly to the light beam is reduced—with respect to that of the incoming flux—by the amount dI(Z, t) =  − A(Z, t) I(Z, t)dZ, where A(z, t) is the attenuation coefficient (absorbance).

In this simple one-dimensional case characterized by the linear coordinate z, if a constant value for the term A is assumed, at the generic time t and with the boundary condition I(0, t) = I₀(t), the solution of the Beer-Lambert equation reads:

The above expression can be generalized to a three-dimensional setting, leading to well-known Beer-Lambert, a first-order linear partial differential equation (PDE): where Ω₀, ∂Ω₀ indicate the domain of the problem and its boundary in the undeformed configuration, respectively, l(X, t) is the unit vector of the incoming light beam, I(X, t) is the light intensity at the position X and time t, ∇ is the gradient operator, and A(X, t) is the local attenuation coefficient (or absorbance) quantifying the light attenuation due to photons that did not travel through the material because of scattering or absorption. The local attenuation coefficient A(X, t) can be written as follows [27]: where θ is the photo-initiator molar absorptivity, C_I is the photo-initiator concentration, A_abs(X, t), A_pol, A_mon are the absorption due to photoabsorbers, converted polymer, and un-polymerized monomers, respectively. It is worth noticing that Eq. (10) is highly nonlinear since the term A(X, t) depends on the light-intensity history I(X, 0 ≤ t ≤ τ) at the location X.

Polymeric materials, such as rubbers, gels, etc., have an amorphous microstructure constituted by an entangled network of long linear chains, coiled and reciprocally joined at discrete points termed as cross-links. The microstructural state of polymers, characterized by thermal fluctuations, and their mechanics, are dominated by the entropic energy of the network, at least for not too large deformation regime [36]. The consolidated and well-known freely jointed chain model [37] —assuming the chains to be made of N rigid segments of equal length b, randomly arranged and connected at their extremities with neighboring chains—is usually adopted to describe the network topology, while the chain’s end-to-end vector r is used to define the deformed state of the chain itself [38]. Due to the disordered nature of the network chains, a statistical description of the end-to-end vectors is suitable to define the chains arrangement within the network. To this end, a distribution function ρ(r) can be conveniently introduced [30], providing the number of network’s chains with a given end-to-end vector r = (r_x, r_y, r_z) in a given state of the material, for instance the stress-free one. For our purpose, it is convenient to write the above-mentioned distribution in the form ρ(r, t) = c_a · φ₀(r, t), where c_a corresponds to the number of reciprocally connected load bearing chains per unit volume (see Sects 2.1 and 2.2), while φ₀(r) is a dimensionless distribution (usually obeying the standard Gaussian law with mean value r = 0 and standard deviation \( b\sqrt{N/3} \) [38]). A further assumption is the so-called affine deformation hypothesis, stating that the deformed chain’s length is given by r = |r| = λr₀, being \( {r}_0=b\sqrt{N} \) the chain’s mean square length and λ the macroscopic stretch of the material. According to the above definitions, within the time interval \( \left(0,\overline{t}\right) \)(note that this time interval follows the curing time interval (0,t_c)), we have: where the operator 〈■〉 indicates the following integral evaluated over the chain configuration space, \( \left\langle \blacksquare \right\rangle ={\int}_0^{2\pi }{\int}_0^{\pi}\left({\int}_0^{Nb}\blacksquare {r}^2 dr\ \right)\sin \theta d\theta d\beta \), with θ, β being the Euler angles of the end-to-end vector r in the 3D space [30, 39]. The mechanical response of the polymer is assumed to take place after the completion of the photopolymerization process, i.e., the micromechanical model enters into play only after printing the material.

It is worth noticing that the above definition of the chain concentration assumes that only the cross-linked chains enter into c_a, i.e., only the chains linked at their extremities with other chains have to be considered for the mechanical properties of the material. The knowledge of the distribution function ρ(r, t), allows us to evaluate the elastic energy density stored in the material once the deformation energy of a single chain ψ(r) is expressed through its end-to-end vector r. According to the above statement, by adding up all the energies coming from the chains contained in a representative volume of material the energy density can be evaluated as:

It is worth recalling here that the derivative with respect to r of the energy per single chain gives the force f existing in the chain, i.e., f(N) =  ∇ ψ = ∂ψ(λ, N)/∂r, where the dependence of ψ on the chain length N has been emphasized. In general, being ψ a non-negative function of |r|, the energy per unit volume of Eq. (13) is nonzero also in the stress-free state of the material. It is thus much useful to define the deformation energy density of the material as ΔΨ(t) = c_a(t)〈[φ(r, t) − φ(r, 0)]ψ〉 where φ(r, 0) is the dimensionless distribution function at t = 0 that is assumed to be in a macroscopically undeformed state (at the end of the photopolymerization process, i.e., when the material is assumed to be in the stress-free state), i.e., when F = 1 (being F = ∂x/∂X the deformation gradient tensor), while φ(r, t) is the corresponding distribution in a generic deformed state of the material induced by mechanical actions. The stress state in the material can be finally be determined as follows [30] : where the true Cauchy stress σ tensor has been used as the stress measure, while π is the hydrostatic pressure (playing the role of a Lagrange multiplier to enforce the proper volume change ratio J = det F accounting for the actual volume change of the material), L is the velocity gradient tensor. For an incompressible material J = det F = 1 and tr L = 1 : L = 0 so the above expression simplifies.

As it can be noticed, the above-described micromechanical model allows determining the polymer’s mechanical properties and thus its response under a mechanical action; this can be done once the chain statistical distribution φ₀(r, t) and the chain concentration are known at any point of the solid domain at the end of the polymerization time, t = t_c. The end of the photopolymerization process is assumed to occur before the mechanical problem takes place. In our case, at a given point inside the initial liquid monomer material, the chain concentration is determined by the chains formation triggered by the light energy; the distribution of the chain end-to-end vectors at the formation time can be assumed to follow the Gaussian function, i.e., ρ₀(r, t = t_c) = c_a · φ₀(r, t = t_c) being φ₀ the standard Gaussian, while the chain concentration is obtainable through the solutions of Eqs. (6) and (8), by using the kinetics of chains formation governed by Eqs. (3)–(5). Finally, according to Eq. (14), once the current chain distribution φ(r, t) determined according to the deformation process applied to the material has been determined, the stress state can be easily evaluated [30].

It is worth recalling that in this work, we apply the above-presented multiphysics model for studying the tunability of the mechanical properties of photopolymerized materials (see Sections. 2.1 and 2.2 for the theory, Sections. 3.1 and 3.2 for the FE implementation, and Section. 4 for the numerical simulations); the study of the mechanical response related to the properties of the photopolymerized polymer, will be considered in future studies.

In order to determine a form of the light diffusion governing equation suitable to be implemented in a computational code, the development of a weak form of the mathematical problem is usually adopted. Such a form can be obtained by introducing a proper functional or by transforming the strong form represented by Eq. (10) into an integral (weak) formulation. By multiplying the Beer-Lambert equations by the weight function w(X) and integrating over the domains where they are defined, yields to:

By assuming that the vector field l(X, t) is constant in space (i.e., the incident light beam is made of parallel rays) and depends only on time t, i.e., \( \boldsymbol{l}\left(\boldsymbol{X},t\right)=\overline{\boldsymbol{l}} \)(t), and by applying the divergence theorem to the first term of Eq. (10) we get: n being the outward normal to the boundary of the domain, while f(X, t) represents the light-intensity distribution over the irradiated boundary of the domain. The term on the right hand side of Eq. (16) can be rewritten as follows: where \( {l}_n=\left|\overline {\boldsymbol{l}}(t)\cdotp \boldsymbol{n}\right| \) is the projection of the light unit vector of the incoming laser beam on the normal to the boundary surface of the domain. By referring to Fig. 3, the incident light beam forms an angle γ with the normal to the outer top surface of the domain; by assuming γ = 0, we have l_n = 1. The above integrals represent the weak form of the governing equation; they can be conveniently rewritten by introducing the domain discretization in which the field variables are interpolated through the corresponding nodal values, i.e.: where the subscript h indicates the values interpolated from the corresponding nodal quantities, whose values are indicated with \( \overset{\sim }{\blacksquare } \), being n_n the number of nodes of the element; \( {\overset{\sim }{I}}_i,\kern0.5em \tilde{w}_{i} \)are the nodal light intensity and the nodal values of the weight function, respectively, while ∇_X represents the gradient operator in the reference (or undeformed) configuration.

In Eq. (10), the presence of only the convective (or transport) term, l · ∇_XI, leads to the so-called boundary layer, i.e., a region—generally arising close to the boundary—where the solution is characterized by large gradients or oscillations: this entails an unstable and meaningless numerical results [40]. These cases are typical of problems whose governing equations have a large Peclet number, defined as the ratio between the rate of advection of a physical quantity to the rate of diffusion of the same quantity. To overcome this problem, a stabilization term can be added to the original equation. This term is usually an artificial diffusion-like contribution term of the form Q h∆I(X, t), where Q is a coefficient and h a characteristic small geometrical size related to the discretized problem in turn [41]. The amended stabilized PDE governing the problem can thus be rewritten as follows:

The introduction of the discretized field quantities of Eq. (18) in the weak form of Eq. (16), written for a generic finite element e having volume V_e and boundary S_e and thanks to the arbitrariness of the weight function w(X) (or, correspondingly, of its nodal values \( \overset{\sim }{\boldsymbol{w}} \)), leads to the problem written in the following matrix form: where E_e is the light gradient matrix, A_e is the depletion matrix of the finite element, D_e is the stabilization matrix (being Q > 1, h_e a coefficient and a characteristic size of the finite element, respectively), while P_e(t) is the vector of the nodal values of the distribution of the incoming light intensity on the boundary and f_n(X, t) = l_n f(X, t) represents the distribution of the of the beam light intensity normal to the boundary of the domain. Upon assembly of the elements’ matrices over the whole discretized domain (\( \boldsymbol{E}={\mathbbm{A}}_{e=1}^{ne}{\boldsymbol{E}}_e,\boldsymbol{A}={\mathbbm{A}}_{e=1}^{ne}{\boldsymbol{A}}_e,\boldsymbol{D}={\mathbbm{A}}_{e=1}^{ne}{\boldsymbol{D}}_e, \) \( \boldsymbol{P}={\mathbbm{A}}_{e=1}^{ne}{\boldsymbol{P}}_e \) with \( \mathbbm{A} \) the assembly operator and ne the number of finite elements), we get: \( \left(\boldsymbol{E}(t)+\boldsymbol{A}(t)+\boldsymbol{D}\right)\ \overset{\sim }{\boldsymbol{I}}(t)=\boldsymbol{P}(t) \).

For the sake of simplicity, we do not consider the effect of light refraction since typically the light is irradiated by the photo-curing apparatus in a direction perpendicular to the surface of the monomer fluid; in cases where the light is not normal to the surface, the light propagation angle may change in space due to the difference of the refractive index, and the propagation direction must be determined accordingly.

In a 2D setting, the above discretized system of equations becomes: being a the element thickness, h_e can be set as \( {h}_e=\sqrt{A_e} \), being A_e the area of the finite element e, while L_e is the length of the finite element edge irradiated by the incoming laser light. The scheme of the light diffusion in the medium, upon irradiation of its top surface by the laser beam source, is illustrated in Fig. 3.

Once the light-intensity distribution has been evaluated by solving the system of Eq. (20) with the corresponding boundary conditions, such a quantity is used to quantify the degree of cure taking place at each Gauss point of the finite elements used to discretize the monomer vat domain. Firstly, the interpolated light-intensity value at the Gauss point is obtained through the standard expression \( {I}_{h, GP}(t)={\sum}_{i=1}^{n_n}{N}_{i, GP}{\overset{\sim }{I}}_i(t) \), where N_{i, GP} indicates the shape function related to node i and evaluated at the specific Gauss point GP, while 0 ≤ t ≤ t_c is the generic time instant. The current value of the photo-initiator concentration is then evaluated by the time integration of the corresponding time rate, i.e.: where Δt = (t − t₋₁) is the time step interval adopted for the integration over the time domain. Once the photo-initiator concentration is known, the concentration of the free radicals can be determined by firstly evaluating its time rate:

Due to the polymerization process, the concentration of the initial liquid monomer (referred, as done before, to the Gauss point positions GP) decreases in time at the rate: while its current value becomes \( {C}_{M, GP}(t)={C}_{M, GP}\left({t}_{-1}\right)+\varDelta t\ {\dot{C}}_{M, GP}(t) \). All the above-mentioned quantities are evaluated through time integration starting from the knowledge of the corresponding values at the beginning of the photopolymerization process, i.e., C_{I, GP}(t = 0), C_R,GP(t = 0), and C_M,GP(t = 0) have to be known at all the Gauss points of the discretized domain.

Finally, the degree of cure can be easily evaluated as follows: and the related chain concentration density can be determined as being \( {\overline{c}}_{a, GP}={c}_{a, GP}\left(t=\infty \right)=\frac{\overline{\mu}}{k_BT} \) while 0 < p_GP(t) ≤ 1 is the coefficient quantifying the degree of developed chains, with respect to the fully cured material obtainable for an exposition to the light source for a theoretically infinite time.

In this section, we consider the effect of the laser beam moving speed on the degree of cure and, consequently, on the chain concentration related to the polymer’s shear modulus. We consider two different cases, with laser speeds equal to v = 10⁻³ m/s and v = 10⁻⁴ m/s, respectively. It is assumed that the curing process ends when the laser covers the whole width of the layer being printed: the time required by the laser to cover such a distance, known in the AM literature as curing time, is defined as t_c = w/v. A dimensionless curing time can conveniently be defined as t^∗ = t/t_c, with 0 ≤ t^∗ ≤ 1. The geometrical dimension of the layer is kept fixed between the simulations (w = 30 mm), so that the curing times are equal to t_c = 30 s and t_c = 300 s for the two velocities considered. Furthermore, the material absorbance is assumed to be constant, so that the light-intensity field within the material is the same, irrespectively of the laser speed. In other words, the light intensity distributions shown in Fig. 4, are the same for both the laser speeds adopted.

For each laser speed, we are interested in determining the characteristics of the material being printed at two different time instants, namely t^∗ = 0.5 and t^∗ = 1, corresponding to two distinct positions of the laser with respect to the layer being printed. The first case corresponds to the laser beam placed at X = w/2, see Fig. 4a showing the light-intensity field within the material at that instant. The second case corresponds to the final position given by X = w, see Fig. 4b showing the light-intensity field within the material at the end of the curing process.

The corresponding distributions of the degree of cure are shown in Fig. 5. In particular, Fig. 5a illustrates the degree of cure within the material when the laser is placed at the middle of the layer being printed for the case characterized by the highest laser speed (or, correspondingly, for the lowest curing time), while Fig. 5b shows the degree of cure when the laser is at the same position but reached with the lowest laser speed (or, correspondingly, for the highest curing time). Since the degree of cure is the results of a kinetic equilibrium (see Section 2.1), a longer irradiation entails a more pronounced, homogenous and deeper degree of cure within the material. Analogous considerations can be made by looking at Fig. 5c and d, representing the distribution of the degree of cure at the end of the curing process.

Figure 6 shows the dimensionless shear modulus, \( \mu (t)/\overline{\mu} \) within the analyzed domain. Such a quantity can be obtained through the relationship with the chain concentration, Eq. (7), leading to the expression \( \mu (t)=\overline{\mu}\cdotp \exp \left[\alpha \left(\varrho \left(\boldsymbol{x},t\right)-1\right)\right] \), where \( \overline{\mu}=267\ \mathrm{MPa} \) is the shear modulus for the fully cured polymer (this value is always assumed henceforth) and α = 3; in other words, the fields of the shear modulus distribution are a straightforward consequence of the degree of cure. As it can be appreciated from Fig. 6c, at the end of the curing process with the highest laser speed ( other parameters being fixed), roughly only one half of the layer is solidified. Moreover, the stiffness of such a cured part is not homogenous, showing a higher shear modulus close to the top boundary, while a nearly not-cured zone (μ(t) → 0) appears at the bottom. On the other hand, the curing process with the lowest laser speed allows to get a more pronounced, homogeneous and higher values of the shear modulus within the domain, Fig. 6d, even if there is still a zone with a low shear modulus value close to the bottom of resin vat. The previous considerations allow us to appreciate the importance of a layer-by-layer process in 3D-printing. Assuming that the component to be printed had the dimension of the liquid vat investigated in this case (h = 10 mm and width w = 30 mm), it would require to be sliced into layers of thickness d < h in order to be optimally solidified. The curing process, here considered for a single layer for sake of simplicity and clearness, must be repeated for all the layers in which the component being printed has been divided. Firstly, the laser solidifies the top layer, irradiating the liquid monomer placed at Y = d (in the presented case one single layer is adopted, i.e., d = h) by moving rightwards in order to cover the whole part width; then, the part already printed is covered by a liquid layer of thickness d and the laser repeats the curing operation until the whole width is covered again. Such an operation must be repeated until the whole height h of the component is achieved, curing successive layers into which the total height has been subdivided. The number of layers (and therefore the proper layer thickness for the curing process) can be evaluated by using the present results with respect to the optimal distribution of the shear modulus. For instance, the optimum setup of printing parameters can be determined in order to achieve \( \mu \left({t}_c\right)/\overline{\mu}\to 1 \) in each material point of the component in the shortest possible curing time, if such a component is desired at the end of the printing.

In order to generalize the present case, it is worth considering what happen if the initial liquid monomer resin would be irradiated by two laser sources at the same time, one operating from above and the other from below. The light intensity at a point of the material can be evaluated through the superposition of the light intensities induced by both sources. However, since the absorbivity generally depends on the degree of conversion of the material (see Sections. 2.2 and 4.3), the light-intensity superposition from the two sources cannot be applied to get the whole degree of cure because the light diffusion process must account for their reciprocal interaction. For not too low laser speeds, this provides a final photopolymerized material with a high degree of cure located close to the upper and lower surfaces of the domain, while the internal region will be the less polymerized.

In the following examples, we investigate the effect of the laser beam light intensity, by keeping all the other parameters fixed. Two values of the peak light intensity are considered, namely I₀ = 25 W/m² and I₀ = 100 W/m². The light-intensity distribution when the laser reaches the middle position of the layer being printed (X = w/2) is shown in Fig. 7a and b, for the two distinct light intensities considered, respectively. As expected, the case corresponding to the higher peak light hitting the vat surface, provides a more pronounced, diffused, and deeper light-intensity field within the medium.

In turn, the curing of the material is significantly affected: Fig. 8 illustrates the degree of cure obtained for the two adopted peak light intensities, corresponding to the laser position X = w/2. The curing process with the lowest peak light intensity (Fig. 8a, A1) provides a less pronounced and shallower degree of cure within the medium, in comparison with the curing process characterized by the highest peak light intensity (Fig. 8a, A2). The corresponding profiles of the resulting degree of cure obtained along the two straight vertical cross-sections 1 and 2 shown in Fig. 8a, are reported in Fig. 8b.

The dimensionless shear modulus distribution, corresponding to the degree of cure shown in Fig. 8, is reported in Fig. 9a for the two adopted peak light intensities. Close to the top boundary, the material that is being cured with the lowest I₀ presents values of the dimensionless shear modulus limited to 0.5, while it achieves greater values if cured with the highest light intensity I₀. The corresponding profiles of the dimensionless shear modulus of the material, evaluated along the two vertical sections 1 and 2 shown in Fig. 9a, are reported in Fig. 9b. It should be noticed that, an increase of the peak light intensity provides a stiffer material at the same curing time (other parameters being fixed). However, it is important to highlight that a too large light intensity may induce a degradation of the material being printed [42, 43]. This aspect is still an open issue in the AM research field and will not be considered in the present study.

In this paragraph, we investigate the effect of the material’s absorbance on the curing process. Three different cases are considered. In the first case we model the curing process of a liquid monomer assumed to have a constant material absorbance, namely A = 600 m⁻¹, which has the magnitude order of a purely liquid monomer. In the second case we assumed the material absorbance still being constant in space, but with an higher value namely A = 2400 m⁻¹, which has instead the magnitude order of an already cured monomer (polymer in a solid state). However, real photopolymerization processes are characterized by a gradual transition from the liquid monomer to the solid polymer. Therefore, the absorbance ranges gradually between two extreme values according to the expression provided by Eq. (11). For this reason, a third case is considered, in which the curing process occurs with the absorbance varying in time according to the above-mentioned relationship. We assume the initiator molar absorptivity to be θ = 30 m²/mol, the absorbance of a fully cured polymer and of a liquid monomer (infinitely transparent) are, respectively, A_pol = 2400 m⁻¹ and A_mon = 0, while no photoabsorbers are considered in the resin, i.e., A_abs(X, t) = 0. From Eq. (11), by taking C_I(X, 0) = 20 m³/mol, it can be noticed that when the material is in a liquid state (before curing, i.e., ϱ(X, 0) = 0) we get A = 600 m⁻¹, which is the value corresponding to the purely liquid material. On the other hand, when the material attains the fully cured condition, the photo-initiator concentration C_I tends to zero while the degree of cure tends to one, leading to A = 2400 m⁻¹, which is the value corresponding to the fully solid polymer. In other words, in the first simulation, we assume to apply the curing process to a material that ideally maintains the absorbance of the original liquid monomer, while in the second case we ideally assume the absorbance to be constantly equal to that of the solid material; finally, in the third case, we adopt a more realistic hypothesis assuming the absorptivity to vary with the degree of cure. We firstly consider the same peak light intensity I₀ = 150 W/m² for all the examined cases and introduce a dimensionless intensity \( {I}_0/{\bar{I}}_0 \) that will be used in the following, being I-₀ = 150 W/m² .

The light-intensity distribution obtained by using a constant value of the absorbance are reported in Fig. 10a, b, for the lowest and the highest material’s absorbance, respectively. The maps of Fig. 10 represent the results corresponding to the laser position X = w (i.e., at the end of the curing process, t^∗ = 1.0); however, similar results can be obtained for other laser beam positions. As expected, the light penetrates deeper and deeper in the liquid monomer with the lowest absorbance, Fig. 10a, while it is more dimmed in the case with the highest absorbance, Fig. 10b.

The degree of cure distributions at the end of the curing process, characterized by the lowest and the highest value of absorbance, are reported in Fig. 11. Due to the higher intensity of the light in the material corresponding to the lowest A, a higher degree of cure is achieved with respect to that with the highest A (see Fig. 11a, A1 and A2, respectively). The corresponding profiles of the degree of cure evaluated along the three vertical sections 1, 2, 3 illustrated in Fig. 11a, A1, A2, are reported in Fig. 11b. It can be observed that ϱ presents a higher gradient in the Y direction for the liquid monomer with the highest A (black lines in Fig. 11b) while its gradient is less pronounced for the material with the lowest A (blue lines in Fig. 11b).

The corresponding dimensionless distributions of the shear modulus are reported in Fig. 12a, with the respective vertical profiles in Fig. 12b. As expected, the stiffness of the material decreases through the thickness according to the degree of cure.

Figure 13a illustrates the degree of cure existing in the material at the end of the curing process, for a monomer with A varying in time according to the solid-liquid fractions in the material throughout the photopolymerization process. As expected, in this case the resulting field is in between those obtained for the lowest constant A (Fig. 11a, A1), and the highest constant A (Fig. 11a, A2). Finally, the time evolutions of the degree of cure and of the dimensionless shear modulus, from the initial instant t = 0 , to the final one, t = 30 s, i.e., for t^∗ = 1, are considered. The results, evaluated at the material point P (placed at the top of the layer being printed, ξ = 0.1) and M (placed in the middle of the layer being printed, ξ = 0.5), are reported in Fig. 13b, c and d, e, for all the cases investigated in this section. Taking the results corresponding to the varying A as reference, Fig. 13b, c shows that the simulation of the curing process with the highest A leads to an underestimation of ϱ, while the simulations performed by adopting the lowest constant A overestimates ϱ. It is worth mentioning that the more realistic simulation using a variable A requires a bigger computational effort due to the non-linearity introduced by the relationship A = A(ϱ). Similarly, Fig. 13d, e reports the evolution of the dimensionless shear modulus during the whole curing process, leading to the same consideration made for the ϱ evolution. The achieved stiffness is underestimated by using the highest A while it is overestimated by using the lowest A, and it stays in the middle ground by means of A varying over time.

The three cases described above have been simulated also by using a lower peak light intensity, namely I₀ = 7.5 W/m², i.e., for \( {I}_0/\overline{I_0}=0.05 \). The evolutions of the degree of cure and of the dimensionless shear modulus during the curing process are reported in Fig. 14. As expected, the curves corresponding to a varying A fall in between the curves corresponding to the highest and the lowest constant absorbance values. However, compared to the analyses with the higher peak light intensity, the curve is now shifted towards the case with the lowest constant absorbance, Fig. 14.

The analyses performed allow comparing the quality of the photopolymerized material according to the simplifying hypotheses on the absorbance values mentioned above. In general, the assumption of the highest value for the absorbance (that of the solid material) underestimates the resulting mechanical properties of the printed elements, while the opposite occurs if the lowest value of the absorbance is used in the simulations. However, as discussed above, such under or overestimations have to be related to the light intensity, laser velocity and spatial-dependency; if the light beam is characterized by a sufficiently high intensity and the traveling speed is low enough, the adoption of the solid-like absorbance should provide reasonably results, especially for a small layer thickness .

In this last section, we investigate a real case of photopolymerization process as reported in the work by Wu et al. [27]. The monomer used is polyethylene (glycol) diacrylate (PEGDA) (with a molecular weight of 250 g/mol, Sigma-Aldrich, St. Louis, USA), and 0.3% weight percent 2,2-dimethoxy-2-phenylacetophenone (Sigma-Aldrich, St. Louis, USA) as the photo-initiator. We use the proposed FE model by assuming a stationary laser beam v → 0 m/s irradiating a single point lying on the resin surface, as made in [27]. The rate constants used for the simulations are given by \( {k}_p=\frac{k_{p0}}{1+\frac{k_{p0}}{k_{p,D0}}{e}^{c\varrho}} \) and \( {k}_t=\frac{1}{\frac{1}{k_{t, SD}}+\frac{e^{c\varrho}}{k_{t, TD0}}}+\frac{C_{RD}\left(1-\varrho \right){k}_{p0}}{1+\frac{k_{p0}}{k_{p,D0}}{e}^{c\varrho}} \) for the propagation and termination rate constants, respectively. Such constants depend on the degree of cure, which is unknown; therefore, the step-by-step numerical procedure previously illustrated requires to update the rate constants at the end of each calculation step, before moving on to the next one. Several parameters are required to quantify the rate constants evolution. These depend on the resin property, environmental conditions etc, and need to be experimentally measured. Following Wu et al. [27], such parameters are: \( {k}_{p0}=1.86048\ \frac{{\mathrm{m}}^3}{\mathrm{m}\mathrm{ol}\ \mathrm{s}} \), \( {k}_{p,D0}=8.994\cdotp {10}^8\frac{{\mathrm{m}}^3}{\mathrm{m}\mathrm{ol}\ \mathrm{s}} \), \( {k}_{t, SD}=4.39\cdotp {10}^3\frac{{\mathrm{m}}^3}{\mathrm{m}\mathrm{ol}\ \mathrm{s}} \), \( {k}_{t, TD0}=10\ 024.43\ \frac{{\mathrm{m}}^3}{\mathrm{m}\mathrm{ol}\ \mathrm{s}} \), β = 8.999 · 10⁻⁴s²/kg,  c= 34.149 and C_RD = 1.0146. Furthermore, the parameters adopted for the estimation of the chains concentration are: E_d = 3.321 MPa, E_c = 1.059 MPa, s = 5.248, and ϱ_gel ≅ 0.15.

The results, expressed as usual in terms of the degree of cure and the polymer chain concentration, are reported in Fig. 15 for a monomer resin cured for t_c = 100 s by using different light intensities. In the figure, the dashed lines correspond to the literature results [27] while the present results are represented by the continuous lines. The chain concentration shown in Fig. 15b has been made dimensionless with respect to that corresponding to the fully cured material, i.e., \( \overline{c_a}={c}_a\left(\varrho =1\right)=9.86\cdotp {10}^{27}{\mathrm{m}}^{-3} \). It can be noticed that the curing process as predicted by our model appears to be faster than that reported in the literature. This might occur because we are neglecting the role played by the oxygen concentration (indeed, the same assumption is often made in literature, see, for instance [28, 44] ). In fact, the oxygen eventually present in the environment may inhibit the polymerization reaction by combining with radicals, thus hindering the propagation of the polymer chains. The specific issue is widely treated in [33]. As clearly demonstrated by our study, neglecting the oxygen role in a kinetic model provides an overestimation of the obtained degree of cure, resulting in a non-conservative prediction of the polymer’s mechanical properties. Such a problem can also be overcome with specific strategies, by means of physical-methods (for instance, curing in an oxygen-free atmosphere thanks to physical barriers or inert gases) or using chemical methods (for instance, by inserting specific additives in the liquid monomer) as shown in [45] .

In conclusion, as already mentioned In Section 2.1, the chemical terms involved in a complex kinetic model like the one here presented, have to be slightly modified and adapted according to the actual condition of the problem under consideration which can vary from case to case, in order to correctly capture all the involved chemical-physics phenomena. For instance, the FE model presented herein can be easily updated in order to capture the role played by the oxygen inhibition: following [27], we modify Eq. (4) by adding the concentration of oxygen, namely C_O(X, t), which affects the evolution of free radicals, i.e., \( {\dot{C}}_R\left(\boldsymbol{x},t\right)=-m\ \dot{C_I}\left(\boldsymbol{x},t\right)-m\ {k}_t\left(\boldsymbol{x},t\right){\left[{C}_R\left(\boldsymbol{x},t\right)\right]}^2-{k}_O{C}_R\left(\boldsymbol{x},t\right){C}_O\left(\boldsymbol{x},t\right) \).

The problem is now even more non-linear, since the production of oxygen depends on the evolution of free radicals themselves; in fact, according to [27] \( {\dot{C}}_O\left(\boldsymbol{X},t\right)=-{k}_O{C}_O\left(\boldsymbol{X},t\right){C}_R\left(\boldsymbol{X},t\right) \) which is the equation that completes the system. The curing process considered above has been re-simulated by taking into account the role played by the oxygen inhibition, for the exemplary case with I₀ = 2 W/m²; the results are plotted in orange solid squares in Fig. 15 (Model (*)). As can be clearly appreciated, by taking into account the oxygen inhibition, the trend of the degree of cure and of the chain concentration evolution, appear to be shifted (delayed in time) with respect to the same curves obtained in the case of oxygen inhibition neglected (continuous line with black squares). Such curves are in good agreement with those obtained in [27] for the same representative case (black dashed line with empty squares).

Finally, it is worth mentioning that in order to enhance the mechanical properties of the material after photopolymerization, additive manufactured components may be subjected to post-curing treatments; to this end several techniques are available, such as the use of an ultraviolet chamber, microwave, ultra-sound or conventional oven where the printed parts has to be placed for a while [46]. Post-curing has the main role of further improving the degree of cure of a material, by completing the polymerization process after printing [28] , which could have not reached the complete conversion during printing; this could be caused, for instance, because of the use of a too low light intensity [3].

The most common post cure techniques are the UV and thermal ones, both aimed at directly polymerize any uncured liquid trapped within the solid by radiation of the component with ultra-violet light or by promoting the development of polymer chains through higher temperatures, respectively.

In the present study, we have developed a model describing the photopolymerization printing process only, while all the possible post-treatments are beyond the scope of the research and have been intentionally neglected. A simplified phenomenological model for the post-curing treatment can be find in [47].