Experimental Investigation and Modelling of the Curing Behaviour of Photopolymers T

Home , Curing (chemistry)

Experimental investigation and modelling of the curing behaviour of photopolymers T. Rehbein, A. Lion, M. Johlitz, Andrei Constantinescu

To cite this version:

T. Rehbein, A. Lion, M. Johlitz, Andrei Constantinescu. Experimental investigation and modelling of the curing behaviour of photopolymers. Polymer Testing, Elsevier, 2020, 83, pp.106356. 10.1016/j.polymertesting.2020.106356. hal-02468304

HAL Id: hal-02468304 https://hal.archives-ouvertes.fr/hal-02468304 Submitted on 5 Feb 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Experimental investigation and modelling of the curing behaviour of photopolymers

T. Rehbeina,∗, A. Liona, M. Johlitza and A. Constantinescub aInstitute of Mechanics, Faculty of Aerospace Engineering, Bundeswehr University Munich, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany bLaboratoire de Mecanique des Solides, CNRS, École Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau, France

ARTICLEINFO ABSTRACT

Keywords: This paper focuses on the experimental investigation and modelling of the crosslinking Photopolymer reaction of photopolymers used in additive manufacturing processes such as digital light Photo-DSC processing and stereolithography. Starting with general mathematical concepts for the Digital Light Processing description of the material behaviour of polymeric materials, the importance of mod- Stereolithography elling the crosslinking reaction is emphasized. In order to characterise the crosslinking Reaction kinetics reaction experimentally, photocalorimetric measurements with varying isothermal tem- Additive manufacturing perature and light intensity are shown. From the exothermic heat flows measured during the crosslinking reaction, the degree of cure can be determined for each experimental scenario as a function of time. It is shown that the test temperature and light intensity have a significant influence on the crosslinking reaction. A modelling approach for the description of the crosslinking reaction incorporating temperature and light intensity is presented. Moreover, parameter identification and a comparison of the proposed model with the experiments are conducted. The indentified model has an excellent match with experimental data, resulting in a least square error smaller than 3 %. The proposed model and identification method opens up several extensions for the modelling of the material behaviour.

1. Introduction in additive manufacturing processes such as digital light processing and stereolithography are challenging tasks. Additive manufacturing (AM) is an innovative tech- For this purpose, the crosslinking reaction in the addi- nology to create three dimensional objects with com- tive manufacturing process must be described correctly plex geometry. A large panel of different technologies by a phenomenological model and adapted to experi- and materials ensures a maximum design freedom for mental data. the production of polymeric and metallic components Many developers of commercial software solutions for leading to a reduced time and lowered expenses for the the numerical analysis of structures (e.g. Ansys Inc., development of new products to be saled. Altair Engineering Inc., Dassault Systèmes SE) are also Current studies in the additive manufacturing of poly- working on numerical tools for the analysis of additive mers cover, for example, the usage of additively manu- manufacturing processes, see [5] and [6]. These include factured lattice structures as energy-absorbing helmet process simulation, part analysis and topology optimi- liners for improved blunt impact protection [1]. Gen- sation, just to name a few fields of application. All of erally, additive manufacturing enables the fabrication the aforementioned software developers have one thing of novel designs consisting of arbitrary arranged lattice in common: they need experimentally based models for structures for energy absorbing components with re- the materials which are analysed in their programs for spect to lightweight construction requirements [2]. Us- additive manufacturing. Instead of the countless printing these technologies, components can be manufac- ing of a desired geometry with variation of the process tured not only at the macro scale but also down to the parameters up to the desired product properties, the nano scale including the addition of fillers [3]. printing process can be optimised on the computer with Furthermore, dental applications such as customised experimentally validated simulation methods even be- crowns, implants and dentures for patients can be made fore the final printing. This paper starts with this goal. precisely with additive manufacturing technologies [4]. Although the technology has been used in industrial The crosslinking of photopolymers is initiated by applications since the 1980s, reliable and experimen- the absorption of light in the UV wavelength range. tally validated models for the design of printed com- Commercially available 3D printers using photopoly- ponents are still missing. Especially the material mod- mer resins work typically at wavelenghts of 365 nm elling and finite element analysis of photopolymers used (atum3D DLP-Station 5-365) and 405 nm (Formlabs ∗Corresponding author Form 3), respectively. Besides that, the process de- thumbnails/[email protected] ( T. Rehbein) pends strongly on the temperature: a higher ambient orcid(s): 0000-0003-0833-2264 ( T. Rehbein)

T. Rehbein et al.: Preprint submitted to Elsevier Page 1 of 8 Short Title of the Article

ing a deep drawing process. Based on these findings, a) Mahnken and Dammann presented a multiscale ap- Uncured photopolymer proach for the curing of epoxy resin systems in the resin transfer moulding process, see [15]. To get a broad knowledge, the interested reader is re- ferred to [16] in which a highly detailed overview is given about the modelling of curing processes of polymers. b) Tryson and Shultz [17] were the first to develop a test UV radiation activates method for the investigation of the crosslinking reaction the crosslinking reaction of photopolymers used in coating applications with the help of a modified DSC device. In addition, empirical model equations for curing were derived and adapted to the experimental data. Maffezzoli and Terzi [18] investigated the influence of light intensity on the crosslinking reaction of a pho- c) topolymer used in stereolithography, but without in- Cured photopolymer vestigating temperature dependence. Like Tryson and Shultz, they developed a test method using DSC with a xenon lamp to study the crosslinking reaction. Figure 1: Simplified sketch of the crosslinking progress of a In [19], a simulation methodology based on the finite photopolymer based on [7]. element method was developed for the propagation of light in liquid photopolymer resins used in stereolithography. The program also includes a material model for temperature leads to faster crosslinking whereby the the description of the degree of cure taking into account same applies to an increasing light intensity. light intensity, temperature and photoinitiator concen- Fig.1 shows a simplified sketch of the crosslinking tration. This study is a purely theoretical/numerical progress of a photopolymer. Photopolymers usually one and does not include an experimental investigation of the crosslinking reaction of photopolymers used in consist of monomers ( ), oligomers ( , ) and additive manufacturing. photoinitiators ( ) whereby it is also possible to de- Wu et al. investigate the evolution of mechanical prop- sign photopolymer resins using renewable raw materials erties during photopolymerisation and build up a model [8]. At first, the components of the liquid photopolymer for the finite element analysis of residual stresses and are not crosslinked (a). The photoinitiators are excited bending strains, see [20]. by the incoming UV radiation (b) and finally crosslink The experimental investigation of different postcuring with the monomers and oligomers (c) to a solid mate- processes and their influence on the thermal and me- rial. chanical properties of a commercially available pho- Digital light processing and stereolithography are two topolymer is the main part of [21]. Calorimetric mea- similar additive manufacturing technologies using pho- surements and thermogravimetric as well as dynamic topolymerisation. The former uses a projector as a light mechanical analyses are conducted incorporating two source and a complete layer is cured simultaneously different postcuring mechanisms (with and without UV whereby the latter one uses a laser as a light source light exposure) and several curing times for printed which irradiates the trajectory to be printed in each specimens. individual layer. Calorimetric measurements with the aid of an addi- The phenomenological modelling of the transition from tional light source proved useful for the experimen- a liquid to a solid polymeric material has often been tal investigation of the crosslinking reaction of pho- used to describe the hot curing of adhesives. Many topolymers [22]. Since the degree of cure directly influ- authors developed material models for the description ences the mechanical, thermal and chemical properties of the curing behaviour of adhesives and adapted their of photopolymers, a validated material model for the models to experimental data, see [9] and [10]. These degree of cure is essential for structural computations models are primarily based on the work of [11] and [12]. of additive manufacturing processes. Furthermore, in [13], a material model for the description of the crosslinking reaction of acrylic bone cements is developed and adapted to experimental data. In [14], Mahnken developed a macroscopic material model incorporating temperature dependent viscoelastic ef- fects during curing processes. As a numerical example, the prognosis ability of the model is shown us-

T. Rehbein et al.: Preprint submitted to Elsevier Page 2 of 8 Short Title of the Article

2. Material modelling and reaction G∞, K kinetics

2.1. Representation by rheological G1 (θ, q) η1 (θ, q) elements In phenomenological modelling of viscoelastic ma- T T terials, it is common practice using an arrangement of G2 (θ, q) η2 (θ, q) standardised rheological elements to represent the ma- αth (q) ∆εch terial behaviour of polymers [23]. Fig.2 shows the rheological model for a viscoelastic polymeric material consisting of mechanical (mech), thermal (th) and chemical GN (θ, q) ηN (θ, q) (ch) parts. The mechanical part consists of an arrangement of springs ( ) and dashpots ( ) which is also E elk E ink called the generalised Maxwell model representing the time dependent material behaviour. Both shear mod- E mech E th E ch uli G k and viscosities ηk can depend on temperature θ and degree of cure q which works as an internal vari- E able. The number of Maxwell elements N is arbitrary and depends on the desired representation quality of Figure 2: Rheological model for the representation of the me- the viscoelastic spectrum. The rate-independent be- chanical, thermal and chemical parts of polymeric materials. haviour is represented by the equilibrium shear modulus G and bulk modulus K. In addition, the bulk ∞ material increases during the transition from a liquid behaviour is supposed to be purely elastic. For small resin to a solid material due to the crosslinking reac- deformations, the total three dimensional strain E is tion. Due to the constant mass, the volume is reduced. split additively into mechanical, thermal and chemical parts [24]: This process is also known as chemical shrinkage ( ) and is assumed to be isotropic, similar to thermal ex- E = E mech + E th + E ch (1) pansion:

For the consideration of curing processes at large de- E ch = q∆εch1 (5) formations, Lion and Höfer [25] and Sain et al. [26] proposed a general finite strain framework for the for- ∆εch is a material parameter and can be determined, mation of the polymer network in a stress-free interme- for example, by measurements according to the Archi- diate configuration. medes’ principle. Chemical shrinkage can induce resid- The mechanical part E mech is again divided into a purely ual stresses which can negatively influence the desired elastic E elk and a viscous part E ink for each Maxwell mechanical properties of the printed product. Besides element k: that, excessive warpage can occur during the printing process. E mech = E elk + E ink with k = 1,...,N (2) Thus, under the assumption of isotropy and for a ho- mogeneous material, the resulting stress T is split into For the inelastic strains E ink , N evolution equations are a volumetric part T V and deviatoric part T D and reads proposed [27]: as follows:

˙ 1 D T = T V + T D (6) E ink = 2G k (θ, q) E elk with k = 1,...,N ηk (θ, q) V (3) T = K ((E − E th − E ch): 1) 1 (7) N X 1 T D = 2G E D + 2G (θ, q) E D − E (8) Here, (.)D = (.) − tr(.)1 denotes the deviatoric part ∞ mech k mech ink 3 k=1 of a second order tensor and 1 is the unit second order tensor. Since the degree of cure inﬂuences the mechanical, ther- If the material is exposed to an increase in tem- mal and chemical properties of the material and has a perature, it expands. This mechanism is represented high impact on the quality of the printed product, it is by the symbol and described mathematically by advisable to start with the experimental characterisa- isotropic thermal expansion: tion and modelling of the crosslinking reaction of the photopolymer. E th = αth (q) ∆θ1 (4) 2.2. Modelling of the degree of cure Generally, the coeﬃcient of thermal expansion (CTE) This contribution takes up the basic work of Ka- αth depends on the degree of cure q. The density of the mal [12] using the consideration of light intensity as

T. Rehbein et al.: Preprint submitted to Elsevier Page 3 of 8 Short Title of the Article proposed by Maffezzoli and Terzi. In addition, the in- light guides fluence of the temperature is also taken into account. The ordinary differential equation for the degree of cure UV radiation q depends on time t, light intensity I (z (t) , t) and ab- lid solute temperature T . Following the work of Kamal, an furnace autocatalytic model of (m + n)th order incorporating the maximum attainable degree of cure qmax is applied for the degree of cure q: crucible sample m q˙ = (k1 (I (z (t) , t) ,T ) + k2 (I (z (t) , t) ,T ) q ) (9) · (q − q)n max platform Here, I (z (t) , t) denotes the time- and location-dependent light intensity. The light intensity must depend on the vertical location z (t) since it evolves during the print- ˙ ing process. The maximum attainable degree of cure h (t) qmax acts as a limit value. The functions Figure 3: Schematic diagram of a photo-DSC measurement cell. I (z (t) , t)b ki (I (z (t) , t) ,T ) = k0i (T ) (10) Iref technology allows the direct measuring of the light in- with the Arrhenius type equations tensity at the reference and sample platforms. For a better understanding, fig.3 shows a schematic Ei k0i (T ) = Ai exp − (11) diagram of a photo-DSC: the light guides point to the RT reference and sample crucibles made of aluminium and thus enable the crosslinking reaction of the liquid pho- are based on the proposal by Maffezzoli and Terzi [18] topolymer in the sample crucible. During irradiation, depending on light intensity I (z (t) , t) and tempera- the photo-DSC measures the incoming or outcoming ture T . If no light source is present, i.e. I (z (t) , t) = 0, specific heat flow h˙ (t) of a sample located in a crucible the crosslinking reaction is stopped. A and E are i i with regard to an empty reference crucible. The quan- pre-exponential factors and activation energies, respec- tity h˙ (t) is related to the mass of the sample and an tively. R denotes the universal gas constant with R = indicator of the degree of cure due to the exothermic 8.314 J/(mol K). The additional parameter b is intro- crosslinking reaction. duced for a better representation of experimental data. The following relationship is used to convert the Furthermore, the auxiliary constant I = 1 mW/cm2 ref specific heat flow h˙ (t) into the degree of cure q (t): is used to convert the light intensity into an unitless value. t R ˙ ˜ ˜ Both isothermal and non-isothermal processes as well h t dt 0 as time dependent light intensity boundary conditions q (t) = (12) h can be taken into account by this model. tot To put it simply, the degree of cure is the accumulated area between the adjusted photo-DSC signal and the 3. Experimental investigation of the abscissa relative to the total specific heat of reaction. curing behaviour Due to the normalisation to the total specific heat of 3.1. Experimental setup reaction htot during the crosslinking reaction, the de- In order to understand and investigate the crosslink- gree of cure assumes values between 0 and 1. R ing process of commercial photopolymers used in addi- The commercial photopolymer LOCTITE 3D 3830 tive manufacturing processes, differential photocalorime- resin was used for the photocalorimetric measurements. try (DPC) measurements are conducted using the TA It is an acrylic compound for prototype development Instruments DSC Q2000 with an additional light source with low elongation at break, high tensile strength and (OmniCure R S2000). The light source contains a high low chemical shrinkage. Both stereolithography and pressure 200 W mercury lamp with a spectral output digital light processing 3D printers can work with this of 320 to 500 nm. UV radiation is conducted directly photopolymer. into the DSC measurement cell via dual quartz light During the photocalorimetry measurement, following guides. The calibration of the light intensity at the steps are performed subsequently for the determination ˙ ˜ reference and sample platforms is carried out via the of the specific heat flow h t in eq. (12): separate height adjustment of the light guides and neu- 1. Equilibration of the measurement cell at a specific tral density filters inside the light guides. The TzeroTM isothermal temperature (−10◦C... 50◦C).

T. Rehbein et al.: Preprint submitted to Elsevier Page 4 of 8 Short Title of the Article

Table 1 0 Maximum attainable degree of cure qmax of all measurements −10 5 mW/cm2 10 mW/cm2 0 ◦ −10 C 0.469 0.593 −20 0 ◦C 0.619 0.685 −10 10 ◦C 0.675 0.775 −30 20 ◦C 0.775 0.845 −20 50 °C 30 ◦C 0.851 0.907 −40 −30 40 °C 40 ◦C 0.904 0.949 30 °C ◦ 20 °C

50 C 0.947 0.997 Specific heat flow [mW/mg] 60 62 64 66 68 70 72 −50 10 °C 0 °C −10 °C 2. Isothermal phase of 60 s. −60 3. Irradiation of the specimen for 300 s at constant 2 2 light intensity (5 mW/cm and 10 mW/cm ). 0 4. Isothermal phase of 60 s. −10 The experimental conditions (temperature, light in- 0 tensity) are chosen in such a way that a suﬃciently −10 −20 large database for the subsequent adaptation of the −20 model is guaranteed. Furthermore, the second isother- −30 −30 mal phase in step four is applied in order to achieve −40 50 °C a clear signal for the evaluation of the experimental −40 40 °C −50 30 °C data. 20 °C

Specific heat flow [mW/mg] 60 62 64 66 68 70 72 After finishing step four, the whole procedure is re- −50 10 °C 0 °C peated with the cured specimen generating the base- −10 °C line. The baseline is subtracted from the heat flow sig- −60 0 50 100 150 200 250 300 350 400 nal of the first measurement to consider only the heat Time [s] flow of the crosslinking reaction. Otherwise, the light energy input would be taken into account. Addition- Figure 4: Measured specific heat flows after substraction of the baseline under isothermal conditions. Top: light in- ally, nitrogen is used as a purge gas with a flow rate 2 tensity I1 (z(t), t) = 5 mW/cm , bottom: light intensity of 50 ml/min. The sample masses are between 3 and 6 2 I2 (z(t), t) = 10 mW/cm . mg.

3.2. Photocalorimetric measurements ◦C/min (red curve in fig.5). Subsequently, the same The specific heat flows after substraction of the base- procedure is repeated with the cured specimen (blue line of all measurements are shown in fig.4. It can be curve in fig.5). clearly seen that the crosslinking reaction proceeds very By using this approach, the area between the red and quickly and that the specific heat flows have reached blue curves represents the total specific heat of reac- their maximum value after only a few seconds. Addi- tion produced during full cure. The evaluation of the tionally, it is obvious that a higher light intensity leads area leads to a total specific heat of reaction of h = to a higher maximum value in the heat flows and that tot −319.73 J/g which is taken as reference value for the a higher temperature accelerates the reaction. conversion of the specific heat during crosslinking reac- Moreover, complete curing is not achieved at low tion into degree of cure for all measurements. temperatures and light intensities and the crosslinking Another interesting feature of the crosslinking reac- reaction is stopped prematurely, cf. fig.7. The max- tion is the temperature evolution of the sample material imum attainable degrees of cure q for all measure- max during and after curing, see fig.6. Initially, when the ments are listed in table1. It can be seen that an almost shutter of the light source is opened at 20 ◦C and the fully cured material is achieved at a temperature of 50 crosslinking reaction starts, the temperature increases ◦C, regardless of the light intensity. This result must briefly and then follows linear heating up to 70 ◦C (red definitely be taken into account when formulating the curve in fig.6). This can also be seen in the slight material model for the degree of cure. inclination to the right of the specific heat flow signal In order to determine the total specific heat of re- of the red curve in fig.5. In the subsequent second run action h during the crosslinking reaction, an addi- tot (blue dotted curve in fig.6), no temperature raise due tional measurement at non-isothermal temperature is to the opening of the shutter is recognised. performed. During 600 s irradiation with 5 mW/cm2, the temperature is increased from 20 ◦C to 70 ◦C at 5

T. Rehbein et al.: Preprint submitted to Elsevier Page 5 of 8 Short Title of the Article

0 Table 2 Identified model parameters for the description of the crosslink- −5 ing reaction 0.0 −10 −0.5 Parameter Identified value −1 −15 A1 0.6606 s −1.0 −1 A2 0.8744 s −1.5 −20 E1 9794 J/mol −2.0 E2 1001 J/mol b 4.554 −25 20 25 30 35 40 45 50 Specific heat flow [mW/mg] m 2.996 1st run n 1.365 −30 2nd run 20 25 30 35 40 45 50 55 60 65 70 Temperature [°C] degree of cure (A1, A2, E1, E2, b, m and n), and the Figure 5: Photo-DSC measurement under non-isothermal con- number of regression points, respectively. For the best ditions and additional irradiation. representation of the experimental data, all 14 measurements of fig.4 are taken into account as objective 70 functions during the parameter identification process which is represented by the iteration index i. A total 60 of M = 1000 regression points are used for the evaluation of each mean squared error. For simplification, 50 the particular maximum attainable degree of cure was specified as a constraint in order to achieve a better 40 agreement with the experimental data.

30 4.2. Comparison with experiments Temperature [°C] The parameters listed in table2 represent the best 20 result of the parameter identification with a summarised 1st run mean squared error of all objective functions of 0.0295. 2nd run 10 Fig.7 shows a comparison between the simula- 0 50 100 150 200 250 300 350 400 450 500 550 600 tions of the degree of cure according to eq. (9) with Time [s] identified model parameters and all measurements con- Figure 6: Measured temperature profiles of the photo-DSC ducted by the photo-DSC. For the simulations, two 2 measurements in fig.5. constant light intensities I1 (z (t) , t) = 5 mW/cm and 2 I1 (z (t) , t) = 10 mW/cm as well as seven different temperatures are used as boundary conditions accord- 4. Results and discussion ing to the experimental procedure which is described in sec. 3.1. 4.1. Parameter identification The degree of cure is evaluated by means of eq. (12) The parameters in eq. (9) are identified using the for all measurements represented in fig.4. The pro- R commercial optimisation program LS-OPT (Livermore posed model is capable of predicting the temperature Software Technology Corporation, Livermore, CA) in and light intensity dependence of the crosslinking reac- R combination with MATLAB ’s ode45 solver for the tion leading to a perfect representation of the experi- solution of the ordinary differential equation (9). LS- mental data. Moreover, only seven model parameters R OPT uses the sequential response surface methodol- are sufficient to consider the influence of the two most ogy (SRSM) for optimisation. For further informations important process parameters (temperature and light see [28]. intensity) on the crosslinking reaction. Both the tremendous increase to the degree of cure in In order to quantify the quality of the identified the first seconds after opening of the shutter and the model parameters, the sum of the mean squared errors decrease of the conversion rate after reaching the peak of all objective functions is calculated: are reproduced by the model in almost error-free agreement. 14  M  X 1 X 2 In dependence of the isothermal temperature, the de- MSE = (q (x) − q )) (13) M ij,sim ij,exp  gree of cure reaches an asymptotic value for both light i=1 j=1 intensities. The vector x and scalar value M denote the design However, it is more important which light energy space vector, i.e. the parameters of the model for the (light intensity multiplied by time) is applied to the

T. Rehbein et al.: Preprint submitted to Elsevier Page 6 of 8 Short Title of the Article

with a smaller effect. 1.0 0.9 The proposed model is capable of predicting the 0.8 temperature and light intensity dependence of the cross- 0.7 linking reaction with only seven model parameters. The R R 0.6 combination of LS-OPT and MATLAB provides an excellent result for the parameter identification and a 0.5 small summarised mean squared error. However, the 0.4 50 °C correlation of the identified parameters with each other 0.3 40 °C Degree of cure [−] 30 °C still has to be investigated in order to reduce the num- 0.2 20 °C 10 °C ber of parameters of the model. 0.1 0 °C 0.0 −10 °C

1.0 6. Outlook 0.9 In this study, the maximum attainable degree of 0.8 cure qmax is specified as a constraint for all 14 measure- 0.7 ments. Forthcoming considerations include the correct 0.6 description of the maximum attainable degree of cure 0.5 as a function of absolute temperature and light intensity: 0.4 50 °C 0.3 40 °C Degree of cure [−] 30 °C qmax = f (T, I) (14) 0.2 20 °C 10 °C 0.1 0 °C The limitation of the maximum attainable degree of 0.0 −10 °C cure can also be described by an additional diffusion 0 25 50 75 100 125 150 175 200 225 250 275 300 factor. For this, Fournier et al. [29] made a significant Time [s] contribution which has been applied in various recent Figure 7: Measurement (—) and simulation (- -) of the de- studies for epoxy resins, see [30], [31] and [32]. gree of cure q with identified model parameters. Top: light Since the glass transition temperature of acrylic pho- 2 intensity I1 (z(t), t) = 5 mW/cm , bottom: light intensity topolymers strongly depends on the degree of cure, con- 2 I2 (z(t), t) = 10 mW/cm . ventional DSC measurements must be performed after irradiation to determine the glass transition temperature for distinct degrees of cure. Subsequently, photopolymer during irradiation. The measurements DiBenedetto’s proposal [33] can also be used to estab- with the lower light intensity lead to the equivalent de- lish a relationship between the maximum attainable degree of cure of the measurements with the higher light gree of cure and the glass transition temperature. intensity when the exposure time is increased. The dependence of the shear moduli Gk (q, θ) and viscosities ηk (q, θ) on the degree of cure q can be deter- 5. Conclusion mined with a dynamic shear rheometer using the same light source as with the Photo-DSC. During irradia- This paper discusses the experimental characteri- tion, a thin photopolymer layer is sheared between two sation as well as the modelling and simulation of the plates and thereby cured. This allows the storage and crosslinking reaction of a commercial photopolymer for loss modulus of the photopolymer to be determined ex- stereolithography and digital light processing 3D print- perimentally, depending on the degree of cure. After- ers. Using a commercial photopolymer as sample ma- wards, a suitable model is adapted to the experimental terial, it is shown that both ambient temperature and data to cover the mechanical properties. It may also light intensity have a strong influence on the crosslink- be sufficient to formulate only the shear moduli or vis- ing reaction. Two conclusions can be drawn from the cosities as a function of the degree of cure. study of the crosslinking reaction of the photopolymer: Finally, the chosen atmosphere during photopolymerisation influences significantly the crosslinking speed. 1. The crosslinking reaction depends strongly on the Previous work shows that without purge gas the crosslink- ambient temperature; the higher the tempera- ing reaction is much slower [34]. Since available 3D ture, the faster the crosslinking reaction and a printers usually work without a protective atmosphere higher degree of cure is achieved at the end of the of nitrogen, photo-DSC measurements must be carried irradiation. out without purge gas. 2. Equivalent to the ambient temperature, an increasing light intensity also has a positive influence on the speed of the crosslinking reaction, but

T. Rehbein et al.: Preprint submitted to Elsevier Page 7 of 8 Short Title of the Article

7. Acknowledgement photopolymerization, Journal of Polymer Science Part B: Polymer Physics 17 (1979) 2059–2075. The financial support of the project "Constitutive [18] A. Maffezzoli, R. Terzi, Effect of irraditation intensity on modeling of UV-curing printed polymer composites" by the isothermal photopolymerization kinetics of acrylic resins the German Research Foundation (DFG) and Agence for stereolithography, Thermochimica Acta 321 (1998) 111– nationale de la recherche (ANR) under the grant num- 121. [19] P. J. da Silva Bartolo, Photo-curing modelling: direct irra- bers LI 696/20-1 and ANR-18-CE92-0002-01 is grate- diation, The International Journal of Advanced Manufac- fully acknowledged. turing Technology 32 (2007) 480–491. [20] J. Wu, Z. Zhao, C. M. Hamel, X. Mu, X. Kuang, Z. Guo, H. J. Qi, Evolution of material properties during free radical References photopolymerization, Journal of the Mechanics and Physics [1] T. Kayhart, S. Shpiner, Analysis of Engineered Polymer of Solids 112 (2018) 25–49. Structures for Blunt Impact Protection, Technical Report, [21] C. Mendes-Felipe, D. Patrocinio, J. M. Laza, L. Ruiz-Rubio, J. L. Vilas, Evaluation of postcuring process on the thermal Predictive Engineering, Inc., 2018. TM [2] J. Mueller, K. H. Matlack, K. Shea, C. Daraio, Energy ab- and mechanical properties of the Clear02 resin used in sorption properties of periodic and stochastic 3d lattice ma- stereolithography, Polymer Testing 72 (2018) 115–121. terials, Advanced Theory and Simulations (2019) 1900081. [22] G. W. H. Höhne, W. F. Hemminger, H.-J. Flammer- [3] D. Gailevičius, V. Padolskyt˙e, L. Mikoli¯unait˙e, sheim, Differential Scanning Calorimetry, 2003. doi:10. S. Šakirzanovas, S. Juodkazis, M. Malinauskas, Additive- 1007/978-3-662-06710-9. manufacturing of 3d glass-ceramics down to nanoscale [23] R. M. Christensen, Theory of Viscoelasticity. An Introduc- resolution, Nanoscale Horiz. 4 (2019) 647–651. tion, 1982. doi:10.1016/B978-0-12-174252-2.X5001-7. [4] A. Bhargav, V. Sanjairaj, V. Rosa, L. W. Feng, J. F. Yh, [24] M. Hossain, P. Steinmann, Degree of cure-dependent mod- Applications of additive manufacturing in detistry: A re- elling for polymer curing processes at small-strain. part view, Journal of Biomedical Materials Research 106 (2018) i: consistent reformulation, Computational Mechanics 53 2058–2064. (2014) 777–787. [5] Spotlight on Additive Manufacturing, Technical Report, [25] A. Lion, P. Höfer, On the phenomenological representation Ansys, Inc., 2018. of curing phenomena in continuum mechanics, Arch. Mech. [6] Additive Manufacturing for Generative Design, Technical 59 (2007) 59–89. Report, Dassault Systèmes, 2017. [26] T. Sain, K. Loeffel, S. Chester, A thermo-chemo- [7] Determining cure profile and post-cure shrinkage of pho- mechanically coupled constitutive model for curing of glassy topolymers using UV accessory on a rotational rheometer, polymers, Journal of the Mechanics and Physics of Solids Technical Report, Malvern Instruments Limited, 2015. 116 (2018) 267–289. [8] E. Skliutas, S. Kasetaite, L. Jonušauskas, J. Ostrauskaite, [27] B. Yagimli, A. Lion, Experimental investigations and mate- M. Malinauskas, Photosensitive naturally derived resins to- rial modelling of curing processes under small deformations, ward optical 3-d printing, Optical Engineering 57 (2018) Journal of Applied Mathematics and Mechanics 91 (2011) 1–9. 342–359. [9] C. Liebl, M. Johlitz, B. Yagimli, A. Lion, Three-dimensional [28] N. Stander, W. Roux, A. Basudhar, T. Eggleston, T. Goel, R chemo-thermomechanically coupled simulation of curing ad- K. Craig, LS-OPT User’s Manual, Livermore Software hesives including viscoplasticity and chemical shrinkage, Technology Corporation, 2015. Computational Mechanics 49 (2012) 603–615. [29] J. Fournier, G. Williams, C. Duch, G. A. Aldrige, Changes [10] M. Hossain, G. Possart, P. Steinmann, A small-strain model in Molecular Dynamics during Bulk Polymerization of an to simulate the curing of thermosets, Computational Me- Epoxide-Amine System As Studied by Dielectric Relaxation chanics 43 (2009) 769–779. Spectroscopy, Macromolecules 29 (1996) 7097–7107. [11] M. R. Kamal, S. Sourour, Kinetics and Thermal Characteri- [30] C. Jochum, M. Arrigoni, M. Boustie, J.-C. Grandidier, A zation of Thermoset Cure, Polymer Engineering and Science cut-off fracture approach for residual stress estimation in 13 (1973) 59–64. thick epoxies, Materials Science & Engineering Technology [12] M. R. Kamal, Thermoset Characterization for Moldability 47 (2016) 530–538. Analysis, Polymer Engineering and Science 14 (1974) 231– [31] R. Landgraf, J. Ihlemann, Thermally Controlled Adhesive 239. Curing during the Production of Piezo-Metal-Compounds: [13] S. Kolmeder, A. Lion, R. Landgraf, J. Ihlemann, Thermo- Finite Element Modeling and Analyses, Advanced Engineer- physical properties and material modelling of acrylic bone ing Materials 20 (2018) 1800411:1–1800411:9. cements used in vertebroplasty, Journal of Thermal Analysis [32] C. Leistner, S. Hartmann, D. Abliz, G. Ziegmann, Modeling and Calorimetry 105 (2011) 705–718. and simulation of the curing process of epoxy resins using fi- [14] R. Mahnken, Thermodynamic consistent modeling of poly- nite elements, Continuum Mechanics and Thermodynamics mer curing coupled to visco-elasticity at large strains, Inter- 29 (2018) 1–24. national Journal of Solids and Structures 50 (2013) 2003– [33] A. T. DiBenedetto, Prediction of the Glass Transition Tem- 2021. perature of Polymers: A Model Based on the Principle of [15] R. Mahnken, C. Dammann, A three-scale framework for Corresponding States, Journal of Polymer Science Part B: fibre-reinforce-polymer curing part i: Microscopic modeling Polymer Physics 25 (1987) 1949–1969. and mesoscopic effective properties, International Journal [34] M. C. Rusu, C. Block, G. Van Assche, B. Van Mele, of Solids and Structures 100–101 (2016) 341–355. Influence of temperature and UV intensity on photo- [16] M. Hossain, P. Steinmann, Continuum physics of materi- polymerization reaction studied by photo-DSC, Journal of als with time-dependent properties: Reviewing the case of Thermal Analysis and Calorimetry 110 (2012) 287–294. polymer curing, Advances in Applied Mechanics 48 (2015) 141–259. [17] G. R. Tryson, A. R. Shultz, A calorimetric study of acrylate

T. Rehbein et al.: Preprint submitted to Elsevier Page 8 of 8